computational multiphase flow thesis

Development of a

Generic Engineering Model

for Packed Bed Reactors

using Computational Fluid Dynamics

Developm

ent of a Generic Engineering M

odel for Packed Bed R

eactors using Com

putational Fluid Dynam

icsB

ouke Tuinstra

Bouke Tuinstra

Development of a generic engineering model

for packed bed reactors

using Computational Fluid Dynamics

Proefschrift

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus, prof.dr.ir. J.T. Fokkema, voorzitter van het College voor Promoties,

in het openbaar te verdedigen op dinsdag 23 december 2008 om 15.00 uur

door

Bouke Fokkes TUINSTRA

scheikundig ingenieur geboren te Cleveland Ohio, Verenigde Staten van Amerika

Dit proefschrift is goedgekeurd door de promotor: Prof.ir. C.M. van den Bleek

Samenstelling promotiecommissie:

Rector Magnificus, voorzitter Prof.ir. C.M. van den Bleek, Technische Universiteit Delft, promotor Prof.dr. R.F. Mudde, Technische Universiteit Delft Prof.dr.ir. G.B. Marin, Universiteit Gent Prof.dr.ir. J.C. Schouten, Technische Universiteit Eindhoven Prof.dr.ir. W.P.M. van Swaaij, Universiteit Twente Dr.ir. H.P.A. Calis, Shell Gas & Power Dr.ir. A.N.R. Bos, Shell Global Solutions

Het in dit proefschrift beschreven onderzoek is gedeeltelijk financieel ondersteund door: Koninklijke DSM n.v. Kiwa Onderzoek & Advies b.v. Stork Comprimo b.v.

Copyright ©c 2008 by B.F. Tuinstra All Rights Reserved

ISBN 978-90-9023662-9

Summary

Most industrial chemical processes are possible only through the use of a catalyst, i.e., a compound that speeds up the reaction but that is not consumed. In most cases, heterogeneous catalysis is used, here the catalyst is a solid material while the reactants are in the liquid or gaseous state. An example of such a reactor is the one that is used in automotive exhaust systems. There the catalyst is fixed on the surface of a honeycomb-like ceramic structure. For larger scale chemical reactors, structured packings like this are relatively expensive. Therefore, in most chemical reactors random packings of porous ceramic catalyst particles are used instead.

One of the key factors in the operation of these packed bed reactors is the exchange of reactants and products between the catalyst and the process fluid. In many cases, heat has to be supplied to or removed from the reactor to get a complete and swift conversion of reactants into desired products. In these cases also the transfer of heat inside the bed and between the reactor wall and the bed is crucial. All these transport processes are in turn influenced by the macroscopic flow profile in the catalyst bed. Maldistribution of flow occurs due to the influence of the wall on the bed packing and, for low pressure drop reactors, due to the geometry of the flow channel upstream and downstream the catalyst bed.

Nowadays, computer models are generally used to design packed bed reactors. There are various approaches to the modelling of packed bed chemical reactors, differing in level of detail and complexity as well as computation time. These models can be thought to be ranging from the ”engineering” approach on one end of the spectrum to the ”scientific” approach on the other.

In the engineering approach, the goal is to estimate the performance of a packed bed with sufficient accuracy to allow the design, rating and optimisation of the reactor. The models are usually based on fit relations for empirical data. Typically, the result is a one-dimensional model that gives a relation between the inlet and the outlet conditions. The advantage of this approach is that the models don’t need much computation time or resources. The disadvantages

i

are that the models are specific for a given reactor and are limited to those geometries, for which empirical relations are available.

In the scientific approach, microscopic first-principle descriptions are used as much as possible. For a packed bed reactor, the flow around the particles and transport phenomena on the scale of a particle could be calculated using Computional Fluid Dynamics (CFD) techniques. This approach leads to generic and detailed models that can be used to better understand the microscopic processes. However, they are usually not very suitable to use as a design tool because of the high level of expertise needed to build and use such a model and because of forbiddingly high computational effort to calculate reactors of commercial size.

The central hypothesis of this work is that a practical alternative approach to the modelling of packed bed reactors can be found that is a combination of the engineering and scientific approaches and that combines the advantages of both. This tool consists of a library of routines that can be linked to a CFD code and that allows the engineer to model a packed bed using engineering rules within the CFD domain. For this, models needed to be developed in several areas and scales.

The important processes in a packed bed reactor are reaction, convection and dispersion. Reaction takes place on the scale of a single particle. For single reaction systems, simple particle geometry and/or simple kinetics, approximation methods are available that are accurate enough and computationally inexpensive. However, in the most important practical cases, multiple reactions play a role. For reaction networks in complex catalyst geometry, no approximation method was available. Therefore, a method was developed to extend the so-called Aris number method to reaction networks with general kinetics in porous catalyst particles with general geometry.

Convection and dispersion take place on the scale of the bed. Therefore they are determined by the structure of the bed. Near walls and solid inserts, the structure of the bed is different from the bulk of the bed; therefore also convection and dispersion are different near a wall and far from a wall. In a CFD model, we need to be able to describe the flow near the wall correctly, based on local values of the bed parameters. However, engineering models for convection and dispersion in packed beds that are available from literature are based on global or average bed parameters. Therefore, new models needed to be developed.

As a basis, a model was developed to describe the bed structure, especially near a wall, in terms of local parameters: the porosity, tortuosity and specific outer particle surface area. To obtain these parameters, a computer program

ii

was written that generates simulated random packings of uniform spherical particles. General rules for the bed structure parameters were extracted from a large number of these computer simulated random packings.

A relation was developed that gives flow resistance in the catalyst bed as a function of the local values of the bed structure parameters only. This model was based on a physical description of the structure of the flow channels in a packed bed. It was found that, for standard values of the bed and flow variables, this model gives identical results as the well-known engineering rules for pressure drop in packed beds. This is remarkable, because our model was not fitted to experimental packed bed pressure drop data. In addition, our model can be used to describe the flow near walls.

The dispersion (mixing) behaviour of the bed was also described in terms of local bed structure parameters (porosity, tortuosity, outer particle surface area). As heat transfer with a packed bed reactor takes place through a wall, the wall region is important for many non-isothermal reactions. Available engineering methods use fitted parameters (for instance, wall heat transfer coefficients) or profiles that are not consistent with the bed structure. Our model was derived directly from the bed structure and was found to describe the literature experimental data for standard cylindrical beds just as well as the literature models, even though no fitting was used. In contrast with literature models, no assumptions regarding the bed geometry were made in our model, so it can in principle also be used for non-cylindrical packed beds.

In the final step, all models were combined with a freely available open source CFD code. The data generated by the simulations is relatively detailed, giving pressure, velocity, temperature and concentrations at every point in the catalyst bed. Therefore, it is hard to find suitable experimental data in literature for a full validation. A limited validation was done using experimental literature data for a laboratory scale ethylene oxidation reactor. Good correspondence was found between the concentration and temperature profiles predicted by the model and the literature data.

To demonstrate the capabilities of the CFD code, two model reactions were simulated: the catalytic reduction of NO with ammonia that is used in flue gas treatment, and the catalytic oxidation of sulphur dioxide that is part of the sulphuric acid production process. Different reactor designs were simulated and compared, showing that the CFD model is a useful tool to simulate and optimise such systems.

In general, application of CFD techniques in chemical reactor design is limited by the high cost of the software and the high level of expertise needed to use it. The packed bed simulation code that was developed in this work is distributed as part of the Comflow CFD code. This package allows engineers

iii

with a basic knowledge of fluid dynamics to simulate packed bed reactors at a detailed level without the need for in-depth expertise the specifics of CFD software. It is distributed as open source software and therefore it is freely available for the reactor engineering community.

We conclude that it has been shown that a CFD tool has been developed which can be used as a design tool for packed bed chemical reactors with complex reactions, non-standard geometry and high interaction of flow and heat transfer. The model gives results that are more detailed than the engineering relations, allowing more detailed optimisations and alternative reactor designs. Therefore, we conclude that the central hypothesis of this work can be accepted.

iv

Contents

1 Introduction 1 1.1 Randomly Packed Beds . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Modelling approaches . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 The engineering approach . . . . . . . . . . . . . . . . . 3 1.2.2 The scientific approach . . . . . . . . . . . . . . . . . . . 3

1.3 Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4.1 Engine . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4.2 Packed bed model . . . . . . . . . . . . . . . . . . . . . . 9 1.4.3 Sub-hypotheses . . . . . . . . . . . . . . . . . . . . . . . 11

1.5 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Chemical reaction and diffusion in a single catalyst particle 15 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2 Theoretical background . . . . . . . . . . . . . . . . . . . . . . . 21

2.2.1 Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2.2 Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2.3 Balance equations . . . . . . . . . . . . . . . . . . . . . . 25

2.3 Reduction of the number of differential equations . . . . . . . . 27 2.3.1 Single reaction . . . . . . . . . . . . . . . . . . . . . . . 27 2.3.2 Multiple reactions . . . . . . . . . . . . . . . . . . . . . . 29

2.4 Approximating the effectiveness factor . . . . . . . . . . . . . . 31 2.4.1 Single reaction systems . . . . . . . . . . . . . . . . . . . 31 2.4.2 Reaction networks . . . . . . . . . . . . . . . . . . . . . 34

2.5 Example: the Sohio process . . . . . . . . . . . . . . . . . . . . 44 2.5.1 Process description . . . . . . . . . . . . . . . . . . . . . 44 2.5.2 Numerical solution . . . . . . . . . . . . . . . . . . . . . 48 2.5.3 Approximation . . . . . . . . . . . . . . . . . . . . . . . 51

2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

v

3 The wall effect 71 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.2 A short review of existing literature . . . . . . . . . . . . . . . . 76 3.3 Bed packing simulation models . . . . . . . . . . . . . . . . . . 87

3.3.1 Potential energy minimisation model . . . . . . . . . . . 87 3.3.2 Ball placement model . . . . . . . . . . . . . . . . . . . . 88 3.3.3 Modelling approach . . . . . . . . . . . . . . . . . . . . . 92

3.4 Bed packing simulation results . . . . . . . . . . . . . . . . . . . 92 3.4.1 Mean bed porosity and thickness effect . . . . . . . . . . 92 3.4.2 Sphere centre density . . . . . . . . . . . . . . . . . . . . 96 3.4.3 Porosity profile . . . . . . . . . . . . . . . . . . . . . . . 100 3.4.4 Particle surface area profiles . . . . . . . . . . . . . . . . 102 3.4.5 Tortuosity profiles . . . . . . . . . . . . . . . . . . . . . 106

3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

4 Flow resistance in randomly packed beds 119 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 4.2 A short overview of existing literature . . . . . . . . . . . . . . . 122 4.3 Flow resistance in infinite packed beds . . . . . . . . . . . . . . 123 4.4 Flow resistance: the wall effect . . . . . . . . . . . . . . . . . . . 136 4.5 Velocity profile . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

5 Dispersion in randomly packed beds 147 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 5.2 Dispersion in packed beds . . . . . . . . . . . . . . . . . . . . . 150 5.3 A short overview of existing literature . . . . . . . . . . . . . . . 156

5.3.1 Experimental methods . . . . . . . . . . . . . . . . . . . 156 5.3.2 The standard model . . . . . . . . . . . . . . . . . . . . 157 5.3.3 The wall heat conduction model . . . . . . . . . . . . . . 158 5.3.4 Other models . . . . . . . . . . . . . . . . . . . . . . . . 161

5.4 Dispersion modelling . . . . . . . . . . . . . . . . . . . . . . . . 162 5.4.1 Correlations for Pe at stagnant conditions . . . . . . . . 163 5.4.2 Correlations for Pe at high Reynolds numbers . . . . . . 170 5.4.3 Dispersion of momentum . . . . . . . . . . . . . . . . . . 176 5.4.4 Dispersion model . . . . . . . . . . . . . . . . . . . . . . 178

5.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

vi

6 Modelling of packed bed reactors using CFD 191 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 6.2 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

6.2.1 Basic flow and transport equations . . . . . . . . . . . . 195 6.2.2 Porosity, specific particle outer surface area and tortuosity197 6.2.3 A note on discretisation . . . . . . . . . . . . . . . . . . 199 6.2.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 200 6.2.5 Flow resistance . . . . . . . . . . . . . . . . . . . . . . . 213 6.2.6 Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . 214 6.2.7 Pellet scale models . . . . . . . . . . . . . . . . . . . . . 215

6.3 Model verification and validation . . . . . . . . . . . . . . . . . 221 6.3.1 Flow in a packed tube . . . . . . . . . . . . . . . . . . . 221 6.3.2 Heat transfer in a steam heated packed tube . . . . . . . 226 6.3.3 Catalytic oxidation of ethane . . . . . . . . . . . . . . . 239

6.4 Demonstration . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 6.4.1 Reduction of NO in a packed tube . . . . . . . . . . . . . 243 6.4.2 Reduction of NO in a radial flow reactor . . . . . . . . . 247 6.4.3 Non-isothermal chemical reaction in a packed tube: ox

idation of SO2 . . . . . . . . . . . . . . . . . . . . . . . . 252 6.4.4 Oxidation of SO2: alternative reactor design . . . . . . . 257

6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

7 Conclusions 273

Samenvatting 281

vii

Chapter 1

Introduction

1.1 Randomly Packed Beds

Most industrial chemical processes are possible only through the use of a catalyst, i.e., a compound that participates in the reaction but that is not consumed. One of the key factors in designing industrial reactors for catalytic processes is that the contact between the process fluid and the catalyst needs to be as intimate as possible during the reaction, while it should be easy to separate the catalyst and the product stream after the reaction. In homogeneous catalysis, the catalyst is part of the reaction mixture so contact is excellent but recovery of catalyst after the reactor requires a separation step. In heterogeneous catalysis, the catalyst is a solid material with active sites on its surface. Therefore, separation of catalyst and process fluid is relatively easy, but transport of mass and heat between the fluid and the catalyst is critical. Although many reactor concepts for heterogeneous catalysis can be conceived, the most common type found in the process industry is the randomly packed bed reactor.

In randomly packed bed reactors, the solid catalyst usually consists of porous particles of an inert (e.g., ceramic) carrier material that is doped or coated with catalytically active material. These particles are placed in the reactor vessel and kept in place by means of, for instance, sieve plates, through which the process fluid enters and leaves the reactor. Randomly packed bed reactors are widely used throughout the chemical process industry, for example in reactors for steam reforming and subsequent reaction steps like water-shift and partial oxidation in the production of hydrogen from hydrocarbons, oxidation reactors in the sulphuric acid production, ethane oxidation, methanol synthesis and in many oil refining processes like hydrocracking and hydrodesulphurisation. Apart from these examples of chemical reactors, randomly packed

1

1. Introduction

beds are also widely used in physical process equipment like mixers, absorbers and scrubbers.

Although many of the topics apply to randomly packed beds in general, the focus in this work is on the packed bed reactor. Such a reactor typically contains a large number of catalyst particles. Reactants are transported through the bed by the flowing fluid and are then transported to the active surface by diffusion. After the reaction has taken place, the products are transported in the reverse order. The flow pattern in a packed bed is complex: the maze-like structure of the flow paths in between the particles causes the flow to continuously change direction, split up into separate side streams and merge with other side streams. This flow pattern causes a relatively high pressure loss in a packed bed, but also provides the high degree of mixing that is often desired.

The wall is an important factor in the operation of a packed bed reactor. The reactor can in principle have any shape, but cylindrical vessels are the most common. In many cases, especially where heat has to be removed from or added to the reactor, the catalyst material is contained in a number of smaller tubes. The space between these tubes is filled with a heating or cooling medium and heat is transferred through the tube wall. Inside the bed, heat is transported by conduction and convection through the fluid, but also by radiation and by conduction through the solid particles. The porosity of the bed, i.e., the fraction of open space in between the particles, plays an important role in these processes. Far from any wall or other solid object, the porosity of a packed bed is about independent of position (if averaged over a volume that is small compared to the reactor but large compared to the particles). When we get near a wall, the structure of the bed changes because the particles cannot penetrate the wall. This change in structure results in a porosity profile perpendicular to the wall. Due to this profile the local flow resistance and transport parameters change compared with the far-fromthe-wall values, causing flow maldistribution and changing concentration and temperature profiles near the wall.

The combination of the practical importance and relative complexity of these phenomena has spurred research from the 1930s onward. Through the years, an increasing detail in both measurements and models can be seen.

1.2 Modelling approaches

There are various approaches to the modelling of packed bed chemical reactors, differing in level of detail and complexity as well as computation time, ranging from the ”engineering” approach on one end of the spectrum to the ”scientific” approach on the other.

2


1.2.1 The engineering approach

In the engineering approach, the goal is to estimate the performance of a packed bed with sufficient accuracy to allow the design, rating and optimisation of the reactor. To be of practical relevance, the engineering models need to be simple. Therefore, a selection has to be made of the processes and phenomena that have the largest impact on the most relevant output parameters for a specific system. This selection depends on the specific features of the process and therefore requires expertise and intimate knowledge of the system. As the model becomes more accurate, taking into account more physical phenomena, the mathematical complexity also increases. As a result, such models often turn out to be not very simple. The parameters used in an engineering model are often obtained from measurements on an existing reactor. Therefore, they are specific for a given design. A disadvantage of this approach is that for each different combination of catalyst, reactor, reactants and process conditions a new model with a different choice of sub-models, needs to be developed. Mathematically, the model is typically a set of 1-dimensional differential equations, but depending on the complexity of the process and on the type of output data required (spatial or temporal variations), the solution of multi-dimensional or time-dependent partial differential equations may be necessary. As output, the engineering model gives predictions for macroscopic variables that can often be measured directly in the actual reactor. Comparison between experimental data and model predictions leads either to acceptance or to adaptation of the assumptions on which the model is based and thus to insight. In addition, the parameters of the model are often obtained by fitting the model results to experimental data. These parameters then contain not only measurement errors but also the effects of all phenomena that were not included (correctly) in the model: therefore these parameters are coupled specifically to the model. This is shown schematically in figure 1.1.

1.2.2 The scientific approach

In the scientific approach, the purpose of the model is to increase the understanding of the different phenomena that take place in a given process. Ideally, the model starts from first principles and general parameters like physical properties that are not specific for the process or geometry. A general schematic of this process is given in figure 1.2. The model describes the physical phenomena on a microscopic level. For example, detailed 2 or 3-dimensional Computational Fluid Dynamic (CFD) models are built to describe the motion of particles in a fluidised bed or the interaction between liquid flow and gas bubbles in a bubble column. In such simulations, the particles or bubbles are

3

1. Introduction

specific parameters

“simple” model

macroscopic prediction

compare experiments

parameter fitting

model adaptation

insightassumptions

Figure 1.1: The typical structure of the engineering approach

modelled as separate entities, where additional microscopic models describe the interaction between two particles and between particles and the fluid. For packed bed reactors, models like these have been proposed to calculate pressure drop, heat and mass transfer parameters (Derckx and Dixon, 1996).

In order to describe the influence of a particle on the flow, the flow equations need to be solved on a length scale smaller than the particle size. Consequentially, these models are computationally intensive. In practice only a limited number of particles — typically much less than a thousand — can be taken into account. For instance, to calculate flow and heat transfer in a ’packed bed’ of only 10 particles already requires a computational grid of more than a million of cells. The volume of an actual chemical reactor is often many cubic meters, and the particle size is usually in the order of one millimetre. Therefore, a typical reactor contains billions of particles. It is clear that it is not practical to compute a complete reactor using these models. Instead, these models should be regarded as research tools for a better understanding of the microscopic processes. A better understanding leads to better macroscopic approximating relations that can be used in improved engineering models.

4


A comparison of model results with experimental data (often from detailed lab-scale experiments) is required to validate the choice of microscopic models, but in contrast with the engineering approach, in the scientific approach insight in the behaviour of reactors on a macroscopic scale is gained mainly from analysis of the results of the model.

general insight

parameters

experiments

complex microscopic macroscopic model results relations

compare modelfirst

adaptationprinciples

Figure 1.2: The typical structure of the scientific approach

5

1. Introduction

1.3 Hypothesis

The central hypothesis of the present work is that a practical alternative approach to the modelling of packed bed reactors can be found that is a combination of the engineering and scientific approaches and that combines the advantages of both.

In this alternative approach, the engineering rules are adapted and applied on a local scale. A simplified ’scientific’ CFD solver is used as the computational engine (see figure 1.3). To be of practical use, the approach needs to give results at a higher accuracy than the engineering method, but with a considerably lower computational effort and requiring a lower level of CFD expertise than the scientific approach. In addition, it should be accessible and easy to use for the reactor engineer. The model should be sufficiently generic to be applied to the majority of reactor designs without major modifications.

complexity / computation time

error

engineering approach

scientific approach

this work

submodels

adapt / improve

engineadapt /simplify

Figure 1.3: Graphical representation of the engineering approach and scientific approach on both ends of the error – computational effort spectrum and the approach developed in this work

6

1.4 Method

1.4 Method

To be able to accept or reject the central hypothesis, we will develop a tool that consists of a packed bed model, implemented in an existing CFD code. The CFD code calculates the balances of mass, momentum, enthalpy and chemical species, while the packed bed model supplies values for a.o., friction force, dispersion coefficients and source terms, as is shown in figure 1.4. The packed bed model consists of four sub-models: the particle model, that describes reaction and diffusion within a porous catalyst particle; the packing simulation that gives the porosity, tortuosity and particle outer surface area at each location in the bed; the friction model and the dispersion model.

CFD

species balance

mass balance

momentum balance

enthalpy balance

particle model

friction model

dispersion model D

T

c

p

v

S(c)

F

packing simulation

ε

a

τ

S(T)

Packed bed model

Figure 1.4: Schematic representation of the CFD and packed bed models; major interactions and variables. The variables shown are the pressure p, the fluid velocity vector v, the temperature T and the concentrations c. The linking coefficients are the friction force vector F , the dispersion coefficients D and the temperature and component sources S(T ) and S(c). The bed packing is characterised by the porosity ε, tortuosity τ and specific particle outer surface area a at every point in the bed.

7

1. Introduction

For use in day to day engineering practice, the tool should have the following characteristics:

• The tool should be generic, so that many different reactor systems can be described without the need to modify the code

• The input of the model should be in terms of parameters known to the reactor engineer, avoiding parameters that are familiar only to the CFD expert

• The tool should run on a standard desktop or laptop PC

• The calculation time should be low enough to simulate several cases in a single day for a typical calculation

• The model should be able to handle single and multiple reactions, complex kinetics and complex particle shapes

• The model should also be valid for packed beds with a low bed-to-particle diameter ratio

• The tool should be freely available

1.4.1 Engine

The engine for the randomly packed bed reactor model will be a simplified version of the scientific CFD codes. To be useful for use in reactor design, the CFD package should also have a preprocessor to set-up a simulation and a postprocessor for displaying and analysing the results. The CFD package selected in this work is Comflow. In contrast with many larger CFD systems that require a flow dynamics expert to be operated, Comflow was developed as a tool for the process engineer. It is simple, in the sense that it supports only a subset of the general CFD functionality: 2-dimensional mostly hexagonal grids and only very simple turbulence modelling. On the other hand, there are comparatively extensive models for physical properties and flow restrictions. Comflow is a pre- and postprocessor for an external CFD solver, and it extends the solver with a library of routines to support the additional models, e.g., tube bundles and perforated plates (Roelofs, 1998).

In the past, Phoenics (http://www.cham.co.uk) has been used as a solver for Comflow, as well as a custom solver based on the (now historic) champion code (see e.g., Pun and Spalding, 1976). For the present work, it was decided to create a new version of Comflow based on a more modern CFD solver. The requirements for the solver, that it should be lightweight and that the source

8

1.4 Method

code should be available, led to the selection of Dolfyn (www.dolfyn.net). This is an open source, 3-D unstructured, single phase CFD code written in Fortran 90 and based on the numerical method described by Ferziger and Peric (2002). Since an open source CFD code is used as engine, and the source code of the packed bed model will also be released as open source, the result of this effort will be a tool that is freely available for every reactor engineer.

Dolfyn is a relatively simple CFD code that does not include many of the facilities needed to simulate packed bed reactors. Therefore, the code will be extended to implement variable physical properties of ideal gas mixtures, custom source terms (allowing the specification of the drag force F and the reaction source terms S) and anisotropic dispersion (D in figure 1.4).

1.4.2 Packed bed model

In a CFD code, the flow domain is divided into a large number of computational cells. The balance equations for mass, momentum, enthalpy and concentrations are linearised, discretised and then solved for each cell. As the resulting algebraic equations for each cell depend on all of the cell’s neighbours, a large (sparse) system of equations is formed. Because of the non-linear nature of the original equations the solution of this system requires an iterative procedure. After each iteration, new estimates for the state variables (pressure, temperature, velocity, temperature and composition) are made for each cell.

The packed bed model interacts with the CFD code through transport parameters (e.g., dispersion) and source terms (e.g., heat source and flow resistance). It provides the values of the parameters that are specific for a packed bed, based on the local values of constants (e.g., porosity) and state variables (e.g., temperature) that are calculated by the CFD code. These values are then used in the CFD code for the next iteration.

The local values of the transport parameters and the reaction rates are evaluated once for each cell for each iteration. Hence they are calculated very often during a simulation. Each sub-model calculation should therefore be computationally inexpensive.

The transport processes in a packed bed are diffusion (dispersion, conduction), convection and radiation. These processes have a limited characteristic length over which information can be passed, typically in the order of a few particle diameters. Therefore, theoretically, the data needed to calculate the transport parameter in a computational cell should depend only on the local values of the flow variables, i.e., the values for the cell itself and its direct neighbours. For instance, the radial dispersion in a cell should be given in terms of the velocity in that cell (and possibly that of a neighbouring cell); it cannot depend directly on the velocity at the inlet or at the centre of the bed.

9

1. Introduction

As another example, the diameter of the bed should not show up in any of the equations used to calculate the transport parameters. Many engineering models, however, use this kind of global information. Therefore, they cannot be used in a CFD model without modification.

In summary, we can conclude that to be able to calculate transport parameters locally in a CFD code, the submodels should have the following characteristics:

• the calculation should be computationally inexpensive

• the parameters should not depend directly on the geometry of the bed

• the parameters should not depend directly on global information

The important processes in a packed bed reactor are reaction, convection and dispersion. Reaction takes place on the scale of a single particle. For single reaction systems, simple particle geometry and/or simple kinetics, approximation methods are available that are accurate enough and computationally inexpensive. However, in the most important practical cases, multiple reactions play a role. For reaction networks in complex catalyst geometry, no approximation method is available. Therefore, this needs to be developed.

Convection and dispersion take place on the scale of the bed. Therefore they are determined by the structure of the bed. Near walls and solid inserts, the structure of the bed is different from the bulk of the bed; therefore also convection and dispersion are different near a wall and far from a wall. In a CFD model, we need to be able to describe the flow near the wall correctly, based on local vales of the bed parameters. Engineering models for convection and dispersion in packed beds that are available from literature are based on global or average bed parameters and therefore need to be adapted.

We need to describe the bed structure, especially near a wall, in terms of local parameters like the local porosity, tortuosity and the local specific outer particle surface area. These parameters are hard to determine experimentally, but easy to calculate for a simulated packing. A computer program can be used to generate simulated random packings and general rules for the bed structure parameters can be extracted from these.

Once the bed structure is known, a relation needs to be developed that describes flow in the catalyst bed (and especially flow resistance) as a function of the local values of the structure parameters (porosity, tortuosity, outer particle surface area) only. This model should not only give the correct pressure drop but also correctly describe the velocity profiles near a wall.

The dispersion (mixing) behaviour of the bed should also be described in terms of local bed structure parameters. As heat transfer with a packed bed

10

1.5 Structure

reactor takes place through a wall, the wall region is important for many non-isothermal reactions. Available engineering methods use fitted parameters (for instance, wall heat transfer coefficients) or profiles that are not consistent with the bed structure.

1.4.3 Sub-hypotheses

In order to fulfil the main hypothesis, existing engineering models need to be re-evaluated or adapted and new models need to be developed. We therefore formulate a set of sub-hypotheses that are conditions for the main hypothesis:

1. It is possible to develop a method to calculate chemical reaction with diffusion limitation inside a catalyst particle that requires a limited amount of computation time and yet is accurate enough to use in a reactor model. An approximation method that handles complex reaction networks, general kinetics and general catalyst particle shape can be developed.

2. The structure of a packed bed near a wall can be characterised such that the main parameters that influence transport processes in the bed (porosity, particle outer surface area and tortuosity) can be described as a function of the location in the bed. These functions can be derived from computer simulations of contained random packed beds.

3. The existing engineering rules to calculate flow resistance of packed beds can be adapted and rewritten in terms of the main parameters (porosity, particle outer surface area and tortuosity) so they can be used on a local scale in a CFD-based packed bed model. These models should correctly predict the pressure drop as well as velocity profiles near the wall, without using global fit parameters.

4. Dispersion models can be developed that describe heat and mass transport in a packed bed based on the main bed parameters in such a way that they can be used locally in a CFD-based packed bed model. These models should be consistent with the bed structure and cannot use superficial fit parameters

1.5 Structure

Each of the following chapters will address one of the sub-hypotheses given in the previous section. We start at the smallest length scale: the processes inside a catalyst particle. In chapter 2, the methods used to calculate the reaction rate in a porous catalyst particle, taking into account diffusion limitation are

11

1. Introduction

evaluated. A method is needed that is valid for general particle shape, general kinetics and also for reaction networks. For simple particle geometry and/or for simple reaction kinetics, analytic, numeric or approximative solutions are available in the form of the Aris number method (Wijngaarden et.al. 1998). However, if multiple reactions take place in a catalyst particle with a complex geometry, the approximation methods cannot be used. In that case, the effective reaction rate in the particle can only be found by integration of a set of partial differential equations. The computational effort for this is higher than we can afford in a CFD packed bed model. To remove this limitation, we will extend the Aris number method so that it can be used for reaction networks.

Chapter 3 focuses on the packing of the bed, especially near a solid wall. It is well known that near a solid surface like the container wall, the particles in a packed bed are no longer distributed in space in a random way. Due to the fact that the wall cannot be penetrated by the particles, a more or less regular pattern occurs near it. This pattern is most clear and well-defined for packings of smooth, identical spheres. For these, the porosity has a value of 1 at the wall, drops to a minimum value of about 0.2 at one radius from the wall, increases again and approaches the mean bed value (usually about 0.4) in a few oscillations, at a distance of five particle diameters from the wall. The profile is less well-defined for non-spherical particles or packings of particles with a broad particle size distribution. This so-called wall effect has been included in packed bed models in the past through an exponential approximation of the porosity profile. Although this may give a reasonable description of the porosity profiles for packings of irregular particles, it does not describe the minimum in the porosity profile found for regular particles. In addition, it is not solely the porosity that is of interest. The specific particle outer surface area and the tortuosity of the flow paths also determine the transfer of momentum, mass and heat between the fluid and the particles. Both these parameters are influenced by the presence of the wall. Therefore a procedure will be developed to determine the distribution of particles near the wall for perfectly spherical, mono-disperse particles through computer simulation of bed packings. The porosity, tortuosity and particle outer surface area profiles can be determined from the simulated particle distribution.

In chapter 4 the porosity, specific outer particle surface and tortuosity profiles from chapter 3 are used to calculate the flow distribution and pressure drop in a packed bed. The basic equation used to calculate flow resistance in a packed bed is the well-known Ergun equation. However, in its most common form, this equation does not contain the specific surface area or the tortuosity explicitly. It is clear that these two parameters have a large influence on the flow resistance. Therefore, a new equation will be derived from a parallel

12

1.5 Structure

twisted tubes model, using common engineering expressions for the pressure loss in tubes, at bends, expansions and contractions. The shape of the tubes is a function of the distance to the wall and related to the bed structure. With this equation, the radial velocity profiles in a packed bed can be calculated. These will be compared to measured packed bed velocity profiles that are available in the open literature.

Chapter 5 builds on the results from chapter 3 and 4 to model the transfer of mass and heat in a randomly packed bed. The effective heat conduction is a combination of molecular or stagnant terms (conduction through the fluid and solid phase) and the mixing that is caused by the random flow path through the bed. Both contributions depend on the porosity and tortuosity profiles, as well as on the velocity profile in the bed. A new dispersion model will be developed that takes these profiles into account.

Each of the sub-models will be verified and/or validated using literature measurement data for packed tubes. The use of the models for non-tubular geometries cannot be validated by lack of experimental data. However, provided the correct particle density distribution or porosity, tortuosity and particle outer surface areas are supplied, there is no fundamental reason why the model would not apply to different geometries.

Finally, in chapter 6 the sub-models developed in the previous chapters will be combined with the CFD code to address the main hypothesis.

Just like the individual sub-models, the resulting combined model will be validated using literature data measured in packed tube reactors. Unfortunately, the number of useful cases in literature even for cylindrical geometry is limited. For other geometries, no literature measurement data is available. Therefore, the application of the model for non-tubular geometries will be demonstrated instead of validated.

The data generated by the CFD simulations is relatively detailed, giving pressure, velocity, temperature and concentrations at every point of the grid. Therefore, it is quite a task to duplicate this resolution in experiments and measurements, and validation of the model on a detailed level in a complex simulation is not possible at this time. This applies to this work in particular, but also more in general as the possibilities for detailed modelling improve rapidly. To make similar improvements in measurement techniques for chemical reactors is outside the scope of this study and will be left as a challenge to future experimentalists.

13

1. Introduction

Literature Derckx, O.R., Dixon, A.G. (1996), Determination of the fixed bed wall heat transfer coefficient using computational fluid dynamics, Num. Heat Transfer part A: Applications, 29(8), 777-794

Ferziger, J.H., Peric, M, Computational methods for fluid dynamics, 3rd Ed., Springer Verlag, 2002

Pun, W.M., D.B. Spalding, A General Computer Program for Two-dimensional Elliptic Flows, HTS 76/2, CHAM Ltd. (1976), as described on http://www.simuserve.com/cfd-shop/hts76-2.htm

Roelofs, H.J. (1998), CFD-programma Comflow is van alle markten thuis, npt procestechnologie, 5 (september-oktober), 27-29

Wijngaarden, R.J., A. Kronberg, K.R. Westerterp (1998), Industrial catalysis: optimizing catalysts and processes, Wiley-VCH, Weinheim

14

Chapter 2

Chemical reaction and diffusion in a single catalyst particle

Summary

The goal of this chapter is to develop a method to calculate chemical reaction with diffusion limitation inside a catalyst particle that requires a limited amount of computation time and yet is accurate enough to use in a CFD reactor model. This method should be able to handle complex reaction networks, general kinetics and general catalyst particle shape.

The reaction rate usually depends in a non-linear way on several reactant (or product) concentrations and on the temperature. Therefore, the mass and heat balances lead to a set of non-linear differential equations, one for each component and one for the temperature. For a single reaction, it is easily found that these equations are not independent. It suffices to solve only the differential equation for a key component.

For reaction networks, the number of degrees of freedom is equal to the number of independent reactions, so the set of differential equations can also be reduced. However, the selection of key components is not always straightforward and not easy to do in a generic way. It is shown here that the number of equations in the set of differential equations can be reduced efficiently to the minimum number by adding a virtual key component to each reaction, with a pseudo concentration zj for each reaction j. The resulting differential equations, one for each virtual key component, can be solved simultaneously. The real concentrations and the temperature can be calculated from the pseudo concentrations through algebraic equations.

For single reactions, the Aris number approximation can be used to calculate the effectiveness factor for any particle shape and kinetics to a claimed

15

∑ ∑ ∣

2. Chemical reaction and diffusion in a single catalyst particle

accuracy of 5-10 % (Wijngaarden and Westerterp, 1994). The method is written here in a slightly more general way in terms of the virtual key component concentration. In this way, a single set of equations can be used to estimate the effectiveness factor for a single, isothermal or non-isothermal reaction with any number of components in any catalyst particle geometry.

For reaction networks, the particle geometry is important. In 1-dimensional particles, the balances lead to a set of ordinary differential equations. Although not trivial, it is feasible to solve these with an acceptable computational effort. For particles with a more complex geometry, the mass and heat balances lead to a coupled set of partial differential equations. The numerical solution of this set involves a relatively large computational effort. It is not feasible to solve this system for each cell and iteration in a CFD computation of a complete chemical reactor. Therefore, an approximation method is developed in this work to estimate the effectiveness factors for reaction networks with general kinetics in general particle geometry.

Our approach is based on the Aris number method. An estimate for the effectiveness factor for each reaction is calculated for the fast reaction regime and for the slow reaction regime; the actual value is then found by interpolation between the two regimes.

For the fast reaction regime, the reaction takes place in a thin shell of the particle. The profiles for the pseudo concentrations become independent of the particle size. Each strictly increases from zero at the particle edge to approach an equilibrium value at a certain penetration depth δj . The equations reduce to a set of ordinary differential equations. This set is integrated to find the zeroth Aris number for each reaction.

For the slow reaction regime, the effectiveness factor for the components will be close to one. The kinetics are linearised around this point. The deviation of the effectiveness factor for a component i from one will depend on the reactions in which component i is involved, but also on other reactions that produce or consume components that appear in the rate equation of each of these reactions. It is shown that the first Aris number An1,i for component i can be approximated by: ( )2 ∣nr nrΓ Vp ∂�j ∣

An1,i ≡ lim(1 − ηi 2) ≈ νi,j �k,s ∣ (2.1)

η→1 �i,s Ap ∂zk zk→0j=1 k=1

Note the cross terms on the right hand side that give the dependence of reaction j on the value of the pseudo concentration of all other reactions k.

To obtain an estimate for the effectiveness factor we need to interpolate between the fast and slow regimes. For single reaction systems, Wijngaarden

16

and Westerterp (1994) propose an implicit interpolation formula to obtain the effectiveness factor from the values of An0 and An1. This approach can not be used for reaction networks. Instead, an explicit method is presented, based on the penetration depth δj of reaction j.

The approximation method is demonstrated for the ammoxidation of propene in the Sohio process. The kinetics for this process are known from literature. The reaction network consists of six reactions. Each reaction is assumed to be first order in its hydrocarbon reactant concentration and includes an Arrhenius type temperature dependence. The effectiveness factors are calculated using the approximation method as well as by direct integration of the reduced equations for a range of slab-shaped catalyst particle and for a ring shaped catalyst particle. The approximation method correctly describes the trend of the effectiveness factor curves as a function of particle size. The curves take the correct values in the limits of low and high reaction rate, and show the correct trend in between. Effectiveness factors higher than one and lower than zero are correctly predicted and in the right order of scale. It is shown that the error made in the approximation is in the order of 5 % under most conditions and for the primary reactions, but may be as large as 50 % for the secondary reactions in conditions where the effectiveness factor is much larger than one. The large errors are probably caused by the fact that the cross terms in the calculation of An1 are taken at bulk conditions. The effect of the cross terms is only large if the reaction rate itself is very low compared to the other reaction rates. Consequently, the larger errors do not have a large effect on the mass balances.

To judge the effect of the errors in a practical calculation, a very simple model of an ideal tube reactor packed with ring-shaped particles is used to calculate the conversion and selectivity towards the desired product and the by-products for the Sohio process. The maximum difference between the selectivity curves based on the approximation and those based on the full numerical solution is less than 3 percent point, which is satisfactory. It is concluded that for the systems that have been modelled, the approximation method can be used with sufficient accuracy to calculate the effectiveness factor. In principle, the method is valid for any particle shape and for general kinetics. Only first order kinetics with Arrhenius type temperature dependence were tested in this work. Application of the approximation method for other particle geometries and other reaction systems is left for future studies.

17


2.1 Introduction

In packed bed reactors, usually porous particles are used as carrier for the catalyst. These particles have a large internal surface area (hundreds of square meters per gram). On this surface, catalytically active sites, for instance nickel, platinum or cobalt atoms, are dispersed. The catalyst particles can have many shapes; spherical catalyst particles are sometimes used but cylindrical particles are more common as the are more easily produced. More complex extruded shapes like rings, trilobes and quadrulobes are often chosen to optimise the pressure drop and conversion for a specific process.

For a heterogeneously catalysed reaction, the following steps can be distinguished (see figure 2.1):

1. diffusion of the reactant from the bulk of the fluid through the hydrodynamic boundary layer to the outer edge of the particle

2. diffusion of the reactant through the pores into the particle to reach an active catalytic site

3. reaction on the active site to form products with a certain reaction rate

4. diffusion of products through the pores to the outer edge of the particle

5. diffusion of products through the hydrodynamic boundary layer into the bulk of the fluid

In this chapter we will only consider the processes inside the particle itself; transport through the boundary layer will be discussed elsewhere in this work.

Inside a porous catalyst particle, there is a balance between transport of reactants and products by diffusion and conversion by the chemical reaction. It is clear that if the particle is large (or if the reaction rate is high compared to the diffusion rate), the reactant concentration will decrease strongly inside the particle. In this case, the reaction will mainly take place in a thin layer on the outside of the particle and the central part of the particle will not be used. This is a waste of reactor space and catalyst material. If the particles are small (or the reaction slow compared to diffusion), the concentration in the particle will be (almost) the same as in the bulk gas and all catalytic sites of the catalyst are used to an equal degree. However, small particles give rise to a high pressure drop over the bed which is usually not desirable. Therefore, packed bed reactors are often designed in a way where the particle size is a trade-off between catalyst usage and pressure drop. In other words, most packed bed reactors are operated in a range where the reaction can be considered neither ’fast’ nor ’small’. In these cases, the effective reaction rate

18

2.1 Introduction

CA,b

CA,s

CB,b

CB,s

1

2

3

4

5 A

B

boundary layer

catalyst particle

Figure 2.1: Schematic representation of the steps involved in chemical reaction in a porous catalyst particle for the reaction A → B and typical concentration profiles.

depends not only on the bulk composition and temperature but also on the profiles inside the catalyst particle.

To calculate the conversion of reactants to products in a packed bed reactor, we need to be able to predict the performance of a single catalyst particle as a function of the bulk conditions. Sometimes, only one reaction takes place between reactants and only the overall conversion needs to be calculated. However, often there are side reactions leading to different products or the desired product can react with a reactant to form an undesired by-product. In those cases, the selectivity of the reactor is an even more important factor than the conversion, because it determines the economy with which reactants are used and the amount of (possibly hazardous) waste that is produced.

Computational Fluid Dynamics (CFD) can be used as a tool in the design of more efficient reactors using heterogeneous catalysis. In a CFD model, the reactor is divided in to a large number (typically thousands to millions) of small cells. The flow equations (mass, momentum and energy balances) are

19


solved for each of these cells. To do this, we need to be able to predict the performance of particles at the local conditions in each cell. Ideally, we should be able to use simple (spherical, cylindrical, slab-shaped) and complex particle shapes (trilobes and other extrudates), for single reaction systems, for parallel and consecutive reactions and for even more complex reaction networks. Also, the computation time available to calculate the effective reaction rate in the particles is small because the calculation needs to be carried out many times (once for each cell for each iteration). The goal of this chapter is to give efficient methods to calculate internal diffusion limited reaction in porous catalyst particles that can be used in such a CFD code.

We will first briefly discuss discuss the basic diffusion-reaction equations for reaction networks. Although the balance equations lead to one (partial) differential equation for each active component in the system, it is known that in fact the number of independent differential equations is in fact equal to the number of (independent) reactions which is always lower or equal. We will introduce a generic way to find the independent differential equations by the introduction of a virtual key component for each reaction. For one-dimensional particles (spheres, long cylinders and large slabs), the balance equations lead to a set of coupled ordinary differential equations. This set of equations may be solved numerically with an acceptable computational effort.

For more complex particle shapes, a system of coupled partial differential equations needs to be solved to calculate the conversion in a single particle. Although this is possible for a single particle, the purpose of this work is to calculate a complete packed bed reactor in a Computational Fluid Dynamics code. It is certainly not desirable to solve a set of partial differential equations for each computational cell and for each iteration in a CFD model.

Even though CFD codes may be able to compute a solution to a high numerical accuracy, the errors made in modelling and determination of parameters is usually in the order of several percents. Therefore. there is no need to spend a lot of effort to compute the reaction rate to a much higher accuracy than that. Therefore, approximation methods to estimate the conversion in a catalyst particle are acceptable for our purposes.

To estimate the conversion in a catalyst particle for simple cases, in Thiele (1939) introduced the notion of the effectiveness factor of a catalyst particle, which is defined as the conversion in the particle as a fraction of the conversion that would be found if no mass transfer limitation would be present. The value of this effectiveness factor can be calculated analytically for single reactions with some specific forms of the kinetic equations (for instance n-th order reactions) in one-dimensional particles. Wijngaarden and Westerterp (1994) have introduced the so-called Aris number method that allows the estimation of the

20

2.2 Theoretical background

effectivity for single reactions with arbitrary single reaction kinetics and arbitrary shapes. For multiple reactions, no approximation methods are available so the only option is to use numerical methods. Therefore, the situation in the existing literature can be summarised as is shown in table 2.1.

Table 2.1: Options available for the calculation of reaction and diffusion in catalyst particles for different particle shapes and kinetics

particle shape

kinetics simple (1-D) complex

single simple analytical solution numerical (PDE) Aris method single complex Aris method

network simple

network complex

numerical (ODEs)

numerical (PDEs) ?

As the solution of partial differential equations is too computationally intensive to be performed within a CFD calculation, there are currently no options for network kinetics in a complex particle shape. To be able to calculate this class of problems in a CFD packed bed simulation, a method will be developed to use the Aris number approach for reaction networks.

The methods will be demonstrated using literature kinetic data for the Sohio process for the ammoxidation of propene. Note that, although the remainder of this work is mainly focused on packed bed reactors, the results of this chapter are equally applicable to other reactor types where catalyst particles are used like fluidised bed reactors.


2.2.1 Diffusion

It is assumed that the diffusion flux of component i inside a porous particle is proportional to the concentration gradient:

∂ciΦ ”

i = −Deff,i (2.2)∂x

21


The proportionality constant is the effective diffusion coefficient Deff,i for component i in the gas mixture in the particle pores. This effective diffusion coefficient differs from the bulk diffusion coefficient of component i in the gas mixture due to three effects:

• part of the space is occupied by solids and therefore not accessible to gas molecules

• the pores in the particle are often not straight. This increases the length of the path a molecule has to travel to penetrate a certain distance into the particle

• the size of the pores is often in the same range as the free path length of the gas molecules, so collisions of the molecules with pore walls (i.e., Knudsen diffusion) needs to be taken into account

The shape of the channels is characterised by the pore tortuosity τc, which is defined as the length of a channel divided by the distance between the start and end points. Commonly, the effective diffusion coefficient is expressed as

εpDeff,i = Dt,i (2.3)

τc

where Dt,i is the diffusion coefficient of component i in a straight pore and εp

the particle porosity. The tortuosity is usually determined experimentally, and typically values in the order of 3 or higher are found (e.g., Fogler, 1986). This is remarkable, as a channel needs to be quite labyrinth-like to get such a high value for the tortuosity. For a porous conglomerate of small solid particles (as the porous materials usually are), one would sooner expect values between 1 and 2.

On closer inspection, it can be concluded that equation (2.3) is not correct. The tortuosity of the channels has two effects: the distance along the length of a channel that needs to be travelled is longer, but also the concentration gradient needs to be taken along the channel axis and not along the coordinate axis. Therefore, the effective diffusion coefficient decreases not by the tortuosity but by its square

εpDeff,i = Dt,i (2.4)

τ 2 c,act

As the general practice is to use equation (2.3) instead of equation (2.4), we will do so here, too. However, it should be kept in mind that the ’tortuosity’ used in that equation is actually the square of the actual, geometric tortuosity

τc = τc,2act (2.5)

22

[ ]


This of course also explains the high values reported for τc.

The error made in the estimation of the effective diffusion coefficients is relatively large. Binary bulk diffusion coefficients are commonly estimated with an error of about 5 %. In addition, most porous particles will contain a distribution of pore sizes, and not all pores may be accessible to all gas molecules. The shape of the pores and the characteristics of the maze structure with interconnected or blocked-off pores are hard to model and the tortuosity factor is often a rough guess. The resulting error in the effective diffusion coefficient may well be in the order of 10% or more. Therefore, replacing the diffusion flux (equation 2.2) by the Maxwell-Stefan equations to account for non-equimolar diffusion (drift flux effects) will in most cases not drastically improve the accuracy, but it will drastically increase computation time.

2.2.2 Kinetics

In this work, we will try to write equations that are independent on the choice of the kinetic or reaction rate equations �. However, there will be some restrictions on the kinetics that can be used, so we will discuss a simple kinetic model here.

In theory a gas phase reaction can be thought to be the result of a collision between reactant molecules. The probability that a molecule of component A and a molecule of component B collide is proportional to the concentration of A and the concentration of B. The energy of the collision should be large enough to overcome the energy barrier; the fraction of collisions that lead to a reaction therefore increases with temperature with an Arrhenius factor. Apart from that, the fraction of successful collision also depends on other conditions like the orientation of the molecules when they collide, so there is also a temperature independent factor. Therefore, for a reaction

νAA + νB B → νC C (2.6)

the reaction rate should in theory be

−EA νA νB−�A = k0 exp c (2.7)A cBRT

All reactions are in principle reversible; for the above example, if νC molecules of C collide with sufficient energy, they could react to reproduce the original number of molecules A and B. The rate at which this happens of course depends on the value of νC : the larger it is the lower the chance that so

23

[ ] [ ]

∏

[ ]


many product molecules collide at (more or less) the same time. The reaction should be written as:

νAA + νB B � νC C (2.8)

with reaction rate

−EA,f νA νB −EA,b −νC−�A = kf,0 exp cA cB − kb,0 exp cC (2.9)

RT RT

Starting from pure A and B, the reaction will continue until there is thermodynamic equilibrium; there is a relation between the equilibrium constant K and the kinetics:

nc [ ] νi

kb,0 EA,f − EA,bK ≡ c = exp (2.10)i kf,0 RT

i=1

For a heterogeneous reaction, the situation is somewhat more complicated. For the reaction to occur, at least some of the reactants need to be adsorbed on the catalytically active site on the surface. The absorption of a reactant is in itself an equilibrium step in series with the actual reaction. The product(s) formed are often also adsorbed on the surface and therefore need to desorb to make the active sites available for another reaction. If one considers these steps, the rate equation in the form of a Langmuir-Hinshelwood equation is found, for instance,

νA νB νCkf c kbcA cB C�i = − (2.11)νA νB νC1 + kf cA cB 1 + kbcC

In reality, there are many complicating factors and it is in general impossible except in a few cases to calculate the reaction rate based on theoretic considerations. Therefore, the kinetic equations are determined by fitting to experimental data. the kinetic equations found often lump together many reaction details like intermediate steps. One possibility for the form of the equation that is often encountered is power law kinetics:

m p −EA�A = k0CAn cB cC exp (2.12)

RT

which is similar to the theoretical gas phase reaction but the exponents n and m are no longer connected with the stoichiometric coefficients and could take any value; usually they are not integer numbers and sometimes they are even

24

( [ ])

( )


negative. The validity of the equations is limited to the range of conditions that were included in the experiment.

For many kinetic equations, the reaction rate will be zero when the concentrations of the (main) reactants are zero. For equilibrium reactions, some finite reactant concentration will be found at equilibrium. For power law kinetics with reaction orders of zero or lower, no equilibrium state exists; the reaction rate is constant or even increases with decreasing reactant concentration. In this case, the kinetic equation (which is usually determined experimentally at higher reactant concentrations) can be modified to tend to zero for lower reactant concentrations, without modifying its behaviour at higher concentrations, e.g., by multiplying with

�∗ = �× 1 − exp − ci

(2.13) ci,low

where ci,low is a low value of the concentration of component i for the specific operating conditions.

The concentration of reactants that are present in a large excess, will only change by a small factor as the reaction takes place. Therefore, the observed reaction rate is almost independent of these concentrations, and the coefficient is almost zero. Usually the reaction rate equation is simplified by neglecting these factors. However, it may be that these concentrations are depleted somewhere in a catalyst particle; as the reaction rate has been made independent of this concentration the reaction will continue and unphysical negative concentrations can occur in the model. To prevent this, the modification shown above can also be applied to these reactants.

2.2.3 Balance equations

The microscopic mass, concentration and and heat balances inside a catalyst particle can be written as a set of partial differential equations that describe the diffusion and reaction processes

∂ci ∂ ∂ ∂ Φ ” Φ ” Φ ” + = � (2.14)i,x + i,y + i,z i

∂t ∂x ∂y ∂z

where �i gives the reaction rate for component i, which may be a non-linear function of the temperature and gas composition in the particle. Under the assumption that molecular transport inside the particle is by diffusion only (no convective transport) and that the effective diffusion coefficient is constant:

∂ci ∂2ci ∂2ci ∂2ci + Deff,i + + = �i (2.15)

∂t ∂x2 ∂y2 ∂z2

25

( )

( )

∑

∑


We will assume stationary behaviour for the particles

∂2 ∂2 ∂2ci ci ciDeff,i + + = �i (2.16)

∂x2 ∂y2 ∂z2

The way in which this set of equations can be solved depends on the geometry of the catalyst particles and the complexity of the reaction scheme. When diffusion inside a particle takes place in one (coordinate) direction only, the set of partial differential equations reduces to a set of ordinary differential equations. This is the case for spherical particles and for cylindrical and slab-shaped particles that are either infinitely long (and wide) or have blocked-off faces (that can be treated as symmetry planes). For brevity, these particles are referred to as one-dimensional particles; in fact the particles are perfectly three-dimensional; it is the balance equations that are one-dimensional.

For one-dimensional catalyst particles, the partial differential equation 2.16 reduces to an ordinary differential equation:

d2ci m dciDeff,i + = �i (2.17)

dx2 1 − x dx

Here m is the power with which the surface area of the particle perpendicular to the transport flux changes as a function of the coordinate x. For infinite slabs, cylinders and spheres, m has the value 0, 1 and 2, respectively. The boundary conditions for equation (2.17) are that the concentration is assumed to be known at the outer edge of the particle and that the concentration is bounded inside the particle:

x = 0 → ci = ci,s (particle edge) dci

x = 1 → = 0 (particle center) dx

For a system with nc components involved in nr reaction, the system is written as: ( ) nrd2ci m dci

Deff,i 2 + = − νi,j �j (i = 1..nc) (2.18)

dx 1 − x dx j=1

The energy balance leads to an additional differential equation that is analogous to equation (2.18)

( ) ncλ d2T m dT −ΔrHj+ = �j (2.19)

ρcp dx2 1 − x dx j=1

ρcp

26

2.3 Reduction of the number of differential equations

The first factor on the left hand side λ/(ρcp) has the same dimensions as the diffusion coefficient (m2/s) in equation (2.18), the factor −ΔrHf /(ρcp) is analogous to the stoichiometric constant νij . Note that the factor ρcp is included to demonstrate this analogy only and cancels out on both sides of the equation. The boundary conditions for equation (2.19) are:

x = 0 → T = Ts (particle edge) dT



For one-dimensional particles the balance equations lead to a set of coupled, usually non-linear, second order ordinary differential equations, where the number of equations is equal to the number of components nc plus (2.18) plus one for the temperature (2.19). The concentrations of the components are not independent of each other since they are coupled through the stoichiometry of the reactions. Likewise, the temperature is coupled to the concentrations through the heat of reaction. For simple, single reaction systems, this is often directly clear from the balance equations. For multi-reaction systems this is not as obvious. In this section a method is presented to write down the system of balance equations in an efficient way so that the number of equations that need to be solved is reduced from the number of different chemical components nc (plus one for the temperature) to the number of stoichiometrically independent reactions nr. Note that for any reaction system, nc ≥ nr.

This procedure is generic in that it can be done without any prior knowledge about the reaction scheme (e.g., key components) and the reaction kinetics and therefore it is ideal for application in a flexible computer code.

2.3.1 Single reaction

Consider a general (isothermal) chemical reaction where nc species are involved:

(−νa,1)A1 + (−νa,2)A2 + ... + (−νa,na)Ana �

νna+1B1 + νna+2B2 + ... + νncBnb (2.20)

The reaction takes place in a spherical particle. Now consider a spherical sub-volume with radius r inside the particle. In a stationary situation, every νa,i

molecules of reactant Ai that enter the sphere are converted to νb,j molecules

27

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

∣ ∣ ∣ ∣ ∣ ∣

( )


of each product Bj . To keep the mass balance, these product molecules must be transported out of the spherical sub-volume. Therefore, the mole fluxes of all species A and B at the radial position r are related:

φ ” i φ ”

j= (i = 1..nc; j = 1..nc) (2.21)

νi νj

or, with diffusion according to Ficks law:

Deff,i dci Deff,j dcj= (2.22)

νi dx νj dx x=r x=r

Hence all concentration gradients are linked to each other by a constant factor. Equation (2.22) can be integrated from the edge of the particle (x = 0) to some position r:

∫ ∫ ∣ r ∣ rr rDeff,i dci Deff,j dcj Deff,i Deff,jdx = dx ⇒ ci = cj (2.23)

0 νi dx 0 νj dx νi 0 νj 0

Since the concentrations at the particle edge ci,s are known as the boundary conditions, there is an algebraic relation between all concentrations at each position inside the catalyst particle:

Deff,i Deff,j(ci − ci,s) = (cj − cj,s) (2.24)

νi νj

Deff,iνj⇒ cj = (ci − ci,s) + cj,s (2.25)Deff,j νi

Now we choose a key component i. With the aid of equations (2.24), the concentrations of the non-key components j can be eliminated from the reaction rate. Thus, the balance equation can be written as a function of the chosen key component i only. For the key component, the mole balance equation (2.18) reduces to:

Deff,i d2ci m dci + = −�(ci) (2.26)

νi dx2 1 − x dx

Depending on the complexity of the rate equation, this equation may be solved analytically or numerically to yield the concentration profile of component i. From the solution, the concentration profiles of the other components (and the temperature profile) can be calculated using equations (2.24). Thus, we have reduced the set (2.18) from nc equations to only one.

28


2.3.2 Multiple reactions

Slab-shaped porous particles

The simplest shape for a one-dimensional particle is a slab; a flat plate that is either infinitely wide and long or has side-faces that are closed for diffusion and thermally insulated. For a slab-shaped particle, the cross-sectional area does not depend on the depth in the particle, so the geometric parameter m is 0. Thus, the equations (2.18) simplify to:

nrd2 ∑ Deff,i

ci = − νi,j �j (i = 1..nc) (2.27)

2dxj=1

For each reaction j we define a variable zj such that

d2zj 2

= −�j (j = 1..nr) (2.28) dx

with boundary conditions

x = 0 → zj = 0 (particle edge) dzj


The variable zj can be interpreted as the (scaled) concentration of an imaginary key component that is produced by reaction j with stoichiometry 1, with a bulk concentration 0 and that is not involved in any other reaction. On the right hand side of the equations, a concentration always appears together with its diffusion coefficient (see for example equation 2.26). For the pseudo concentration, we will incorporate the diffusion coefficient into the variable zj , which consequently has units [mol/m3 × m2/s] or [mol/m/s]. Note that the zj

are just a mathematical construction; it is irrelevant whether a key component actually exists for a reaction j or not.

The reaction rates �j are functions of the concentrations ci. If the concentrations can be rewritten as functions of the variables zj , then the reaction rates can be rewritten as functions of the zj only and the set of nr equations (2.28) can in principle be solved. In order to find this relationship between the zj and the ci, we substitute equation (2.28) into equation (2.27):

nrd2 ∑ d2ci zjDeff,i = νi,j (i = 1..nc) (2.29)

dx2 dx2 j=1

Equation (2.29) is essentially a sum of second derivatives to x which can be integrated between the center of the particle (x = L) and certain x: ∫ nr ∫ x d2 ∑ x d2ci zj

Deff,i dx = νi.j dx (2.30)2 2dx dxx=L x=Lj=1

29

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∑

∑

∑

∑

∑


so

x nr xdci dzj

Deff,i = νi,j (2.31)dx dx x=L x=Lj=1

and with the boundary conditions at x = L:

dcinr dzj

Deff,i = νi,j (2.32)dx dx

j=1

This equation simply states that the flux of each chemical species is related to the flux of the ’key component’ of each of the reactions that takes place through the stoichiometry. Equation (2.32) can be integrated once more, this time from the edge of the particle (x = 0) to a certain depth x:

∫ x ∫ x ∑ xdci dzjDeff,i dx = νi,j dx (2.33)

dx dxx=0 x=0j=1

which, with the boundary conditions at x = 0 leads to:

nr

Deff,i (ci − ci,s) = νi,j zj (2.34) j=1

or

1 nr

ci = ci,s + νi,j zj (2.35)Deff,i j=1

The concentration of component i at a location inside the particle is seen to depend on the boundary concentration and a contribution from each reaction that takes place. Equation (2.35) again has an equivalent for the temperature:

1 nr

T = Ts + −ΔrHj zj (2.36)λ

j=1

The reaction rate equations �j are non-linear functions of the concentrations ci

and the temperature T . Using equations (2.35) and (2.36), the reaction rates can be expressed in the variables zj only. Thus the set of (nc + 1) differential equations (2.18, 2.19) with (nc + 1) unknowns can be reduced to the set of nr

differential equations (2.28) that we used to define the variables zj .

30

2.4 Approximating the effectiveness factor

Cylindrical and spherical particles

The procedure described above can be generalised to spherical and cylindrical particles. Hereto, we define the variable zj by:

d2zj m dzj 2

+ ≡ �j (2.37)dx 1 − x dx


x = 0 → zj = 0 dzj

x = 1 → = 0 dx

so that

( ) nr ( ) d2 ∑ d2ci m dci zj m dzj

Deff,i + = − νi,j + (i = 1..nc)dx2 1 − x dx dx2 1 − x dx

j=1

(2.38)

Equations (2.28) and (2.29) for slab-shaped particles are special cases of equation (2.37) and (2.38). It can easily be seen that the solution reached for a slab-shaped particle (equation 2.35) is also a solution of equation (2.38) and conforms to the boundary conditions of equation (2.37).


Although the set of differential equations has been reduced to the minimum size, the solution still requires quite some computational effort. Especially for more complex particle shapes, for which a set of non-linear partial differential equations needs to be solved numerically. This usually involves defining a 2 or 3-dimensional grid of (typically thousands of) nodes to describe the geometry, discretisation of the equations and an iterative solution procedure. As it is not desirable to do this for each computational cell and each interation of a CFD model, we will be looking at approximation methods to reduce the computational load.

2.4.1 Single reaction systems

Wijngaarden and Westerterp have introduced generalised formula for the approximation of the effectiveness factor for a single reaction with arbitrary kinetics in a catalyst particle of any shape (Westerterp and Wijngaarden, 1992;

31

( ) ∣

∣ ∣

( ) ∣ ( ) √ ∫


Wijngaarden and Westerterp, 1994; Wijngaarden et.al., 1998). In this section, this so-called Aris number approach is rewritten along the same lines, making use of the imaginary key component z as was introduced above.

Wijngaarden et.al. define the zeroth and first Aris number as:

1 ∣ ( )∣ ∣ 2 ∣An0 ≡ An1 ≡ 1 − η (2.39)2 ∣ η→1η η→0

An0 is the square of a Thiele modulus that (by definition) brings together the effectiveness curves for all geometries for fast reactions. For fast reactions, only a thin layer of the catalyst particle contributes in the reaction. If this layer is thin enough, it can be considered as a flat plate, i.e., a slab. Hence, the reaction-diffusion process can be described by:

d2z 2

= � (2.40)dx

or, after some manipulation: ( )2dz

d = �dz (2.41)dx

At the edge of the particle (x = 0), the values of z is known to be 0 (by definition). Since the reaction takes place only in the outer layer of the particle with thickness δ, the composition will approach thermodynamic equilibrium at some distance X > δ from the edge. Since every possible composition is characterised by a single value of z, the equilibrium value zeq exists that corresponds to the equilibrium composition. This value can be found by setting the reaction rate to zero:

�(zeq) = 0 (2.42)

The concentration gradients will approach zero at x = X. We can integrate equation (2.41) between x = 0 and x = X:

( dz

)2 ∣∣ ( dz

)2 ∣∣ ∫ zeq ∣ − ∣ = 2 �dz (2.43)dx ∣ dx ∣ 0

x=X x=0

After some rearrangement:

− dz ∣ 2 zeq �dzA pdx x=0η| ≡ p

= A 0

(2.44)η→0 Vp �| Vp �|z=0 z=0

32

( )

∣ ∣ ∣

∣ ∣

∣ ∣

∣ ∣ ∣ ∣

[ ]


The integral on the right-hand side can be calculated (either analytically or numerically) for well-behaved reaction kinetics. The definition of An0 can be rewritten as: ( )∣ ( )2

1 ∣ Vp (�| )2 ∣ z=0An0 = = ∫ (2.45)η2 ∣ 0

η→0 Ap 2 zeq

�dz

Due to the definition of zj this expression for the zeroth Aris number can be used for both isothermal and non-isothermal reactions with any number of components.

The first Aris number is defined at low reaction rates, where the effectiveness factor differs only slightly from 1.

An1 ≡ 1 − η2 η→1

(2.46)

Note that An0 is the square of the Thiele modulus, so η approaches one as An0

approaches zero. We use a Taylor expansion around An0 = 0 to approximate the effectiveness factor at slow reaction rates:

∂η ∣ η|η→1 ≈ 1 + An0 (2.47)

∂An0 An0→0

To calculate the derivative, we linearise the reaction rate around the rate at bulk composition

∂� ∣ �∗ = �| + z (2.48)z=0 ∣∂z z=0

where the star superscript indicates a value for the linearised kinetics. We can write the first-order relation for An ∗

0: ( )2 ∣ Vp ∂� ∣

An ∗ 0 = (2.49)

Ap ∂z z=0

Now we can write the definition as: ( )2 ∂η∗ ∣ ∂η∗ ∣

An1 = − An ∗ − 2 An ∗ (2.50) ∣ 0 ∣ 0∂An ∗ ∂An ∗ 0 An ∗

0→0 0 An ∗ 0 →0

For small values of An0, we can neglect the quadratic term, so that

∂η∗

An1 = −2 An ∗ (2.51)∂An ∗

00

33

∣ ∣

√

∣ ∣ ∣ ( )


The derivative of the effectiveness factor for the linearised reaction rate on the right hand side depends on the particle shape only and is called the geometry function Γ:

dη∗

Γ ≡ −2 (2.52)dAn ∗

0

The geometry function can be calculated for any particle geometry; for a slab it has the value 2/3; for a cylinder 1 and for a sphere 6/5. In this way, the first Aris number can be calculated as the product of a part that depends on geometry only and a part that depends on reaction kinetics only: ( )2 ∣

Vp ∂� ∣ An1 = Γ (2.53)

Ap ∂z z=0

The effectiveness factor for fast and slow reactions can be approximated using the definition of An0 and An1 respectively:

1 η ≈ √ (fast reaction) (2.54)

An0

η ≈ 1 − An1 (slow reaction) (2.55)

For practical use, Wijngaarden and Westerterp (1994) recommend the following (implicit) interpolation formula for the entire An0 range:

1 η ≈ √ (2.56)

1 + ηAn1 + (1 − η)An0

The flux in to the particle for each of the chemical components i can be calculated from the reaction effectiveness as:

dci VpΦ ”

i = −Deff,i ∣ = η νi �|z=0 (2.57)

dx x=0 Ap

for the heat flux: ( )

Φ ” H =

λ ρcp

dT dx

= η Vp

Ap

ΔrH ρcp

�|z=0 (2.58)

2.4.2 Reaction networks

If more than one reaction is taking place at the same time, the situation is inherently more complex than in the single reaction case. For a reaction network, distinction needs to be made between effectiveness factors for reactions and

34

∣


effectiveness factors for components. The effectiveness factor for components may depend on several reactions, some fast and some slow. The effectiveness factor for a reaction in a reaction network only depends on the rate of one reaction. The effectiveness factor ηj for reaction j is identical to the effectiveness factor for pseudo concentration zj .

dzj ∣Ap dx ∣ ηj =

Vp �j |x=0 (2.59)

zj =1

Of course, the reaction rate �j probably depends on at least one reactant concentration and that reactant may be involved in other reactions in the network. The other reactions influence the concentration profiles and therefore the rate and effectiveness factor of this reaction.

No general approximation method is available for reaction networks. Therefore, we will develop one here. The approach will be the same as for the single reaction Aris number method: we will calculate both Aris numbers for each reaction and interpolate between the fast reaction estimate for the effectiveness factor based on An0 and the slow reaction estimate based on An1. The interactions between the reactions need to be taken into account in the calculation of the Aris numbers.

For a given process, the reaction kinetics and diffusion coefficients are given and cannot be changed. The only parameter that we are in direct control of is the particle size. It will be useful to think of the slow reaction regime as the limit for small particles and the fast reaction regime as the limit for large particles.

The zeroth Aris number

In the fast reaction regime, the concentration profiles inside the particle can be quite complex: in contrast with the single reaction situation where all profiles are strictly increasing or decreasing, for reaction networks maxima and minima can appear in the concentration profiles. This occurs if a component is e.g., produced by one reaction and consumed by another. In contrast, the profiles for the pseudo concentrations are all strictly ascending as the zj are involved in only one reaction by definition. Therefore, we will base the calculation of An0 on the z-profiles.

We cannot use the regular equation (2.45) to calculate An0,j because we cannot calculate the integral in the denominator as it depends on the values of all other z. Therefore, the basis for the calculation of An0,j is the intermediate

35

( ) ∣


result of equation (2.44):

dzj ∣Ap −

dx ∣ x=0,η→0

η| ≡ (2.60)η→0 Vp �j |zj =0

The term in the numerator gives the gradient of zj at the edge of the particle when the effectiveness factor approaches zero, e.g., for a very large particle. If we take a slab-shaped particle with closed sides and increase the thickness of the slab, the concentration of the reactants in the center will decrease (the concentration zj will increase) and the conversion in the particle will increase; therefore the gradient at the edge of the particle must also increase. However, the effect of increasing the particle size will become less pronounced when the concentration in the center becomes low (or: close to the equilibrium value). At a certain point the concentration profile will be located for the largest part near the edge of the particle and increasing its size only adds more volume to the inactive central part of the particle. Hence, the total conversion, and therefore the gradient at the particle edge, will not change if the particle size is increased any further. In practice, this gradient at the edge quite rapidly approaches the one shown in the numerator of equation (2.60).

Of course, the size of the particle where this occurs depends on the rate of the reaction; for fast reactions this will be at a smaller size than for slow reactions. However, there is a certain minimum particle size that is large for all reactions. In the center of such a particle, all zj are at their equilibrium values which can be calculated by setting all component reaction rates to zero simultaneously (provided the rate equations are well posed in the sense explained earlier).

To calculate the gradient at infinitesimal effectiveness, we need to solve the differential equations

d2zj 2

= �j (2.61)dx


x = 0 → zj = 0 dzj

x = X → = 0 dx

In fact, there is an additional known boundary condition: zj = zj,eq at the particle center, where the equilibrium value of zj can be calculated by setting all reaction rates to zero. However, it is probably easier to solve the differential

36

∣ ∣


equations to obtain the value of zj,eq instead. Of course, we need an estimate of X, but this is usually not hard to give and the result is not sensitive to the actual value provided that X is chosen large enough; this does not need to influence the calculation time or accuracy if an adaptive grid method is used.

The solution of the differential equations and then taking the gradient at the surface is equivalent to the integral calculated in the original Aris number approach (equation 2.45); however, for multiple reactions all differential equations are solved simultaneously so that the influence of one reaction on the other is taken into account in a natural way.

Apart from the gradient at the particle surface, the solution of the differential equations give us the equilibrium values zeq,j as well as an estimate of the thickness of the reaction zone for each reaction.

Once we have the particle edge gradient at zero effectiveness, we can easily calculate An0: ⎛ ⎞2

Vp �j |⎜ z=0 ⎟An0,j = ⎝ ∣ ⎠ (2.62)

dzj ∣Ap dx ∣

z=0,η→0

Note that as before, An0 does not depend on the actual particle shape.

The first Aris number

The first Aris number is defined as (1 − η2) for effectiveness factors close to one. In the limit for very small particles, all reactions in the network will be slow in this sense; all zj will be close to zero so all concentrations will be close to the bulk fluid concentrations. To simplify the expressions, the slow reaction regime is based on the effectiveness factor for components instead of reactions; of course once these are known, the numbers for the individual reactions can also be calculated.

As a starting point we will take the single reaction equation for An1 (2.53):

( )2 ∣ Vp ∂� ∣

An1 = Γ (2.63)Ap ∂z z=0

In the slow reaction regime the form of the kinetic equation is not important, so it may be expected that a similar form should also exist for each component in a multi-reaction system. The derivative in the last factor should of course take into account the effect of all reactions in which a given component is involved. Furthermore, there are cross terms because of the influence of reaction i on the

37

∑ ∑ ∣ ∣

∑


rate of reaction j (because reaction i may consume or produce components that occur in the rate equation of reaction j). Without rigourous proof we propose the following relation:

( )2 ∣nr nrΓ Vp ∂�j ∣ An1,i = νi,j �k,s (2.64)�i,s Ap ∂zk zk→0j=1 k=1

As can be seen, equation (2.64) reduces to the single reaction form (2.53) when nr = 1. The derivatives of the reaction rates �j to the z values are weighed according to the influence of each reaction in the overall bulk rate of component i (note that �i,s = j νi,j �j,s).

To demonstrate the accuracy of equation 2.64, consider the following small reaction system:

(−νa,1)A + (−νb,1)B → C (2.65)

(−νa,2)A + (−νb,2)B → D (2.66)

where

�1 = k1ca (2.67)

�2 = k2cb (2.68)

For this system, equation 2.64 expands to ( )2 { [ ] Vp �1,s ∂�1 �2,s ∂�1

An1,i = Γ νi,1 + + Ap �i,s ∂z1 �i,s ∂z2 [ ]} (2.69)�1,s ∂�2 �2,s ∂�2

νi,2 + �i,s ∂z1 �i,s ∂z2

The derivatives can be calculated by substituting equation (2.35) in to the kinetic equations

∂�1

∂zk =

νa,kk1

Da (2.70)

∂�2

∂zk =

νb,kk2

Db (2.71)

so

An1,i = Γ �i,s

( Vp

Ap

)2 {

νi,1

[

�1,s νa,1k1

Da

νi,2

[

�1,s νb,1k2

Db

+ �2,s νa,2k1

Da

]

+

+ �2,s νb,2k2

Db

]} (2.72)

38

[ ]


Table 2.2: Parameter values used in the simple reaction network given by reaction equations (2.65) and (2.66).

parameter symbol value unit bulk concentration A bulk concentration B effective diffusion coefficient A effective diffusion coefficient B reaction rate constant reaction rate constant stoichiometry stoichiometry stoichiometry stoichiometry

ca,s

cb,s

Da

Db

k1

k2

νa,1

νa,2

νb,1

νb,2

15 5 2.71e-6 3.14e-6 0.5657 0.12 -2 -3 5 -7

[mol/m3] [mol/m3] [m2/s] [m2/s] [1/s] [1/s] [–] [–] [–] [–]

To demonstrate the accuracy of equation (2.64), the set of differential equations for the reaction system is solved using a 1-dimensional partial differential equation solver for a slab-shaped particle for several particle sizes between 0.01 and 50 mm. The (arbitrary) bulk and transport parameters used are listed in table (2.2).

The values of An1 for both active species A and B calculated with equation (2.64) are compared to the value of (1 − η2) calculated numerically (see figure 2.2). It can be seen that the equation for An1 is a good estimate for (1 − η2) in the slow reaction (small particle size) regime. The equation has been compared to numerical data for a wide range of the parameter values and was found to be correct for the fast reaction regime for all parameter values, both for parallel reactions (νb,1 < 0) and for serial reactions (νb,1 > 0).

As the first Aris number is based on linearised kinetics at bulk conditions, the validity of equation (2.64) can be extended from the simple first order kinetics used in the example to general kinetics. For first order Arrhenius type kinetics:

−EA,j�j = k0,j exp ci (2.73)RT

the cross derivatives can be calculated from ∂�j

∂zk = k0,j exp

[ −EA,j

RT

]( νi,k

Di +

ΔrHkEA,j

λRT 2 ci

)

(2.74)

Similar equations can be written for n-th order or general power law kinetics. However, as these expressions become more elaborate, it may be advantageous (and even more accurate) to calculate the derivative by numerical methods instead.

39


0.0001

0.001

0.01

0.1

1

(1-2 ),

An 1

[-]

An1 estimate (a,b)

1 a (numerical)2

21 b (numerical)

0.01 0.1 1 10 dp [mm]

Figure 2.2: Value of An1 calculated by equation (2.64) compared to the value of (1 − η2) calculated numerically for component A and B of the reaction system given by reaction equations (2.65) and (2.66).

Interpolation

In the previous sections we have given a method to determine the value of the zeroth Aris number for each reaction in a reaction network and we have shown that we can use equation (2.64) to calculate the first Aris number for each component in a reaction network. The zeroth Aris number can give us an estimate of the effectiveness factor for the fast reaction regime and the first Aris number can be used to calculate the effectiveness factor for the slow reaction regime. The effectiveness in the intermediate regime is calculated by an interpolation scheme.

To be able to do the interpolation, we need both Aris numbers either for each reaction or for each component. The best results are obtained if the interpolation is done for the reaction effectiveness, since the reaction rate for a component may depend on several reactions that may be fast or slow.

We can calculate the slow reaction estimate of the effectiveness factor for each reaction from the first Aris number we calculated for each component

40

√

∑


by solving the mass balances for the slow reaction regime. As a first step, he slow reaction regime estimate of the effectiveness factor for component i can be calculated from the first Aris number. However, because we want to be sure all reactions are well within the slow reaction regime, we write the Aris numbers for a small particle. This particle size should be small enough that the effectiveness factors are close to one, but not so small that round-off errors become important. A factor 10-100 smaller than the actual particle size will usually suffice. The effectiveness factors can be estimated from:

1 η1∗ ,i = √ (An1

∗ ,i ≥ 0) (2.75)

1 + An ∗ 1,i

η1∗ ,i = 1 − An1

∗ ,i (An1

∗ ,i < 0) (2.76)

where the star superscript denotes that these are values at decreased particle size. The low rate estimate for the reaction effectiveness factors can now be obtained from:

nr

η1,i�i,s = η1,j νi,j �j,s (i = 1..nc) (2.77) j=1

which is always possible but has a slight complication that there are usually more components than reactions so some equations of the system are not independent. The first Aris number An1,j for reaction j can then be calculated from the effectiveness factor estimate

An ∗ 1,j = 1 − (η1

∗ ,j )

2 (2.78)

Finally, the actual Aris numbers are retrieved by scaling the particle sizes ( A∗ )2

An1,j = An ∗ 1,j V ∗

pVp (2.79)

p Ap

The calculated effectiveness factors of the reactions of the simple reaction network given above are compared with the low reaction rate region estimates (based on the calculated value of An1) and the high reaction rate estimates (based on the calculated value of An0) in figure 2.3. Initially, the effectiveness factor for the second reaction increases when the particle size is increased, to values above 1.5. The reason for this is that the bulk concentration of component B is relatively low; because of mass transfer limitation the concentration of B inside the particle rises above the bulk value as it is produced by reaction 1. As a consequence, the rate of reaction 2 can exceed the rate at bulk conditions.

41


The low rate estimate for the effectiveness factor of reaction 2 correctly follows the calculated curves. To calculate the effectiveness factor for the complete regime, an interpolation between the high and low reaction rate regions is used.

0.1

1

10

Effectiveness

factor

[-]

Reaction 1 (numerical)

Reaction 2 (numerical)

High rate estimate

Low rate estimate

0.001 0.1 2 10 1 1 Particle size [mm]

Figure 2.3: Values of the effectiveness factor for reaction 1 and 2 of the reaction network given by reaction equations (2.65) and (2.66) calculated by numerical integration of the balance equations compared to the low reaction rate and high reaction rate region estimates as a function of the particle size for a slab-shaped particle.

For single reaction systems, Wijngaarden and Westerterp (1994) suggest an implicit interpolation formula that can be solved iteratively:

1 η ≈ √ (2.80)

1 + ηAn1 + (1 − η)An0

Such a scheme can not be used for multiple reaction systems as it does not scale correctly with the value of the reaction rate at the particle edge. Also, it does not handle very high values of the effectiveness factor as the part under the root sign may become negative.

42


We will propose an alternative, explicit interpolation method. The correct scaling of the predicted effectiveness factors will be ensured as long as a linear combination of the fast and slow regime effectiveness factor estimates is used.

For a given reaction system and bulk conditions, the effectiveness factor depends on the particle size only. For large particles, the reaction is fast compared with mass transfer and the reactive zone is limited to a shell at the outer edge of the particle. The effectiveness factor can be estimated by an approximation based on An0. When the particle size is decreased, the thickness of the reactive shell remains nearly constant until the particle size approaches the same order of magnitude as this ’penetration depth’. Further decreasing the particle size has a large influence on the concentration profiles in the particle; this is the intermediate region where both the reaction rate and the mass transfer rate are important for the overall conversion. When we decrease the particle size further so that it is much smaller than the penetration depth, the concentration profiles become relatively flat and the slow reaction approximation based on An1 can be used.

Based on these considerations, it seems logical to use an estimate of the penetration depth as a scaling factor for the interpolation between fast and slow reaction regimes. For each reaction, in the fast reaction regime the pseudo concentration z strictly increases from 0 at the particle edge to the equilibrium value zeq at the end of the reaction zone. We will define the penetration depth δj of reaction j as the distance from the edge of the particle where the value of zj has reached 99% of its equilibrium value. This value can easily be obtained from the profiles calculated for the calculation of An0.

We can then compare the actual particle size with the penetration depth. If the particle size is much smaller than δj , this reaction is in the slow regime, if the particle size is much larger this reaction is in the fast regime. For intermediate values of the penetration depth, an interpolation between the two regimes is needed. A simple strategy that seems to work well is to take the slow rate estimate for the effectiveness factor if the particle size is smaller than δ and take the fast rate estimate elsewhere. The values of δ1 and δ2 are shown for the example case in figure 2.3.

The effectiveness for components can be calculated from the effectiveness factor for reactions 1 and 2 through

∑ nr j=1 νi,j ηj �j,s

ηi = ∑ nr (2.81)

j=1 νi,j �j,s

The result for components A and B is shown in figure 2.4.

43


0

0.2

0.4

0.6

0.8

1

0.001 0.01 0.1 1 10

Effectiveness

factor

[-]

a estimate b estimate a calculated b calculated

Particle size [m]

Figure 2.4: Effectiveness for components A and B of the simple reaction network given by reaction equations (2.65) and (2.66) as a function of particle size. The dots give the results of the numerical integration of the balance equations; the lines give the estimated values.

2.5 Example: the Sohio process

2.5.1 Process description

The ammoxidation of propene to acrylonitril is selected to demonstrate the method described above. In this process, acrylonitril (CH2=CH–CN) is produced from propene (CH2=CH–CH3), oxygen and ammonia in a complex reaction network. As side-products, acetonitril (CH3–CN) and acrolein (CH2=CH– CHO) are formed. In addition, the products can be oxidised further to carbon dioxide and hydrogen cyanide (see figure 2.5).

The kinetics for this reaction system have been published for reaction over a Bi–O–Mo catalyst (Hopper et.al., 1993 and Mleczko, 1996). The reaction equations are:

1. CH2=CH–CH3 + NH3 + 1.5O2 → CH2=CH–CN + 3H2O 2. CH2=CH–CH3 + O2 → CH2=CH–CHO + H2O

44

[ ( )]


acrolein

propene (C3)

(C3O)

acrylonitril (C3N)

hydrogen cyanide (HCN)

2

6

1

3

4

5

acetonitril (C2N)

Figure 2.5: Reaction network for the Sohio process (after Mleczko, 1996)

3. CH2=CH–CH3 + NH3 + 2.25O2 → CH3–CN + 0.5CO2 + 0.5CO + 3H2O 4. CH2=CH–CHO + NH3 + 0.5O2 → CH2=CH–CN + 2H2O 5. CH2=CH–CN + 2O2 → CO2 + CO + HCN + H2O 6. CH3–CN + 1.5O2 → CO2 + HCN + H2O

All reactions are of first order in the organic reactant concentration:

−rj = kj cCH,j (2.82)

where the reaction rate coefficient depends on the temperature according to an Arrhenius expression:

−EA,j 1 1 kj = k0,j exp − (2.83)

R T T0

Here T0 = 470 ◦C is the temperature at which k0,j is determined. The kinetic parameters for each reaction are listed in table 2.3. All six reactions are exothermic. The heat of reaction depends only slightly on the temperature and is therefore taken as a constant. It was calculated at the bulk temperature (400 ◦C) from data given by Hopper et.al. (1993).

In industrial practice, this this reaction is usually carried out in a fluidised bed reactor with particle sizes in the order of 0.1 mm where internal mass transfer limitation is not an issue. Therefore, the published kinetic relations and parameters can be assumed to approximate the intrinsic behaviour of the catalyst.

45


Table 2.3: Kinetic parameters for the ammoxidation of propene (Hopper et.al., 1993 and Mleczko, 1996)

reaction 1 2 3 4 5 6 CH C3 C3 C3 C3O C3N C2N k0 [1/s] 0.4056 0.0097 0.0174 0.6813 0.1622 0.073 EA [kJ/mol] 79.5 79.5 29.3 29.3 82.9 29.3 ΔrH [kJ/mol] -47.31 -332.37 -862.54 -178.99 -780.77 -590.78

Table 2.4: Assumed catalyst properties (typical values) parameter values particle diameter dp [m] thermal conductivity λs [W/m/K] porosity ε [-] pore diameter dc [m] tortuosity τc [-]

0.01 0.1 0.65 1.0 × 10−6

3

For this example it is assumed that the reaction takes place in a spherical porous catalyst particle. The catalyst properties are listed in table 2.4. The effective diffusion coefficient inside the catalyst particles for each component can be estimated from the binary molecular diffusion coefficients, Knudsen diffusion coefficients and the tortuosity. Diffusion coefficients are calculated at the bulk composition (table 2.5). The values for the effective diffusion coefficients used are listed in table 2.5.

Table 2.5: Effective diffusion coefficient component Deff × 106

propene (C3) oxygen (O2) ammonia (NH3) acrylonitril (C3N) acrolein (C3O) acetonitril (C2N) hydrogen cyanide (HCN) carbon monoxide (CO) carbon dioxide (CO2) water (H2O) nitrogen (N2)

2.61 3.21 4.22 2.34 2.28 2.79 2.79 3.39 2.70 4.24 3.34

46


Table 2.6: Composition at the catalyst particle edge at reactor entrance conditions (after Hopper et.al., 1993)

component mole fraction propene oxygen ammonia nitrogen others

0.07 0.18 0.07 0.68 0.01

The composition at the outer edge of the catalyst particle is taken as given in table 2.6. The temperature at the outer edge of the particle is Ts = 500 ◦C. Note that external mass transfer limitation is not taken into account in the model presented here.

The concentrations of the intermediate products (C3N, C3O and C2N) in the reaction feed are taken as zero in the literature. This means that initially, the bulk rates of reactions 4, 5 and 6 are also zero, and therefore that the effectiveness factor of these reactions is infinitely large. In a real reactor this condition could only occur in the first layer of particles in the bed, and even there, the value at the edge of the particles will be somewhat higher than the bulk fluid value because of external mass transfer limitation. Therefore, it seems justified to put a lower limit on the concentrations of the intermediate products; here we will take a value of 0.01 mole %.

47

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )


2.5.2 Numerical solution

The concentration and temperature profiles inside the catalyst particle are computed in two ways: once by integration of the full set of differential equations (2.18 and 2.19) and once by integration of the reduced set (2.37). For the ammoxidation process the full set of equations is:

d2cC3 2 dcC3Deff,C3 + = −k1cC3 − k2cC3 − k3cC3dx2 1 − x dx d2cC3N 2 dcC3N

Deff,C3N + = k1cC3 + k4cC3O − k5cC3Ndx2 1 − x dx

d2cC3O 2 dcC3ODeff,C3O + = k2cC3 − k4cC3O

dx2 1 − x dx d2cC2N 2 dcC2N

Deff,C2N + = k3cC3 − k6cC2Ndx2 1 − x dx

d2cO2 2 dcO2Deff,O2 + = −1.5k1cC3 − k2cC3dx2 1 − x dx

− 2.25k3cC3 + −0.5k4cC3O

− 2k5cC3N − 1.5k6cC2N

d2cNH3 2 dcNH3Deff,NH3 + = −k1cC3 − k3cC3 − k4cC3O (2.84)dx2 1 − x dx

d2cHCN 2 dcHCN Deff,HCN + = k5cC3N + k6cC2N

dx2 1 − x dx d2cCO 2 dcCO

Deff,CO + = 0.5k3cC3 + k5cC3Ndx2 1 − x dx d2cCO2 2 dcCO2Deff,CO2 + = 0.5k3cC3 + k5cC3N + k6cC2N

dx2 1 − x dx d2cH2O 2 dcH2O

Deff,H2O + = 3k1cC3 + k2CC3 + 3k3cC3dx2 1 − x dx

+ 2k4cC3O + k5cC3N + k6CC2N

with the temperature equation

d2T 2 dT λ + = ΔrH1k1cC3 + ΔrH2k2cC3dx2 1 − x dx

(2.85)+ ΔrH3k3cC3 + ΔrH4k4cC3O

+ ΔrH5k5cC3N + ΔrH6k6cC2N

48

∑


Due in part to the simple form of the individual rate equations (2.82), it may be suspected that the system could be solved without integrating all of the equations (2.84). In fact, the first four equations may be solved together with the last one independently of the other six since only four concentration variables appear on the right hand side of an equation. However, for problems in which the individual reaction rates cannot be described by first-order behaviour, this may not be as clear. The procedure described above gives a way to reduce the number of equations in a straightforward way, regardless of the complexity of the rate equations. From equation (2.37) it follows that the reduced set of equations is:

d2z1 2 dz1 + = −k1cC3dx2 1 − x dx

d2z2 2 dz2 + = −k2cC3dx2 1 − x dx

d2z3 2 dz3 + = −k3cC3 (2.86)

dx2 1 − x dx d2z4 2 dz4

+ = −k4cC3Odx2 1 − x dx d2z5 2 dz5

+ = −k5cC3Ndx2 1 − x dx d2z6 2 dz6

+ = −k6cC2Ndx2 1 − x dx

(2.87)

with (equation 2.35)

1 cC3 = cC3,s + (−z1 − z2 − z3)

Deff,C3

1 cC3O = cC3O,s + (z2 − z4)

Deff,C3O

1 cC3N = cC3N,s + (z1 + z4 − z5) (2.88)

Deff,C3N

1 cC2N = cC2N,s + (z3 − z6)

Deff,C2N

61

T = Ts − ΔrHj zjλ

j=1

The relatively compact representation of equations (2.86) and (2.88) compared to the full system (2.84) is a direct advantage of the introduction of the

49


pseudo concentrations. Only the concentration profiles that are required to solve the system are shown in equation (2.88). Once the profiles for the variables zj are known, the other concentration profiles can be calculated using equation (2.35) as needed.

For the conditions given in table 2.6, the full system and the reduced system are integrated using a finite difference formulation. The profiles for the variables zj and the corresponding concentration profiles are shown in figure 2.7 and 2.6, respectively. It can be seen that while the profiles for the zj strictly increase from the boundary value 0 to a value in the centre of the particle, the concentration profiles are more complex. The profile for the intermediate products acrylonitril (C3N) and acrolein(C3O) even shows a maximum inside the particle. Since it is mathematically equivalent, the solution of the full set of equations naturally leads to the same set of concentration profiles.

z3

0

1

2

3

4

5

6

7

z j [

mol

/m3 ]

z j [

mol

/m3 ]

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

z1

z5z2

z4

z6

0 1 2 3 4 5 x [mm]

Figure 2.6: Profiles inside the catalyst particle for the pseudo concentrations zj for the Sohio reaction system in a 10 mm porous catalyst particle.

50

6 0.06


5

4

3

2

1

0

C3 C3N O2

NH3 C3O C2N 0.05

0.04

0.03

0.02

0.01

0

c i [m

ol/m

3 ]

c i [m

ol/m

3 ]

0 1 2 3 4 5 x [mm]

Figure 2.7: Concentration profiles inside the catalyst particle for selected components for the Sohio reaction system in a 10 mm porous catalyst particle.

2.5.3 Approximation

Fast reaction regime

We will now use the approximation method to estimate the effectiveness factor for the Sohio reaction in a porous catalyst particle. First we will calculate the concentration profiles for the fast reaction regime. Since the particle geometry will not influence the results, we will do this for a slab-shaped particle. To ensure that the reaction only takes place in a thin region near the edge of the particle, a large particle size is chosen for this calculation: 50 mm.

The kinetic equations given by (2.82) are valid only for an excess amount of oxygen and ammonia. However, for the chosen conditions oxygen will be depleted inside a large catalyst particle. Therefore, the reaction rate equation is modified slightly:

−rj = k0,j exp

[ −EA,j

R

( 1 T

− 1 T0

)]

cCH,j ×

(

1 − exp

[ cO2

cO2,low

])

(2.89)

where cO2,low is taken as cO2,s/1000. This ensures that the reaction rate approaches zero if the oxygen is depleted, but does not alter the kinetics at normal oxygen levels.

51


The resulting profiles for the most interesting components are shown in figure 2.9 and the z-value profiles are shown in 2.8. From the latter we obtain the gradients at the particle edge, the equilibrium value for z for each reaction and the penetration depth δ. The resulting values are given in table (2.7).

0

1

2

3

4

5

6

0 5 10 15 20 25 x [mm]

z j [

mol

/m3 ]

× 1

0 6

z j [

mol

/m3 ]

× 1

0 6

0

0.05

0.1

0.15

z1 z3 z5

z2 z4 z6

Figure 2.8: Profiles of the z values for each reaction of the Sohio process in a very large porous catalyst particle

The effectiveness factor for reactions 4, 5 and 6 is considerably higher than one, due to the fact that the bulk rates for these reactions is very low. However, when the interpolation is done in a consistent way, the bulk reaction rate is just a scaling factor and will not influence the final result. It also shows that the chosen parameter values give something of a worst case situation for the estimation of the effectiveness factor; as the reaction progresses, the bulk concentrations of the (intermediate) products C3N, C3O and C2O will increase, the bulk rates will increase and the effectiveness values will decrease to values closer to one.

Slow reaction regime

The next step is to calculate the value of An1,i for each component in the slow reaction regime using equation (2.64). For this we need the derivative of each

52

∑

4

5

6 0.06


C3 C3N O2

NH3 C3O C2N

0 5 10 15 20 25 x [mm]

0.05

0.04

0.03

2 0.02

1 0.01

0 0

Figure 2.9: Profiles of the most important components of the Sohio process in a very large porous catalyst particle

reaction rate �j to each pseudo concentration zk, which for the given kinetics (equation 2.82 and 2.83) can be calculated by equation (2.74). The estimates of the first Aris numbers for the components are now used to calculate the first Aris numbers for the reactions. Since the estimates are valid only in the slow

c i [

mol

/m3 ]

reaction regime, we decrease the particle size for which the Aris numbers are determined until all component effectiveness factors are close to one; in this case, decreasing the particle size by a factor 100 is sufficient. We calculate the slow reaction regime effectiveness factors from the Aris numbers using the definition:

An† = 1 − η†2 (2.90)1,i 1,i

where the daggered variables are taken at the limit of small particle size and the subscript ”1” designates that this is the estimate for the slow reaction regime. The effectiveness factor for the reactions in the slow reaction regime is now calculated from the component mass balances:

nr

η† η† i �i,s = j νi,j �j,s (2.91)

j=1

c i [

mol

/m3 ]

3

53


Table 2.7: Calculated values for the fast reaction regime for the Sohio process for the given boundary conditions.

reaction zj,eq ∂zj /∂x|x=0 δ # [mol/m3] [mol/m4] [mm] 1 5.38e-06 3.04e-03 5.25 2 1.28e-07 7.26e-05 5.25 3 1.29e-07 8.02e-05 5.0 4 1.29e-07 4.44e-05 5.63 5 4.78e-06 1.35e-03 5.75 6 6.59e-08 1.73e-05 5.63

Table 2.8: Estimated values for the slow reaction regime for the Sohio process, for a 10 mm slab-shaped particle.

reaction An1 η1

1 19.481 0.221 2 19.577 0.22 3 18.505 0.226 4 -8.1 3.017 5 -1350.074 36.757 6 -33.539 5.877

and an equivalent expression for the temperature. These balances form a set of nc +1 = 11 equations with nr = 6 unknowns (ηj ) so the set is over-determined. This is of course caused by the fact that some equations are linear combinations of two or more of the other equations. In this case, it is not very easy to spot the dependencies. With some manipulation it can be seen that the equations for C3, C3N, C3O, C2N, O2 and the temperature together form a complete set of 6 equations with 6 unknowns. The reaction effectiveness factors are obtained from this set by matrix inversion. An alternative would be to use numerical methods to find the vector of ηj that best fits equation (2.91).

We now have the first Aris number for an arbitrary but small sized particle. To obtain the value for a given particle size is a simple scaling operation, since the first Aris number is proportional to the square of the particle volume to area ratio (Vp/Ap). The values are shown for a 10 mm particle in table (2.8).

Interpolation

From the values of the zeroth and first Aris numbers we can calculate estimates for the effectiveness factor for the fast reaction and slow reaction regimes, respectively. The actual estimate of the effectiveness factor is an interpolation

54


between these values. In order to get a smooth transition between the slow and fast reaction regimes, the following interpolation function is used:

f = 1 d/δj ≥ 1

(d/δj )2

f = d/δj < 1 (d/δj )2 + (δj /d)2 − 1

where the effectiveness factor is calculated from:

η = (1 − f)η1,j + fη0,j (2.92)

Figure 2.10 shows the estimated effectiveness values for each of the six reactions (for the reaction numbering, see figure 2.5) as well as the value calculated by numerical integration of the set of differential equations.

The first three reactions show relatively standard behaviour, and the prediction of the estimation procedure is quite accurate (within a few percent from the actual value). The low rate estimate overlaps the actual curve for a large part of the domain. For reaction 4, effectiveness factors higher than one are predicted, as the acrolein (C3O) concentration increases above the bulk level because it is produced by reaction 2. The low rate regime curve now is increasing at higher particle size, which is correct, although it gives slightly too high values.

Reactions 5 and 6 show maximum effectiveness factors much higher than one (up to about 30 for reaction 5, and 1.8 for reaction 6). The maximum is predicted in the right order of scale, but there is a tendency to overestimate the peak values. This may be improved by using a different type of interpolation procedure.

For reaction 5 the maximum is nearly at the actual value, although this is probably a coincidence. The curve for the low reaction rate regime is lower than the actual values; this causes the estimated effectiveness factor for the smaller particle sizes (0.5-2 mm) to be (10-30 %) lower than the actual values. The deviation of the low rate curve is caused by the fact that the relation used to calculate An1 (equation 2.64) only takes into account the direct influence between reactions and not the secondary interactions. For instance, for reaction 5, the fact that reaction 4 produces acrylonitril which will increase the rate of reaction 5 is taken into account, but not the fact that reaction 2 produces acrolein which will speed up the rate of reaction 4. Especially when the conversion in the bulk gas is low (i.e., at near the entrance of the catalyst bed), as is the case for these simulations, the primary reactions 1, 2 and 3 will be relatively fast; the product concentrations will be low and therefore the secondary reaction rates 4, 5 and 6 will be low, so the indirect influence

55


0.01

0.1

1

10

estimate low rate estimate high rate estimate

numerical


numerical(1) (2)

0.01

0.1

1

10

0.1 1 10 100 0.1 1 10 100

10


numerical

(3) 10

1 1

0.1 0.1

0.01 0.01

(4)


numerical

0.1 1 10 0.1 1 10 100

Figure 2.10: Estimated effectiveness factor for the Sohio process as a function of particle size (in mm). The estimated value is compared to the result of a direct numerical solution of the set of differential equations; also shown are the low rate and high rate regime estimates between which is interpolated.

estimate low rate est. high rate est.

numerical


numerical

(5) (6)

1

100

10

0.1

10

1

0.1 1 10 100 0.1 1 10 100

56


will be relatively high. A similar secondary influence takes place through the temperature: for each reaction the effect of every other reaction on the temperature is taken into account, but these reaction rates are taken at the bulk temperature and not at the increased temperature inside the particle. To increase the accuracy of the estimation procedure in these conditions, it may be possible to extend equation (2.64) to include the secondary effects (although it appears that this will make it implicit). As we move down a catalyst bed toward the exit, the reactant concentrations in the bulk will decrease and the (intermediary) product concentrations will increase, so the difference in reaction rates will decrease and the secondary interaction effects will become less important.

Figure 2.11 shows the estimate as well as the value obtained from direct numerical integration of the effectiveness factor for selected components as function of the particle size for a slab-shaped particle at reactor entrance conditions. These curves are obtained by linear combination of the reaction effectiveness factors according to the stoichiometry. In contrast with the reaction effectiveness actors, the component effectiveness factors can become zero or negative. Therefore these plots are given on a linear scale instead of the customary log-log scales.

It can be seen that the effectiveness factor for the consumption of propene (C3) and for the production of acrylonitril (C3N) and acetonitril (C2N) follows more or less the standard curve: starting at one for small particle size (slow reaction regime) and then gradually decreasing into an exponential decay towards zero. For these components (and others that are only produced or consumed but are not shown: ammonia, oxygen and water) the estimation is quite good (within 10 % of the true value). The largest errors occur at intermediate particle sizes.

The curve for acrolein (C3O) shows peculiar results: the effectiveness factor starts at one and with increasing particle size decreases, drops below zero and then slowly increases towards zero. This is caused by the fact that C3O is produced by one reaction (2) and consumed by another (4). At bulk conditions there is a net production of C3O, but inside the particle the concentrations and temperature change; apparently, the consumption reaction benefits more from these conditions than the production reaction. The result is a net consumption of C3O above a certain particle size. It can be seen that the estimation shows the correct trend, including the prediction of negative effectiveness factors in the right order of scale, but the accuracy in the transition zone is poor.

A third class of curves is represented by the hydrogen cyanide plot but also includes other components that are produced by follow-on reactions that have a very low rate at bulk conditions (CO2 and CO). Consequently, the effectiveness factor for these components can become much larger than one.

57


0.1 1 10 100

C3O estimate low rate estimate high rate estimate

numerical

1 C3

numerical estimate low rate estimate high rate estimate

1

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0 0

C3N


0.1 1 10 100 0.1 1 10 100

C2N

numerical

low rate estimate estimate

high rate estimate

11

0.8 0.5

0.6

0

0.4

-0.5 0.2

-10 0.1 1 10 100

50


numerical HCN 5

40 4

30 3

20 2

10 1

0 0

numerical T

low rate estimate estimate

high rate estimate

0.1 1 10 100 0.1 1 10 100

Figure 2.11: Estimated effectiveness factors for selected components and the temperature for the Sohio process in a slab-shaped catalyst particle as function of the particle size (in mm) for the given bulk conditions, compared with the results obtained from numerical integration of the set of differential equations.

58


Table 2.9: Assumed composition at the catalyst particle edge at 68 % conversion)

component mole fraction propene (C3) oxygen (O2) ammonia (NH3) acrylonitril (C3N) acetonitril (C2N) acrolein (C3O)

0.024 0.084 0.024 0.035 0.026 0.001

In the case of HCN it is about 25 at its maximum in this case. The estimation procedure also predicts this behaviour; for particle sizes near the maximum and larger the accuracy is satisfactory (within 10-15 %), although on the small particle size flank the accuracy is lower. This is caused by the inaccuracies in the effectiveness curves for reactions 5 and 6.

The errors made in the effectiveness factor estimates at the entrance of the bed (as given above) are worst case values as the effectiveness of the secondary reactions will be much larger than one when the product concentrations in the bulk gas are still low. In practice, the effect of the errors (e.g., on the mass balance of a reactor) will be relatively small as the highest errors occur for reactions with a very low absolute rate. As we move down the reactor, the effectiveness factors will have values closer to one. As an example of this, the effectiveness factors for the Sohio reaction system at 68 % conversion are shown in figure 2.12. The temperature of the bulk gas depends on the cooling rate; an arbitrary value of 560 ◦C is taken for this plot. The bulk composition also depends on the effectiveness factor upstream in the reactor; the values taken are given in table 2.9. The estimate in this case gives excellent values with errors less than one percent except in the transition area where maximum the error is less than 8 % except for reaction 4 which has a maximum error of about 15 %.

The maximum error occurs in the transition region between the fast and slow reaction regimes. The maximum error may be further reduced with a more elaborate interpolation method.

Complex geometry

We have shown that the approximation method developed here gives reasonable results for the Sohio reaction system in slab-shaped catalyst particles. However, the advantage of the approximation lies primarily in the estimation of the effectiveness factor for reaction networks in catalyst particles with a

59


10


numerical

(1) 10

1

0.1 0.1

0.01 0.01

(2)


0.1 1 10 100 0.1 1 10 100

10


numerical

(3) 10

0.1 0.1

0.01 0.01

(4)


0.1 1 10 100 0.1 1 10 100

10


numerical

(5) 10

0.1 0.1

0.01 0.01

(6)


0.1 1 10 100 0.1 1 10 100

Figure 2.12: Effectiveness factor for the Sohio process as a function of particle size (in mm) at a bulk gas composition corresponding to 68 % conversion (see table 2.9). The estimated value is compared to the result of a direct numerical solution of the set of differential equations; also shown are the low rate and high rate regime estimates.

60


more complex geometry. For ”one-dimensional” particles like slabs, infinite cylinders and spheres, the numerical solution involves the integration of a set of ordinary differential equations. The approximation method also requires the integration of a set of ODEs, so there is little computational advantage (although one advantage could be that the approximation method gives results for all particle diameters and shapes with just one integration of the ODEs at given boundary conditions).

For complex geometry particles, the direct numerical solution requires the integration of a set of coupled partial differential equations. Here the approximation method has a larger advantage because it still requires only the integration of a set of ODEs. For a given particle shape, also the geometry factor Γ needs to be determined, but this depends on geometry only and needs to be determined only once for a given geometry. As a demonstration, the Sohio reaction system is modelled in a ring-shaped catalyst particle (see figure 2.13). Wijngaarden and Westerterp (1994) give the geometry factor for this particle shape as a function of the inner diameter, outer diameter and length; in this case Γ = 1.12.

L = 16

Ri = 4

Ro = 8

Figure 2.13: Geometry of ring-shaped catalyst particle

The computational effort needed to to calculate the effective reaction rate for this ring-shaped particle using the approximation method is compared to that needed to solve the full balances. For this, a general PDE solver (Flex-PDE) with automatic grid adaptation is used. On a typical desktop computer, this solver needs about 100 s and 1580 nodes to solve the full balances for one

61


Table 2.10: Axial concentration profiles for the Sohio process in a simple cooled tube reactor packed with ring shaped catalyst particles (numerical solution).

node conversion T ◦C

C3 mol/m3

C3N mol/m3

C2N mol/m3

C3O mol/m3

O2 mol/m3

0 0.0% 400 2.207 0.032 0.032 0.032 5.674 1 3.4% 420 2.131 0.101 0.038 0.030 5.551 2 8.2% 440 2.025 0.196 0.044 0.028 5.375 3 14.7% 460 1.883 0.321 0.051 0.027 5.132 4 22.9% 480 1.701 0.472 0.058 0.026 4.805 5 32.9% 500 1.481 0.642 0.064 0.025 4.378 6 44.3% 520 1.229 0.810 0.069 0.025 3.840 7 56.3% 540 0.963 0.949 0.074 0.025 3.191 8 68.1% 560 0.704 1.031 0.076 0.023 2.447 9 78.5% 580 0.475 1.034 0.077 0.022 1.645 10 86.8% 600 0.292 0.954 0.076 0.019 0.837

particle to an accuracy of 0.01%. The approximation method for the same conditions requires about 7 seconds and 93 nodes.

As an example, the approximation method is used to calculate the conversion in a wall-cooled ideal tube reactor packed with ring shaped catalyst particles. The goal here is to show the accuracy of the approximation method at different conditions, so the reactor model is relatively simple: plug flow is assumed and the external mass transfer limitation is not taken into account. The cooling of the tube is assumed to be arranged in such a way that the temperature increases linearly from the entrance temperature (400 ◦C) to the exit temperature (600 ◦C). This ensures that both low and high rate conditions occur in the reactor; in a real Sohio reactor the temperature will be kept well below the latter value to limit oxidation of acrylonitril. The tube reactor is modelled by a series of 10 sections in which the bulk conditions and effectiveness factor are assumed to be constant. The conditions at the inlet are the same as those used previously, the concentrations of all components are calculated from the reaction rates at bulk conditions and the calculated effectiveness factors. The resulting axial concentration and temperature profiles are given in table 2.10.

The numerical solution for the concentration and temperature fields inside the catalyst particle are obtained by solving the partial differential equations using a general PDE solver (FlexPDE). The effectiveness factors calculated from the numerical solutions are compared to the values obtained with the approximation method in figure 2.14. It can be assumed that the error made in the numerical method is much smaller than the errors made in the approxima

62


tions. The curves for reactions 2 and 3 are very similar to that for reaction 1 and therefore are not shown. Because of the assumed temperature profile, the reaction rates increase along the reactor and the effectiveness factors decrease. The curves for the numerical solution and the values predicted by the approximation are reasonably close together for most of the domain. For reaction 5, the difference at the reaction entrance is relatively large (as was noted earlier), but at higher conversions the difference is quite small. For reaction 6 there is a larger difference at location 9; this is caused by the fact that in the approximation the transition between the high and low reaction rate regimes occurs at those conditions and therefore the effectiveness factor is overestimated.

0

0.3

0.6

0.9

1.2

1.5

Eff

ecti

vene

ss f

acto

r [-

]

0

1

2

3

4

5

Eff

ecti

vene

ss f

acto

r [-

]

0 1 2 3 4 5 6 7 8 9 10 axial coordinate [arbitrary units]

reaction 1 approx. reaction 4 approx. reaction 5 approx. reaction 6 approx.

reaction 4 numerical reaction 5 numerical reaction 6 numericalreaction 1 numerical

Figure 2.14: Effectiveness factors obtained from numerical integration of the governing set of partial differential equations compared to the results from the approximation method for the Sohio reaction system in a ring shaped catalyst particle

The most important results of a reactor simulation for reaction networks is the selectivity. The selectivity determines the amount of reactants that is converted to useful products and the amount of waste that is produced. In the case of the Sohio process, the by-products are not easy to separate from the product and hazardous; a careful reactor design can reduce waste production and pollution (Hopper et.al., 1993). Figure 2.15 shows the conversion and

63


the selectivity for the main product (acrylonitril) and by-products acetonitril and cyanic acid as calculated using effectiveness factors determined from the numerical integration of the full PDEs and using the approximation. The selectivity for acrolein (C3O) is not shown as the amount produced is very small (in fact, the assumed inlet bulk concentration is somewhat higher than the equilibrium concentration and there is a small net consumption of acrolein). It can be seen that the difference between approximation and the numerical is only a few percent point, which is acceptable for practical purposes. Note that in practice a reactor for the Sohio process would never be designed like this, as the reactant is converted for 50 % to the hazardous by-product cyanic acid.

1

0.8

0.6

0.4

0.2

0

Sele

ctiv

ity

[-],

Con

vers

ion

[-]

0 2 4 6 8 10 axial coordinate [arbitrary units]

S C3N approx. S C2N approx. S HCN approx. X approx.

S C3N numerical S C2N numerical S HCN numerical X numerical

Figure 2.15: Conversion of propene and selectivity towards products and by-products in a simple 10-compartment tube reactor model for the Sohio process for ring-shaped particles; results using numerical solution of the PDEs to obtain the effectiveness factor compared to results using the approximation method.

64

2.6 Conclusions

2.6 Conclusions

We have introduced the pseudo concentration z of a virtual key component in the balance equations that describe reaction and diffusion in a catalyst particle. For a single reaction system his simplifies the balance equations to a single ordinary differential equation for z and a set of algebraic equations to calculate the concentrations of all components involved and the temperature. When applied to a reaction network, a straightforward method is found to reduce the number of differential equations from one for each component and the temperature, to one for each reaction.

The Aris number approximation (Wijngaarden and Westerterp, 1994; Wijngaarden et.al., 1998) is rewritten here in terms of the pseudo concentration. This gives one set of equations for the zeroth and first Aris numbers for a single isothermal or non-isothermal reaction with any number of components. With these parameters, the effectiveness factor can be estimated for any kinetics and particle shape.

For reaction networks, a new approximation method based on pseudo concentrations and the Aris number approach was developed. In principle this method is valid for general kinetics and general geometry. The method reduces the computational load compared to a direct numerical integration of the balance equations, especially for complex particle shapes. Instead of the integration of a set of partial differential equations, integration of a set of ordinary differential equations suffices. It has been shown that the computation time reduction using the approximation method can be in the order of a factor 10 using a general PDE solver.

The approximation method is tested for simple reaction networks in slab shaped particles. It correctly describes the trend of the effectiveness factor curves as a function of particle size (or, equivalently, zeroth Aris number). The curves take the correct values in the limits of low and high reaction rate, and show the correct trend in between. Effectiveness factors greater than one and less than zero are predicted and in the right order of scale, although somewhat larger errors are found in these cases.

It is shown that the error made in the approximation is in the order of 5 % under most conditions and for the primary reactions, but may be as large as 50 % for the secondary reactions in exceptional cases where the effectiveness factor is much larger than one. The large errors are probably caused by the fact that the cross terms in the calculation of An1 are taken at bulk conditions. These cross terms correct for the influence of one reaction on the rate of another in the slow reaction regime. The effect of the cross terms is only large if the rate of the reaction is very low compared to the other reaction rates. Consequently, these do not have a large effect on the mass balances. It may be possible to

65


reduce the errors by modification of the estimation formula for the first Aris numbers, at the cost of some computational effort.

The effectiveness factor is calculated by an interpolation between the fast and slow reaction regimes. A relatively simple interpolation strategy is used, based on the penetration depth of the reaction system in a large particle. If the equivalent particle size is smaller than the penetration depth, the slow reaction regime estimate is taken, otherwise the fast reaction regime estimate. By and large, this method works well for the entire range of reaction rates, but it gives relatively large errors if the particle size and penetration depth are comparable. The errors in the effectiveness factor values predicted by the approximation method for particle sizes close to the penetration depth δ may be reduced by using a more refined interpolation method.

In the model calculations, the maximum errors in the effectiveness factor only occur when the reaction rate is very low. This is the case for reactions whose rate depends on the concentration of a product component at the beginning of the reactor where the bulk concentration of product is low. Consequently, relatively large error is made in a small contribution. Therefore, these errors have a very small impact on the overall results (predicted conversion and selectivity) of the reactor model calculation. For the demonstrated cases, the error made in the overall mass balance when using the approximation method is in the order of a few percent.

It is concluded that for the systems that have been modelled, the approximation method can be used with sufficient accuracy to calculate the effectiveness factor. In principle, the method works for any particle shape and for general kinetics. However, only networks of first order kinetics with Arrhenius type temperature dependence were tested in this work. Extension to other kinetics and particle geometries is left for future research.

With the extension of the Aris number approach to reaction networks, it is possible to calculate the effective rate of reaction in catalyst particles with general shape, for reaction networks and with general kinetics with a relatively low computational effort but sufficient accuracy so that it can be used in a packed bed CFD model.

66

2.6 Conclusions

Nomenclature

Roman Symbol units Variable Ap m2 Particle outer surface area An0 - Zeroth Aris number, as defined by equation (2.39) An ∗

0 - Zeroth Aris number defined with linearised kinetics An1 - First Aris number, as defined by equation (2.39) An†

1 - First Aris number at the limit of small particle size cCHj mol/m3 Concentration of the main organic reactant of reac

tion j in the Sohio process ci mol/m3 Concentration of component i ci,s mol/m3 Concentration of component i at the outer edge of

the particle cp J/kg/K Specific heat dc m Diameter of the pores in a porous catalyst particle dp m Particle diameter Deff,i m2/s Effective diffusion coefficient of component i inside

the particle Dt,i m2/s Effective diffusion coefficient of component i in a

straight pore EA J/kg Activation energy ΔrHj J/mol Heat of reaction of reaction j k0 (varies) pre-exponential reaction rate constant L m Distance from the particle edge to the symmetry

plane, axis or point L m Length of ring shaped or cylindrical particle m - Geometrical parameter (power with which surface

area increases with particle size) nc - Number of components in the reaction system nr - Number of stoichiometrically independent reactions R J/kg/K Universal gas constant (8.3144 J/kg/K) Ri m inner radius of ring shaped particle Ro m outer radius of ring shaped particle �i mol/m3/s Rate of consumption of component i �j mol/m3/s Rate of reaction j �∗ mol/m3/s Linearised reaction rate

67


Symbol t

units s

Variable Time

T Ts

Vp

x X y yi

z zj

zeq

K K m3

m m m -m mol/m3

mol/m3

Temperature Temperature at the outer edge of the particle Particle volume Space coordinate Large distance from particle edge Space coordinate Mole fraction of component i in the reaction mixture Space coordinate Pseudo concentration: independent variable in the reduced set of equations Pseudo concentration at thermodynamic equilibrium

Greek Symbol units Variable δj m Penetration depth of the reaction into a large particle εp - Porosity of the catalyst particle η - Effectiveness factor: actual conversion divided by

conversion without mass transfer limitation ηi - Effectiveness factor for component i ηj - Effectiveness factor for reaction j ηi † - Effectiveness factor for component or reaction i at

the limit of small particle size Γ - Geometry function defined in equation (2.52) Φ ”

i,e mol/m2/s Molar flux of component i in coordinate direction e λ W/m/K Thermal conductivity inside the catalyst particle νi,j - Stoichiometry for component i in reaction j ρ kg/m3 Density of the catalyst τc - Tortuosity of the pores in a catalyst particle (equals

τ 2 )p,act

τc,act - Geometric (actual) tortuosity of the pores in a catalyst particle

68

2.6 Conclusions

Literature

FlexPDE, Finite element software, web page: www.pdesolutions.com

Fuller, E.N., P.D. Schettler, J.C. Giddings (1966), A new method for the prediction of gas-phase diffusion coefficients, Ind. Eng. Chem., 58, 19

Fogler, H.S. (1986), Elements of chemical reactor engineering, Prentice-Halll

Hopper, J.R., CL Yaws, T.C. Ho, M. Vichailak (1993), Waste minimisation by process modification, Waste Management, 13, 3

Mleczko, L. (1996), Einfluss der Hydrodynamik in einem Wirbelschichtreaktor industriellen Mastabs auf Selektivitt und Ausbeute zu Acrylnitril bei der Ammoxidation von Propen, Chemische Technik, 48 (3), 131

Reid, R.C. , Prausnitz, J.M., Poling, B.E. (1939), The properties of gases and liquids, McGraw-Hill, 1987

Thiele, E.W. (1939), Relation between catalytic activity and size of particle, Ind. Eng. Chem., 31, 916

Westerterp, K.R., R.J. Wijngaarden (1992), Principles of Chemical Reactor Engineering, Ullmann’s Encyclopedia of Industrial Chemistry, Vol. B4, 5

Wijngaarden, R.J., K.R. Westerterp (1994), Generalized formulae for the calculation of the effectiveness factor, University of Twente, Enschede, the Netherlands

Wijngaarden, R.J., A. Kronberg, K.R. Westerterp (1998), Industrial catalysis: optimizing catalysts and processes, Wiley-VCH, Weinheim

69

Chapter 3

The wall effect

Summary

The structure of a packed bed near a wall differs from the structure far from any wall. Therefore, the local bed parameters also differ. In this chapter, we will look at the structure of the packed bed near a wall using an approach based on separate particles. The bed structure is characterised by functions that describe parameters such as porosity, particle outer surface area and tortuosity as a function of the position in the bed. These functions can then be used in a pseudo-continuous CFD model to describe the whole reactor.

The limited distance over which the porosity is influenced by the wall (about 5 particle diameters), suggests that the wall effect is only important for very low vessel-to-particle-diameter reactors. However, due to the importance of the wall zone in transfer of heat to or from the bed and because a relatively large fraction of the flow may pass through the wall zone due to bypassing, the wall effect is important for larger beds, up to D/d = 40. Hence, the wall effect can have a considerable impact on the performance of many packed bed reactors. To be able to make more efficient designs for these reactors, a model is needed that predicts the influence of the wall on the reactor performance.

Most of the discussion in this chapter is limited to packings of identically sized, spherical particles. In real-life applications, spherical particles are not very common. We use uniform spheres in order to be able to simulate and understand the important features of the wall effect. For non-spherical particles, the same mechanisms play a role but the ordering effect of the wall will probably be less pronounced. In principle, the approach described here can be extended to complex particle shapes. However, the computational effort involved will be orders of magnitude higher.

In existing literature, the wall effect is commonly described in terms of the bed porosity. The porosity profiles have been measured directly by fixating

71

3. The wall effect

particles in a bed and cutting and weighting parts of the bed. Fit curves have been given for the oscillating porosity profiles found as a function of the bed-to-particle diameter ratio (e.g., Vortmeyer and Schuster, 1983, Mueller, 1991). Critical inspection of the experimental data indicates that there is no evidence that the shape of the porosity profile depends on the bed-to-particle diameter ratio for D/d > 5.

Govindarao and Froment (1986) and Mariani et. al. (2001) fitted the experimental porosity profiles using a particle centre distribution, i.e., the density of particle locations as a function of distance from the wall. Mariani et. al. (2002) claim that with this approach the particle outer surface area profile near the wall can also be calculated. However, many different particle centre distribution curves may give similar porosity profiles but very different particle outer surface profiles, so this claim is not valid. Computer models have been used to synthesise packed beds. Since as a result, all particle positions are known, all bed parameters can be calculated for these simulated bed packings. Spedding and Spencer (1994) give methods for these calculations, but do not take into account the vessel wall. Mueller (1997) does take into account the wall but only gives the porosity profiles.

Two computer models are developed for the simulation of a random packing of uniformly sized spheres: a numerical potential energy minimisation method and a deterministic ball placement scheme. For the simulated packings, the position of all sphere centres is readily available and parameters that would be difficult or impossible to measure in a real packed bed can be computed with relative ease. The bed packing simulation models are compared to existing measurements to verify that they give realistic packed bed structures. The simulated bed packings have a somewhat higher bulk porosity than experimental bed packings; this is probably caused by the fact that the particles are not allowed to move under the weight of the particles that are placed on top of them. From the simulated beds, a particle centre distribution is calculated; this distribution corresponds to a realistic packing, and therefore can (in contrast to a distribution that is fitted to a porosity profile) be used to calculate not only the porosity profiles but also other parameters like the particle outer surface area profile.

The porosity profiles calculated form the simulated packings conform well to the experimental porosity profiles known in literature. The experimental data suggests that in real packings, the wall effect is somewhat stronger than in the simulated packings. This will at least in part be caused by the higher overall porosity in the simulated packings, which allows more space for variations in the particle layers near the wall.

The results show that the wall effect does not depend very strongly on the vessel-to-particle diameter for the range considered (D/d > 5). This is

72

remarkable since most earlier researchers tried to quantify the wall effect as a function of D/d. The average packing density will have more influence on the wall effect than the influence of D/d.

It is obvious that the tortuosity is an important parameter in most transport processes in a packed bed, since it determines the length over which transport through the continuous phase takes place. However, since it is hard to measure tortuosity in a real packed bed, it is usually incorporated in more or less lumped parameters such as ”effective” transport coefficients. It is clear that the tortuosity is also strongly influenced by the presence of a wall. The tortuosity profile is calculated from simulated packings by dropping infinitely small particles through the packing using the potential energy minimisation method. It is shown that, as a first approximation, the local axial tortuosity is inversely proportional to the local porosity.

73

3. The wall effect

3.1 Introduction

The goal of this work is to develop a tool to model chemical reactors with solid catalysts using computational fluid dynamics (CFD). A packed bed reactor of industrial size can contain millions to billions of catalyst particles. Therefore, it is in general feasible nor desirable to take into account each particle separately in a CFD model of a complete reactor. In this chapter, we will look at the structure of the packed bed, especially near a solid wall, based on separate particles. This structure will be characterised by functions that describe parameters such as porosity, particle outer surface area and tortuosity as a function of the position in the bed. These functions can then used in a pseudo-continous CFD model to describe the whole reactor.

It is well known that near a solid surface like the container wall, the particles in a packed bed are no longer positioned randomly in space; due to the fact that the particles cannot penetrate the wall surface, a (more or less) regular pattern occurs near it. Starting at the wall, we find a first layer of particles that are lined up against the wall. The next layers of particles will be increasingly less ordered and far from the wall surface, the packing will be completely random. This so-called wall effect occurs near the outer wall of a packed tube, but also near inserts 1 , e.g., heat exchanger tubes used in highly exothermic or endothermic processes. Measurements in cylindrical RPBs have shown that the radial porosity profile shows complex behaviour; it starts at a value of 1 at the wall, drops to a minimum value of about 0.25 at approximately one particle radius from the wall and oscillates to a mean value (the bulk porosity) in about 5 particle diameters (Roblee et. al., 1958; Benenati and Brosilow, 1962; Goodling et.al., 1983).

The change in structure of the packed bed near the wall will influence the important transfer processes in this region of the bed. Not only will it cause a change in the porosity near the wall, but it will also influence other bed parameters, e.g., specific particle outer surface area, gas path tortuosity, number of particle to particle contact points. These parameters will have their effect on the transfer processes. It is well known that the pressure drop over a packed bed is very sensitive to the porosity, so the local velocity will be influenced strongly by the wall. Also mass transfer between the fluid and the particles and heat transfer between particles, fluid and the wall will be modified in the proximity of the wall. To be able to model flow and transport of mass and heat in the region near the wall, we need to know parameters like

1This is true even for the particles themselves; a radial porosity profile is observed around the surface of each particle in the bed. (Spedding and Spencer, 1995)

74

3.1 Introduction

the porosity, particle outer surface area and tortuosity of the gas path as a function of the distance from the wall.

The limited distance over which the porosity is influenced by the wall (about 5 particle diameters), suggests that the wall effect is only important for very low vessel-to-particle-diameter reactors. However, since the flow resistance is very sensitive to the packing density, the effect causes bypassing of flow along the wall. This may well influence the flow in packed beds up to 40 of particles in diameter. In addition, especially for highly exothermic or endothermic reactions like the Fischer-Tropsch, classical steam reforming or ethene oxidation processes (that are typically carried out in relatively small diameter packed tubes, the flow near the tube wall is very important for the heat transfer from or to the catalyst particles inside. Hence, the wall effect can have a considerable impact on the performance of many packed bed reactors. To be able to make more efficient designs for these RPB reactors, a model is needed that predicts the influence of the wall on the reactor performance. Therefore, not only the porosity profile but also the behaviour of the particle outer surface area and tortuosity near the wall needs to be included in the model.

Most of the discussion in this chapter is limited to packings of identically sized, spherical particles. In real-life applications in packed bed reactors, all sorts of particle shapes are usually applied — for instance, cylinders, rings or quadrulobes. Spherical particles are not very common since they are not very easy to produce, and where they are used, they are usually not of a uniform size. In this chapter, we restrict ourselves to uniform sphere packings, in order to be able to understand the important features of the wall effect. For non-spherical on non-uniform packings, the same mechanisms will play a role, but the ordering effect of the wall will probably be less pronounced. Therefore, the wall effect will in general be less important than for spherical packings; however, there remain many cases in which detailed understanding of the transfer processes near the wall is needed.

In principle, the approach described here can be extended to complex particle shapes and (more easily) to spherical particles with a particle size distribution. However, for less symmetrical particles, the number of degrees of freedom is higher: a particle is not only characterised by its position in the bed but also by its orientation. Therefore, the computational effort involved will be orders of magnitude higher, not only because placing a single particle is more difficult, but also because more repeated simulations would have to be done to get statistically reliable data.

The wall effect has been included in RPB models in the past through an exponential approximation of the porosity profile. Although it is convenient in the computation, this profile does not capture the oscillating nature of

75

3. The wall effect

the actual profile, so these models cannot accurately predict the flow near a wall in a packed bed of spherical particles. In addition, it is not only the porosity near the wall that is of interest, but also the relative amount of particle outer surface area and the tortuosity of the flow channels that determine the transfer of momentum, mass and heat to the catalyst particles. Particle centre distribution models (Govindarao and Froment, 1984; Delmas and Froment, 1988; Mariani et.al., 2002) are fitted to actual measured porosity profiles and do capture the oscillating nature of the porosity near the wall. With this approach, the particle outer surface area profile near the wall can also be calculated, but it is questionable whether a particle distribution that fits the porosity profile necessarily gives a correct particle outer surface profile. In addition, the particle centre distribution does not allow the calculation of the tortuosity profile.

In order to further study the wall effect, two numerical models are developed for the simulation of a random packing of uniformly sized spheres. Since of these simulated packings the position of all sphere centres is readily available, parameters that would be difficult or impossible to measure in a real packed bed can be computed with relative ease. The bed packing simulation models will be compared to existing measurements and literature models to verify that they give realistic packed bed structures. The models are used to determine relations for the porosity, particle surface area and tortuosity as a function of distance to a wall. These relations can be used to model flow, heat and mass transfer in packed beds in general, without the need to simulate a bed packing for each specific case.

3.2 A short review of existing literature

The wall effect has been known for quite some time and was frequently studied using experimental techniques and different numerical models. The so-called integral approach concentrates on the effect of the vessel diameter on the mean bed porosity and flow resistance of the bed (e.g., Carman, 1937; Fand and Thinakaran, 1990; Eisfeld and Schnitzlein, 2001), usually in cylindrical packed beds of uniform spherical particles. This approach leads to expressions for the mean porosity of the bed as a function of the vessel-to-particle diameter ratio:

Vb − Vpεb = = f(D/d) (3.1)

Vb

Also, the flow resistance of the bed, usually described by some form of the Ergun equation (Ergun, 1952) , is found to be a function of the vessel size:

dp = Av + Bv2 A = f(D/d); B = f(D/d) (3.2)

dz

76


In addition to the effect of the bed diameter, the bed height is taken into account in the so-called thickness effect (Zou and Yu, 1995). The influence of the wall effect on heat transfer between the fluid and the wall was also considered (e.g. Hennecke and Schlunder, 1973; Dixon, 1988). These models are very useful for describing experimental data but they do not give a deeper understanding of the wall effect. Consequently, application of these models to packed beds with different geometries is difficult.

For very slender beds (D/d < 5), Computational Fluid Dynamics models have been used to calculate the flow in the channels around a limited number of spheres and near a wall, to study heat transfer between the wall and the bed (Logtenberg and Dixon, 1998). The packing behaviour of these slender beds where the wall zone thickness is of the same order of scale as the bed radius is very different from that of larger beds and therefore the results cannot be extrapolated.

In the so-called differential approach, the structure of the bed and the flow pattern near the wall is studied in more detail; here the porosity is treated as a local variable inside the bed and is determined as a function of the distance from the container wall.

The porosity profile near the wall has been measured directly by filling the open space of a cylindrical packed bed with a fixing agent that has a density that differs from the density of the particles. After the fixing agent had solidified, successive layers of the cylinder are cut away. The mean porosity in the removed volume can be computed from the weight of the removed material through a simple material balance. Roblee et. al. (1958) were among the first who used this technique, using o.a. spherical particles made of cork and hot wax as fixing agent. Later Benenati and Brosilow (1962) used uniform lead spheres and epoxy resin. Goodling et. al. (1982) used polystyrene spheres and an epoxy resin that was mixed with finely ground iron to increase its density. The results indicate that the porosity starts at a value of 1 at the wall, drops to a minimum value of about 0.25 at approximately one particle radius from the wall, and oscillates to a constant value after about 5 particle diameters for uniform spherical particles. For packed beds consisting of spheres or different sizes of non-spherical particles the effect is also found but not as pronounced.

Legawiec & Ziolkowsky (1994) measured the distance to the wall directly for the first layers of a packing using a depth gauge. They found that 98% of the particles that are close to the wall (with their centres between 1 and 2 particle radii from the wall) actually touch the wall. About 10-12 % of the analysed particles was found at a distance of 1.8 to 2.2 radii from the wall.

For use in analytical or numerical models, the porosity profile has been simplified to a function that decreases exponentially from a maximum value at

77

[ ( )]

( )

3. The wall effect

the wall to a bulk porosity value at infinite distance from the wall (Vortmeyer and Schuster, 1983). To avoid numerical problems in some schemes, the porosity at the wall is set to a smaller value. A parameter C is introduced in the equation to allow this; this parameter has to be adjusted to get the required porosity at the wall for each value of the porosity at large distance of the wall (εb):

x ε(x) = εb 1 + C exp 1 − (3.3)

rp

Here x is the distance from the wall and rp the particle radius. If the porosity is allowed to reach its real value at the wall, (ε(0) = 1) the

parameter C can be eliminated and the relation becomes:

−2x ε(x) = εb + (1 − εb) exp (3.4)

dp

However, this profile does not take into account the oscillatory nature of the porosity profile, e.g., the fact that the porosity has a minimum value at a distance of about one particle radius near the wall. Therefore, the flow near the wall in a packed bed cannot be accurately predicted when the wall effect is modelled using equation 3.4. Mueller (1991) formulated an equation for the porosity profile that does include the oscillating nature:

ε = εb + (1 − εb)J0(ax)e −bx 2.61 ≤ D/d (3.5)

where

12.98 a = 8.243 − 2.61 ≤ D/d ≤ 13.0

(D/d + 3.156) 2.932

a = 7.383 − 13 < D/d (D/d − 9.864)

0.724 b = 0.304 −

(D/d)

He also proposes a correlation for the bulk porosity, that is the porosity far from the wall, as a function of the vessel-to-particle diameter ratio:

0.087 εb = 0.379 + (3.6)

D/d − 1.8

For beds with D/d > 10, this bulk porosity is about constant at a value of 0.38 to 0.39, as would be expected as the bulk porosity should be independent of the vessel size. For lower vessel-to-particle diameter ratios, εb increases to

78

5


1

0.8

0.6

ε [-

]

correlation D/d ≤ 13

correlation D/d ≥ 13

0.4

0.2

0 0 1 2 3 4

x/dp [-]

Figure 3.1: Porosity profile as predicted by the correlation of Mueller (1991), using the two different expressions for a vessel-to-particle diameter ratio of 13.

a value of 0.475 at the lower limit. For D/d < 10, there is no zone in the bed that is not affected by the wall; hence the dependence of εb of D/d, which can be seen as a correction for the influence of the opposite wall on the porosity.

One obvious problem with these empirical correlation is that there is a step in the value of a at D/d = 13, from 7.45 at the lower limit to 6.44 at the higher limit, which causes a sudden change of the profile. Figure (3.1) shows the profile for the two different values of a at D/d = 13. The predicted profiles in the high region (D/d > 13, figure 3.2) and, to a somewhat lesser extent, in the low region (D/d between 4 and 13, figure 3.3) among themselves are quite similar, especially in the first few particle diameters from the wall. In light of the error in the experimental data used (including the stochastic nature of a packed bed) and the magnitude of the differences between the empirical relation and the experimental data (between 2.8% and 13.4%, as reported by Mueller (1991)), the significance of the dependency of the parameter a in equation (3.5) on D/d can be put in doubt.

Govindarao and Froment (1986) proposed to describe the structure of the bed near the wall by means of a particle centre distribution. It has been shown

79

3. The wall effect

D/d = 4 D/d = 6 D/d = 8 D/d = 10 D/d = 12

ε [-

]

0 1 2 3 4 5 6

1

0.8

0.6

0.4

0.2

0

x/dp [-]

Figure 3.2: Porosity profile as predicted by the correlation of Mueller (1991), for vessel-to-particle diameter ratios below 13.

that the packing properties near a wall are caused by ordering of the particles. This ordering can be described by the particle centre distribution, that gives the relative number of particle centres as a function of the distance to the wall. None of the particles can have their centre in the zone between the wall and one particle radius from the wall. At a distance of one particle radius from the wall, there is a high concentration of particle centres, representing all the particles that touch the wall. After this, there will be few particle centres until somewhat less than three particle radii from the wall where the second layer of particles is located. Since this layer is less ordered, the peak in the particle centre distribution curve will be somewhat wider than that of the first peak. The third peak will be even less pronounced, until after about ten particle radii the particle distribution will be approximately flat. From the particle centre distribution, other profiles — e.g., the porosity profile — can be calculated (Mariani e.a., 2001).

Govindarao and Froment (1986) used an approximate expression to derive the particle centre distribution from measured porosity profile data from four earlier sources. They divided the bed in cylindrical shells with a uniform width

80


1

0.8

0.6

ε [-

]

0.4

0.2

0

x/dp [-]

Figure 3.3: Porosity profile as predicted by the correlation of Mueller (1991), for vessel-to-particle diameter ratios above 13.

of an integer fraction of the particle radius (ΔR = rp/m where m typically has the value 3) and found that only the contributions of the first two particle layers, at a distance rp and approximately 3rp from the wall (shells m + 1 and 3m), depended on the bed-to-particle diameter ratio. Later, Delmas and Froment (1988) improved the equations by taking into account the decreasing volume of the cylinder shells ( for m = 3).

1.25 n4 = 5.25(1 − )n4 (3.7)

(D/d)2

1.25 n9 = 4.65(1 − )n9 (3.8)

(D/d)2

7.4 nj = 2.1(1 + ) ¯ j = 13, 14 (3.9)nj

(D/d)2

Here, ni is the number fraction of particle centres in shell i (number of particles in the shell relative to the total number of particles in the bed); ni is the ratio of the volume of shell i to the volume of the bed. Within each layer, the particle centres are assumed to be distributed uniformly. The correlations are

D/d = 14 D/d = 16 D/d = 18 D/d = 20 D/d = 22

0 1 2 3 4 5 6

81

3. The wall effect

said to be valid for D/d > 5. The curvature of the vessel wall is not taken into account in these equations since its influence is small (less than 5%) for D/d > 5. In contrast to the earlier equations (Govindarao and Froment, 1986), the relations of Delmas and Froment (1988) are valid only for m = 3, as can be seen from the fact that nm+1 should be independent of m, while nm+1 clearly decreases with decreasing m.

The authors base their correlations on porosity profile data taken from previous literature (Roblee et. al. (1958), Benenati and Brosilow (1962), Ridgway and Tarbuck (1966) and Lerou et. al. (1980)). It has to be noted that these measurements are all at higher D/d ratios (between D/d = 12.7 to D/d = 20.3) except a single data point at D/d = 5.6. As can be seen, the term that contains the contribution of D/d quickly decreases with increasing D/d. In fact, for the first two peaks, at D/d = 10, the contribution of this term is only 1.25%. If the data point at D/d = 5.6 is removed from the data set, a correlation where ni depends only on ni and not on D/d may be just as adequate as the inverse quadratic relation given by Delmas and Froment (equation 3.9). In other words, it appears that there is little evidence that the volume specific number fraction of particles in layer ni/ni depends on D/d at all. In fact, if we look at the factor before ni in equation (3.7), the difference between its value at D/d = 5 (4.84) and at D/d = 20 (5.23) is less than 8%, so this decrease may well be caused at least in part by the fact that the curvature of the vessel is not taken into account (an effect estimated by the authors to be less than 5%). The dependence of the particle centre distribution on the vessel-to-particle diameter ratio will become important only when D/d becomes so small that the first particle layers from both sides of the vessel influence each other, which will be the case if D/d < 5. In this range, one can hardly speak of a randomly packed bed; the packing will become very sensitive to the diameter ratio; these cases fall outside the scope of this chapter. For D/d < 10, particle layers further from the wall will influence each other, but the importance of these layers on the bed properties is limited.

On the basis of these considerations, the particle density model by Delmas and Froment (1988) can be simplified to a model without D/d dependence:

nm+1

nm+1 =

n4

n4 = 5.25 (3.10)

n3m

n3m =

n9

n9 = 4.65 (3.11)

n4m+1

n4m+1 =

n13

n13 = 2.1 (3.12)

n4m+2

n4m+2 =

n14

n14 = 2.1 (3.13)

82


Figure (3.4) shows that for the first two particle diameters from the wall there is only a small difference in the predicted porosity profiles for the full model and the simplified equations above. For the full model, the last oscillation between 2 and 2.5 particle diameters from the wall has a lower minimum value than the minima closer to the wall, which is not in line with the expected damped oscillatory behaviour. In the original publication (Govindarao and Froment, 1986) this behaviour is also visible. In later publications (Delmas and Froment, 1988, Papageorgiou and Froment, 1995) it appears that the porosity profile is broken off just after the second peak (at about 2.1 particle diameters from the wall) and set to decrease in a linear fashion to the bulk porosity in one particle diameter.

0 0 1 2 3 4

0.2

0.4

0.6

0.8

1

ε [-

]

Delmas & Froment

Delmas & Froment simplified

x/dp [-]

Figure 3.4: Calculated porosity profile (average porosity for concentric rings with a width of 0.1 particle diameter) for D/d=8.56 for the full particle centre density profile proposed by Delmas and Froment (1988) compared to the simplified model (equations 3.10 to 3.13).

The model used by Delmas and Froment (1988) assumes that the particle centres are distributed uniformly within each layer. Experimental results have shown that for the first layer, this may not be a valid assumption (Legawiec and Ziolkowski, 1994): most of the near-wall particles actually touch the wall.

83

3. The wall effect

If a uniform distribution of the particle centres is assumed in the first layer, in fact none of the particles will touch the wall.

Mariani et. al. (2002) describe the particle centre distribution as a impulse function at one radius distance from the wall (group 1), a small number of particles uniformly distributed at a distance between 2 and 2.2 particle radii from the wall (group 2), a larger group between 2.42 and 2.97 radii from the wall (group 3) and the rest of the particles at a distance larger than 3.47 particle radii from the wall (group 4). The number of particles in the first and third group is fitted to the porosity profiles of Benenati and Brosilow (1962), Ridgway and Tarbuck (1966) and Goodling et.al. (1983). The numbers are given in terms of the maximum possible number of particles in a close packed cylinder shell NC ; it is assumed that NC is a function of D/d, while the fraction to which this close packing is realised in a bed is independent of the bed to particle diameter ratio. A disadvantage of this approach is that the predicted packing density near the wall does not depend on the bulk packing density, i.e., the wall zone is not affected by loose or dense packing (compacting) of the bed. Mariani et.al. find that the first layer contains 82.5% of the maximum number of particles and the second layer contains 71.0% of the maximum value, independent of bed porosity. The second group is assumed to contain 10.6% of the number of particles in the first group. The fourth zone contains the rest of the particles, so that sum of the number of particles in the four groups corresponds to the total number of particles in the bed.

The number of particles in the first particle layer (at one particle radius from the wall) and the second layer (at about 3 particle radii from the wall) for the model by Delmas and Froment (1988) and the model by Mariani et.al. (2002) is compared in figure 3.5. It can be seen that the difference between the two estimates is not very large, despite the considerable difference in the model layer structure and position.

Mariani et.al. (2002) claim that with this approach, the particle outer surface area profile near the wall can also be calculated. However, there may be many particle centre distributions that fit the experimental porosity data just as well as the one they propose, but give a different particle outer surface distribution. In other words, it is not demonstrated that a particle distribution that fits the porosity profile necessarily gives a correct particle outer surface profile.

Several so-called sequential packing models have been used to create computer models of randomly packed beds. In these models, a packed bed is simulated by adding new particles one by one at stable positions to a set of previously determined particle positions. Generally, stable positions are considered where a particle has three contact points, touching three other particles

84


0

0.2

0.4

0.6

0.8 F

ract

ion

part

icle

s in

laye

r [-

]

Layer 1; Mariani et.al.

Layer 1; Delmas & Froment



5 10 15 20 D/d [-]

Figure 3.5: Number of particles (as a fraction of the total number of particles in the bed) in the first and second layer from the wall, as predicted by the correlations of Froment and Delmas (1988) and Mariani et.al. (2002), as a function of the bed to particle diameter ratio.

or two other particles and a container wall. The models differ in the way these stable positions are found. A distinction can be made between stochastic models in which a random number generator determines which of the stable positions a particle will take, and deterministic models where the outcome is predetermined (for a given initial or seed set of particles).

Spedding and Spencer (1994) have studied the structure of an infinite packed bed, but have not included wall effects in their simulations. They used a dynamic model where each new particle was given a random velocity and direction from a set position above the bed. The movement of the particle was calculated taking into account the effects of impact, rolling and sliding of the particle on the bed. They found that the resulting packing was not satisfactory and rejected the complex model in favour of models that did not take into account momentum effects. In drop-and-roll models, a particle is released from a random position just above the top of an existing bed. From there, it moves straight down until it touches a particle or the bottom wall of the container. In the first case, the particle continues its way down by rolling along

85

3. The wall effect

the surface of the second particle until it reaches either a third particle or the container wall or drops off. The simulation is continued in this way, calculating the path of the particle along the surface of another particle, in the groove between two other particles or between another particle and the container wall, until it reaches a stable position. These models involve a relatively large number of complex geometrical calculations to determine the particle path, which leads to a complex and computationally intensive simulation procedure. The particles are considered to be sticky i.e., a particle placed earlier will not move when another particle is placed on top of it. Therefore, in theory, unphysical (unstable) structures can occur in the bed and the packing density will usually be less than that of real packings.

An alternative method to simulate a bed packing is the particle placement model. Here, starting with an initial layer of particles, all possible stable positions on top of the bed are calculated (e.g., Mueller, 1997). These positions are either those where the new sphere touches three existing spheres or those where the new sphere touches two existing spheres and the container wall. The new sphere is placed at (e.g.) the lowest of these positions, after which the list of stable positions is updated to reflect the changes caused by the new particle. This scheme will give the highest packing density possible. It does need a seeding plane of particles to start the calculation. Mueller (1997) compares different procedures for placing either wall spheres (touching the wall) and inner spheres (not touching the wall) and compares the results to measured porosity data. He finds the best fit of the measurement data is the packing method that gives the highest density; this is achieved by running the model several times and finding the fraction of wall to inner spheres that gives the highest packing density.

From the literature review, it can be concluded that there is no satisfactory fit function that accurately describes the porosity profile, particle outer surface area profile and tortuosity near the wall. The particle centre density distribution approach gives reasonable porosity profile predictions, but it is not certain that the particle outer surface area is predicted correctly. Furthermore, there is no method to predict the influence of the wall on the tortuosity. Numerical bed simulation models can give the opportunity to study all important parameters: porosity, specific surface and tortuosity near the wall. However, none of the publications on simulated beds give these parameters.

Therefore, two bed simulation models will be developed here. A potential energy minimisation model and a particle placement model. The results of the different models can be compared; it is expected that the former will give somewhat higher porosity beds than the latter one.

86

3.3 Bed packing simulation models


In this section, two packed bed simulation models are developed; one stochastic model where the stable position of a new particle is found by numerical minimisation of the potential energy and a deterministic-stochastic model that is a variant of the ball placement model. The purpose of this effort is to claculate the porosity, outer particle surface area and gas path tortuosity as a function of the distance from the wall, and if possible as a function of the bulk porosity of the bed.

3.3.1 Potential energy minimisation model

In this computer model, equal-sized spherical particles are released with zero speed at a random position just above the top of the bed in a cylindrical container with radius R. The random positions are distributed uniformly over the top surface of the bed. The movement of the particles is followed as a function of time by minimisation of a function that is equivalent to the potential energy. The spheres are allowed to overlap a distance Δl with other spheres and the container wall. When a sphere overlaps with another sphere, it will feel a force directed away from the other sphere, along the line connecting both spheres. When a sphere overlaps with the side wall of the cylindrical container, it will feel a force directed toward the axis of the container. When a sphere overlaps with the flat bottom of the container, it will feel a force directed upward. The potential energy is the summation of the effect of gravity and these repelling forces.

Each iteration, the particle is moved a short distance Δl in the direction in which the potential energy decreases most rapidly, that is, in the direction of the resultant force. The force that the new particle (n + 1) with radius rp

feels at position (x, y, z) is given by:

Fr = Fg + Fw + Fp

Fg =

⎛

⎝ 0 0 -1

⎞

⎠

Fw = f(r − z)

⎛

⎝ 0 0 1

⎞

⎠ + f ( √

x2 + y2 + r − R )

√ x2 + y2

⎛

⎝ −x −y

0

⎞

⎠

Fp = n ∑

i=1

f (2r − δi) δi

⎛

⎝ x − xi

y − yi

z − zi

⎞

⎠

(3.14)

(3.15)

(3.16)

(3.17)

87

√

{

3. The wall effect

where δi is the distance between the new particle and particle i

δi = (x − xi)2 + (y − yi)2 + (z − zi)2 (3.18)

The function f(x) is the force that occurs when a particle is compressed a distance x; for simplicity a linear dependence was chosen:

0 x ≤ 0 f(x) = (3.19)

Kx x > 0

The constant K and the step size Δl are parameters of the model. When the values are chosen so that KΔl � 1 (the contribution of gravity), the spheres will not be compacted by more than a distance Δl at any time. The step size should be chosen to be much smaller than the particle size. The number of iterations or steps per particle is chosen large enough to ensure that the particles reach a stable position. In a simulated packing, it is unlikely that the distance between the lowest reachable position and the top of the highest particle will be more than 5 particle diameters, hence a safe choice for the number of steps is

√ nΔl > 10 2r (3.20)

This algorithm can be programmed quite efficiently; it only takes about 50 lines of code in the C programming language and simulating a bed of 5000 particles takes about 30 minutes on a standard personal computer.

Although it is implemented here for a uniform sphere packing, this method could also be used to generate packings of non-uniform spheres.

3.3.2 Ball placement model

In the ball placement model, a list of all possible stable positions at the top of the bed is maintained. Stable positions are positions where a sphere either touches three other spheres or two other spheres and the container wall. If a sphere with centre point p and radius r touches three other spheres with centre points p1, p2 and p3, it must follow that point p is an intersection point of three spheres at p1, p2 and p3 with radius 2r. These points, if they exist, are given by: ⎧ ⎨ ‖p − p1‖ = 2r

‖p − p2‖ = 2r (3.21) ⎩ ‖p − p3‖ = 2r

The analytical solution is an unwieldy second order polynomial; the solution can be simplified by a change of coordinate system so that sphere 1 is in

88

′ ′ ′ ′

′ ∥ ∥ ∥ ∥

′ ∥ ∥ ∥ ¯ ∥

′ ∥ ∥ ∥ ∥

′

√

( )


the origin, sphere 2 on the x-axis and sphere 3 in the x, y plane: ⎛ ⎞ ⎛ ′ ⎞ ⎛ ⎞ 0 x x

′ 2 3 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠p = 0 p = 0 p = y (3.22)1 2 3 3

0 0 0

It is easy to see that x2 ′

is simply the distance between spheres p1 and p2 and that x3

′ is the distance between p3 and the line through p1 and p2. Hence

x2 = d12

¯x = ∥ d

¯12.d13 d12

∥ (3.23)3 ∥ ¯ ∥d12.d12

y3 = d12

where

d12 = p2 − p1 (3.24)d13 = p3 − p1

The two intersection points can be calculated easily in the local coordinate system:

′ x x = 2 (3.25)

2 ′ 2 ′ ′ ′ 2

′ x − x2x3 + y y = 3

′ 3 (3.26)

2y3

′ 2 ′ 2 z ′ = ± 1 − x − y (3.27)

Transformation to the global coordinate system gives the coordinates of two intersection points of which only the one with the higher z coordinate is a candidate for a stable position. We now have one sphere p that touches three other spheres. The question remains whether this position will be stable under gravity. The stability criterion may be formulated as follows: form a triangle T by a projection of p1, p2 and p3 on the x, y plane of the global axis system (i.e, along the gravity vector). Position p is stable if (and only if) the projection pxy of p on the x, y plane is inside the triangle T (see figure 3.6). It can be easily verified that this criterion is met if the dot products of the vector from a corner of T to pxy and from a corner to the next corner of T :

xy xy xy xy(p − p ) . p − p (i, j) = (1, 2); (2, 3); (3, 1) (3.28)i i j

have the same sign for each given i, j.

89

3. The wall effect

Figure 3.6: Three spheres forming a stable (left) and unstable (right) position for a fourth sphere. The projections on the (x, y) plane are drawn as well as the triangle T and the projection of the fourth sphere (dashed) and its centre point.

To further generalise the stability criterion, we realise that the three existing spheres and the new sphere p form a tetrahedron with p at the top and p1, p2, p3 as base. If the vertical projection of p falls inside the vertical projection T of the base triangle, the point p also falls inside the vertical projection of any triangle that is formed by connecting any three points in each of the ribs p − pi

of the tetrahedron. For instance, we could take the points where the sphere p touches each of the other spheres (halfway each of the ribs of the tetrahedron). The only important parameter for the stability of the sphere p is the location of the three points on which it rests; the shape of the objects on which it rests is not an issue. Therefore, we can state more generally that a sphere is in a stable position under gravity if (and only if) the vertical projection of its centre point falls inside the vertical projection of the triangle between the three points on which it rests.

To find the stable points that involve two spheres p1 and p2 and the wall, we again take two spheres with double radius 2r and take a cylinder with radius R − r. The candidate for a stable point is the intersection point between both spheres and the cylinder, if these points exist, with the highest z coordinate.

90

( ) (

′ ′ ′ ′


The intersection between both spheres will be a circle with radius rc in the plane V halfway between both spheres and normal to the vector p1−p2. Normal projection of this circle on the x, y plane will give an ellipse with primary radius rc and secondary radius rc cos(α) where α is the angle between the plane V and the x, y plane. If the spheres are smaller than the cylinder, which will be the case in a packed bed, the ellipse will intersect the cylinder in at most two points. These points can be found by rotation around the axis of the cylinder to align the primary axis of the ellipse with the x axis, as given by the equations:

( )2 ( ) x

′ − xc + y ′ − yc = R2

′ ′ )2 x y

+ = 1 (3.29) rc rc cos(α)

x1 + x y1 + y2 2 xc = yc = 2 2

The resulting fourth order polynomial is solved numerically; after transformation the x and y coordinates of the intersection points (if any) are found. The z coordinates can be found by entering the values in the equations that describe the spheres; for each sphere two roots are found, the intersection points are given by the roots that are the same for both spheres. The candidate for a stable position is the root with the highest z coordinate. The stability criterion for a wall point is equivalent to that found for a sphere supported by three other spheres.

The ball placement model must start with an initial layer of particles. This can be either some regular packing or a layer of particles generated by another packed bed simulation algorithm. A list of stable positions is calculated for this initial layer; the number of stable positions is in the same order of magnitude as the number of particles in the first layer; for a hexagonal layer there are two stable positions per particle, for a random layer less. A new sphere is placed at the lowest stable position. If more than one stable position is at exactly the same height (which will be very exceptional unless the existing bed has a regular packing), one of these position is chosen at random. This will ensure that the order in a bed will decrease with height in a random way, not in a coıncidental way. After the particle has been placed, the list of stable positions is updated by

1. deleting any stable positions that are less than 2r removed from the new sphere and therefore no longer available, and

2. adding any new stable positions that involve the new sphere.

91

3. The wall effect

After this, a new position can be chosen et cetera. Since only stable positions in the direct vicinity of the new sphere are affected, this is a very efficient operation.

Since of all stable positions, the lowest one is chosen, the ball placement algorithm will create random packings with the highest density. In order to be able to create bed packings with different density, a parameter P is added to the model. Instead of the lowest stable position, the model takes at random one of the stable positions for which

z ≤ z0 + P (zmax − z0) 0 ≤ P ≤ 1 (3.30)

For P = 0, the model is completely deterministic (for a random first layer) and will have a high packing density; for P = 1, the model is completely stochastic and will have a — relatively — low packing density.

3.3.3 Modelling approach

Calculations were performed with both packing models for five different vesselover-particle diameter ratios. To rule out any influence of the initial ’seed’ layer of particles needed for the ball placement model, a seed layer was constructed for each case as follows: a small number of spheres (50 for the smaller tubes, 100 for the largest) were dropped at random in an empty tube; then all balls not touching the vessel bottom were removed, after which another batch of spheres were dropped, et cetera. This procedure was repeated until, on visual inspection, no sphere could be added so that it would touch the vessel bottom. This ’seed layer’ was then used as initial condition for all further simulations. In each case, beds were simulated by adding spheres to the existing number in the bottom layer until the height of the bed was 110 particle diameters. For the stochastic models, the results presented here are averaged results of 10 simulations starting with the same seed layer.

An example of the first part of a simulated bed is shown in figure (3.7).

3.4 Bed packing simulation results

3.4.1 Mean bed porosity and thickness effect

One of the most obvious parameters that can be determined from a simulated packed bed is the mean bed porosity. This is simply defined as the volume of open space divided by the volume of the bed. The mean bed porosity will depend on the size of the vessel and will be higher than the bulk porosity of the packing far away from any walls, since it includes the effect of the wall. This in

92


Figure 3.7: A 3-D plot of the first 200 particles in a ball placement simulation (D/d=8.56).

principle includes the effect of the cylindrical wall as well as the bottom and top boundaries of the bed. Since most literature data concerns only dependence on the side wall, we will exclude the bottom and top sections of the bed. In order to determine the extent of these sections, we will first look at the influence of the top and bottom bed boundaries on the mean bed boundaries, the so-called thickness effect.

For simulated as well as most real packings, there is a difference between the bottom of the bed and the top. The bottom will be confined by a solid (albeit usually perforated) wall where the packing will be influenced in (approximately) the same way as near the side walls of the container. Near the wall, the mean coordination number, i.e., the number of contact points per sphere, will be approximately six. In other words, a particle will rest on three contact points (either other spheres or the wall) and will on average support three other spheres. The top of the bed however, is more or less a free surface,

93

3. The wall effect

where the coordination number is much lower. As such, the top of the bed is not as well-defined as the other bed boundaries. The most convenient way to define it is as hmax, the top of the topmost particle. As there may well be peaks and valleys on the top surface of the bed with amplitudes of several particle diameters, the porosity of the bed defined this way increases rapidly near the top of the bed. To study the thickness effect in the simulated packing, the mean bed porosity between height h1 and h2 is defined as:

∑ n i=1 Vi

εm(h1, h2) = (3.31)πR2(h2 − h1)

where Vi is the volume of particle i that lies inside the slice of the bed considered. This includes sphere segments of particles whose centres lie outside the range h1, h2 but are closer than one particle radius to the edge and excludes sphere segments of particles whose centres lie inside the range but are closer than one particle radius to the edge (see figure 3.8)

Vi = 1/3π (r + zi − h1)2 (2r − zi + h1) h1 − r ≤ zi ≤ h1

Vi = 4/3πr3 − 1/3π (r − zi + h1)2 (2r + zi − h1) h1 ≤ zi ≤ h1 + r

Vi = 4/3πr3 h1 + r ≤ zi ≤ h2 − r

Vi = 4/3πr3 − 1/3π (r + zi − h2)2 (2r − zi + h2) h2 − r ≤ zi ≤ h2

Vi = 1/3π (r − zi + h2)2 (2r + zi − h2) h2 ≤ zi ≤ h2 + r

Vi = 0 elsewhere (3.32)

The mean bed porosity is influenced not only by the effect of the cylindrical side-wall of the bed but also by the top and bottom boundaries. In order to be able to exclude the effect of the top and bottom of the bed, the effect of these boundaries on the mean bed porosity is investigated. The mean bed porosity is calculated for a fixed top cut plane of 5 particle radii from the top of the bed and a bottom cut plane varying between 0 and 10 particle radii from the bottom of the bed and vice versa. The result can be seen in figure (3.9) for a bed to particle diameter of 8.56. From this figure it can be concluded that the effect of the top and bottom bed boundary can be ruled out by disregarding the part of the bed extending 5 particle diameters from the top and bottom.

Zu and You (1995) have given a formulation for the mean bed porosity, based on their own experimental data and that of Carman (1937) and Dixon

94


h2

h1

r

iz

iz

iz

iz

i 2z +r-h

i 1z +r-h

h -z +r2 i

ih -z +r

1

Figure 3.8: The different contributing volumes to calculate the mean bed porosity between two planes h1 and h2.

(1988), for loose and dense packed beds as a function of the bed to particle diameter that excludes the thickness effect:

10.68/(D/d) − 1)ε(D/d) = 0.400 + 0.010(e (D/d) > 3.91 (3.33)

for the loose random packing and

15.30/(D/d) − 1)ε(D/d) = 0.372 + 0.002(e (D/d) > 3.95 (3.34)

for the dense packing. Mariani et.al. (2002) propose a hyperbolic relationship:

0.355 ε(D/d) = 0.375 + (3.35)

D/d

The mean bed porosity calculated from the simulated packings and the values predicted by equations (3.33), (3.34) and (3.35) are compared in figure 3.10. The simulated packings are somewhat looser than the experimental packings. This is a known feature of bed packing simulations that can probably be ascribed to the fact that the particles are not allowed to move under the weight of particles that are placed on top of them. In other words, the fact that positions that are stable for a single sphere may become unstable

95

3. The wall effect

0.44

0.438

0.436

ε m [

-]

top cut

bottom cut

0.434

0.432

0.43 0 2 4 6 8 10

Cut position [particle radii]

Figure 3.9: The effect of the top and bottom of the bed on volume average bed porosity: the bulk porosity as a function of the distance of the cut plane from the top of the topmost particle (top cut) and from the bottom of the bed (bottom cut).

when other spheres are placed on top of it. The models follow the trend of the experimental curves quite well. For the low range tube diameters, the correlations by Zu and You predict a sharper rise in mean bed porosity than is observed in the simulated packings.

It was expected that it would be possible to increase the porosity of the simulated packings by changing the way particles are placed in the ball placement method (i.e., by not always choosing the lowest position available). It was found that with the methods described it was not possible to increase the porosity of the packing by more than 2 percent point.

3.4.2 Sphere centre density

The number of particles in the first two layers near the wall of a number of simulated ball placement packings is compared to the literature models in figure (3.11). For the simulated bed, all particles between 2 and 3 radii

96


0.5

0.38

0.4

0.42

0.44

0.46

0.48

ε m [

-]

Yu & Woo loose Yu & Woo dense Mariani Place Drop

0.36 4 6 8 10 12 14 16 18 20

D/d [-]

Figure 3.10: The mean bed porosity εm as a function of the vessel-to-particle ratio D/d as predicted by the deterministic ball placement model, the random ball dropping model and the empirical relations for dense and loose packings of Zu and You (1995) and Mariani et.al. (2002).

from the wall are included in the second layer. It can be seen that there is a quite good agreement between the literature models and the ball placement packings; although the first layer of the simulated packings contains somewhat less particles than the literature models predict.

The packing density of a typical packing from the ball placement model of this work is compared to the packing density in the model of Froment and Delmas (1988) and the model of Mariani et.al. (2002), for a bed to particle diameter ratio of 10. The simulated packing contains about 4300 spheres and has a height of 110 particle diameters; a slice with a height of 5 particle diameters is removed from the top and bottom of the packing to remove any bed thickness effects. The particle centre density was calculated with a uniform bin size of 1/6 of a particle radius; the width of the first layer of particles was investigated separately and found to be very small (less than 0.01 particle diameters wide).

The resulting accumulative particle density is shown in figure (3.12). The

97

3. The wall effect

0

0.2

0.4

0.6

0.8

Fra

ctio

n pa

rtic

les

in la

yer

[-]





Layer 1; place model

Layer 2; place model

5 10 15 20 D/d [-]

Figure 3.11: Number of particles (as a fraction of the total number of particles in the bed) in the first and second layer from the wall, as predicted by the correlations of Froment and Delmas (1988) and Mariani et.al. (2002) and as calculated from the ball placement model as a function of the bed to particle diameter ratio.

figure gives the number of particle centres as a fraction of the total number of particles in the bed that can be found between the wall and the chosen wall distance. Figure (3.13) gives the density function, i.e., the slope of the curves in figure (3.12). The slope decreases to 0 at higher wall distances because the radius and therefore volume of the remaining cylindrical bed core becomes small. For the model of Mariani et.al. as well as for the simulated bed packing, a fraction of the particles is at a position exactly one radius from the wall, leading to a step change in the accumulated density function.

It can be seen that the number of particles in the first layer near the wall in the simulated packing is lower than for the literature correlations. Also, the literature models expect no particles in the zones between layers, while in the simulated packing the packing density never reaches zero. These two phenomena are connected as a lower packing density near the wall leaves more open space for intermediate particles. Also, the notion that there are no particles in

98


0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

frac

tion

par

ticl

es [

-]

Delmas & Froment

Mariani et.al.

Place model

x/dp [-]

Figure 3.12: Cumulative particle density as function of the distance from the wall for a simulated ball placement packing, the model of Delmas and Froment (1988) and the model of Mariani et.al. (2002).

between the designated layers is an assumption that was used in the fit procedures of both literature correlations, but that does not necessarily follow from the experimental data. The main features of the literature correlations (very narrow first layer of particles at one particle radius from the wall, somewhat wider layer of particles at about 3 particle radii from the wall) are observed in the simulated packing.

When we consider the packing of particles in the first two layers from the wall, it is clear that the overall packing density is a much more important parameter for the degree of order in the bed than the curvature of the vessel wall. The higher the packing density, the more the packing will resemble a close packing. Hence there will be less space in between the particles where other particles — belonging to intermediate layers — fit. Therefore, the peaks in the density distribution will be sharper at higher densities. In the bed simulations, the density is somewhat lower than that of the experimental beds, leading to less close packing near the wall and less sharp peaks in the density distribution.

99

3. The wall effect

0

0.2

0.4

0.6

0.8

1 pa

rtic

le c

ente

r de

nsit

y [-

]

Delmas & Froment

Mariani et.al.

place model

0 1 2 3 4 5 x/dp [-]

Figure 3.13: Particle density as function of the distance from the wall for a simulated ball placement packing, the model of Delmas and Froment (1988) and the model of Mariani et.al. (2002).

3.4.3 Porosity profile

Mariani e.a. (2001) give analytical expressions to calculate porosity profiles from sphere centre distribution data. They are here rewritten to calculate the profiles for a simulated bed (i.e., e set of sphere positions) instead of a sphere centre distribution.

The mean porosity in a cylindrical shell of the bed with width Δr can be written as

∑ n i=1 vi(r + Δr) − vi(r)

ε(r, Δr) = 1 − (3.36)πh ((r + Δr)2 − r2)

where vi(r) is the volume of particle i left inside a cylinder of radius r from the centre of the bed:

⎧ 4 ⎨ 3 πrp

3 ri < r − rp

vi(r) = 4 r3 (EV (r, ri) + θ(r, ri)) |ri − r| < rp (3.37)3 p ⎩ 0 ri > r + rp

100

√

[


where ri is the radial position of particle i

ri = x2 i + yi

2 (3.38)

and ⎧ ⎨ π if ri < r θ(r, ri) = π/2 if ri = r (3.39) ⎩

0 if ri > r

The term EV is defined by:

1 Δ EV = (αV + βV φB + )F

(φM − φB)2 φA ] (3.40)1 Δ

+ (φA − φB )( J + βV D)3 φ2 k2

A

where

φA = min(1, φS ) ( )2 ri + r

φS = rp (

ri − r )2

φB = rp

φM = max(1, φS )

αV = 1(σ + Δ2 − 6Δ − 3)

31

βV = (4 + 3Δ − 2σ)3ri 2 − r2

Δ = 2rp

2 2r + riσ = 2 2rp

and the Carlson standard elliptic integrals (e.g., Press e.a., 1992):

F = RF (0, k2 , 1)

D = RD(0, k2 , 1)

J = RJ (0, k2 , 1, p)

101

3. The wall effect

where

φM − φAk2 =

φM − φB

φB k2 p =

φA

The use of equation 3.40 assures that the curvature of the cylindrical surface is taken into account.

The porosity profiles predicted by the particle placement model and the particle drop model are compared to experimental data in figures 3.14 and 3.15. The model curves are for a bed-to-particle diameter ratio of 5, 10 and 15; the experimental data are for five D/d ratios of 5.6 and higher. As can be seen, neither the experimental data nor the model shows a clear dependence of the porosity profile on D/d. The bulk porosity level of the experimental data is lower than that of both models. Also, the amplitude of the oscillations is slightly under-predicted by the model. These two observations are related, since a higher porosity in a particle layer will create holes that can be partially filled with particles that are somewhat offset from the layer.

The calculated porosity profiles for the particle centre density distribution proposed by Mariani et.al. (2002) and for the simplified model based on the distribution proposed by Delmas and Froment (1988) are compared to experimental data and results of the particle placement model in figure (3.16). For the model curves, the average porosity in concentric cylinder shells of 0.1 particle diameter is plotted, since the experimental procedures also determine the porosity in thin shells of the bed. For the ball placement model results, the porosity profiles for ten simulated beds with D/d from 5 to 20 were averaged. It can be seen that there is quite good agreement between the computed and measured porosity profiles. Both particle centre distribution models predict the porosity somewhat better than the ball placement model, but this is not surprising since these are based on curve fits on the experimental data points, while the ball placement model is built up from first principles.

3.4.4 Particle surface area profiles

Many processes in a packed bed, e.g., mass- and heat transfer or friction, involve the outer surface of the particles. Often, for instance for chemical reactors with cooled or heated walls, especially the rate of transfer near a wall is important. Therefore, it is important to have a good estimate of the particle surface area near the wall. In a randomly packed bed, the outer particle surface

102

∑


1

0.8

0.6

0.4

0.2

0 543210

x/dp [-]

ε [-

]

Goodling, D/d=7.35 Goodling, D/d=8.41 Goodling, D/d=8.56 Goodling, D/d=10.7 Goodling, D/d=16.77 Benenati, D/d=5.6 Benenati, D/d=14.1 Benenati, D/d=20.3 Benenati, D/d=∞ Roblee, D/d=8.82 Roblee, D/d=13.7 place model, D/d=5 place model, D/d=10 place model, D/d=15

Figure 3.14: Porosity profile as predicted by the particle placement model (for D/d=5,10 and 15) compared to experimental data (Goodling et.al. (1983), Benenati and Brosilow (1962); Roblee et.al. (1958)).

area in a sufficiently large volume can be calculated from:

Ap 6(1 − ε) sv ≡ = (3.41)

V dp

Near the wall, the volume of the particles in the ordered layers decreases quite rapidly (leading to an increasing porosity to a value of 1.0 at the wall), while the surface of the particles does not decrease at quite the same rate. In fact, the particle outer surface per unit volume does not decrease to zero (as does the particle volume per unit volume) , but to a finite value. Hence equation 3.41 can not be used to calculate the particle surface area from the porosity profile near a wall.

Mariani et.al. (2001) devised a method to calculate the particle outer surface area profile from a given particle centre density analytically. These equations are rewritten here to allow calculation of the surface area profile from a simulated bed packing. The particle out surface area per unit of volume in a cylindrical shell between r and r + Δr can be written as:

n Ai(r + Δr) − Ai(r) a(r, Δr) = i=1 (3.42)

πh ((r + Δr)2 − r2)

103

[

3. The wall effect

0

0.2

0.4

0.6

0.8

1

ε [-

]

Goodling D/d=7.35 Goodling, D/d=8.41 Goodling, D/d=8.56 Goodling, D/d=10.7 Goodling, D/d=16.77 Benenati, D/d=5.6 Benenati D/d=14.1 Benenati, D/d=20.3 Benenati D/d=∞ Roblee, D/d=8.82 Roblee, D/d=13.7 drop model, D/d=5 drop model, D/d=10 drop model, D/d=15

0 1 2 3 4 5

x/dp [-]

Figure 3.15: Porosity profile as predicted by the particle drop model (for D/d=5,10 and 15) compared to experimental data (Goodling et.al. (1983), Benenati and Brosilow (1962); Roblee et.al. (1958)).

where Ai(r) is the outer surface of spherical particle i that is left inside a cylinder with radius r: ⎧ ⎨ 4πRp

2 ri < r − rp

Ai(r) = 4R2 (EA(r, ri) + θ(r, ri)) |ri − r| < rp (3.43)p ⎩ 0 ri > r + rp

where ri is the radial position of particle i and the term EA is defined by:

1 Δ EA = (αA + βAφB + )F+

(φM − φB)2 φA ] (3.44)1 Δ (φA − φB)( J + βAD)

3 φA2 k2

with the definitions given for equation 3.40 in the previous section and:

αA = −Δ − 1

βA = 1

104


0

0.2

0.4

0.6

0.8

1

ε [-

]

Goodling, D/d=7.35 Goodling, D/d=8.41 Goodling, D/d=8.56 Goodling, D/d=10.7 Goodling, D/d=16.77 Benenati, D/d=5.6 Benenati, D/d=14.1 Benenati, D/d=20.3 Benenati, D/d=∞ Roblee, D/d=8.82 Roblee, D/d=13.7 Mariani et.al. Delmas & Froment simplified place model

0 2 4 6 8 10

x/dp [-]

Figure 3.16: Measured porosity profiles (Goodling et.al. (1983), Benenati and Brosilow (1962); Roblee et.al. (1958)) compared to computed porosity profiles from particle density distribution models by Mariani et.al. (2002) and equations (3.10 - 3.13) and results from the ball placement model.

The calculated particle surface area profiles for the sphere dropping model and the sphere placing method are shown in figure (3.17). The curve for the sphere placing method gives the average curve of six such curves for simulated beds of D/d=5 to D/d=19. There is a large difference near the wall between both literature correlations, even though the porosity profiles that are predicted by these models are very similar (see figure 3.16). The fact that in the model by Delmas and Froment the particles in the first layer are distributed uniformly over al shell with a thickness of dp/6 causes the specific surface area to drop to zero near the wall. In contrast, the specific surface predicted by the distribution of Mariani et.al. is (nearly) constant near the wall since it is influenced only by the spheres with centres at one particle radius from the wall. The ball placement model shows a more gradual decline near the wall, due to the fact that in the simulated bed there are some particles close to, but not touching the wall. Figure (3.17) shows clearly that the assumptions made about the near-wall packing strongly influence the specific surface area near the wall, even for models that give similar porosity profiles. Therefore, a

105

3. The wall effect

particle distribution fitted to the porosity profile, as proposed by Delmas and Froment and Mariani et. al. can not generally be used to calculate other bed packing parameters.

0

0.5

1

1.5

2

s v /s

vb [

-]

Delmas & Froment Mariani et.al. Ball placement

0 1 2 3 4

x/dp [-]

Figure 3.17: Specific surface profiles (divided by the bulk specific surface) as calculated from the computed porosity profiles from particle density distribution models by Mariani et.al. (2002) and by Delmas and Froment (1988) compared to the results from the ball placement model.

3.4.5 Tortuosity profiles

The radial profile of the axial tortuosity can be studied using the potential energy minimisation model described in section 3.3.1, when the diameter of a tracer particle that is released is set to zero. Such a particle will not settle and will instead find a path through the flow channels to the bottom of the bed. In fact, the path of such a particle is a streamline for creeping axial flow (flow without recirculations, turbulence or inertia).

The radial tortuosity profile is generated from a large number (400) of such tracer particle paths, starting at random positions just above a simulated bed. Each tracer particle path is divided into stretches in with a height of of 1/2 bed particle radius; for each of these stretches, the length of the path and therefore

106

∑


the tortuosity is calculated. The bed is divided into a number of thin shells. Each stretch of a path is attributed to a shell depending on its centre radial position. For each shell, the average tortuosity of all stretches that have their centre point at a radial position within that shell is calculated. The volume average tortuosity in the bed, i.e.,

1 n

R2 (ri

2 − ri2 −1)τi (3.45)

i=1

is about 1.43, which is close to the expected value for randomly packed beds. The influence of the bed-to-particle ratio on the volume averaged tortuosity is not very large because the volume of the near-wall zone is not very large compared with that of the whole bed.

1

1.2

1.4

1.6

4 6 8 10 12 14 16

τ m [

-]

D/d [-]

Figure 3.18: Calculated bed-average tortuosity for simulated beds with different bed-to-particle ratios. For D/d=5 and D/d=8.56, points for multiple different packings are shown.

The preference of the flow paths for the near-wall zone becomes visible when the number of one radius particle path stretches in each cylindrical shell is plotted (figure 3.20). It can be seen that there are many flow paths near the

107

3. The wall effect

wall and only a few flow paths that pass at a distance of one particle radius from the wall. The reason for this is that the channels between the particles in the first layer are all approximately horizontal, so that there is often no driving force for the tracking particle to cross the channel.

2.4 D/d=8.56( 1) D/d=8.56(2) D/d= 8.56(3) D/d=8.56(4)2.2 D/d=8.56(5) D/d=5 D/d=7 D/d=9

2 D/d=11 D/d=15

1.8

τ [-

]

1.6

1.4

1.2

1 0 1 2 3 4

x/dp [-]

Figure 3.19: Local tortuosity as a function of distance from the wall, for several simulated packings with different bed to particle diameter ratio D/d.

There is quite a lot of spread in the raw tortuosity data. This is a result of the stochastic nature of the data; the random factors in the bed packing and the way the tortuosity is determined. The spread is especially large in regions where the tortuosity is high, i.e., where the number of tracer particle stretches in each shell is low.

From figure (3.19) and figure (3.14) it can be seen that the tortuosity profile is roughly the inverse of the porosity profile: when the porosity increases, the tortuosity decreases and vice versa. Therefore, it seems practical to try to write the tortuosity as a function of porosity:

τ(x) = f(ε(x)) (3.46)

108

4


n/n t

rack

s [-

]

D/d=8.56(1) D/d=8.56(2)

D/d=8.56(3) D/d=8.56(4)

D/d=8.56 (5) D/d=5

3 D/d=7 D/d=9

D/d=11 D/d=15

2

1

0

0 1 2 3 4

x/dp [-]

Figure 3.20: Number of tracer particle path stretches as a function of distance from the wall, divided by the number of tracks, for several simulated packings with different bed to particle diameter ratio D/d. Note that the first point (at x=0) is not shown since it has a value 10 times the maximum of the graph.

where the function should have the following boundary behaviour:

ε ↑ 1 → τ ↓ τw (3.47)

ε = εb → τ = τb (3.48)

where τb is the tortuosity far from the wall and τw is the tortuosity at the wall (which is close to one). The tortuosity at the wall is not exactly equal to zero because a small volume around the point where the particles touch the wall is not accessible for gas-phase transport since the space is too narrow. In the model the size of this volume is determined by the step size with which the flow paths are calculated. In reality there will also be excluded volume, but the magnitude will depend on the transport process at hand. For instance, for diffusion processes the size will be in the order of the free path length of the gas molecules. In general the excluded volume will be small.

It should be noted that in theory, tortuosity and porosity are independent properties; if one would drill a flow path into a block of solid material, both

109

3. The wall effect

the porosity and the tortuosity can be chosen. However, in a fixed randomly packed bed, the properties of the flow channel are of course not independent, e.g., compaction of the bed will not only lead to a higher porosity but also to a change (increase) in tortuosity. In this case, the increase in tortuosity is caused by ordering of the particles near the wall, which may not lead to the same correlation between porosity and tortuosity as for instance compaction.

There are two classes of simple functions that comply to the restraints listed above and give the observed inverse behaviour: a linear function and a reciprocal function:

1 − ε(x)τ(x) = τw + (τb − τw) (3.49)

1 − εb

1 − 1/ε(x)τ(x) = τw + (τb − τw) (3.50)

1 − 1/εb

As can be seen in figure (3.21), both functions give quite similar results except for low porosities, where the linear function has a maximum (1.66 for the chosen bulk values) and the reciprocal function becomes infinite, which means that the flow paths are horizontal. The low porosity values near the wall in a randomly packed bed are caused by the fact that the particles are neatly stacked near the wall, all right on top of each other. Therefore, it is probable that nearly horizontal flow paths and therefore very high values of the local tortuosity to occur in this zone. The reciprocal correlation (3.50) therefore is thought to describe the relation better.

In figure (3.22) the reciprocal correlation (3.50) is compared with the tortuosity profile. The latter was obtained by averaging the tortuosity profiles of five different simulated beds (ball placement method). The bulk porosity was taken as 1.43, the bulk porosity was 0.43; the error bars give plus and minus one standard deviation of the five averaged tortuosity values. The correlation fits the calculated tortuosity profile quite well, although it seems that the flanks of the first peak are a bit steeper than the correlation suggests. The peak of the correlation is higher than most data points, the data set even shows a dip in the centre of the first peak. However, this is caused by the fact that the high tortuosity flow paths are not taken by the tracing particles.

110


τ [-

]

2.2

2

1.8

1.6

1.4

1.2

1

Linear Reciprocal

εb

τb

0 0.2 0.4 0.6 0.8 1

ε [-]

Figure 3.21: Comparison of a linear and reciprocal function for correlating the porosity ε and the tortuosity τ

111

3. The wall effect

τ [-

]

2

1.8

1.6

1.4

1.2

bed simulation

correlation

0 1 2 3 4 x/dp [-]

1

Figure 3.22: Local tortuosity as a function of wall distance; results from a series of bed simulations (D/d=8.56) and correlation

112

3.5 Conclusions

3.5 Conclusions

Two different models were used to simulate a randomly packed bed of spherical particles in a cylindrical container: a deterministic ball placement model and a stochastic random ball dropping model. Both models were compared to each other and to experimental data and models from literature concerning the overall bed porosity, the particle centre density distribution and the porosity profile.

Both models show very similar behaviour, despite the large differences in bed simulation algorithm. As was also found by other authors using similar models (e.g., Spedding and Spencer, 1995), the simulated bed packings have a somewhat higher bulk porosity than experimental bed packings. This is probably caused by the fact that the particles, once they are settled, are frozen in their position and are not allowed to move under the weight of the particles that are placed on top of them.

The porosity profile at the wall of the vessel for both models is similar. The experimental data suggests that in real packings, the wall effect is somewhat stronger than in the simulated packings. This will at least in part be caused by the higher overall porosity in the simulated packings, which allows more space for variations in the particle layers near the wall. It cannot be ruled out that realistic packings are not quite random in some way, e.g. because particles are poured in at the centre of the bed, roll down a slope at some speed, skipping positions that would be stable due to their momentum and not stopping until they hit the wall. In this way, the wall positions could be filled preferentially, leading to a higher density near the wall and a somewhat lower density further from the wall.

Both models show that the wall effect does not depend very strongly on the vessel-to-particle diameter for the range considered (D/d > 5). Critical inspection of the experimental data found in literature (e.g., Goodling et.al. (1986)) indicates that there is no evidence that the shape of the porosity profile depends on the bed-to-particle diameter ratio for D/d > 5. This is remarkable since most earlier researchers (Mueller, 1990; Delmas and Froment (1988)) tried to quantify the wall effect as a function of D/d.

It is expected that the packing density and the packing and compacting method used to create a packed bed will have more influence on the wall effect than the influence of D/d. The experimental conditions given in literature are usually not sufficient to calculate the bulk packing density with a good accuracy. There is no dependable procedure to create random packings experimentally with a given porosity. On the other hand, it is not easy to control the porosity of a simulated bed packing; for the ball placement method, the addition of a stochastic parameter that causes a fraction of the particles to be

113

3. The wall effect

placed at random in one of the stable positions instead of in the lowest stable position increased the bulk porosity only by about 2% point. It would be interesting for future research to modify the method (for instance, by allowing a fraction of the particles to settle in metastable positions) to be able to simulate a wider range of bed porosities and investigate the effect of the bulk porosity on the wall effect.

Apart from the porosity, the simulated packing also allow the calculation of other bed parameters. The fact that the position of all spheres in a simulated packing is known, allows the direct calculation of the specific particle outer surface as a function of the distance from the wall. Mariani et.al. (2001) give a procedure for calculating this value based on a particle centre distribution; however, this distribution cannot be measured directly in a real packing but is determined by fitting measured porosity profiles. Therefore it depends on the choices made in the fitting procedure. For instance, the particle centre distributions proposed by Mariani et.al. (2002) and Delmas and Froment give quite similar porosity profiles (which is not surprising since they are partly based on the same porosity measurements), but very dissimilar particle surface area profiles.

A third parameter that is determined in the simulated packings is the tortuosity. It is obvious that the tortuosity is an important parameter in most transport processes in a packed bed, since it determines the length over which transport through the continuous phase takes place. However, since it is hard to measure tortuosity in a real packed bed, it usually does not appear explicitly in the transport equations. Instead, it is incorporated in more or less lumped parameters such as ”effective” transport coefficients. Although most literature concerned with the wall effect focuses on the porosity profile or the particle surface area, it is clear that the tortuosity is also strongly influenced by the presence of a wall. It has been shown that as a first approximation, the local tortuosity is inversely proportional to the local porosity.

114

3.5 Conclusions

Nomenclature

Roman Symbol units Variable A Pa/s constant in equation (3.2) Ap m2 outer surface area of particles Ai m2 outer surface area of particle i a − fit parameter B (Pa/s)2 constant in equation (3.2) b − fit parameter C ... constant in exponential porosity profile D m tube diameter dij m distance between centre points of sphere i and j dp m particle diameter EA − particle surface function defined by equation (3.44) EV − particle volume function defined by equation (3.40) Fg arbitrary gravitational force vector Fp arbitrary inter-particle repulsion force vector Fw arbitrary wall repulsion force vector f ... generic function hi m vertical position of a cut plane J0 − Bessel function of the first kind and order 0 k − parameter in equation (3.40) Δl m step size for sphere drop model n − number of steps ni − fraction of particles in layer i ni − fraction of particles in layer i with no wall effect m − number of concentric layers per particle radius p Pa pressure; parameter in equation (3.40) pi − centre point of sphere i pxy − projection of point p on the x, y plane r m space coordinate in radial direction rp m particle radius sv m2/m3 specific outer particle surface sv,b m2/m3 specific outer particle surface in the bulk of the bed Vb m3 volume of the bed (fluid space + particles) Vp m3 volume of the particles vi m3 volume of particle i x m spatial coordinate; distance from the wall y m spatial coordinate z m spatial coordinate (in axial direction)

115

3. The wall effect

Greek Symbol units Variable α - angle αA - parameter in equation (3.44) αV - parameter in equation (3.40) βA - parameter in equation (3.44) βV - parameter in equation (3.40) Δ - difference, parameter in equation (3.40) ε - porosity εm - mean bed porosity (volume available for fluid / bed

volume) εb - bulk porosity (porosity of the packing without wall

influences) φi - parameter in equation (3.40) τ - tortuosity of the bed (path length / distance) τb - bulk tortuosity of the bed (tortuosity of the packing

without wall influences) τi - average tortuosity in concentric layer i τm - average tortuosity of the bed τw - tortuosity at the wall θ - angle σ - parameter in equation (3.40)

Miscellaneous Symbol units Variable F - Carlson standard elliptic integral in equation (3.40) D - Carlson standard elliptic integral in equation (3.40) J - Carlson standard elliptic integral in equation (3.40)

116

3.5 Conclusions

Literature

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Dixon, A.G. (1988), Wall and particle-shape effects on heat transfer in packed beds, Chem. Eng. Commun. 71, 217-237

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Goodling, J.S., R.I. Vachon, W.S. Stelpflug, S.J. Ying, M.S. Khader (1983), Radial porosity distribution in cylindrical beds packed with spheres, Powder Technol. 35, 23-29

Govindarao, V.M.H., G.F. Froment (1986), Voidage profiles in packed beds of spheres, Chem. Eng. Sci. 41 (3), 533-539

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Kubie, J. (1988), Influence of containing walls on the distribution of voidage in packed beds of uniform spheres, Chem. Eng. Sci. 43, 1403-1405

Legawiec, B., D. Zilkowski (1994), Structure, voidage and effective thermal conductivity of solids within near-wall region of beds packed with spherical pellets in tubes, Chem. Eng. Sci. 49, 2513-2520

Logtenberg, S.A., A.G. Dixon (1998), Computational Fluid Dynamics studies of fixed bed heat transfer, Chem. Eng. Proc. 37, 7-21

Mariani, N.J., G.D. Mazza, O.M. Martnez, G.F. Barreto (2000), Evaluation of radial voidage profile in packed beds of low aspect rations, Can. J. Chem. Eng. 78, 1133-1137

117

3. The wall effect

Mariani, N.J., O.M. Martnez, G.F. Barreto (2001), Computing radial packing properties from the distribution of particle centres, Chem. Eng. Sci. 56, 5693-5707

Mariani, N.J., et.al. (2002), Packed Bed structure: evaluation of radial particle distribution, Can. J. Chem. Eng. 80(2) 186-193

Mueller, G.E. (1991), Prediction of radial porosity distributions in randomly packed beds of uniformly sized spheres in cylindrical containers, Chem. Eng. Sci. 4692), 706-708

Mueller, G.E. (1992), Radial void fraction distributions in random packed fixed beds of uniformly sized spheres in cylindrical containers, Powder Technol. 72, 269-275

Mueller, G.E. (1997), Numerical simulation of packed beds with monosized spheres in cylindrical containers, Powder Technol. 97, 179-183

Papageorgiou, J.N., G.F. Froment (1995), Simulation models accounting for radial voidage profiles in fixed-bed reactors, Chem. Eng. Sci. 50(19), 3043-3056

Roblee, L.H.S., R.M. Baird, J.W. Tierney (1958), Radial porosity variations in packed beds, AIChE J. 4(4), 460-464

Ridgway, K., K.J. Tarbuck (1968), Voidage fluctuations in randomly packed beds of spheres adjacent to a contaning wall, Chem. Eng. Sci. 23, 1147-1155

Spedding, P.L., R.M. Spencer (1994), Simulation of packing density and liquid flow in fixed beds, Computers Chem. Engng 19(1), 43-73

White, S.M., C.L. Tien (1987), Analysis of flow channeling near the wall in packed beds, Wrme- und Stoffbertragung 21, 291-296

Wijngaarden, R.J., K.R. Westerterp (1992), The statistical character of packed-bed heat transport properties, Chem. Eng. Sci. 47(12), 3125-3129

Zou, R.P., A.B. You (1995), The packing of spheres in a cylindrical container: the thickness effect, Chem.Eng.Sci. 50 (9), 1504-1507

118

( ) ( )

Chapter 4

Flow resistance in randomly packed beds

Summary

To calculate flow in a packed bed, a model is needed to relate the fluid velocity to the pressure drop. Traditionally, equations like the Ergun equation are used, that relate pressure drop to the (mean) bed porosity and particle size. These equations cannot bed used for a detailed flow model near the wall, because they are based on the assumption of a uniform random packing.

One of the ways to look at flow through a packed bed is as a collection of interconnected, twisted channels. In this chapter a detailed model for the flow through such channels is presented that is based on first principles and engineering relations. With this model, the pressure drop can be written as a function of velocity with the porosity ε, the specific particle outer surface area sv and tortuosity τ as parameters. The result is:

Δp νf s2 v

[ ] sv 1 = 2τ 2 ρf w0 + ktτ 3 − τ ρf w0|w0| (4.1)

Δz bed ε3 4ε3 2

where kt is a term that accounts for wall roughness, bends and expansion/ contractions. This equation shows the same dependency on the porosity and particle size as the (primarily empirical) Ergun equation. In addition, when the coefficients are fitted to the Ergun equation, very reasonable values are obtained for the properties of the twisted channel (i.e., about one bend and one expansion/ contraction for each particle passed, a tortuosity of 1.44 and a wall roughness of 0.04).

In contrast with Ergun and similar empirical relations, the equation above can be applied locally in the bed near a wall if the local porosity, particle surface area and tortuosity are known.

119

4. Flow resistance in randomly packed beds

The equation found is applied in a two-dimensional model using the modified Brinkman equation to calculate the radial velocity profiles in packed tubes, using the bed parameter profiles from the previous chapter. They compare well to experimental velocity profiles from literature.

120

4.1 Introduction

4.1 Introduction

Randomly packed beds are used very widely in a broad range of equipment for many types of solid-fluid contacting processes. Consequently, there has been a large interest in the modelling of the important processes in randomly packed beds. Traditionally, randomly packed beds are applied as packed tubes. Therefore, the majority of measurements have been carried out using this cylindrical bed geometry. Flow resistance models for packed beds have been in use for a long time and are very successful in describing these standard packed beds. Even though no specific bed shape is implied, the experimental data on which they are based is mostly for packed tubes. In the classical approach, the effect of the tube wall is not taken into account; the tube is assumed to be large compared to the particle diameter. The flow resistance equation of Ergun (1952) is still very widely used and is relatively accurate in predicting the pressure drop in packed tubes of small particles, even though there is a significant spread in the experimental values. In more recent years, the wall effect has been taken into account in more detailed models (e.g., Vortmeyer and Schuster, 1983; Delmas and Froment (1988), Bey and Eigenberger, 1996). These models are in general limited to cylindrical beds.

Although the cylindrical packed tube is the most common geometry for a packed bed reactor, other geometries may be more suitable for specific applications. For low-pressure drop, shallow packed beds, the packed tube is usually not the most efficient arrangement, and radial flow, lateral flow or parallel passage reactors may be employed (Calis, 1995). Also for the development of compact installations, non-standard geometries may be necessary for ease of construction, accessibility and heat integration. An example of such an application is a mobile reforming installation for fuel cell applications.

It has been shown that the presence of a wall has a large influence on the local structure of a randomly packed bed. The influence of the wall extents up to five particle diameters into the bed. Therefore, unless the bed diameter is very large compared to the particle diameter the wall will have an effect on the flow resistance of the bed. As a rule of thumb, the wall effect needs to be taken into account for beds with a diameter smaller than 40-50 particle diameters.

For cylindrical bed geometry, the wall effect may be taken into account by adding a tube-to-particle diameter ratio dependence to the classical equations. However, such an approach would only give a prediction of the pressure drop over a packed tube, and not give any information about the flow near the wall. For packed beds where the wall is important for the flow resistance, the wall will also be important for other transport processes i.e., the transfer of heat and mass. Detailed information about the flow near the wall will be needed to predict the performance of such a packed tube as a heat exchanger

121


or a chemical reactor. Furthermore, for non-cylindrical bed geometries, or if there are internal walls in the bed (e.g., heat exchanger tubes), the wall effect cannot be described by a simple tube-to-particle diameter ratio dependence. Therefore, it is necessary to look in more detail to the processes that determine the flow near the wall of a packed bed.

In the previous chapter relations were derived for the porosity, tortuosity and specific outer particle area as a function of the distance to the wall. In this chapter those relations are used to create a model that describes the flow resistance of a randomly packed bed as a function of the distance from the wall. Based on microscopic models for the flow in the packed bed, relations are derived that describe the influence of the porosity, specific outer particle area and tortuosity on the flow resistance. This leads to a local-homogeneous model, that is identical to the classical (e.g., Ergun) approach far from the wall, but takes into account the change in bed structure near the wall.

In the following chapters, models will be derived for dispersion and chemical reaction in randomly packed beds. These will be integrated with the results of this chapter into a general Computational Fluid Dynamics based randomly packed bed model.

4.2 A short overview of existing literature

Among the first to acknowledge that the velocity profile in packed beds is not uniform were Schwartz and Smith (1953). They measured the radial velocity profile two inches downstream of primarily packed beds of cylinders and found velocity profiles with a single maximum near the wall. As at the time there were no porosity profile measurements available, they calculated the porosity profile from their velocity profile measurements and found profiles that, starting from the centre of the bed, are almost flat until a distance of up to two particle diameters from the wall and then increase monotonously towards the wall.

Later studies employed the reverse path. The porosity profile was measured by direct methods (see chapter 3 for details), and this was used in various models to calculate the velocity profile inside the bed. Although the porosity measurements showed an oscillatory decrease of porosity from the wall towards the centre of the bed, early and even some recent flow models employ an exponential approximation of the porosity profile to simplify the equations (e.g., Vortmeyer and Schuster, 1983; White and Tien, 1987; Tsotsas and Schlunder, 1988; Cheng and Vortmeyer, 1988; Winterberg et al., 2000).

Detailed measurements of the radial velocity profile using optical techniques (McGreavy et al., 1986; Giese et al., 1998) or using a monolith section to conserve the flow profile downstream the bed (Bey and Eigenberger, 1997)

122

4.3 Flow resistance in infinite packed beds

show oscillations of the velocity as a function of the radial position near the wall. However, the amplitude of the oscillation and the distance in terms of number of particle diameters into the bed in which it is damped out differs considerably between authors.

Models to calculate the flow inside the bed are usually based on the Brink-man equation, that includes viscous friction in the fluid as well as friction caused by the particles of the bed (Vortmeyer and Schuster, 1983; White and Tien, 1987; Delmas and Froment, 1988; Cheng and Vortmeyer, 1988; Giese et al., 1998; Subayago et al., 1998; Winterberg et al., 2000). Some use a two-zone model in which the bed is divided into a wall zone and a core zone. In these models there is no viscous term and wall influence is limited to the wall zone (Schwartz and Smith, 1953; McGreavy et al., 1986). Cheng and Yuan (1997) calculate the hydraulic diameter of flow channels as a function of the distance to the wall. They use the Ergun equation to compute the flow profile and therefore do not take into account radial (viscous) transport of momentum. The surface of the vessel wall iss incorporated in the hydraulic diameter profile. However, they do not take into account the difference in distribution of volume (porosity) and surface of the particles near the wall.

Also true 2-dimensional models are used, where the momentum balances are solved in both the axial and radial directions (Papageorgiou and Froment, 1995; Bey and Eigenberger, 1996; Tierney et al., 1998). For all these models, the porosity profile chosen has a large influence on the resulting velocity profiles, and are thus a large factor in the comparison with measured profiles. Giese et al. (1998) used laser-Doppler velocimetry in a transparent packed bed and determined the porosity from the locations where the velocity was zero. Their model is very successful in describing their own measured velocity profiles, but there is a large difference with other literature.

Many of these models use Ergun type friction coefficients that are implicitly based on the assumption that the bed is a uniform randomly packed with spherical particles (e.g., Vortmeyer and Schuster, 1983; White and Tien, 1987; Giese et al., 1998; Winterberg et al., 2000) although near the wall this assumption is not valid. Delmas and Froment (1988) take into account both the specific surface area and the tortuosity profile; however, the way the tortuosity is taken into account is based on empirical arguments and they do not take into account the effect of the particle outer surface area profile correctly.


In a random packed bed, the void space in between the particles forms a continuous network of twisted channels. When a fluid flows through a randomly

123


packed bed, these channels will cause a flow resistance. The pressure drop across a packed bed has long been known to be proportional to the velocity at low flow rates (Blake-Kozeny equation) and proportional to the square of the velocity at high flow rates (Burke-Plummer equation). Ergun (1952) has shown that an expression for the whole range of Reynolds numbers can be obtained by summation of the expressions for the low and high Reynolds number range. This lead to his well-known formulation, which is adequate for dense packings (0.2 < ε < 0.7) of compact particles without holes:

Δp (1 − ε)2 1 − ε − = 150ηf w0 + 1.75ρf w0|w0| (4.2)Δz ε3d2 ε3dp p

For the purpose of application of a similar equation in a detailed packed bed model, we will look into the basic assumptions underlying this equation (e.g., Bird et al., 2002). For the derivation of equation 4.2, we can either look at the packed bed as a set of separate particles or as a set of twisted interconnected flow channels. Since both approaches should lead to the same result, we follow the latter path, which seems more physically correct for dense beds.

For low to intermediate velocity flow in a straight tube with constant cross-sectional area, the pressure drop is caused primarily by wall friction. The wall friction is characterised by the friction factor f , which is defined as the ratio between the wall shear stress and the dynamic head of the fluid:

τwf ≡ (4.3)1 ρf v|v|2

A force balance over a length Δz of a tube with cross sectional area A⊥ and circumference W :

ΔpA⊥ = τwWΔz (4.4)

leads to an expression for the pressure drop per tube length:

Δp 4f 1 = ( ρf v|v|) (4.5)

Δz d 2h

where dh is the hydraulic diameter:

4Adh ≡ ⊥

(4.6)W

which is defined so that for cylindrical tubes dh equals the tube diameter. For laminar flow of a Newtonian fluid in a smooth straight cylindrical tube

of constant diameter, the steady-state velocity profile can be solved analytically: the friction factor f is found to be inversely proportional to the Reynolds number (Poisseuille relation):

64 4f = (4.7)

Re

124


For fully turbulent flow in not perfectly smooth tubes the friction factor is independent of the Reynolds number and is a function of wall roughness.

The channels in the randomly packed bed differ from these perfect tubes in a number of ways:

• the combined cross sections of all channels is a factor ε smaller than the cross-section of the tube

• the cross-section of the channels is not circular

• the channels are not straight

• the channel walls are not perfectly smooth

• the channels are interconnected

Although the geometry of the channels is too complex to derive an expression for the friction factor based on theoretical arguments only (parameters fitted to experimental data will be needed in the end), it does provide insight to follow this road a little further.

In order to apply the equations derived for cylindrical tubes to the flow channels in a packed bed, we need to determine the hydraulic diameter of the tubes. For the twisted channel of a packed bed with porosity ε and specific surface sv, we can multiply numerator and denominator of equation (4.6) by the channel length to obtain:

4ε dh = (4.8)

sv

This gives an idea of the size of the channels in a typical packed bed: for random packings of spheres with a porosity of 0.37, the hydraulic diameter is about 0.4 times the particle diameter.

The corresponding Reynolds number is calculated as:

vdhReh = (4.9)

νf

Here v is the mean velocity in the channel, so for a given mass flow, v increases inversely proportional with the porosity.

The concept of a hydraulic diameter is based on the assumption that the influence of the channel wall is limited to a thin layer near that wall. Therefore, the curvature of this layer can be ignored and the influence depends only on the size of the wall surface. For fully developed laminar flow, the velocity profile spans the whole cross section, so the walls of a channel influence the velocity

125

( )

ε


in every point of the cross-section. Therefore, strictly spoken, the concept of a hydraulic diameter can not be applied to laminar flow. However, in a packed bed the changes in channel shape appear so often and suddenly that the flow will not be fully developed anywhere in the bed (except for very small values of the Reynolds number, say Re < 0.1). Therefore, we will use the hydraulic diameter also in laminar conditions.

The tortuosity τ is used to characterise the measure of straightness of the flow channels in a packed bed:

path length length of channel τ = = (4.10)

distance height of bed

For randomly packed beds, the value is usually in the order of 1.4. For a length Δl of a channel, the pressure drop can be written to depend

on the following factors:

Δp = (ζ) (f(Re)) 1 ρv2 Δl (4.11)

2

Here ζ depends on the channel shape and f(Re) gives the dependence on flow regime. The tortuosity influences the pressure drop in the channels in three ways:

• As the tortuosity increases, so does the length of the channel (for a given bed height), leading to higher pressure losses through a change in Δl

• As the channel length increases with tortuosity, for a given mass flow through the bed, also the velocity in the channel must increase, leading to an increase in pressure loss through a change in v

• A highly tortuous channel will (usually) have more bends and shape changes, leading to an increase in pressure loss through a change in ζ

The porosity, specific volume and tortuosity of a packed bed are independent properties that can be varied separately from each other, as is shown in figure 4.1.

In the laminar regime, we can write for a slice of a (cylindrical) channel of length Δl: ( )

Δp Δl l

= 64

Rehdh ( 1 2 ρf v|v|) (4.12)

The average distance travelled in the axial direction is obviously Δl/τ . The velocity v in the channel is related to the superficial velocity w0 as:

w0τ v = (4.13)

126

( )

( )

√


= 0.25 sv = 2/d= 1

d

= 0.5 = 0.5 sv sv = 2/d sv = 1/d= 1 = 1

= 0.5 sv = 2/d= 2

Figure 4.1: Independent variation of porosity, tortuosity and specific channel surface area, illustrated for a square 2-dimensional grid.

Hence, with equation (4.8) and (4.9), the laminar term can be written as:

Δp νf sv 2

= 2τ 2 ρf w0 (4.14)Δz l ε3

For packed beds with a uniform random packing of spheres, the specific outer particle surface area can be written as a function of the porosity and particle diameter:

6(1 − ε) sv = (4.15)

dp

so that

Δp νf (1 − ε)2

= 72τ 2 ρf w0 (4.16)Δz d2ε3

l p

Comparison with the first term of the Ergun equation (equation 4.2) reveals that in the latter, τ has a value of 150/72 = 1.44, which is in excellent

127

( ) ( )

( ) ( )


accord with the expected value. In other words, the pressure drop of an actual packed bed under laminar conditions is quite similar to laminar flow through a bundle of twisted cylindrical tubes.

Now consider the fully turbulent regime. As the channels cannot be considered to be smooth, the friction factor of the channels will be independent of the Reynolds number for turbulent flow. Therefore, in the fully turbulent flow regime, equation 4.5 can be written as

Δp 4f 1 = ρf v|v| (4.17)

Δl dh 2t,wall

where (4f) is a value that will depend on the channel roughness. The lower limit for (4f) will be given by that for smooth tubes, about 0.01; for very rough tubes, (4f) increases to about 0.1 (e.g., Bird et.al., 2002). With equations (4.8) and (4.9), we can write (4.17) in terms of the bed porosity, tortuosity and specific surface:

Δp τ 3sv 1 = 4f ρf w0|w0| (4.18)

Δz 4ε3 2t,wall

To compare this result with the Ergun equation (4.2), we can substitute sv by equation (4.15) which is valid for uniform randomly packed beds of spherical particles: (

Δp Δz

)

t,wall

= 4f 3τ 3

2 (1 − ε) ε3dp

( 1 2 ρf w0|w0|

)

(4.19)

To comply with the Ergun equation, 34 (4f)τ 3 has a value of 1.75, so that τ has

a value between 3 and 6, depending on the value of 4f . Hence for turbulent flow, the straight tube model leads to unphysically high values of τ . In other words, the effective friction factor in a packed bed for turbulent flow (based on the Ergun equation) is much higher than the wall friction factor for the straight tube model (between 0.01 and 0.1). Apparently, in the turbulent region, the contribution of friction at the particle wall to the pressure drop in packed beds is small and pressure drop is mainly caused by other mechanisms, i.e., shape drag due to bends and expansion/contractions in the channels.

The pressure drop caused by bends in the channel can be estimated when we make a few assumptions. For turbulent flow in tube systems, the pressure lost in one sharp (i.e., not smooth) bend is usually given as a flow resistance coefficient ζ that is tabulated for different bend angles. The pressure loss in a bend is then calculated as:

1 Δp = ζ ρf v|v| (4.20)

2

128

( ) ( )


Here we need the pressure loss in the bend as a function of the bend angle β (see figure 4.2). From the literature values, the following behaviour can be observed: for a straight channel (β = 0), the bend pressure loss is of course zero, while for a 90 degrees bend, almost all kinetic energy of the flow is lost (ζ ≈ 1). For even sharper bends, the pressure drop increases to about twice this value. A simple function that shows this behaviour is ζ = 1 − cos(β). Without proof it is postulated here that the bend pressure drop can be estimated by:

1 Δp1 = (1 − cos(β)) ρf v|v| (0 ≤ β ≤ π) (4.21)

2

where β is the angle of the bend. It can be seen (figure 4.3) that this function fits the commonly used resistance factor values for sharp bends quite well, while its mathematical form is very convenient.

To estimate the mean value of the bend angle, we can write:

α 1 sin( ) = (4.22)

2 τ

For equation (4.21) we need an expression for cos( β). Since cos(β) = − cos(α) and cos(α) = 1 − 2 sin2(α/2),

cos(β) = 2 − 1 (4.23) τ 2

Let δ be the average distance between two bends in a channel; then the pressure drop due to bends per length of bed is: (

Δp Δz

)

t,bends

= 2τ δ

(1 − 1 τ 2

)

( 1 2 ρf v|v|

)

(4.24)

The distance between bends depends of course on the particle size, but also on the distance between two particles in the packing. Therefore, let us suppose that the distance δ between two bends is proportional to the hydraulic diameter dh:

δ = nbdh (4.25)

so that nb is the number of hydraulic diameters of straight channel between two bends. For a standard packed bed (porosity 0.37), the hydraulic diameter is about 0.4 times the particle diameter so since we expect something in the order of one bend per particle, the number of hydraulic diameters between bends (nb) should be between, say, 2 and 3. For the pressure drop due to bends, in terms of the superficial velocity (equation 4.13), it follows that:

Δp 2(τ 3 − τ) 1 = ρf w0|w0| (4.26)

Δz ε2nbdh 2t,bends

129

( ) ( )


dp

zz

Figure 4.2: Schematic representation of a simple zigzag model to determine the mean number of bends and bend angle in the flow channels of a packed bed

or, eliminating the hydraulic diameter dh with equation (4.8):

Δp (τ 3 − τ)sv 1 = ρf w0|w0| (4.27)

Δz 2ε3nb 2t,bends

For a uniform randomly packed bed of spherical particles we can apply equation (4.15): (

Δp Δz

)

t,bends

= 3(τ 3 − τ)

nb

(1 − ε) ε3dp

( 1 2 ρf w0|w0|

)

(4.28)

Hence, if the pressure drop in packed beds in the turbulent regime was caused solely by the bends in the channels, the factor 3(τ 3 − τ)/nb should have the value 3.5 (compare the Ergun equation (4.2)). With the value of 1.44 for τ found in the laminar regime for the standard packed bed, it follows that the number of hydraulic diameters between bends nb is 1.33. Of course, as the pressure drop is not caused by the bends alone, this number will be somewhat

130


0

0.4

0.8

1.2

1.6

2 [

]

1-cos( )

[1] smooth

[1] rough

[2]

[3]

0 30 60 90 120

[º]

Figure 4.3: Pressure drop coefficient ζ in sharp bends at high Reynolds numbers as a function of bend angle β (see figure 4.2) for different literature (handbook) sources and the fit function (equation 4.21). Literature data: [1] Leijendekkers et al. [red.] (1998); [2] Janssen and Warmoeskerken (1987); [3] Blerins (1984).

higher in reality, but it is clearly in the right order of scale. The main result, however, is that the pressure drop due to bends depends on the porosity and particle diameter in a way that is consistent with the Ergun equation, and increases linearly with τ 3 − τ .

The pressure drop due to a sudden expansion from cross-sectional area A1

to A2 can be estimated from (e.g., Leijendekkers et.al., 1998)

( A2

)2 1

Δp = − 1 ρf v 2 (4.29)A1 2 2

where the velocity v2 is taken at the downstream position. To estimate the expansion in a packed bed we look at a close packing of four spheres with radius R. The sphere centres form a regular tetrahedron with sides equal to the sphere diameter. We define a z axis perpendicular to one of the faces of tetrahedron, so three of the sphere centres have z = 0. The fourth, top sphere √ has its centre at z =

34 3R. Now we look at the area occupied by the solids in

131

{


cross sections perpendicular to the z axis, bounded by the prism that is formed by the equilateral triangle that connects the bottom three sphere sections in the plane z = 0 and the z-axis. This prism encloses a single — vertical — segment of a flow channel; each channel is formed by a large number of these √ segments in different directions. The cross sectional area of the prism is 3R2 . The cross section Ab of the three bottom spheres consists of three 60 ◦ circle segments:

1 π(R2 − z2) −R ≤ z ≤ R Ab(z) = 2 (4.30)

0 elsewhere

Likewise, the contribution of the top sphere to the cross section At is { √ √ √ π(R2 − (z − 4 3R)2) 4 3R − R ≤ z ≤ 4 3R + R

3 3 3At(z) = (4.31)0 elsewhere

3

2

1

1 2 3

Figure 4.4: Cross-sectional solid (white) and empty (black) space in a close packing arrangement of four spheres. At plane 1, the three lower spheres are cut through their centre points at a (local) maximum solid area; at cut plane 2, still only the lower three spheres are cut, at about the minimum solid area. At plane 3, all four spheres are cut.

132

( )

( )


1

0

2

3

A/R

2 [-

]

0 1 2 3 4

z/R [-]

Figure 4.5: Cross-sectional surface of four close-packed spheres and a plane parallel to the plane through three sphere centres, as a function of the position of the plane (black line). Dashed lines give the cross-section of the three bottom spheres (Ab) and of the top sphere At; grey lines show surface that falls outside the prismatic volume considered (see figure 4.4).

A plot of the total solid cross-sectional area is given in figure (4.5). Clearly, the maximum cross sectional solid area occurs at z = 0. The minimum cross sectional area of the channel is:

√ A1 = 3 −

1 π R2 (4.32)

2 √

The minimum cross sectional solid area occurs at z = (43 3 − 1)R, so the

maximum cross sectional area of the channel is: √ √

A2 = 3 − 1 π(1 − (

4 3 − 2)2) R2 (4.33)

2 3

Therefore, the expansion factor becomes: √

A2 π(43 3 − 2)2

= 1 + √ (4.34)A1 2 3 − π

133

( ) ( )

( ) ( )

( ) ( )


or ( √ )2 π(4 3 − 2)2A2 3ka = ( − 1)2 = √ ≈ 0.87 (4.35)

A1 2 3 − π

If we assume that the interstitial velocity in the bed is the mean value of the velocity upstream and downstream of the expansion, the downstream velocity v2 equals 2/3v. If we now assume that there is one expansion for ne

hydraulic diameters of channel length, the pressure drop due to expansions can be estimated as:

Δp 4kaτ 1 = ρf v|v| (4.36)

Δz 9nedh 2t,expansion

where the factor τ appears because the pressure drop is given per unit bed height z instead of per unit channel length. In terms of the superficial velocity w0 and specific surface sv (equation 4.8, 4.13) this becomes:

Δp kaτ3sv 1

= ρf w0|w0| (4.37)Δz 9neε3 2t,expansion

When there are expansions in the channel, there should also be contractions; since the pressure drop in a sudden contraction is only a fraction of that of a sudden expansion, and the dependence on bed parameters will be equal, the contractions can be taken into account by a slight increase in the parameter ne. For a uniform random packing of spheres, the specific surface can be eliminated (equation 4.15):

Δp 2kaτ3 (1 − ε) 1

= ρf w0|w0| (4.38)Δz 3ne ε3dp 2t,expansion

For a standard packed bed (tortuosity 1.44), to comply with the Ergun equation (equation 4.2), there should be 0.5 hydraulic diameters between two expansions if the pressure drop was attributed to expansions alone. This is about 0.2 particle diameters for a standard packed bed, which is in the right order of magnitude since obviously the pressure drop in a packed bed is not caused channel expansions alone. The main result is that the pressure drop due to expansion/contractions is proportional to τ to the third power.

For the pressure drop in the turbulent region, we can add the contributions of bends and expansions/contractions together with the wall friction. To reduce the number of degrees of freedom, we will assume that the number of bends is equal to the number of expansions (nb = ne = n). ( ) [ ] ( )

Δp τ 3 (τ 3 − τ ) kaτ3 sv 1

= (4f)t + + ρf w0|w0| (4.39)Δz 4 2n 9n ε3 2t

134

( ) ( )

( ) ( )


To determine the values for the remaining parameters n and (4f)t, we once more compare this result to the Ergun equation (4.2). For uniform randomly packed beds of spherical particles we can write: (

Δp Δz

)

t

=

[

(4f)t 3τ 3

2 +

3(τ 3 − τ) n

+ 2kaτ

3

3n

] (1 − ε) ε3dp

( 1 2 ρf w0|w0|

)

(4.40)

so that, to comply with the Ergun equation,

1 2

[

(4f)t 3τ 3

2 +

3(τ 3 − τ) n

+ 2kaτ 3

3n

]

= 1.75 (4.41)

If we take (4f)t to be 0.1 (for very rough walls) and τ = 1.44 as found for laminar flow, it follows that n equals 2.1. If we take (4f)t to be 0.01 (for not so rough walls), the value of n becomes 1.85. For ease of calculation, we take n = 2. This means that there is one bend and one expansion for every 2 hydraulic diameters or about 0.8 particle diameters. With this value ( ) [( ) ] ( )

Δp 2ka sv 1 = (4f)t + 1 + τ 3 − τ ρf w0|w0| (4.42)

Δz t 9 4ε3 2

We will replace the factor before τ 3 by the single parameter kt, hence,

Δp [ ] sv 1 = ktτ

3 − τ ρf w0|w0| (4.43)Δz t 4ε3 2

If we require the equation to be identical to the Ergun equation (4.2) at τ = 1.44, kt has a value of 1.26 (this corresponds with a value of 0.0415 for the wall friction factor (4f)t).

The pressure drop in a packed bed over the full Reynolds number range can now be written (equations 4.14, 4.43) as:

Δp νf s2 v

[ ] sv 1 = 2τ 2 ρf w0 + ktτ 3 − τ ρf w0|w0| (4.44)

Δz bed ε3 4ε3 2

Note that since equation 4.15 has not been not used to eliminate the specific outer particle surface, the assumption that the bed consists of uniform spheres that is implied there, is no longer needed. The packing is completely characterised by the porosity, tortuosity and specific outer particle surface area, so it can easily be applied to packings of non-spherical particles or particle size distributions (the original restrictions to particle shape still apply). For the special case of a uniform random bed with a tortuosity of 1.44, equation (4.44) reduces to the Ergun equation (4.2) if an equivalent diameter based on the particle outer surface area is used as the particle diameter:

6(1 − ε)dp,eqA = (4.45)

sv

135


If we look at two random packings of particles with the same size but with a different bed porosity, it is clear that the tortuosity in the loose packing will be lower than in the dense packing. Hence, tortuosity will increase with decreasing porosity. This implies that the pressure drop in a packed bed, as predicted by equation (4.44), depends more strongly on the porosity than is suggested by Ergun (equation (4.2). However, it is important to realise that the change in tortuosity between dense and loose random packings differs in nature from the change in tortuosity near a wall due to the ordering of particles.

4.4 Flow resistance: the wall effect

For finite random packed beds, the porosity cannot be treated as a constant value throughout the bed, but is a function of the distance x from the container wall. The specific particle outer surface area sv is also a function of x. Equation 4.15 is not a very good estimate of the local specific particle surface area if the packing is not fully random. Using the equations presented in the previous chapter, both the particle volume profile and the particle surface profile can be calculated from the particle centre distribution. In addition, the tortuosity will change from a mean value far from the wall to a value of 1 at the wall; a relation for the tortuosity profile was also presented in chapter 3. We can write equation 4.44 as a function of the distance to the container wall x:

Δp νf [sv(x)]2

=2[τ(x)]2 ρf w0Δz [ε(x)]3 ( ) (4.46) ( ) sv(x) 1

+ kt[τ(x)]3 − τ(x) ρf w0|w0|4[ε(x)]3 2

Near the wall, the porosity will increase and the tortuosity and specific surface area will decrease. According to equation 4.46, the increase in porosity and the decrease of tortuosity near the wall will have a strong diminishing effect on the local flow resistance, while the decrease in specific surface area will have a somewhat smaller effect. Figure (4.6) shows the profiles for porosity, tortuosity and specific area for a typical packed bed, according to the results of the previous chapter.

4.5 Velocity profile

The one-dimensional flow in a packed bed is often described by the modified Brinkman equation (e.g., Vortmeyer and Schuster, 1983), in cylindrical

136

( )

4.5 Velocity profile

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6 ε

[-],

τ

[-]

, s v /

s v,b

[-

]

ε τ sv /sv,b

0 1 2 3 x/d [-]p

Figure 4.6: Porosity ε, tortuosity τ and specific outer particle surface area sv

(normalized by its bulk value sv,b) as a function of the distance to the wall (in particle radii) for a typical randomly packed bed (based on the results of chapter 3).

coordinates: ∂p 2 ηeff ∂ ∂w0

= −f1w0 − f2w0 + r (4.47)∂z r ∂r ∂r


wall : r = dt/2 → w0 = 0 ∂w0

(4.48)centre : r = 0 → = 0

∂r

Here f1 and f2 are the laminar and turbulent friction terms, often taken from the Ergun equation. When the pressure gradient in the z direction is fixed, equation (4.47) reduces to an ordinary differential equation and can easily be solved to yield a radial (superficial) velocity profile.

When we take equation (4.46) to describe the flow resistance of a packed bed as a function of the distance from the wall, it follows that

νf [sv(x)]2

f1(x) = 2[τ(x)]2 ρf (4.49)[ε(x)]3

137


Table 4.1: References and conditions for data points in figure (4.7). symbol w0,m [m/s] dp [mm] Rep D/d note

1 2 3 4 5

0.5 0.25-1.5 0.25-1.5

0.5 1

8.0 268 4.5 76-450 9.8 165-987 7.0 235 9.8 658

6.3 11.1 5.1 7.1 5.1

a a a a a

6 7 8 9

0.004 0.076 0.102 0.526

8.6 4 8.6 77 8.6 103 8.6 532

9.30 9.30 9.30 9.30

b b b b

0.014 0.14 1.4

12.7 10 12.7 100 12.7 1000

9 9 9

c c c

a) Bey and Eigenberger, 1997. Fluid: air; velocity profiles measured downstream of a packed bed supported by a 3.5 mm long monolith to preserve the flow profile. b) Giese et.al, 1998. Fluid: special liquid mixture (viscosity 8.5×10−6m2/s); velocity measured inside bed by Laser-Dopper velocimetry. c) This work. Results of 1-D computational model for superficial velocity inside the bed; fluid properties correspond to air, effective viscosity is a function of wall distance according to equation (4.51).

and

[ ] sv(x)ρff2(x) = kt[τ(x)]3 − τ(x) (4.50)

8[ε(x)]3

The third term of equation (4.47) gives the dispersion of momentum in the bed. The effective viscosity ηeff is not known a priori and is usually a fit factor that is a function of the average Reynolds number. Here we have chosen a profile based on the analogy between the dispersion of mass and momentum, which will be described in more detail in the next chapter:

√ νeff

= 1 − √

1 − εm + Rep τ 2 − 1

(4.51)νf 8

The superficial velocity profiles computed according to equation (4.47) are compared to measured data from literature in figure (4.7) below.

It is clear that there is quite a large difference between the two different sources for experimental velocity profiles. The measurements inside the catalyst bed show a much more pronounced oscillation than the measurements just

138

4.6 Conclusions

0 0 1 2 3 4

0.5

1

1.5.

2

2.5

3

3.5

v 0 /v

m [

-]

1 2 3

4 5 6

7 8 9

Rep = 10

Rep = 100

Rep = 1000

x/dp [-]

Figure 4.7: The radial superficial velocity profile as computed from the modified Brinkman equation (4.47) with the porosity, tortuosity and specific outer particle surface area shown in figure (4.6) for different values of the particle Reynolds number (drawn lines), compared to literature data (dots, see table 4.1).

downstream of the bed. The data of Bey and Eigenberger show no evidence of a dependence of the flow profile below the bed on the Reynolds number. The calculated velocity profile fits the measured profiles quite well, especially considering that the model is based on physical arguments and reasonable assumptions only.

4.6 Conclusions

The Ergun equation (4.2) for flow resistance in packed beds can be almost entirely be derived from a parallel twisted flow channel model. The laminar term can be derived from Poiseuille flow in a twisted channel where the twistedness of the channel is characterized by the tortuosity (length of the channel per length travelled). The result is identical to the Ergun equation when a tortu

139


osity of 1.44 is chosen, a value that corresponds very well to the values that are usually proposed in literature. The turbulent term of the Ergun equation can be derived from a model that includes wall friction at the channel walls and pressure loss in bends, expansions and contractions of the channels. If the wall friction factor for the flow channels is taken as 0.1 (for very rough walls) and it is assumed that there is one bend and one channel expansion for roughly each particle diameter of channel length (which seems reasonable), the model (equation 4.44) is nearly identical to the Ergun equation for standard packed beds (tortuosity 1.44). Since the model is not based on the assumption of identical, spherical particles, it also applies to non-homogeneous packed beds of non-spherical particles, provided the porosity, tortuosity and specific outer particle area are known.

In contrast with the classical (empirical) equations, the relation derived here can be used locally in a packed to calculate the flow near a wall based on the porosity, particle outer surface area and tortuosity profiles.

The flow resistance model was included in a modified Brinkman equation (4.47) model to calculate velocity profiles in a cylindrical packed bed. The profiles for the porosity, tortuosity and specific outer particle area were taken from the numerical bed packing experiments described in an earlier chapter. The calculated velocity profile fits the measured profiles quite well, especially considering that the model is based on physical arguments and reasonable assumptions only.

140

4.6 Conclusions

Nomenclature

Roman Symbol units Variable A1 m2 channel cross section upstream of an expansion A2 m2 channel cross section downstream of an expansion Ab m2 cross section area At m2 cross section area d m characteristic dimension dh m hydraulic diameter dp m particle diameter dp,eqA m particle surface equivalent diameter dp,eqV m particle volume equivalent diameter dt m tube diameter f - friction factor f1 k/m3/s flow resistance factor in the laminar flow regime f2 k/m4 flow resistance factor in the turbulent flow regime H m height of bed ka - flow resistance constant for expansion effects kt - constant in turbulent flow term pressure drop equa

tion L m length of tube n - number of hydraulic diameters between two bends or

two expansions nb - number of hydraulic diameters between bends ne - number of hydraulic diameters between channel ex

pansions p Pa (absolute) pressure r m space coordinate in radial direction for cylindrical

coordinates R m radius of a sphere sv m2/m3 specific outer particle surface sv m2/m3 mean specific surface in a subvolume of the bed sv,b m2/m3 specific outer particle surface in the bulk of the bed v m/s velocity v2 m/s velocity upstream of a sudden expansion w0 m/s superficial velocity w0 m/s mean superficial velocity in a subvolume of the bed x m spatial coordinate (perpendicular to the wall) z m spatial coordinate (in axial direction)

141


Greek Symbol units Variable α - bend angle β - bend angle δ m distance between two bends ε - (local) porosity (volume open to flow / total volume) εb - bulk porosity, porosity far from a wall εmb - mean bed porosity (volume in bed open to flow /

volume of bed) ηf Pas dynamic viscosity of the fluid ηeff Pas effective dynamic viscosity νf m2/s kinematic viscosity of the fluid ρf k/m3 density of the fluid τ - tortuosity of the bed (path length / distance) τw k/m/s2 wall shear stress

Dimensionless groups

Symbol definition Variable Re vd Reynolds number (inertia forces / viscous forces)

νf vdhReh νf

Reynolds number based on hydraulic diameter vdpRep νf

Reynolds number based on particle diameter

142

4.6 Conclusions

Literature

Bey, O., G. Eigenberger (1996), Bestimmung von Stromungsverteilung und Warmetransportparametern in schuttungsgefullten rohren, Chemie-Ing.-Techn. 68(10), 1294-1299

Bey, O., G. Eigenberger (1997), Fluid flow through catalyst filled tubes, Chem. Eng. Sci. 52(8), 1365-1376

Bird, R.B., W.E. Stewart, E.N. Lightfoot (2002), Transport Phenomena, second edition, John Wiley and Sons, New York

Blevins, R.D. (1984), Applied Fluid Dynamics Handbook, Van Nostrand Reinhold company, New York

Calis, H.P.A. (1995), Development of dustproof, low pressure drop reactors with structured catalyst packings : the bead string reactor and the zeolite covered screen reactor, PhD dissertation, Delft University of Technology, Delft, the Netherlands.

Cheng, P., D. Vortmeyer (1988), Transverse thermal dispersion and wall channeling in a packed bed with forced convective flow, Chem. Eng. Sci. 43(9), 2523-2532

Cheng, Z.-M., W.-K. Yuan (1997), Estimating radial velocity of fixed beds with low tube-to-particle diameter ratios, A.I.Ch.E. J. 43(5), 1319-1324


Ergun, S. (1952), Fluid flow through packed columns, Chem. Eng. Progress 48(2), 89-94

Eisfeld, B., K.Schnitzlein (2001), The influence of confining walls on the pressure drop in packed beds, Chem. Eng. Sci. 56, 4321-4329

Epstein, N. (1989), On tortuosity and the tortuosity factor in flow and diffusion through porous media, Chem. Eng. Sci. 44(3), 779-781

Fand, R.M., Thinakaran, R. (1990), The influence of the wall on flow through pipes packed with spheres, J. Fluid Eng. 112, 84-88

Giese, M., K. Rottschafer, D. Vortmeyer (1998), Measured and modelled superficial flow profiles in packeds beds with liquid flow, A.I.Ch.E. J. 44(2), 484-490

143


Givler, R.C., S.A. Altobelli (1994), A determination of effective viscosity for the Brinkman-Forchheimer flow model, J. Fluid Mech. 258, 355-370

Goodling, J.S., R.I. Vachon, W.S Stelpflug, S.J. Ying (1983), Radial porosity distribution in cylindrical beds packed with spheres, Powder Technology 35, 23-29

Jakobsen H.A., H. Lindborg, V. Handeland (2002), A numerical study of the interactions between viscous flow, transport and kinetics in fixed bed reactors, Comput. Chem. Eng. 26(3), 333-357

Janssen, L.P.B.M., M.M.C.G. Warmoeskerken (1987), Transport phenomena data companion, Delftse Uitgevers Maatschappij, Delft

Leijendeckers, P.P.H., J.B. Fortuin, F. van Herwijnen, H. Leegwater [red.] (1998), Polytechnisch zakboekje, Koninklijke PBNA b.v., Arnhem

MacDonald, I.F., M.S. El Sayed, K. Mow, F.A.L. Dullien (1979), Flow through porous media — the Ergun equation revisited, Ind. Eng., Chem. Fundam. 18(3), 199-208

McGreavy, C., E.A. Foumeny, K.H. Javed (1986), Characterization of transport properties for fixed bed in terms of local bed structure and flow distribution, Chem. Eng. Sci. 41(4), 787-797

Papageorgiou, J.N., G.F. Froment (1995), Simulation models accounting for radial voidage profiles in fixed bed reactors, Chem. Eng. Sci. 50(19), 3043-3056

Puncochar, M., J. Drahos (1993), The tortuosity concept in fixed and fluidized bed, Chem. Eng. Sci. 48(11), 2173-2175

Roblee, L.H.S., R.M. Baird, J.W. Tierney (1958), Radial porosity variations in packed beds, A.I.Ch.E. J. 4(4), 460-464.

Subagyo, S., N. Standisch, G.A. Brooks (1998), A new model for velocity distribution of a single-phase fluid flowing in packed beds, Chem. Eng. Sci. 53(7), 1375-1385

Schwartz, C.E., J.M. Smith (1953), Flow distribution in packed beds, Ind. Eng. Chem. 45(6), 1209-1218

Tierney, M., A. Nasr, G. Quarini (1998), The use of proprietary computational fluid dynamics codes for flows in annular packed beds, Sep. Purif. Techn. 13, 97-107

144

4.6 Conclusions

Tsotsas, E., E.U. Schlunder (1988), Some remarks on channelling and on radial dispersion in packed beds, Chem. Eng. Sci. 43 (5), 1200-1203

Vortmeyer, D., J. Schuster (1983), Evaluation of steady flow profiles in rectangular and circular packed beds by a variational method, Chem. Eng. Sci. 18(10), 1691-1699

White, S.M., C.L. Tien (1987), Analysis of flow channeling near the wall in packed beds, Warme-Stoffubertrag. 21, 291-296

Winterberg, M., E. Tsotsas, A. Krischke, D. Vortmeyer (2000), A simple and coherent set of coefficients for modelling of heat and mass transport with and without chemical reaction in tubes filled with spheres, Chem. Eng. Sci. 55, 967-979

145

Chapter 5

Dispersion in randomly packed beds

Summary

The goal of this chapter is to find a method to model dispersion in a packed bed in such a way that it can be used in a pseudo-continuous Computational Fluid Dynamics (CFD) model for packed beds.

Dispersion determines, together with convection, the rate at which heat and mass are transported in the packed bed and therefore is one of the most important factors in a packed bed model. Historically, dispersion has been used to describe the deviation of a real packed bed from a, usually one-dimensional, ideal model. As a consequence, in such models, the dispersion coefficient is a container for all non-idealities, so its value and meaning depend strongly on the model that is chosen. More recently, two-dimensional models have been developed, among which the so-called standard or wall heat transfer (WHT) model and the wall heat conduction (WHC) model are the best known. The WHT model defines a heat transfer coefficient at the wall to quantify a temperature jump at the wall boundary. Although this model is used with some success, the physical correctness of the temperature jump is doubtful. The WHC model uses the wall temperature as a more natural boundary condition. The porosity profile is used (together with some parameters that were fitted to experimental data) to describe the heat transfer resistance at the wall.

None of these literature models is completely satisfactory for use in a CFD code, because data is used that may not be defined or available in a general-geometry bed (e.g., the Peclet number at the centre of the bed). Also it is questionable whether these models can be extrapolated to non-cylindrical beds. Therefore, an alternative model is developed here.

147

√

5. Dispersion in randomly packed beds

It is clear that the tortuosity of the flow channels will play an important role in the mixing behaviour. Our model uses not only the porosity profile but also the tortuosity profile to estimate the dispersion coefficients. As it is based on plausible assumptions and bed packing simulation results for the tortuosity profiles, the model does not have parameters that are fitted to experimental data. Therefore, it is by nature more general than literature models like the WHC, WHT and similar empirical models. The result for radial dispersion of heat is:

1 ε (1 + εaR) + (τ 2 − ε) a ∗ τ 2 − 1 =

τ 2 r s + a,r

(5.1)Per,h r Pef,h 10

This equation consists of two terms, the first one for the stagnant contribution to conduction and the second for the convective (high flow) contribution. Her aR is the stagnant radiation contribution, as

∗ is the solid conduction contribution (as defined by Zehner and Schunder, 1972); τr is the bed tortuosity in radial direction and τa,r is the radial component of axial tortuosity. All parameters are taken locally in the bed. For axial dispersion and dispersion of mass and momentum, similar equations are given.

For cylindrical bed geometries, the results of our model are, as they should be, comparable to the best literature models. The real power of our model is in the modelling of non-standard, 2 and 3-dimensional bed geometries. These cases cannot be calculated by a simple finite difference method as was done for the cases in this chapter, but require a finite volume (CFD) code. This will be described in the next chapter.

148

5.1 Introduction

5.1 Introduction

Randomly packed beds are used for many types of solid-fluid contacting processes. Consequently, there has been a large interest in the modelling of the important processes in randomly packed beds. Traditionally, models for packed beds are focused on packed cylindrical tubes. The flow in the tube is described by a model that treats the bed as a continuum. The simplest of these models is the ideal tube model that assumes a flat velocity profile (plug flow), perfect radial mixing and no axial mixing. More sophisticated models include axial and radial dispersion coefficients to account for non-ideal behaviour.

Although the cylindrical packed tube is the most common geometry for a packed bed reactor, other geometries may be more suitable for specific applications. For low-pressure drop, shallow packed beds, the packed cylinder is usually not the most efficient arrangement, and radial flow, lateral flow or parallel passage reactors may be employed (Calis, 1995). Also for the development of compact installations, for instance mobile reforming installations for fuel cell applications, non-standard geometries may be necessary for, e.g., ease of construction, accessibility or heat integration.

The goal of this work is to build a model for packed beds that can be used in computational fluid dynamics (CFD) code to simulate packed bed reactors of any geometry. In a CFD model, the flow domain is divided into a large number of subvolumes (cells). In a subvolume of a packed bed, the transfer of momentum, heat and mass is determined only by local forces, gradients and bed structure. Therefore, a physically correct model of a packed bed should use only local information (temperatures, concentrations, porosity) in the transport balance equations. However, most of the generally accepted models for heat conduction and dispersion in packed beds use global parameters such as the Reynolds number at the bed inlet, the Peclet number at the inlet or at the axis of the bed and the bed-to-particle diameter ratio. The goal of this chapter is to create a model to calculate dispersion of mass and heat in packed beds where only local information is used to calculate the mixing behaviour locally in the bed. Such a model is by nature more generic than models that are based on averaged parameters.

In the previous chapters, the structure of a randomly packed bed, especially near a solid wall, was described by profiles of porosity, specific particle outer surface area and tortuosity. An expression was derived for the flow resistance of a packed bed based on local bed properties. In this chapter, the model will be extended to include axial and radial dispersion of momentum, heat and mass inside a packed bed.

The general theory of dispersion in packed beds will be given and a brief overview of the extensive existing literature is presented and discussed. It is

149


found that, although the models give a reasonable description of the experimental data, none of the literature models is satisfactory for application in a CFD model. Therefore, a new dispersion model is developed that takes into account the profiles of porosity and tortuosity near the wall. This model will be based on a physical description of the processes in the bed, using the bed structure from simulated packings and will not be fitted to experimental data. For cylindrical beds, the model should give results that are comparable to the existing literature models. Therefore, the results will be compared to state-of-the-art models and published experimental data for cylindrical packed beds.

The real power of our model is in the modelling of non-standard, 2 and 3-dimensional bed geometries. These cases cannot be calculated by a simple finite difference method as will be done for the cases in this chapter, but require a finite volume (CFD) code. This will be described in the next chapter.

5.2 Dispersion in packed beds

One of the simplest models of a packed bed is the ideal tube model, where there is no axial mixing and are no radial gradients. The value of a given intensive quantity (i.e., heat or the concentration of a tracer) φ is given at any time and place by the differential equation:

∂φ ∂φ = w0 (5.2)

∂t ∂z

where w0 is the superficial velocity of the fluid. In real packed beds, the assumptions of the ideal tube model are often not valid. Dispersion models can be seen as a way to describe deviations from the ideal tube model. The mixing behaviour of a packed bed can be considered at three levels of sophistication (see figure 5.1). At the coarsest level, the question is in what degree the packed bed works as a static mixer: how will temperature or concentration profiles be evened out as the flow passes the bed The bed itself is treated as a black box. The corresponding model is a 1-dimensional tube model with axial dispersion:

∂φ ∂φ ∂2φ = w0 − Da,φ,1D (5.3)

∂t ∂z ∂z2

As a result, a step in the value of φ at the inlet will give a sigmoid response at the outlet of the tube. One of the most common experimental methods to determine the axial dispersion coefficient is to fit the axial dispersion model to a recorded response curve. This dispersion coefficient is a lumped parameter

150


that contains every non-ideality in the bed, including real axial mixing of the flow but also influence of the solid phase and vessel wall (adsorption, heat conduction, radiation), radial mixing and flow maldistribution (wall effects). As a result, the dispersion coefficient depends not only on operating conditions (the velocity, fluid properties) and packing geometry (packing density, particle shape) but also on vessel geometry (tube-to-particle diameter ratio).

For non-adiabatic tubes, the heat transferred from or to the fluid is written as:

Φ ” h = αw,1D(T − Tw) (5.4)

where Φ ” h is the heat flux per unit tube surface area and αw,1D is the wall heat

transfer coefficient. The latter is a lumped parameter that compensates for all radial heat transfer limitations since in the plug flow model, there are no radial temperature gradients. The wall heat transfer coefficient can be determined by fitting the axial dispersion model to steady-state measurements on a wall cooled or heated packed bed. In these procedures, the axial heat dispersion coefficient and the wall heat transfer coefficient are determined simultaneously; however, since transient effects are not part of the system, the axial dispersion coefficient determined in this way may be different from that determined from a temperature response curve. In addition, any errors caused by the relative coarseness of the wall heat transfer model will also find their way into the axial dispersion coefficient.

The one-dimensional model does not give information about the radial profile of φ inside the bed. To calculate this profile, a quasi two-dimensional model can be defined, where variable φ is calculated as a function of both axial coordinate x and radial coordinate r, but the parameters (w0, Da,φ,2D, Dr,φ,2D) are constant:

∂φ ∂φ ∂2φ ∂2φ = w0 − Da,φ,2D − Dr,φ,2D (5.5)

∂t ∂z ∂z2 ∂r2

When we compare the 1-dimensional model with the two-dimensional model, it is clear that the physical meaning (and value) of the dispersion coefficients Da,φ,1D and Da,φ,2D is not the same:

∂2φ ∂2φ ∂2φ Da,φ,1D = Da,φ,2D + Dr,φ,2D (5.6)

∂z2 ∂z2 ∂r2

The boundary conditions for quantities that cannot penetrate the wall are:

151


I

II

III

Figure 5.1: Dispersion models with different levels of detail: 1-D tube model with axial dispersion, quasi 2-D model with constant parameters and real 2-D model with bed characteristics.

z = 0 φ = φ0 − Da,φ,2D

w ∂ ∂z

φ

∂ z = L φ = 0

∂z ∂

r = 0 ∂r

φ = 0 (5.7)

∂ r = R φ = 0

∂r

The boundary condition for the temperature equation at the vessel wall is somewhat different since there can be heat transport through the wall:

∂ r = R − λeff,r T = αw,2D(T − Tw) (5.8)

∂r

The wall heat transfer coefficient αw,2D now does not correct for the existence of radial temperature profiles, since these are taken into account in the model.

152


However, it is still needed to correct for radial profiles of bed packing and velocity that are caused by the reactor wall. The boundary condition at the wall causes a jump in the temperature at the wall from the fluid temperature to the wall temperature; this temperature jump is not physical. Since the wall effect also influences flow in the core of the bed, the dispersion coefficients are still a function of the tube-to-particle diameter ratio.

To accurately describe the packed bed as a chemical reactor, the mixing behaviour has to be known at each point in the bed, taking into account porosity, particle surface area and tortuosity profiles near the wall. In empty tubes, the heat transfer resistance near the wall is associated with a stagnant fluid layer. In packed tubes, as we have seen in chapter 4, the heat transfer resistance near the wall coincides with a high-velocity region. There will of course be a stagnant fluid layer at the wall, but the thickness of this layer will be small compared to the particle diameter; the heat transfer resistance caused by this layer will be in the same order as the particle-to-fluid heat transfer resistance. The influence of this layer on the temperature profile is small enough to disappear in the experimental error, which is considerable as the temperature profiles inside packed beds are by nature not easy to measure (Wijngaarden and Westerterp, 1992). Therefore, the heat transfer resistance in packed beds at a wall must be caused by a decrease in the dispersion coefficient near the wall, caused by the structure of the bed.

∂φ ∂w0φ ∂ ∂φ ∂ ∂φ = − Da,φ,2D − Dr,φ,2D (5.9)

∂t ∂z ∂z ∂z ∂r ∂r

When the dispersion coefficient is taken as a function of the distance from the wall, the wall heat transfer coefficient (and temperature jump near the wall) is no longer needed to describe measured temperature profiles, and the physically correct boundary condition T = Tw can be used at the wall. A disadvantage of this model is that there are many possibilities for the choice of the dispersion coefficient (and velocity) profiles.

Dispersion is not only a way to model deviation from ideal behaviour, but also a physical phenomenon. A random packed bed can be described as a network of twisted flow channels with uneven cross-section, where the flow of neighbouring channels is repeatedly mixed and split. Even though the bed packing may be regarded as isotropic (if averaged over a large enough volume and far from a wall), dispersion is not an isotropic phenomenon. Two directions of dispersion can be distinguished: radial and axial dispersion, i.e., in the radial and axial directions of a packed cylindrical tube. For general bed geometry, it would be better to relate the dispersion to the direction of the net flow instead of to the tube geometry, but since they are so generally used, these terms

153


are retained here. However, one should bear in mind that radial dispersion is really dispersion normal to the direction of the flow and axial dispersion is parallel to the flow. As a consequence, close to a wall, radial dispersion will be perpendicular to the wall and axial dispersion parallel to the wall.

Close to a wall, the bed packing will not be isotropic: there will be a difference between the tortuosity of the fluid channels parallel to the wall and perpendicular to the wall. Therefore, there will be three different types of dispersion near a wall: axial dispersion which is parallel to the flow and parallel to the wall; radial dispersion perpendicular to the flow and perpendicular to the wall and what we will call transversal dispersion parallel to the wall, but perpendicular to the flow.

Axial dispersion (parallel to the flow) is caused by differences in residence time of different flow paths in the bed, i.e., by profiles of the axial velocity. Three different mechanisms for axial dispersion can be distinguished. The first mechanism is molecular transport (conduction and diffusion) through the fluid and through the solid phase. The second mechanism operates on the particle scale: the axial velocity profile in the channels will not be flat due to the fact that the velocity at the particle edge will be zero; since the channels are not straight but twisted and of uneven cross-sectional area. In addition, due to the random character of the bed packing, local differences in packing density on the scale of several particles can lead to preferential flow paths and thus to local differences in axial velocity. The third mechanism operates on the scale of the bed: axial velocity profiles on the bed scale are caused by the effect of the reactor wall on the packing. Only the first two mechanisms should be included in a true 2-dimensional (or 3-dimensional) local dispersion model. The third should be modelled as part of the geometry, not as part of the dispersion.

Radial dispersion (perpendicular to the wall and the direction of the flow) is caused by exchange between different flow lines. This is caused by molecular transport (diffusion, conduction) and by convective mixing due to the interconnection of the flow channels. The bed structure influences the radial dispersion through porosity and the tortuosity of the fluid channels. Since radial dispersion takes place in channels perpendicular to the wall, the tortuosity to be used differs from that used for axial dispersion.

Transversal dispersion (perpendicular to the flow but parallel to the wall) is caused by the same mechanisms as radial dispersion: splitting and mixing of flow channels. However, for transversal dispersion these flow channels will be parallel to the wall, so the tortuosity of these channels is the same as that used for axial dispersion. Far from a wall, where the bed structure is isotropic, transversal and radial dispersion will be identical.

154


At low flow rates, both axial and radial dispersion are important factors for the transfer of mass and heat; conductive mechanisms prevail. As the flow rate approaches zero, the difference between axial and radial direction vanishes and both will be equally important. In this conduction mode (Balakrishnan and Pei, 1978b), mass is transported by diffusion though the fluid phase only; diffusion through the particle phase is either non-existent (in the case of solid particles) or very small (in the case of porous catalyst particles). Heat, however, can be transported by conduction through the stagnant fluid, through the solid phase and also by radiation, so that at low flow rates, the effective heat conduction term differs from the effective mass dispersion term.

As the flow rate increases, the mixing will become more intense. The balances of mass and heat become dominated by convective transport (caused by the net flow) and convective mixing (random exchange of fluid where two adjacent channels are mixed and split). In axial direction, the convective transport term will increase faster than the mixing term, so that the relative importance of axial mixing will decrease. In radial direction the net flow is zero by definition, so the radial transport will be dominated by radial convective mixing. In the high flow rate regime, heat and mass will be transported by the same mechanism, so that heat conduction can be treated completely analogously to mass dispersion.

For packed bed flow systems, the dispersion is usually given in terms of the dimensionless particle Peclet number: the ratio of convective transport over dispersive transport, where the particle diameter dp is taken as a characteristic length:

w0dpPei,φ = (5.10)

Di,φ

Here w0 is the superficial velocity, Di,φ is the dispersion coefficient of quantity φ and subscript i designates the axial (in the direction of w0) or radial (perpendicular to w0) direction. In addition, a stagnant (conduction) Peclet number can be defined:

w0dpPe0 = (5.11)i,φ D0

φ

where Dφ 0 is the dispersion coefficient at stagnation conditions (i.e., the effective

diffusion coefficient) and a turbulent Peclet number

w0dp∞Pe (5.12)= i ∞Di ∞ iwhere D is the dispersion coefficient in the limit of infinite Reynolds numbers.

Note that this depends on the flow only so the subscript φ has been dropped.

155


Apart from these three Peclet numbers that depend on bed structure, the molecular Peclet number can be defined:

w0dpPef,φ = (5.13)

Df,φ

where Df,φ is the molecular, unconfined diffusion coefficient (or heat conduction coefficient) for quantity φ.


5.3.1 Experimental methods

From the early 1930s on, a large amount of research has been performed regarding the transport of mass and heat in randomly packed beds. The main purpose of the bulk of this research is to determine relationships for the dispersion fluxes in the bed. Thereto, models of different sophistication are developed in which the dispersion coefficients are adjustable parameters. These are determined by fitting the model to data from different experimental setups. Balakrishnan and Pei (1978a,b,c) and Tsotsas and Martin (1987) give extensive reviews of earlier literature.

The model and the experimental method are usually closely linked. In one type of experiment a signal (step, pulse or wave) is set on the inlet tracer concentration or temperature and the response (mixing cup tracer concentration or temperature) is recorded at the outlet of the bed. The axial dispersion coefficient is then calculated by fitting the model results to this profile. More detailed experiments inject a tracer at the centre of the bed and record the concentration profile at different radial positions downstream of the injection point (Foumeny et al., 1992), thus determining the axial and radial dispersion coefficients simultaneously. Votruba et.al. (1972) determined the axial dispersion coefficient alone, in the absence of radial temperature profiles by heating a packed bed at the downstream side with an infrared lamp and recording the steady-state axial temperature profile. A more or less standard setup is a packed bed with a heated or cooled wall, where the results include not only dispersion in both directions, but also transfer of heat from the vessel wall. Yagi and Kunii (1960) studied an annular bed between a cooled and a heated tube, while Bey and Eigenberger (1996) studied a packed bed between two flat walls with each a different temperature. More common setups of this type use a single cooled or heated wall (Olbrich and Potter, 1972a; Specchia et.al, 1980; Dixon, 1988; Tsotsas and Schlunder, 1990; Vortmeyer and Haidegger, 1991; Borkink and Westerterp, 1992; Freiwald and Paterson, 1992; Martin and Nilles, 1993; Dixon and Van Dongeren, 1998). Equivalent methods for mass

156


transfer use for instance porous walls from which water evaporates (Hennecke and Schlunder, 1973) or walls from which mercury sublimates (Olbrich and Potter, 1972b). Tsotsas (1992) measured the average outlet concentration of a compound that sublimates from the surface of the bed particles, thus combining mass dispersion and particle to fluid mass transfer.

It is clear that the different experimental procedures that include transient effects (heat capacity of particles, absorption of tracer on or in particles), wall-to-fluid transfer effects or particle-to-fluid transfer effects, combined with the models with different levels of sophistication lead to different correlations for (and in fact, meaning of) the dispersion parameters. Experimental errors as well as modelling errors find their way in to the values of the fitted parameters. As an example, Foumeny et. al. (1992) fitted the same model to transient concentration injection responses and steady-state values. They obtained different correlations for the dispersion coefficients for the transient and steady cases, indicating that some transient effects that were present in the experiments, were not taken into account in the model and ended up in the value of the dispersion coefficients.

5.3.2 The standard model

Radial and (sometimes) axial dispersion are taken into account in a pseudo-homogeneous two-dimensional plug flow model that is commonly called the standard or wall heat transfer (WHT) model (equations 5.5, 5.7 and 5.8). This model is most widely used in engineering practice and recommended in handbooks (e.g., Tsotsas, 1997). Here, the velocity is taken as constant while concentrations and temperature have profiles in axial and radial directions (e.g., Hennecke and Schlunder (1973), Specchia et.al., 1980; Dixon (1988), Freiwald and Paterson (1992), Tsotsas (1992), Dixon and Van Dongeren (1998)). The reduced transport of heat and mass near the wall is described by a wall heat transfer coefficient αW , that is commonly calculated from a Nusselt relation, e.g, Tsotsas (1997):

Nuw = Nuw,0 + aRem 0 Prn (5.14)

For low flow conditions, the temperature drop near the wall will become less steep; the profile will extend further into the bed and the assumption of a temperature jump at the wall will become less accurate. Therefore, the wall heat transfer models tend to become less accurate for lower Reynolds numbers (Re0 < 100). The minimal Nusselt number Nuw,0 is used to improve the predictions in the low-flow regime; it usually depends on the bed-over-particle

157

( )


diameter ratio D/dp, e.g., Martin and Nilles (1993)

5 λ0

Nuw,0 = 1.3 + (5.15)D/dp λf

As a refinement of the standard model, a heterogeneous model is presented by some authors, where the particle temperature is solved as a separate variable from the fluid temperature (Dixon and Cresswell, 1979; Wijngaarden and Westerterp, 1993; Westerterp e.a., 1993). However, it seems that this is only necessary when there is reason to expect a considerable temperature difference between the solid and the fluid, such as in the case of an exothermal chemical reaction inside a porous catalyst particle. Since the reaction does not influence the dispersion itself (Wijngaarden and Westerterp, 1989), this should not alter the meaning or value of the dispersion coefficients. Hence, the (quasi) homogeneous model can be used to determine the dispersion coefficients in the absence of chemical reaction, and these dispersion coefficients can later be used to model packed beds with reaction. Due to the added complexity (chemical reaction kinetics, particle-to-fluid heat and mass transport), it does not seem practical to employ a reactive system to determine the dispersion coefficients.

5.3.3 The wall heat conduction model

The standard or WHT model was extended by taking into account the radial porosity and velocity profiles (Delmas and Froment, 1988; Vortmeyer and Haidegger, 1991; Winterberg and Tsotsas, 2000; Winterberg et.al, 2000). In this so-called wall heat conduction (WHC) model, the radial effective heat conductivity varies with the distance to the wall, and the need for a wall heat transfer coefficient αw and wall temperature jump is removed. The boundary condition for the temperature at the wall becomes: T (r = R) = Tw. The radial variation in radial effective heat conductivity is calculated by taking into account the radial profiles of porosity and velocity. Winterberg et.al (2000) have compared the predictive performance of the WHC model with the wall heat transfer model and conclude that the WHC mode is more accurate than the WHT model. Therefore, it can be concluded that the WHC model is one of the best currently available models for heat and mass transfer in packed beds with fluid flow. Our modelling efforts will be compared to the WHC model according to Winterberg et.al (2000), which will be described here in some more detail.

In the wall heat conduction model, the wall effect is taken into account through the porosity profile. The assumed porosity profile is usually an exponential expression, however the parameters chosen vary considerably. For

158

( [ ])

( [ ])

( )


example, Winterberg et.al. (2000) choose an expression attributed to Giese (1998):

R − r ε(r) = εb 1 + 1.36 exp −5.0 (5.16)

dp

while an earlier expression proposed by Vortmeyer and Schuster (1983) is also widely used (e.g. Vortmeyer and Haidegger, 1991):

R − r ε(r) = εb 1 + 0.55 exp 1 − 2

dp ( [ ]) (5.17)R − r

= εb 1 + 1.495 exp −2.0 dp

Both these expressions are claimed to be valid for packed beds of particles with small deviations from the spherical shape. The main difference between the expressions is the factor 5 respectively 2 inside the exponent. In equation (5.16) the porosity decreases to within 1% of its bulk value at one particle diameter from the wall, while in equation (5.17) this value is not reached until 2.5 particle diameters distance from the wall (5.2). Both equations have the disadvantage that the predicted porosity at the wall depends on the value of the bulk porosity, so strictly they are valid only for a given bulk porosity value (for the latter: 0.4).

The radial velocity profile of the WHC model is calculated from the extended Brinkman equation

∂p ρf νeff ∂ ∂w0(r) = f1w0(r) − f2[w0(r)]

2 + r (5.18)∂z r ∂r ∂r

where f1 and f2 are the laminar and turbulent factors from the Ergun equation (see the previous chapter for details). The effective viscosity νeff is approximated by:

νeff = 2.0 exp(CRe0) (5.19)

νf

where C = 2.0 × 10−3 . This equation is attributed to Giese et.al. (1998), but there a value of C = 3.5×10−3 is proposed. The Reynolds number Re0 is based on the average superficial velocity in the bed. It seems somewhat inconsistent that a constant value is used for the effective viscosity (dispersion of momentum) while considerable effort is put into the development of a relation for the effective conductivity (dispersion of heat) that depends on the wall distance. The mechanism for effective transport of momentum in a packed bed is very

159


0

0.2

0.4

0.6

0.8

1

[-]

Winterberg et.al.

Vortmeyer and Schuster

This work

0 1 2

x/dp [-]

Figure 5.2: Porosity profile used in the WHC model by Winterberg et.al. (2000), equation (5.16), compared to that proposed earlier by Vortmeyer and Schuster (1983), equation (5.17) and a typical porosity profile determined from a bed packing simulation in this work.

similar to that for effective transport of heat and it also depends on local bed structure.

The temperature profiles are calculated from a 2-dimensional quasi-homogeneous heat transport model. In steady-state and without chemical reaction, this reads:

1 ∂ [

∂T ]

∂2T ∂T 0 =

r ∂r λeff,r(r)r

∂r + λeff,a(r)

∂z2 − w0rρf cp,f

∂z (5.20)

The effective radial heat conduction λeff,r is calculated from

λeff,r(r) = λ0(r) + K1,hPef,cf(R − r)λf (5.21)

where λ0 and the function f depend on the distance from the wall. The stagnant effective heat conductivity is calculated according to Zehner and Schlunder (1970, 1972) with the radial porosity profile given by equation (5.16). For the convective mixing term of the heat conductivity, a parabolic profile near

160

( )

( )


the wall is proposed: ( R − r

)2 R − r

f(R − r) = 0 < < K2,h (5.22)K2,hdp dp

At the wall, the heat conductivity is equal to the conductivity of the fluid; it increases quadratically until a distance K2,h. At a distance further than K2,h

particle diameters from the wall, f is taken as 1.0, so the bulk value for the effective heat conduction is used. Hence, there is a sharp transition between the wall zone and the bulk zone. An empirical correlation is proposed for the parameter K2,h:

Re0K2,h = 0.44 + 4 exp − (5.23)

70

so that the distance over which the wall influences the heat conductivity depends on the Reynolds number, from 0.44 particle diameters at high Reynolds numbers to 4.44 particle diameters at low Reynolds numbers. Later (Winterberg and Tsotsas, 2000), a dependence on the Peclet number instead of the Reynolds number was proposed, in order to describe liquid/solid packed bed systems in addition to gas/solid systems:

PefK2,h = 0.44 + 4 exp − (5.24)

50

A similar model is used for mass transfer in packed beds. However, for mass transfer, the value of K2,m is the constant value 0.44. In other words, for mass transfer the thickness of the wall zone with decreased convective mixing is always 0.44 particle diameters, while for heat transfer it depends on the Reynolds number and can be between 0.44 and 10 times that value. The physical background of this is not clear, because at high Reynolds numbers, the mechanism for mass transfer and heat transfer is the same. It appears that there must be a physical phenomenon related with heat conduction through the solid or radiation that is not taken into account correctly in this model. The error in the heat conduction term is then corrected for in the convective mixing term.

5.3.4 Other models

Although the majority of literature follows the pseudo-homogeneous approach, there are some other approaches that are worth mentioning. Kufner and Hof-man (1990) developed a packed bed model based on a mixing cell model with a porosity profile near the wall. Logtenberg and Dixon (1997) made a model

161


of a small bed of eight spheres in a commercial CFD code and determined the velocity and temperature fields. The wall heat transfer coefficient (or NuW ) was calculated by fitting the CFD results to a formulation of the WHT model. The number of spheres in the CFD model and the bed-to-particle diameter (D/dp = 2.86) are too low to allow extrapolation of their results to a general randomly packed bed. Furthermore, it seems that they use a relatively low number of computational cells in the CFD model, and they did not investigate the effect of mesh refinement on their results. However, they find some interesting phenomena. In the flow field, they observe the development of eddies between the spheres at higher Reynolds numbers (Re > 39). Also, they find that the qualitative features of the flow field do not change when laminar flow equations are used instead of turbulence models.

5.4 Dispersion modelling

Although a number of dispersion models can been found in literature, none of these is completely satisfactory for use in a CFD code. None of the models adheres to the rule of locality: that the flow parameters should depend on local parameter and variable values only and not on far removed or average data (like the average bed porosity or the Peclet number at the centre of the bed). Furthermore, all models use parameters that are fitted to experimental data; as the experimental data is almost exclusively for tubular geometry, this limits the general applicability of the modes. For these reasons, a new dispersion model will be developed here. An attempt is made to derive this model as much as possible from physical mechanisms based on local bed structure.

Despite the differences in models and experimental setups, there is a reasonable agreement about the form of the correlation for the axial and radial Peclet number in unconfined packed beds (Tsotsas and Schlunder, 1988). The basic concept is that the stagnant contribution and the convective mixing contribution can be added together to get the Peclet number at intermediate conditions:

1 1 1 = + (5.25)

φ Pe∞Pei,φ Pe0 i

for direction i (radial or axial) and quantity φ (e.g., temperature or a concentration). Therefore, the dispersion at intermediate flow conditions can be calculated if the dispersion at stagnant conditions and high-flow conditions are known. Equation (5.25) will be the basis for the dispersion model developed here.

162


5.4.1 Correlations for Pe at stagnant conditions

At stagnant conditions, a difference must be made between the dispersion (conduction) of heat and the dispersion (diffusion) of mass. It is assumed that the contribution of the solid phase in mass transport is negligible. For non-porous particles this is a natural assumption; for porous catalyst particles, mass may diffuse through the particles as well as through the fluid in the voids between the particles. However, since the effective diffusion coefficient in a typical porous catalyst particle is at least an order of scale lower than that in an unconfined gas space, the contribution of the solid phase will not be taken into account for mass transport on the bed scale. On the other hand, in most cases the heat conduction through the particles will be of the same order of scale as the heat conduction through the fluid, so the solid phase needs to be taken into account in the heat transport term..

Dispersion of mass at stagnant conditions is caused by diffusion. The effective diffusion coefficient of a component φ in the stagnant fluid in a packed bed (Dφ

0 ) differs from the (molecular) diffusion coefficient in an unconfined stagnant fluid (Df,φ) because mass cannot pass through the particles. It depends on bed porosity and the tortuosity τ (mean path length through the fluid per net distance) of the channels in the bed packing. If we look at the bed as a bundle of parallel channels, we can define three different diffusion fluxes. The diffusion flux inside the channels can be given by:

dφ Φ ”

p = −Df (5.26)d�

where � is the length coordinate along the channel and Df the molecular unconfined diffusion coefficient (assuming the diameter of the channels is much larger than the mean free path length of the gas molecules). In order to transport a molecule a distance Δx the distance travelled through the channel is � = τΔx. Hence, the interstitial diffusion flux inside the bed is given by:

1 Df dφ Φ ”

i = Φ ” p = − (5.27)

τ τ d�

The superficial diffusion flux is based on the empty tube area instead of the cross-sectional area of the pores, hence it differs a factor ε. Therefore, with d� = τdx,

Df − ε Φ ”

s = εΦ ” i = −

τ 2

d

d

φ

x (5.28)

b

If we now define

Φ ” ≡ −D0 dφ (5.29)s eff dx

163

√


it follows that the effective diffusion coefficient in a stagnant packed bed is related to the molecular diffusion coefficient as:

εD0Deff = f (5.30)

τ 2

The bed tortuosity appears in the denominator to the second power, and not to the first power as is sometimes suggested (Foumeny et.al., 1993; Berger et.al., 2002, see also Epstein, 1989).

It should be noted that the tortuosity used here is the tortuosity of the channels of the bed (unrelated to the flow), in the direction of the transport. Therefore, even though there is no difference in mechanism between axial and radial dispersion under stagnant conditions, there is a difference between dispersion perpendicular and parallel to the wall because the bed packing is not isotropic near the wall. This radial tortuosity, perpendicular to the wall, is quite different from the axial tortuosity, parallel to the wall. The radial tortuosity will be one up to a distance of at least rp, since the fluid is always in √ direct view of the wall in the wall zone. In the zone from rp to 3rp, a decreasing fraction of the fluid is in direct view of the wall, while further away, the tortuosity will be approximately equal to the bulk porosity, see figure 5.3. The axial tortuosity profile will be that obtained from particle tracking simulations in chapter 3.

For the radial tortuosity, we will assume the following radial profile:

x < r τ = 1p r

rp < x < 2rp τr = 1 + (x/rp − 1)(τr,b − 1) (5.31)

x > 2r τ = τp r r,b

The value of the tortuosity far from the wall (τb) depends on the bulk porosity of the bed: a lower porosity will (for random packings of sphere-like particles) give a higher tortuosity. Zehner and Schlunder (1970), based on experimental data for the effective diffusion coefficients for several types of packing and porosities, propose the following correlation

0 √Deff = 1 − 1 − εb (5.32)Df

This correlation is very widely used (Tstotsas and Martin, 1987; Bauer, 1988; Vortmeyer and Haideggeer, 1991; Winterberg et.al., 2000; Jakobsen et.al., 2002). The corresponding tortuosity according to equation (5.30) can be found:

√ τb = 1 + 1 − εb (5.33)

164


1 3

1

x/rp

Figure 5.3: Sketch of the radial tortuosity near the wall. Nearly all of the particles in the first layer touch the wall (Legawiec and Ziolkowski, 1994), so that the radial tortuosity is exactly one up to one particle radius from the √ wall. Between 1 and 3 to 2 radii from the wall, some of the fluid is in the channel perpendicular to the wall (dashed area). Beyond this area the radial tortuosity is assumed equal to the bulk value.

For standard packed beds (ε ≈ 0.4), this equation gives values for the tortuosity that are somewhat lower than the expected value of about 1.4 (it reaches a √ maximum of 2 as εb → 0). This is even more apparent when it is compared with other models for the bed tortuosity, e.g. according to Puncochar and Drahos (1993), see figure (5.4):

1 τb = √

εb (5.34)

It is important to realise that these correlations (equations 5.33 and 5.34) give values for the bulk or average tortuosity of packings with different porosities, which is not necessarily the same as the tortuosity in a given packing as a function of the distance to the wall. Winterberg et.al. (2000) propose to use

b

165


1

1.5

2

[-]

0.2 0.4 0.6 0.8 1.0

[-]

Figure 5.4: Comparison of the bed tortuosity τb as given by Puncochar and Drahos (1993) and as implied by the model of Zehner (1970) as a function of the bed porosity ε.

the local porosity in equation (5.32), which implies that the (ordered) packing with high porosity near the wall in a dense bed is equivalent to the (random) packing far from the wall in a less dense bed; it is obvious that this is not true. It is clear that in the shell up to one particle radius from the wall, the effective diffusion coefficient should be proportional to the porosity only, not proportional to the root of the porosity.

As equation (5.32) describes the measured data quite well, the model selected here should tend towards this equation far from the wall. However, near the wall, the effective diffusion coefficient changes according to equation (5.30) with the radial tortuosity profile given by (5.31). Hence,

w0dp w0dp τr(x)2 τr(x)2

Pe0 (x) = = (5.35)m = Pef,m D0 ε(x) ε(x)eff Df

For the conduction of heat, the situation is more complex than that for the diffusion of mass. Several mechanisms contribute to the transport of heat:

166


conduction through the fluid, conduction from the fluid to a particle, conduction inside a particle and between two particles (although this mechanism is disputed since the contact between two particles is a point unless the particles can be deformed appreciably; Wijngaarden and Westerterp, 1989) and radiation between particles and between fluid patches. However, for non-conducting solids and in the absence of radiation, the heat conduction should be equivalent to the mass diffusion in equation (5.35).

Yagi and Kunii (1957) proposed the following relation (if we disregard radiation effects):

λ0 eff β(1 − ε)

= λf

(5.36)λf φ + γ

λs

where λf is the heat conductivity of the fluid, λs that of the solid. The parameter β is the dimensionless conduction length between two particle centres, which is usually taken as 1.0. This relationship does not take into account conduction of heat through the fluid phase; if the solid phase conductivity becomes zero, the overall heat transfer coefficient will also approach zero.

Specchia et al. (1980) elaborate on a similar correlation by Kunii and Smith (1960)

λ0 β(1 − ε)eff = ε + (5.37)λf φ + γ λf

λs

Here φ and γ are dimensionless parameters describing the packing. Specchia et.al. suggest a value of 2/3 for γ and 0.22ε2 for φ. Note that in the limit of non-conducting solid (which is equivalent to the mass transfer situation), comparison of equation (5.37) with equation (5.30) leads to the conclusion that τ equals 1, i.e., it does not take into account the tortuosity of the path along which heat is transported in this situation.

Zehner and Schlunder (1970) base their correlation on a two-dimensional cylindrical cell model where the heat transport takes place along parallel lines. This model is widely accepted in literature (e.g., Bauer and Schlunder, 1978; Tsotsas, 1997). In this model, the shape of the particles is modified in the model to account for the fact that in reality the heat flow lines are not parallel. The shape of the particles is described by the relation

2zr 2 + = 1 (5.38)

(B − (B − 1)z)2

Here B is an adjustable shape parameter; for B = 1, the shape is a sphere (see figure 5.5). For random packings, the parameter B depends on the porosity

167

(

ε

ε ε


and can be estimated as: )10/91 − ε

B = 1.25 (5.39)

B=0.01

Particle 1

Particle 2

B=0.1

B=1.0

B=2.0

dp

Figure 5.5: Shape of the contact area between two particles as a function of shape parameter B according to equation 5.38.

Part of the heat conduction goes through the fluid phase (with conductivity λf ) only; the rest goes through the solid phase at some stage. The relative size of these parts must be such that if the solid conductivity is zero (effectively blocking any heat flow lines that pass through the solid), the result must be the same as for mass diffusion (equation 5.35):

λ0 eff ∗ = (1 + εaR) + (1 − )as (5.40)

τ 2 τ 2λf r r

Where a ∗ s is the effective heat conductivity factor of flow lines that pass through

the solid and aR the radiation contribution. Note that this formulation differs from that of Zehner and Schlunder (1970):

λ0 √ √ eff ∗ = (1 − 1 − ε) (1 + εaR) + ( 1 − ε)as (5.41)

λf

168

[ ] ( )

[ ( )


Of course, the value is the same for bulk conditions provided equation (5.33) is used to estimate the radial tortuosity τr. Winterberg et.al. (2000) use equation (5.41) with the local value of the porosity to estimate the effective heat conductivity near the wall. For non-conducting solids and in the absence of radiation, the heat conductivity should be equivalent to the effective mass diffusion. Therefore, in the shell up to one particle radius from the wall, the effective heat conductivity should differ from the molecular heat conductivity by a factor ε only. It is clear from figure (5.6) that our equation (5.40) with radial tortuosity profile (5.31) shows this behaviour while equation (5.41) does not.

The effective heat conduction through the solid is calculated by integration over the particle cross section normal to the direction of the conduction. The result is (Zehner and Schlunder, 1970):

2 B(as − 1) as B − 1 B + 1 a ∗ = ln − + (5.42)s N asN2 B N 2

where

λsN = 1 − B/as as = (5.43)

λf

For higher temperatures (usually above approximately 200 ◦C), radiation becomes a factor in the transfer of heat. Since many heterogeneous chemical reactors operate at temperatures well above this value, radiation of heat needs to be taken into account. This is done by adding an additional (parallel) heat transfer term through the fluid phase (Zehner and Schlunder, 1973):

4σT 3dpλR = (5.44)

2/εR − 1

Here σ is the Stephan-Boltzmann constant (56.7051 × 10−9W/m2/K4) and εR

the emissivity of the particles (a value between 0.5 and 1.0 for many materials). With this additional term, equation (5.40) becomes

λ0 ε εeff ∗ = ( ) (1 + εaR) + (1 − )a (5.45)λf

sτr 2 τr

2

with

2 B(as + aR − 1) as + aR B − 1 as ∗ = ln −

N asN2 B N ] (5.46)B + 1 − (aR − B)2B

169

[ ]


where

λRN = 1 − (B − aR)/as aR = (5.47)

λf

Therefore, the stagnant Peclet number for heat can be written as:

τ 2 τ 2w0dp r r ∗ Pe0 h =

λ0 = Pef,h (1 + εaR) + (1 − )as (5.48) eff /(ρcp) ε ε

0.4

0.2

0

x/rp [-]

Figure 5.6: Stagnant bed conductivity factor kbed = λeff,bed/λf in radial direction for a typical packed bed (D/dp = 8.65) without radiation and solid heat conductivity according to the literature model and according to this work . For the latter, the conductivity equals the porosity for x/rp < 1, which is the correct behaviour. Far from the wall, both approaches give similar results.

5.4.2 Correlations for Pe at high Reynolds numbers

At high values of the Reynolds number (or, more accurately, at high values of the stagnant Peclet number Pe0

φ), dispersion of mass and heat in a packed bed is dominated by convective mixing. Therefore, there is no difference in the

0

k bed

[-],

[-]

1

0.8

0.6

Porosity

Conductivity

Winterberg et.al. This work

Porosity

Conductivity

0 1 2 3 4 5

170


treatment of heat and mass. However, there is a difference between dispersion in the axial and radial direction.

The value of the Peclet number in the radial direction at high Reynolds numbers can be derived theoretically from a mixing cell model (Schlunder, 1966). In two dimensions, this model is quite straightforward (figure 5.7). Two types of mixing cells can be distinguished: axial nodes above each particle (points) and radial nodes between two particles (open circles).

j = 0

1

1'

2

2'

x

flow

z

i = 1 1' 2 2' 3 3' 4

Figure 5.7: Rectangular grid mixing cell model of a packed bed in two dimensions, after Schlunder (1966).

The theory is based on the stationary convection-diffusion equation:

∂φ ∂2φ w0 = Dr,φ (5.49)

∂z ∂x2

In terms of the discrete mixing points:

w0Δx2

(φi,j+1 − φi,j ) = (φi−1,j − φi,j ) − (φi,j − φi+1,j ) (5.50) DΔz

If the flow is equally distributed over the mixing points, it follows that

φi−1,j + 2φi,j + φi+1,jφi,j+1 = (5.51)

4

Hence,

φi−1,j − 2φi,j + φi+1,jφi,j+1 − φi,j = (5.52)

4

171


or, with (5.50):

w0Δx2

= 4 (5.53) DΔz

If the assumption is made that Δx = Δz = dp, then it follows that for this 2-dimensional geometry:

Pe∞ (5.54) r,2D = 4

For a rectangular 3-dimensional network, Schlunder assumes that at the radial nodes (between two particles) two flows join while at the axial nodes (nodes above each particle) four flows from the closest radial nodes join. Accordingly, with Δx = Δy = Δz = dp it is found that:

Pe∞ (5.55) r,3D = 8

As, far from the wall, each particle in a packed bed of spheres rests on three others, a rectangular network as used by Schlunder does not seem the to be the best representation of a real random packed bed. In fact, the bed could equally well or better be represented by a hexagonal close packed structure. For such a structure (figure 5.8), flow from three sources is mixed at each node that is located above each particle (see figure 5.9).

Figure 5.8: Sphere packing corresponding to the mixing cell structure in figure 5.9.

172

( )


a

b

cd

e

f

A

B

C

P

p

Figure 5.9: Mixing cell structure for a hexagonal close packed structure. The top cell P depends on three lower mixing cells A,B,C that in turn depend on the seven cells at the bottom (numbered a-f in a hexagonal arrangement with p at the centre).

It is clear that the concentration φi at a node i is the mean value of the concentrations in the three nodes below it. Hence, we can write

Δφ φP − φp 1 = √ = √ (φa + φb + φc + φd + φe + φf − 6φp) (5.56)

Δz 3d0 9 3dp

In order to avoid complications, we assume that the concentration gradient is parallel to the line e-b in figure 5.9. Hence,

1 φa = φc ≈ (φb + φp)

21

φf = φd ≈ (φe + φp)2

and Δφ 1

= √ (2φb + 2φe − 4φp) (5.57) Δz 9 3dp

The second derivative of concentration expressed in the node concentrations is:

Δ2φ 1 φb − φP φP − φe = − (5.58)

Δx2 dp dp dp

173


and therefore √

Pe∞ = 6 3 (≈ 10.4) (5.59)r,3D

If, however, we assume that the concentration gradient is perpendicular to the line e-b in figure (5.9), the result is

Pe∞ = 9r,3D (5.60)

For concentration gradients that do not happen to be parallel or perpendicular to one of the lines (d-a),(e-b) or (f-c), the corresponding Peclet number will be in between these values. For randomly packed beds, this number will differ from these values, but will probably be somewhat higher than the value suggested by Schlunder. In literature models, values between 7 (Bauer, 1988) and 11 (Gunn, 1987) are used. The values found experimentally are in this same range: Borkink and Westerterp (1992) find values between 8.8 and 10.9 for spheres and lower values for cylinders and Raschig rings (7.6 and 4.2 respectively); Dixon (1988) finds values between 7 and 10 for spheres (between 4 and 8 for cylinders); Foumeny et al. (1992) found a value of 12.5. A value of 10 seems to be an acceptable average for random packings of spheres (e.g., Dixon and Cresswell, 1979; Delmas and Froment, 1988).

The derivation above is based on the assumption that the packing is regular. Although it is shown that there is some dependence on the structure of the bed, it can be accepted that it is approximately correct for a random packing. Near the wall the bed is not random, and therefore the mixing behaviour will be different. The change in bed structure will have an effect on the distribution of mixing points in space and on the flow rates between nodes.

The assumption in e.g., equation (5.54) that Δx = Δz implies that the gas paths make an angle of 45 ◦ with the axis of the bed, or in other words, that √ the tortuosity equals 2. For other values of the tortuosity, we can write (see figure 5.10):

√Δx = τ 2 − 1 (5.61)

Δz

Then if we take Δx = dp, (5.54) becomes

Pe∞ = √ 4

(5.62)r,2D τ 2 − 1

For the three-dimensional case we need to take into account the radial and tangential dispersion near the wall. We define a radial and tangential

174

√

√

√


x

z

flow

Figure 5.10: Graphical representation of the relation between Δx, Δz and the tortuosity τ in a rectangular mixing cell model of a packed bed in two dimensions

component for the tortuosity of the axial flow paths:

la 2 + lr

2

τa,r ≡ (5.63)la

l2 + l2

τa,t ≡ a t (5.64)la

where la, lr and lt are the axial, radial and tangential components of a section of a flow path. Of course, the three-dimensional tortuosity is defined as

l2 + l2 + l2 a r tτa ≡ (5.65)

la

The structural correction for the bed tortuosity profile near the wall becomes:

Pe∞ = √ 10

(5.66)r,3D τ 2 − 1a,r

for the radial dispersion. The tortuosity that needs to be used in this equation is the radial tortuosity component of the axial flow paths, where only flow variations in radial direction (i.e., perpendicular to the wall) are taken into

175


account. Near the wall, the flow will be parallel to the wall, which means that the tortuosity has a value of 1. This will be the case at distances up to one third particle radius from the wall. Fluid in this region will flow in a zigzag manner around the particles, but the zigzags will be primarily in the tangential plane at constant wall distance. In contrast, at one particle radius from the wall, the fluid will flow almost exclusively in radial direction, so the radial component of the axial tortuosity will be approximately equal to the 3D tortuosity. Therefore, we will assume that the τa,r will be 1.0 up to a distance of rp/3 from the wall, and then increase to the maximum value of the 3D axial tortuosity at one particle radius from the wall (see figure 5.11).

Since it is quite exceptional for strong temperature and concentration gradients to occur parallel to a wall, the effect of the tangential dispersion will usually be negligible. Therefore, we will, use the same value for the radial and tangential tortuosity.

The model is quite sensitive to the shape of the tortuosity profile used to calculate the radial dispersion term, especially in the first particle radius from the wall. The curve used is an estimate of the actual profile. Since near the wall the bed is quite ordered, it would not be very difficult to build a CFD model for the first layer of particles near the wall and calculate the flow and (through a tracer particle simulation) the actual tortuosity profile at high flow rates. However, this is outside the scope of this work.

For axial dispersion at high flow rates, there is not as much data available because axial dispersion in unconfined packed beds becomes a less important transport mechanism at higher flow rates. The value used in most previous investigations (e.g. Dixon,1988, Jakobsen et.al.,2002) is

Pe∞ a = 2 (5.67)

Foumeny et.al. (1992) found values of about 1.9 in mass transfer experiments; Gunn (1988) found a value of 2.0 for spheres and values between 1 and 2 for cylinders, based on older experimental data.

5.4.3 Dispersion of momentum

The flow pattern inside the packed bed not only changes the transport of heat and mass, but also of momentum. Again, we can make a difference between axial and radial dispersion. Axial dispersion of momentum will lead to a force exerted by high-velocity fluid on low-velocity fluid in the axial direction on a streamline. This force will be negligible compared to the force exerted on the fluid due to the friction in the bed. Therefore, only radial dispersion of momentum will be taken into account. At very low velocities, the transfer takes

176


2

[-],

[-]

[-]

a [-] a,r [-]

r [-]1.8

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

0 0 1 2 3 4 5 6 7 8

x/rp [-]

Figure 5.11: Profiles of the three different tortuosities as a function of the distance from the wall; the 3-dimensional axial tortuosity τa that is used in flow resistance calculations, the radial component of axial tortuosity τa,r that is used in radial convective mixing calculations and the radial tortuosity τr

that is used in stagnant radial dispersion calculations

place on a molecular scale. Momentum is transferred when a high velocity stream and a low velocity stream share a single channel, by viscous effects. In this regime, transfer of momentum between two neighbouring streams may be less intense than in an open space since they are separated by solid particles for a significant fraction of the way. In the high-velocity regime, we can look at the packed bed as a number of mixing volumes with connecting channels. The balance of momentum must then be satisfied for each mixing volume. Therefore, if a high velocity stream enters from one channel and a low velocity stream from another, the exit streams will have an intermediate velocity. In effect, the stream on the low-velocity side will be accelerated at the cost of the high-velocity side momentum. Therefore, high velocity gradients (on a macroscopic scale) inside the bed will be suppressed.

Not much literature is available on the subject of dispersion of momentum in packed beds. The mechanisms for transfer of momentum are very similar to

177

√

√

√


the mechanisms for the transport of mass, so an analogous model is proposed here. Increased transport of momentum in turbulent flow is often represented as an increase in effective viscosity. We will here introduce an equation for the effective viscosity in radial direction instead of the equivalent of a Peclet number for momentum (which would be an effective Reynolds number).

5.4.4 Dispersion model

The overall dispersion model is formed by adding together the contributions for stagnant conditions and at high Reynolds numbers (equation 5.25). For dispersion of mass:

1 ε 1 = + (5.68)

τ 2Pea,φ r Pef,φ 2

τ 21 ε a,r − 1 = + (5.69)

τ 2Per,φ r Pef,φ 10

For dispersion of heat:

1 ε (1 + εaR) + (τr 2 − ε) as

∗ 1 = + (5.70)

τ 2Pea,h r Pef,h 2

1 ε (1 + εaR) + (τr 2 − ε) as

∗ τ 2 − 1 = + a,r

(5.71)τ 2Per,h r Pef,h 10

For radial dispersion of momentum:

Rep τ 2 − 1νeff,r ε a,r = + (5.72)

νm τr 2 10

In these equations, Re and Pe are based on the local superficial velocity and properties.

5.5 Results

In order to compare the dispersion model presented here with literature data and models, a 2-dimensional heat transfer model is presented for flow and heat conduction in a wall-heated packed tube. The porosity, tortuosity and specific surface profiles are given in chapter 3. The velocity profile inside the packed

178

[ ]

5.5 Results

bed is calculated using the relations given in chapter 4, with the effective viscosity according to equation (5.72). The transfer of heat is calculated from the partial differential equation 5.20:

1 ∂ ∂T ∂2T ∂T 0 = λeff,r(r)r + λeff,a(r) − w0rρf cp,f

r ∂r ∂r ∂z2 ∂z


∂T r = 0 = 0 r = R T = Tw

∂r ∂T

z = 0 T (r) = T0 z = 1 = 0 ∂z

The effective radial and axial heat heat conductivity are calculated as a function of the wall distance using the equations (5.68)-(5.71). Physical properties of air are calculated as a function of the local temperature. Figure (5.12) gives the results of this model, compared with the results of the model of Winter-berg et.al. (2000) and measured temperature profile points given by the same authors. The parameters for the calculations are summarised in table 5.1.

Table 5.1: Values of the parameters used for figure 5.12 Quantity symbol value Bed diameter Bed length Particle diameter Bed-to-particle diameter ratio Bed bulk porosity Bed bulk tortuosity Axial wall tortuosity

D L dp

D/dp

εb

τb

τw

0.075 m 0.2 m 9.5 mm 8 0.38 1.41 1.1

Particle material Particle conductivity Particle emissivity

λs

εR

ceramic 1.2 W/m/K 0.85

Fluid Molar mass Pressure Reynolds number at inlet

M p Re0

air 0.028 kg/mol 105 Pa 480

Figure 5.13 compares the results of the model with the wall heat conduction model for different flow conditions. Note that the results of both models are quite similar. In Winterberg e.a. (2000) temperature profiles for these cases are also shown and compared to measured values after Dixon (1988). The curves

179


40

50

60

70

80

90

100 T

[°C]

whc model

this work

experimental

0 1 2 3 4 5 6 7 8

x/rp [-]

Figure 5.12: Temperature profile for a steam-heated tube filled with ceramic particles; comparison of model results of this work, results from the wall heat conduction (WHC) model according to Winterberg et.al. (2000) and measured data after Dixon (1988), as presented by Winterberg e.a.; see table 5.1 for the parameter values used.

appear to fit the measured values quite well. However, with our models (both the detailed profile model and the WHC based model) we cannot reproduce the curves presented by Winterberg et.al. without significantly changing the parameters (e.g., the conductivity of the ceramic material) compared to the case presented in figure 5.12.

In the original article by Dixon, the raw data for the temperature profiles is not given. Instead, the author presents derived parameters, like the radial Peclet number for different packings of nylon, ceramic and steel particles and for different D/d ratios. The packings with 6.3 and 9.5 mm particles give similar effective radial Peclet numbers for each material. However, for the packing with 12.7 mm particles which corresponds to the cases in figure 5.13, anomalous behaviour is found. The ceramic packing gives a higher conductivity than expected, which even roughly coincides with the steel particle packing, whereas the nylon particle packing gives a much lower conductivity than expected. The steel packing seems to be in line with the other measurements.

As the radial thermal conductivity is strongly dependent on the solid con

180

5.6 Conclusions

ductivity (at least for Re < 1000), it is hard to understand how a ceramic packing (λs = 1.2 W/m/K) could have the same effective radial conductivity as an identical steel packing (λs = 50 W/m/K). Therefore, we must conclude that the data for the 12.7 mm ceramic spheres contains some unknown effect and therefore cannot be used to validate the dispersion model.

Table 5.2: Values of the parameters used for figure 5.13 Quantity symbol value Bed diameter Bed length Particle diameter Bed-to-particle diameter ratio Bed bulk porosity Bed bulk tortuosity Axial wall tortuosity

D L dp

D/dp

εb

τb

τw

75 mm 200 mm 12.7 mm 5.9 0.38 1.41 1.1


λs

εR



M p Re0

air 0.028 kg/mol 105 Pa 165, 270, 730

5.6 Conclusions

It can be seen that the model given in this work follows the wall heat conduction (WHC) model proposed by Winterberg et.al. quite closely. This is remarkable since the WHC model contains several parameters that have been fitted to a large number of experimentally determined temperature profiles, while the model in this work was based on physical properties of random packed beds obtained from bed packing simulation results. Figure (5.13) shows that this model is able to describe the temperature profile for different values of the Reynolds number as well as the WHC model.

The decrease in thermal conductivity near the wall of a packed bed is attributed here to the bed structure and more specifically to the bed tortuosity. Since there is a high velocity zone between the wall and one particle radius from the wall, a high level of convective mixing would be expected there. However, since the tortuosity is also close to one in this region, the high velocity does not lead to intensive mixing. Due to these two strong opposing effects, the

181


40

50

60

70

80

90

100

T

[°C]

whc model

this work

Re0 = 730

Re0 = 270

Re0 = 165

0 1 2 3 4 5

x/rp [-]

Figure 5.13: Temperature profiles for a steam-heated tube filled with ceramic particles; comparison of model results of this work with results from the wall heat conduction (WHC) model according to Winterberg et.al. (2000); see table 5.2 for the parameter values used.

model is quite sensitive to the value of the tortuosity near the wall. A CFD calculation for the first layer of particles near the wall could give more detailed information about the actual tortuosity in the flow paths in this zone.

Our model is based on physical mechanisms and estimates for the structure of the bed; therefore it is by nature more general than literature models like the WHC, WHT and similar, empirical models. For 2-dimensional cases, the results of our model will (and should) be comparable to the best literature models. The real power of our model is in the modelling of non-standard, 3-dimensional bed geometries. In fact, there are many 3-dimensional cases for which models like the WHC cannot be used, for instance because the ’centre of the bed’ (at which the Peclet number is needed) is not defined for an asymmetrical bed. These 3-dimensional cases cannot be calculated by a simple finite difference method as was done for the 2-D cases in this chapter, but require a finite volume (CFD) code. This will be described in the next chapter.

6

182

5.6 Conclusions

Nomenclature

Roman Symbol units Variable as − solid phase relative heat conductivity (=λs/λf ) aR − relative contribution of radiation to heat transfer

(=λR/λf ) as ∗ ... auxiliary solid phase heat transfer variable in the

transfer equation by Zehner (1970) B − parameter in particle shape function cp, f J/kg/K fluid heat capacity at constant pressure d m distance to the wall dp m particle diameter D m diameter of the bed Dφ

0 m/s2 effective diffusion coefficient inside the bed for φ in the stagnant fluid

D∞ 2 i,φ m/s effective diffusion coefficient for φ in direction i at

high Re numbers Da,φ,1D m2/s axial dispersion coefficient (parallel to the flow) for

quantity φ, in 1-dimensional model Da,φ,2D m2/s axial dispersion coefficient (parallel to the flow) for

quantity φ, in 2-dimensional model Dr,φ m2/s radial dispersion coefficient (perpendicular to the

flow) Df,φ m2/s molecular diffusion coefficient for φ in a stagnant un

confined fluid f − function f1 ... Ergun coefficient for laminar flow f2 ... Ergun coefficient for turbulent flow kbed − relative contribution of stagnant bed conductivity

(=λ0/λf ) K1,h − Constant in high Reynolds heat dispersion coefficient

correlation K1,m − Constant in high Reynolds mass dispersion coeffi

cient correlation K2,h − Constant in high Reynolds heat dispersion coefficient

correlation K2,m − Constant in high Reynolds mass dispersion coeffi

cient correlation

183


Symbol units Variable � m coordinate along length of channel L m length of tube M kg/mol molecular mass N ... auxiliary variable in heat transfer equation by Zehner

(1970) p Pa (absolute) pressure r m space coordinate in radial direction for cylindrical

coordinates rp m particle radius R m radius of tube sv m2/m3 specific surface sv m2/m3 mean specific surface in a subvolume of the bed t s time T K temperature Tw K wall temperature w0 m/s superficial velocity x m spatial coordinate (perpendicular to the wall) z m spatial coordinate (in axial direction)

184

5.6 Conclusions

Greek Symbol units Variable αw,1D W/m2/K wall heat transfer coefficient, 1-dimensional models αw W/m2/K wall heat transfer coefficient 2-dimensional models β − dimensionless conduction length between two parti

cle centres according to Yagi and Kunii (1957) γ − parameter in heat transfer relation of Yagi and Kunii

(1957) ε − (local) porosity (volume open to flow / total volume) εb − bulk porosity, porosity far from any walls etc. εR − particle emissivity λeff,q W/m/K effective axial heat conductivity λeff,r W/m/K effective radial heat conductivity λf W/m/K heat conductivity of the fluid λs W/m/K heat conductivity of the solid λ0 W/m/K heat conductivity of the bed at stagnant conditions λR W/m/K heat conductivity contribution of radiation νf m2/s molecular kinematic viscosity of unconfined fluid νeff m2/s effective kinematic viscosity of fluid φ − dimensionless parameter describing the packing in

heat transfer relation of Yagi and Kunii (1957) φ [φ] generic intensive quantity (momentum, heat, concen

tration) Φ ”

φ [φ]/m2/s flux of quantity φ Φ ”

h W/m2 heat flux ρf kg/m3 density of fluid σ W/m2/K4 Stephan-Boltzmann constant (56.7051 × 10−9) τ − tortuosity (path length / distance) τa − tortuosity for paths in axial direction τa, r − tortuosity in axial direction, radial component τb − tortuosity at bulk conditions (far from the wall) τr − tortuosity of the bed for paths in radial direction

185


Dimensionless groups

Symbol definition Variable Nuw

Nuw,0

Pef

Pef,c

αwdp

λf

αw,0dp

λf w0dp

Df

w0,cdp

Df,c

Nusselt number (total heat transfer/diffusive heat transfer) at the wall Minimum value of Nusselt number at the wall

molecular Peclet number (convective transfer / diffusive transfer) molecular Peclet number at conditions at the centre of the bed

Pei,φ w0dp

Di,φ Particle Peclet number for direction i and quantity φ

Pe0 φ

Pe∞ i

Pr

w0dp

D0 φ

w0dp

D0 i

ν a

Stagnant particle Peclet number for quantity φ

Turbulent particle Peclet number for direction i

Prandtl number (hydrodynamic / heat transfer

Re0 w0dp

νf

boundary layer) Reynolds number (inertia forces / viscous forces)

186

5.6 Conclusions

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189


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Votruba, J., V. Hlavacek, M. Marek (1972), Packed bed axial conductivity, Chem. Eng. Sci. 27, 1845-1851

Westerterp, K.R., W. de Jong, G.H.W. van Benthem (1993), Comments on discrimination of three approaches to evaluate heat fluxes for wall-cooled fixed bed chemical reactors, Chem. Eng. Sci. 48(14), 2669-2670

Winterberg, M., E. Tsotsas (2000), Modelling of heat transport in beds packed with spherical particles for various bed geometries and/or thermal boundary conditions, Int. J. Therm. Sci 39, 556-570


Wijngaarden, R.J., K.R. Westerterp (1989), Do the effective heat conductivity and the heat transfer coefficient at the wall inside a packed bed depend on a chemical reaction? Weaknesses and applicability of current models, Chem. Eng. Sci. 44(8), 1653-1663

Wijngaarden, R.J., K.R. Westerterp (1992), The statistical character of packed-bed heat transport properties, Chem. Eng. Sci. 47(12), 3125-3129

Wijngaarden, R.J., K.R. Westerterp (1993), A heterogeneous model for heat transfer in packed beds, Chem.Eng.Sci. 48(7), 1273-1280

Yagi, S., D. Kunii (1957),A.I.Ch.E.J. 3, 373

Yagi, S., D. Kunii (1960), Studies on heat transfer near wall surface in packed beds, A.I.Ch.E. J. 6(1), 97-104

Zehner, P., E.U. Schlunder (1970), Warmeleitfahigkeit von Schuttungen bei massigen Temperaturen, Chemie-Ing.-Techn. 42(14), 933-941

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190

Chapter 6

Modelling of packed bed reactors using Computational Fluid Dynamics

Summary

In this chapter, the models developed in the previous chapters are implemented in a Computational Fluid Dynamics (CFD) code. The CFD tool used in this work is Comflow, which is a code that is developed as a tool for the process engineer, in contrast to many larger CFD systems that require a flow dynamics expert to be operated. For the current work, it was decided to create a new version of Comflow based on Dolfyn (www.dolfyn.net), which is an open source, single phase flow code. Dolfyn does not include the models needed to simulated packed bed reactors. Therefore, the code was extended with models for variable physical properties of ideal gas mixtures and the packed bed model including flow resistance, dispersion and heterogeneously catalysed chemical reactions.

The models used for the simulation of flow and reaction in the catalyst bed are those that were developed in the previous chapters. The porosity and the specific outer particle surface area of the packed bed are treated as constant (but not uniform) fields to enable the calculation of flow resistance, dispersion and chemical reaction rates locally in the bed.

The equations used to evaluate gas mixture properties are generally those advised by Reid et.al. (1987). The density is calculated according to the ideal gas law, with local composition, temperature and pressure. It is usual in CFD practice to treat the fluid as incompressible when the velocities are low (well below the speed of sound). It is shown that this is only true for systems where the mechanical energy is (mainly) conserved. For highly dissipative systems

191

6. Modelling of packed bed reactors using CFD

like packed beds, the density needs to be calculated as a function of local pressure.

The dispersion coefficients are calculated according to the theory developed in chapter 5. In the standard CFD codes, dispersion is usually treated as an isotropic phenomenon, while in a packed bed it depends on the direction. In the direction of the flow there is axial dispersion and perpendicular to the flow there is radial dispersion, and the dispersion coefficients for these directions differ. Relations are developed to calculate the dispersion coefficient for each face of a cell, taking into account the orientation of each face with respect to the local flow.

The model is completed with (Maxwell-Stefan based) relations to calculate the mass transfer limitation from the bulk of the fluid to the edge of the particle, and the chemical reaction model described in chapter 2.

The CFD code is verified by comparing simulations of simple tubular geometries with hand calculations. In addition, the sensitivity to the grid density is assessed; it is found that the number of radial cells needs to be more than about 50 to have sufficient resolution of the wall zone. The calculated radial velocity profiles are compared with literature data and the correspondence is found satisfactory. The heat transfer model is validated by comparison of the model result with experimental literature data for a steam-heated tube; the model fits the measurements as well as may be expected. Finally, a laboratory reactor for the catalytic oxidation of ethane is modelled; good correspondence is found between the axial and radial temperature and concentration profiles predicted by the model and those measured by Vortmeyer and Haidegger (1991).

To demonstrate the capabilities of the CFD code, two model reactions are simulated: the catalytic reduction of NO with ammonia and the catalytic oxidation of SO2. The deNOx reactor is usually placed in the exhaust of a process (e.g., a burner or an engine) where little pressure drop can be allowed. The performance of a tubular reactor is compared with that of a radial flow reactor. The pressure drop of the radial flow reactor is much lower than that of the simple tube, but there is some reduced conversion due to maldistribution of flow over the bed. As a first step, measures are taken to improve the flow distribution and increase the conversion. With these improvements, the radial flow reactor shows a conversion that is three times as high as the tubular reactor, at one third of the pressure drop. The SO2 oxidation reaction is an exothermic equilibrium reaction with a relatively complex dependence of the reaction rate on temperature and concentrations. Industrially, this process is usually performed in a staged packed bed reactor with interstage cooling. Two alternatives are simulated: a wall-cooled tubular reactor and a tube-in-tube

192

6.1 Introduction

design where the heat removed from the reaction zone is used to heat up feed stream. The results of the simulation of the tubular design are compared to a 1-dimensional literature model. It is believed that in this case, the CFD model gives better results than the ’simple’ model. The tube-in-tube design could be an interesting energy and space saving alternative for the conventional staged bed design. The performance of such a reactor would be hard to predict, let alone optimise, without the aid of a relatively complex CFD model, due to the strong coupling between the heat transfer from the reaction zone to the feed flow and the kinetics of the oxidation reaction.

The data generated by the simulations is relatively detailed, giving pressure, velocity, temperature and concentrations at every point of the grid. Therefore, it is quite a task to duplicate this resolution in experiments and measurements, and validation of the model on a detailed level in a complex simulation is not possible at this time. This is valid for this work in particular, but also more in general as the possibilities for detailed modelling rapidly improve. It is a challenge for future experimentalists to make similar improvements in measurement techniques for chemical reactors.

6.1 Introduction

In this chapter, the models developed in the previous chapters are implemented in a Computational Fluid Dynamics (CFD) code. The CFD package used in this work is Comflow, which is a code that is developed as a tool for the process engineer, in contrast with many larger CFD systems that require a flow dynamics expert to be operated. Although there are certainly cases where the flow in e.g. a chemical reactor is very critical and a highly accurate solution of the flow field is needed, in many (if not most) situations, a faster, less expensive and less accurate solution is just as useful. This is especially the case for internal flows that are often determined in large part by the flow channel geometry (as opposed to free streams around an object), and if there are obstructions in the flow (e.g., tube bundles or packed beds). Also, in many situations even the purpose of the simulation is to compare two cases, e.g., to see if a change in geometry is an improvement and the absolute quantitative accuracy is less important as long as the solution is qualitatively correct. For this purpose, Comflow uses only a subset of the general CFD functionality: 2-dimensional mostly hexagonal grids and only very simple turbulence modelling. On the other hand, there are comparatively extensive models for physical properties and flow restrictions, of which the packed bed module developed in this work is the most elaborate example.

193


Strictly speaking Comflow is just a pre- and postprocessor for an external CFD solver, although the solver needs to be extended with a library of routines to support the additional models used by Comflow. As a solver, Phoenics (http://www.cham.co.uk) has been used in the past, as well as a custom solver based on the (now historic) champion code (see e.g., Pun and Spalding, 1976).

For the current work, it was decided to create a new version of Comflow based on a more modern CFD solver. The requirements for the solver, that it should be lightweight and that the source code should be available, lead to the selection of Dolfyn (www.dolfyn.net). This is an open source, unstructured CFD code written in Fortran 90 and based on the numerical method described by Ferziger and Peric (2002). Dolfyn is a plain, single phase flow code and does not include many of the the models needed to simulated packed bed reactors. Therefore, the code was extended with models for variable physical properties of ideal gas mixtures and the packed bed model including flow resistance, dispersion and heterogeneously catalysed chemical reactions.

In this chapter, the CFD packed bed model will be described, validated and demonstrated. The basic flow equations solved by Dolfyn will be presented, followed by the equations used to calculate the physical properties of the gas mixture as a function of temperature, pressure and composition; these are based on the relations recommended by Reid, Prausnitz and Poling (1987) but some specific issues for CFD codes are discussed as well. Then the specific packed bed models for flow resistance, dispersion and chemical reaction and transport of heat and mass in the catalyst particle are described. The full details of these models are given in the previous chapters of this work, therefore in this chapter the focus is on the relation between the CFD code and the packed bed models.

The CFD packed bed code will be verified using a basic geometry that can be easily computed with hand calculations; the effect of the density of the computational grid is assessed. Subsequently the code will be validated by comparison to literature models and experimental data. Again, the underlying packed bed models were verified and validated in the previous chapters so the main focus in this chapter will be on the effect of combination of the effects in a 2-dimensional model.

Finally, the use of the code will be demonstrated for two relevant chemical processes: the selective catalytic reduction of nitric oxide by ammonia and the catalytic oxidation of SO2. For these reactions, packed tube designs will used to compare the packed bed CFD code with conventional 1-dimensional calculations. Subsequently, the CFD code will be used to simulate and improve a more advanced reactor concept for both demonstration processes.

194

∫ ∫ ∫ ∫

∫ ∫ ∫ ∫ ∑ ∑

6.2 Models

For this work, the Comflow preprocessor is used throughout, so only stationary calculations have been performed using 2D Cartesian grids. However, the same code can be used for 3D calculations and — with some restrictions1

— for transient calculations. Unstructured, arbitrary cell shape grids can be used in principle (as far as Dolfyn can handle them), although the interpolation routines for the bed packing profiles near the wall will be somewhat less accurate for non-cartesian grids.

6.2 Models

6.2.1 Basic flow and transport equations

The basic flow equations and the balance equations for concentrations and temperature are solved using the Dolfyn solver. Dolfyn is a plain, single phase flow code based on the numerical method described by Ferziger and Peric (2002). A short description of the method will be given below so that we are able to show where the packed bed model intervenes in the Dolfyn code. A full description of the flow model can be found in the original text.

The Dolfyn code uses a finite volume method to solve the flow equations. The basis of this method is an integral representation of the generic balance equation for a intensive quantity φ, for a finite (sub-) volume Ω bounded by the surface S:

∂ ρφdΩ + ρφ�v · �ndS = Γφ∇φ · �ndS + qφdΩ (6.1)

∂t Ω S S Ω

where �n is the normal of surface S. The terms are easily identified as the (transient) accumulation term, the convective term, the diffusive term with diffusivity Γ and a volumetric source term q. Note that each term in this equations has units [φ]kg/s.

The equations are solved on a colocated unstructured grid of volume elements made up of an arbitrary number of faces:

∂ ρφdΩ + ρφ�v · �ndS = Γφ∇φ · �ndS + qφdΩ (6.2)

∂t Ω Sk Sk Ωk k

1The enthalpy and material balances inside a catalyst particle are not solved in the time domain, but the (macroscopic) flow phenomena can be calculated as a function of time. This is valid provided that the transport processes on the particle scale are much faster than the macroscopic processes

195

∫

∫

∑ ∑


Fluxes of quantities through each face Sk are approximated using the midpoint rule, where the face value fk is calculated by linear interpolation.

fdS ≈ fkSk (6.3) Sk

and the volume integrals are approximated by the cell volume ΔΩ multiplied by the cell centre value qp:

qdΩ ≈ qP ΔΩ (6.4) Ω

so that for each cell with centre P and faces k

∂ (ρP φP ΔΩ)+ (ρkφk�vk · �nkSk) = (Γφ,k(∇φ)k · �nkSk)+qφ,P ΔΩ (6.5)

∂t k k

where φP is the value of φ at the cell centre, φk is the value at the centre of cell face k, �vk is the velocity vector at the face centre, �nk is the unit normal vector of face k, Sk is the surface area of face k and ΔΩ is the volume of the cell. Γφ,k is the (effective) diffusion coefficient for quantity φ at cell face k.

The face value φk is estimated by linear interpolation between the two bordering cells (designated P and N), where λk is the relative distance from cell centre P to face centre k on the line PN :

φk = λkφP + (1 − λk)φN (6.6)

The gradient at the face centre (∇φ)k is calculated from the cell centre gradients in the same way.

∇φk = λk(∇φ)P + (1 − λk)(∇φ)N (6.7)

The gradient at the cell centre is calculated from the cell centre value difference between the current cell centre and that of all neighbours by solving for (∇φ)P

the set of equations:

φN − φP = (∇φ)P · (�xN − �xP ) (6.8)

where there is one equation for each neighbour N . Since there are usually more than three neighbour cells and only three coordinate directions in ∇φ, the set is usually over-specified; the gradient is calculated by using a least squares method.

In addition, there are corrections for non-orthogonal cells (i.e., where the centre of the face between cell P and cell N does not lie on the line PN), but as in this study only orthogonal grids are used, these are not described here.

196

∑

∑

6.2 Models

6.2.2 Porosity, specific particle outer surface area and tortuosity

The porosity and the specific outer particle surface area of the packed bed are treated as constant (but not uniform) fields to enable the calculation of flow resistance, dispersion and chemical reaction rates locally in the bed.

The porosity is calculated at initialisation time from a particle centre distribution as a function of the distance to the closest wall, according to the equations developed in chapter 3 of this work. For simplicity it is assumed that cell faces and walls are aligned with the co-ordinate axes. Catalyst beds can have wall effects activated for one or more of their sides, since for sides where there is no solid wall and for symmetry boundaries, no wall effect is expected. The porosity of a single cell is calculated by a weighed average of the porosities expected for each of the distances to active walls alone. The weighing factor is the inverse of the wall distance, i.e.

εi/xi,pε = ∑ i (6.9)

i(1/xi,p)

where xi.p is the distance from the cell centre to the wall and i loops over all walls with wall effect enabled. The term εi is the cell porosity that would arise if only wall i would be active; this is calculated from the cell dimensions and centre location. For instance, for a wall on the low x (’west’) side of the cell, it is calculated by integration of the particle centre distribution between the west and east side of the cell (x = xi,w and x = xi,e), see Figure (6.1). This approach will ensure that there is a smooth transition from the porosity profile at one wall to the porosity profile at another (opposite or adjoining) wall. However, it should be kept in mind that the approximations are valid only in regions where walls are further apart than 5 particle diameters.

The particle outer surface area is calculated in an analogous way as the porosity profile from the particle centre distribution, i.e.

svi /xi,psv = ∑ i (6.10)

i(1/xi,p)

There are three different tortuosities that are needed to calculate the bed flow resistance and the effective dispersion coefficients. For the flow resistance we need the tortuosity of a flow path, i.e., in axial direction. This was shown in Chapter 5 to be

197


xW,p

xW,w

xW,e

xE,p

xE,w

xE,e

Figure 6.1: Different cell wall distances for calculation of porosity profiles

(1/ε − 1)τ3D,a = τw + (6.11)

(1/εb − 1)(τb − τw)

We also need the radial tortuosity, which is assumed to be equal to the bulk tortuosity in most of the bed, and decreases to 1.0 between 2 and 1 radii from the wall. The wall distance used here is the distance from the cell centre to the closest active wall. The tortuosity that needs to be used to calculate the radial dispersion coefficient is the radial (2D) tortuosity of flow paths in axial direction (τ2D,a). This tortuosity differs from the 3D axial tortuosity that is used for the flow resistance calculations in that only flow variations in radial direction (i.e., perpendicular to the wall) are taken into account; tangential (parallel) flow does not lead to additional dispersion. Near the wall, the flow will be parallel to the wall, which means that the tortuosity has a value of about 1. This will be the case at distances up to one half particle radius from the wall, where the velocity peak is located. Fluid in this region will flow in a zigzag manner around the particles, but the zigzags will be mainly in the tangential plane at constant wall distance. In contrast, at one particle radius

198

6.2 Models

from the wall, the fluid will flow primarily in radial direction and hardly in tangential direction, so the 2-D tortuosity will be approximately equal to the 3D tortuosity. Therefore, we will assume that the 2-D tortuosity will be one up to a distance of rp/3 from the wall, and then increase to the maximum value of the 3D tortuosity at one particle radius from the wall.

6.2.3 A note on discretisation

In a CFD simulation, the packed bed will be divided into a large number of computational cells. Since we use a pseudo-homogeneous approach, each cell will get average values for the porosity, tortuosity and other bed parameters. Strictly speaking, these average values are only valid if each cell contains many catalyst particles. However, in many cases, the cells may have a size that is comparable to the particle size or even smaller. This is still correct if the parameter and variable values do not change rapidly in any direction. Of course, we cannot expect to find details of the flow on the particle scale in the simulation, but the simulation will give an accurate estimate of the average flow field.

The actual flow field in a packed bed depends on the random position of the particles. Therefore, if we were able to measure the flow variables locally in the bed, the measurement values at each point would be influenced by the coincidental positions of the particles near that point. Each time we reshuffle the particles in the bed, we will get a (very) different measurement value at the same point. If we reshuffle the particles many times, and average the measurement values for each bed realisation, we can suppress the stochastic nature of the randomly packed bed and get a clearer picture of the actual process. The packed bed model predicts precisely these averaged values.

Near the wall, the situation is slightly different. There the cell size will usually be smaller than the particle size, at least perpendicular to the wall, to catch the flow pattern in the wall zone. Here the parameters and variables change rapidly within one particle diameter, so strictly we cannot use averaged values. However, since the flow is almost parallel to the wall, we can expect to get on average correct values over longer stretches of the wall. Of course, this is only true if we use the actual particle outer surface area and tortuosity near the wall, taking into account the wall effect.

For the calculation of chemical reaction in the particles, there is an additional complication. The calculation of the conversion in the catalyst particles is based on the assumption that the temperature and concentrations in the gas phase at the edge of the particle is constant. Far from the bed, this is

199


not problematic since the concentration and temperature gradients are usually not steep compared to the particle size. However, near the wall we have many particles that have concentration and temperature gradients along their surface. For these particles, the assumption of constant surface conditions are not valid and hence we strictly cannot use the calculation methods presented in chapter 2.

Luckily the error made is not as large as it may seem. To calculate the reaction rate in the bed, we calculate the effectiveness factor for a particle assuming it has a constant particle edge (or actually, hydrodynamic boundary layer edge) concentration that corresponds to the local values in the cell. Using the effectiveness factor, we can calculate the fluxes into the pellet per unit of pellet surface. The source term for the conservation balances in the bed is that number multiplied by the amount of pellet outer surface area in the cell, not the pellet volume in the cell. In this way, we will get approximately correct values e.g., near the wall. The error made is because the effectiveness factor used is based on the assumption of symmetric concentration profiles in the particle. In reality, if e.g. the concentration of a reactant is low on one side of a particle an high on another, the high side flux will increase compared to the uniform case since some reactant will diffuse through the particle to the low-concentraion side. Therefore, our model will slightly underestimate the concentration profiles caused by the wall effect.

6.2.4 Properties

In a chemical reactor, the temperature, pressure and composition can vary significantly from point to point. Therefore, to get accurate flow simulation results, the fluid properties should be calculated as a function of the local temperature, pressure and composition. Here the fluid properties are recalculated at the beginning of each outer iteration loop (i.e., with the variable values from the previous iteration). This leads to approximately (first order) correct results even though this approach may not be correct in a strict (mathematical) sense, because for instance gradients of transport coefficients are not taken into account. For the cases evaluated here, the accuracy is sufficient and it is certainly an improvement over the mathematically correct but physically incorrect constant property assumption. The equations used to evaluate properties are generally those advised by Reid et.al. (1987); these are reproduced here for sake of completeness.

200

∑

6.2 Models

Density

To calculate the density of the fluid as a function of the local composition, temperature and density, an equation of state is needed. For systems where the fluid is a liquid, the density can (usually) be taken independent of the pressure. For (real) gas flow systems, specific equations of state have been developed that can be used e.g., at elevated pressures; such powerful but complex models fall outside the scope of this work. For many processes that operate at intermediate pressure and temperature, the ideal gas law can be used with sufficient accuracy:

(xiMi) pρf = i (6.12)

RT

In CFD calculations, usually two flow regimes are discerned: compressible and incompressible. Compressible flow simulations take into account the effect of the pressure on the density, while in incompressible simulations, the density is not a function of pressure (but could be a function of temperature and composition); in the latter case a constant, reference pressure should be used in equation 6.12 instead of the local static pressure. The general rule is that for low velocity flows (Mach number much smaller than 1), incompressible flow can be assumed. For process engineers, this is somewhat counter-intuitive as considerable pressure loss usually is found in process equipment. Therefore, we will look into this issue at some more detail.

There are two mechanisms by which the static pressure can decrease, through dynamic effects (i.e., acceleration or stagnation of the flow) where mechanical energy is conserved and through dissipation (e.g., friction at walls) where mechanical energy is dissipated. The dynamic effects become important for high-speed (near sonic) flows. For isentropic flow, the conservation of mechanical energy leads to the following well-known relation between pressure and velocity (e.g., Shapiro, 1953):

p p0

=

[

1 + γ − 1

2 M2

] −γ γ−1

(6.13)

where M is the Mach number, p0 the total (stagnation) pressure and γ the specific heat ratio. For most gases, γ has a value between 1 and 1.6. For low velocities, the energy balance simplifies to the Bernouilli equation

p 0 = p +1 ρv2 (6.14)

2

201

[ ]


The relation between the velocity (Mach number) and the gas density for an isentropic ideal gas can be written as:

−1

ρ γ − 1 γ−1

= 1 + M2 (6.15)0ρ 2

For M < 1 the influence of γ is small. The dependence of the density on the Mach number is shown in Figure 6.2. It can be seen that for Mach numbers below 0.3, the decrease in density is less than 5%. Note that the error made in neglecting the effect of static pressure on the density will lead to changes in the velocity field; the mass balance will be kept for each cell. Therefore, for low speed flows (less than 100 m/s for atmospheric air), the decrease in density due to dynamic effects can be neglected unless accuracies higher than 95% in the velocity field are desired. For process equipment, the gas flows rarely exceed values above 10-20 % of the speed of sound.

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

/

0

[-]

= 1.6

= 1.01

M [-]

Figure 6.2: Relative change in density as a function of Mach number for isentropic flow, for different values of γ.

However, this result alone does not mean that the gas density can be taken as independent of pressure. There will also be a pressure loss due to dissipation of kinetic energy (either in turbulent eddies or at the wall). This is especially important for flows found in the process industry, where gas passes through

202

6.2 Models

restrictions like for instance tube bundles or packed beds. Due to friction, mechanical energy is not conserved, but some is turned into thermal energy. Therefore, the total pressure will decrease (and the total temperature will rise). In non-isentropic, subsonic flows, the relative decrease in pressure can easily be much larger than the dynamic pressure effects. The decrease in pressure will lead to a decrease in density, which at the same mass flow will lead to an increase in velocity. The higher velocities will lead to different values of the mass, heat and momentum transport coefficients. As an example, figure 6.3 gives flow values for a simple packed bed in a tube where the variable density is taken into account. The density at the outlet is lower than that at the inlet due to the decrease in pressure and increase in temperature. The superposition principle for packed beds indicates that if the pressure drop is not small compared to the inlet pressure, the Ergun equation should be integrated over the height with variable density. In figure 6.4, the difference between a constant density and a variable density calculation is shown for the tube of figure 6.3; it shows that in this case the pressure drop increases by almost 10%. I can be concluded that the effect of non-isentropic pressure loss on the density of the fluid should be taken into account for flow simulations of process equipment in general and packed beds in particular.

pin = 1.6 bar Tin = 293 K

in = 1.9 kg/m3

vin = 1 m/s

= 1.8e-5 Pa s cp = 1005 J/kg/K

dp = 1 mm = 0.4

pout = 1 bar Tout = 335 K

out = 1.04 kg/m3

vout = 1.8 m/s

Figure 6.3: Pressure drop over a typical packed bed with variable density. Outlet pressure is assumed atmospheric, inlet pressure is adjusted to get 1 m/s superficial velocity. Note that this example is based on the traditional Ergun equation, integrated over the bed height.

203


0.0

0.5

1.0

1.5

2.0

p

[bar],

v

[m/s]

p constant density

p variable density

v

0.0 0.2 0.4 0.6 0.8 1.0

h/H [-]

Figure 6.4: Axial pressure and velocity profile for the typical packed bed of figure 6.3 with constant and variable density.

In the CFD code, the fact that the density is a function of local static pressure does not necessarily change the flow from incompressible to (completely) compressible. The influence of the pressure on the density is not as direct as in truly compressible flow. To prevent a strong coupling between the density and the pressure, which could lead to instabilities, the density field is updated only at the end of an iteration step and is kept constant during an iteration step. For higher velocity incompressible flows, it may be needed to base the density calculation on the local stagnation pressure instead of the static pressure to prevent instabilities, but for the flows considered here, the dynamic head is always much smaller than the total pressure so this would not make a difference.

204

(

6.2 Models

Viscosity

The pure component gas viscosity is estimated at a function of temperature using the approach of Lucas (as described in Reid et.al., 1987)

η = 0.807T 0.618 r − 0.357 exp [−0.449Tr]

+0.340 exp [−4.058Tr] + 0.018) F 0

p

ζ

(6.16)

with

ζ = 0.176

( Tc

M3p4 c

)1/6

(6.17)

μr = 52.46 μ2pc

T 2 r

(6.18)

and

⎧ ⎪ 1 0 ≤ μr < 0.022 ⎪

F 0 P =

⎪ ⎨

⎪ ⎪

1 + 30.55(0.292 − Zc)1.72

1 + 30.55(0.292 − Zc)1.72

0.022 ≤ μr < 0.075 (6.19) ⎪ ⎩ × |0.96 + 0.1(Tr − 0.7)| μr ≥ 0.075

where Tc K critical temperature pc bar critical pressure M kg/kmol molar mass μ D dipole moment

According to Reid et.al. (1987), the expected errors are between 0.5 and 1.5 % for non-polar compounds and 2-4 % for polar compounds. The correlation is not suitable for highly associated gases like acetic acid and is only valid at low pressure (well below the critical pressure). For flexibility, in the computer code the pure gas viscosity is calculated by partwise linear interpolation between tabulated values (see figure 6.5).

The viscosity of a gas is caused by mechanical energy transfer through collisions between gas molecules. In a gas mixture, these molecules can have different masses, while in a pure gas, only collisions between similar molecules occur. Therefore, the mixture viscosity can not be calculated by simply adding together the relative contributions of all species. Instead, the mixing rule of

205


5.0E-05

4.0E-05

3.0E-05

[Pa

s]

N2

NO NH3

H2O O2

2.0E-05

1.0E-05

0 200 300 400 500 600 700 800

T [K]

Figure 6.5: Dynamic viscosity of pure gases for some gas components, according to equation 6.16

Herning and Zipperer is used (as given in Reid et.al, 1987).

η = ∑

i

yiηi(T ) F (i)

(6.20)

√

F (i) = ∑

j

yj Mj

Mi (6.21)

Here yi is the mole fraction of component i, Mi is the molar mass in (kg/mol) and ηi(T ) the dynamic viscosity in (Pa s) at the local temperature of pure i. Note that the dynamic gas viscosity is a very weak function of the pressure but the kinematic viscosity is coupled with the local pressure through the density. The shear stress depends on the dynamic viscosity, and therefore the influence of the pressure on the viscous term in the Navier-Stokes equation will be small.

206

∑

6.2 Models

Thermal conductivity

The pure gas thermal conductivity can be estimated by the method of Chung, as described by Reid et.al. (1987)

λM 3.75Ψ = (6.22)

ηcv cv/R

where M is the molar mass (in kg/mol), R is the universal gas constant and cv is the heat capacity at constant volume. For an ideal gas

cp − cv = R (6.23)

The value of parameter Ψ is calculated from

0.215 + 0.28288α − 1.061β + 0.26665Z Ψ = 1 + α (6.24)

0.6366 + βZ + 1.061αβ cv 3

α = − (6.25)R 2

β = 0.7862 − 0.7109ω + 1.3168ω2 (6.26)

Z = 2.0 + 10.5Tr 2 (6.27)

As the thermal conductivity is estimated using viscosity data, the error in the former is usually larger than that made in the viscosity correlation; according to Reid et.al. it is about 5-7 % maximum for non-polar compounds. For flexibility, the temperature dependence is represented by a piecewise linear approximation in the computational code, see figure 6.6.

The heat conduction in a gas is caused by thermal energy transfer through collisions between gas molecules. Therefore, the molecular thermal conductivity of the gas mixture is calculated using an approach that is analogous to the viscosity mixing rule

yiλi(T )λf = (6.28)

F (i)i

where λi(T ) is the pure gas thermal conductivity of species i at the local temperature T (in W/m/K).

Heat capacity

Pure component heat capacity at constant pressure is estimated using third order polynomial interpolation functions from the physical properties database in Reid et.al. (1987). These functions are very widely used, even though no accuracies or temperature boundaries are given by the authors. For flexibility,

207

∑


0

0.02

0.04

0.06

0.08

0.10

0.12 [W/m/K]

N2

NO NH3

H2O O2

200 300 400 500 600 700 800

T [K]

Figure 6.6: Thermal conductivity of pure gas species as a function of temperature for several gas components

the curves are represented by piecewise linear approximation in the computation code, see figure 6.7.

The mixture heat capacity is calculated by simply adding the contributions of all components.

cp = yicp.i (6.29) i

where cp,i is the heat capacity in J/mol/K of pure species i. There is a complication when using a composition dependent heat capacity

in CFD simulations. The general transport equation (6.1) for scalar quantities needs to be modified slightly when it is written for the enthalpy. This is caused by the fact that convective transport of heat is governed by the enthalpy of the flow, while the diffusion of heat (conduction) is governed by the temperature gradient dT/dx, not the enthalpy gradient dH/dx. If the heat capacity is not a function of temperature, the enthalpy and temperature are related by

H = cp(T − T0) (6.30)

208

6.2 Models

0

500

1000

1500

2000

2500

3000

3500c p

[J/kg/K]

N2

NO NH3

H2O O2

200 300 400 500 600 700 800

T [K]

Figure 6.7: Piecewise linear interpolation scheme of the pure species heat capacity of several gas components

where T0 is a chosen reference temperature. If we take T0 = 0, the (stationary) enthalpy balance can be stated in the standard way (φ = cpT in equation 6.1):

∫ ∫ ∫ ρ(cpT )�v · �ndS = ΓT ∇(cpT ) · �ndS + qT dΩ (6.31)

S S Ω

If there is a gradient in the heat capacity (due to a change in gas composition), the gradient of cpT in the first right hand term can be nonzero even if the temperature is constant. Therefore, heat could be conducted from a high cp

zone to a low cp zone and a temperature gradient would appear spontaneously, which is in violation of the second law of thermodynamics. It is obvious that it is no solution to choose φ = T , since that would introduce a similar error in the convective term.

Another complication occurs when the heat capacity is a function of the temperature. In this case equation (6.30) is not valid, and the definition of the

209


specific heat should be used to relate enthalpy and temperature: ∫ T

H(T ) = cp(T )dT (6.32) T0

0

0.4

0.8

1.2

1.6

2

H, c

p T

[M

J/kg

]

N2 N2

H2O H2O

NH3 NH3

Hcp T

200 300 400 500 600 700 800 900 T [K]

Figure 6.8: Comparison of the enthalpy calculated using the integral (6.32) and approximation (6.30) for some gas compounds.

The difference between the simple approach (equation 6.30) and equation 6.32 is shown in figure 6.8. For many gas compounds (in this example oxygen, nitrogen, nitrous oxide), the heat capacity is a weak function of temperature, and the approximation is quite satisfactory. However, for other compounds (e.g., water and ammonia), the difference is larger. As can be seen in figure 6.8, the error in the enthalpy can be quite large (about 20%). If one would calculate the temperature from an estimated enthalpy for ammonia at the higher range of the graph, the error in the temperature value would be around 80 K.

Of course, figure 6.8 was calculated with a reference temperature of 273 K. The error that is made can be reduced significantly if the reference temperature is taken at an average of the temperatures found in the flow. For instance, for

210

∣ ∣ ∣ ∣ ∣ ∣

∫ ∫ ∫

∑ ∑

6.2 Models

ammonia if the temperature of the fluid is between 273 and 873 K and T0 is set to 573 K, the error is reduced by about a factor 2 compared with the case above, where T0 = 273K.

It can be concluded that the approximate equation (6.30) can be used for flow problems where the temperature differences in the domain are not too large and where the temperature dependence of the specific heat of the main gas compounds is not too strong, i.e.,

∣ cp(T0)(Tmax − Tmin) ∣ ∣ 1 − ∫ ∣ < 5% (6.33)Tmax cp(T )dTTmin

Here a fairly large error of 5 % is proposed since usually the specific heat data calculated from engineering correlations will not be much more accurate.

The correct steady state enthalpy balance can be written as:

ρH�v · �ndS = ΓT cp∇T · �ndS + qT dΩ (6.34) S S Ω

in discrete form:

(ρkHk�vk · �nkSk) = (ΓT,kcp,k∇Tk · �nkSk) + qT,P ΔΩ (6.35) k k

To solve this equation, we can either use the enthalpy as a variable and calculate the temperature when the coefficients are needed, or vice versa. Since it is easier to calculate the enthalpy if the temperature is known than the other way around, the temperature is chosen as a field variable.

The face value of the enthalpy Hk is estimated by linear interpolation between the two bordering cells:

Hk = λkHP + (1 − λk)HN (6.36)

or, when the approximation for the enthalpy is used,

Hk = λkcp,P (TP − T0) + (1 − λk)cp,N (TN − T0)

= λkcp,P TP + (1 − λk)cp,N TN − cp,kT0 (6.37)

The temperature gradient at the face centre ∇Tk is calculated from the cell centre temperature gradients in the normal way:

∇Tk = λk∇TP + (1 − λk)∇TN (6.38)

The temperature gradient at the cell centre is calculated from the temperature difference between the current cell centre and that of all neighbours by solving

TN − TP = (∇T )P · (�xN − �xP ) (6.39)

211

∑ ∑ ( ) ( )

√

∑


Of course, we can divide everything by the reference heat capacity cp,0,

ρkHk cp,k qT,P �vk · �nkSk = ΓT,k ∇Tk · �nkSk + ΔΩ (6.40)

k cp,0

k cp,0 cp,0

Hence, compared to the constant cp case, we need a modified face centre temperature

λkHP + (1 − λk)HNT ∗

k = (6.41) cp,0

and a modified diffusivity cp,k

Γ ∗ = (6.42)T,k ΓT,k cp,0

Diffusion coefficients

The binary diffusion coefficients for all combinations of gas species are calculated using the semi-empirical method of Fuller (Reid et.al (1987))

4.522 × 10−8T 1.75

DAB = ( )2 (6.43) ¯ 1/3 1/3

p MAB VA + VB

2MAB = (6.44)

1/MA + 1/MB

where

DAB m2/s Binary diffusion coefficient of A in B (or vice versa) T K Local temperature p Pa Local pressure Vi m3/mol Molecular diffusion volume of compound i MAB kg/mol Average molar mass MA kg/mol Molar mass

The diffusion coefficient of each component in the mixture is then calculated according to Blancs law (Reid et al (1987))

1 yj= (6.45)

Di,m j Dij

This is only accurate for diffusion of a dilute component in a homogeneous mixture, but there seems to be no better simple way to estimate gas mixture diffusion coefficients. The most correct way would be to implement the Stefan-Maxwell diffusion equation (instead of Ficks law) in the CFD code, but that would increase the complexity and computational intensity of the calculation considerably, while the accuracy of the flow solution would not increase dramatically in any but a few exceptional cases.

212

∫ ∫ ∫ ∫

[ ]

( )

6.2 Models

6.2.5 Flow resistance

The flow resistance of packed beds is computed according to the equations developed in Chapter 4 and is introduced in the flow equations as a momentum source term. In general the flow resistance is given by an equation of the form

dp = Av + Bv|v| (6.46)

dh

where A and B depend on local porosity, tortuosity, specific surface, density and viscosity. The force exerted by the fluid on the particles (and vice versa) in a subvolume ΔΩ of the bed with cross sectional area A⊥ and length h is therefore

Fd = A⊥h (Av + B|v|v) = ΔΩ (Av + B|v|v) (6.47)

The stationary momentum equation has the form (in direction i)

ρvi�v · �ndS = τij�ij · �ndS − p�i · �ndS + qvi dΩ (6.48)

S S S Ω

where each term has the units of force. After discretisation, the source term has the form

kgQφ = Sφ + Sφ

′ φ [φ] (6.49) s

where the first term contains the explicit part of the source term and the second term the implicit part. The friction term is written as an implicit source:

Qui = (A + B|vk−1|) vk (6.50)

where vk−1 is the velocity from the previous iteration and vk for the current iteration. The coefficients A and B are calculated from

2τ 2 3D,aμs2

vA = (6.51)

ε3

ktτ3 ρsv3D,a − τ3D,a

B = (6.52)ε3

Note that, although the flow resistance is introduced as a source (sink) term in the momentum equations, the influence of a homogeneous packed bed on the velocity field in e.g. a simple tube is small. The major influence of the resistance will be on the pressure field. The dissipated kinetic energy must be balanced by a transformation of potential energy.

The kinetic energy that is dissipated in the bed is of course turned into heat, so there should be a source term for the enthalpy balance equal to

Qh = vk 2 (A + B|vk−1|) ΔΩ (6.53)

213

∫

∫ ∫

∫ ∫

∫ ∫


6.2.6 Dispersion

The dispersion coefficients are calculated according to the theory developed in chapter 5. In the CFD code, the dispersion coefficients enter the equation in the first right hand term of equation 6.1. However, in the standard approach, dispersion is treated as an isotropic phenomenon, while in a packed bed it depends on the direction; in the direction of the flow there is axial dispersion and perpendicular to the flow there is radial dispersion, and the dispersion coefficients for these directions differ.

It should be noted that in the majority of packed bed reactor designs, axial dispersion is not an important factor and the solution of a computation would not be different if the axial dispersion coefficient would be set to zero or set to be equal to the radial dispersion coefficient. This would simplify the equations in this section rather drastically. However, there are cases (for instance at low flow rates or in low flow rate regions of the domain) where axial dispersion does become important.

The task at hand is to calculate an ’average’ dispersion coefficient for the centre of face S of the control volume, for a given face average velocity vector and axial and radial dispersion coefficient.

The dispersion term in integral form for a face of a control volume is:

Γ∇φ · ¯ (6.54)ndS S

which can be written in Cartesian (global) coordinates as:

∂φ ∂φ ∂φ ndS = nx nzΓ∇φ · ¯ Γx + Γy ny + Γz dS (6.55)

S S ∂x ∂y ∂z

or, alternatively in the local coordinate system of face S:

∂φ ∂φ ∂φ Γ∇φ · ¯ Γ1 n1 + Γ2 n3dS (6.56)ndS = n2 + Γ3

S S ∂x1 ∂x2 ∂x3

where x1 is the coordinate parallel to n and x2 and x3 are parallel to face S. Since obviously n2 = 0 and n3 = 0 and n1 = 1 by definition,

∂φ Γ∇φ · ¯ Γ1 dSndS = (6.57)

S S ∂x1

Therefore, what we need to know is the dispersion coefficient parallel to n(normal to face S). This dispersion coefficient depends on the angle between the velocity vector and the face normal. If the velocity is normal to the face, the dispersion coefficient is equal to the axial dispersion coefficient; if the

214

√

√

√

6.2 Models

velocity if parallel to the face, the dispersion coefficient is the radial dispersion coefficient. Furthermore, the dispersion coefficient should always be between the values of the radial and axial dispersion coefficients. If Dr = Da, then the dispersion coefficient should be independent of the direction of the velocity.

Take velocity vector v. Then the axial dispersion vector will be:

vda = Da (6.58)|v|

Now define two vectors perpendicular to v: ⎛ ⎞ vy − vz ⎝ ⎠v⊥,1 = vz − vx (6.59) vx − vy ⎛ ⎞

vy(vx − vy) − vz(vz − vx) ⎝ ⎠v⊥,2 = v × v⊥,1 = vz(vy − vz) − vx(vx − vy) (6.60) vx(vz − vx) − vy(vy − vz)

Then we have two radial dispersion vectors,

v⊥,1 v⊥,2¯ ¯dr,1 = Dr dr,2 = Dr (6.61)|v⊥,1| |v⊥,2| and the dispersion in each coordinate direction can be calculated from the dispersion in the local coordinate system (da, d⊥,1d⊥,2): ⎛ ⎞ ⎛ ⎞ d2 + d2 + d2

a,x ⊥,1,x ⊥,2,xΓx ⎜ √ ⎟ ⎜ ⎟¯ ⎝Γ = Γy ⎠ = ⎜ d2 + d2 + d2 ⎟ (6.62)a,y ⊥,1,y ⊥,2,y ⎝ ⎠Γz d2 + d2 + d2

a,z ⊥,1,z ⊥,2,z

Some calculus shows that this is equal to: ⎛ ⎞ Av2 + D2

x r D2 √ − D2 ¯ ⎝ ⎠Γ = √ Avy

2 + Dr 2 A = a r (6.63)|v|2

+ D2Avz 2

r

6.2.7 Pellet scale models

Chemical reaction

Chemical reaction in the catalyst pellets give rise to source terms for component mass balance equations and the enthalpy balance in the CFD simulations. These sources are calculated based on the local porosity, specific outer particle

215


surface area, temperature, composition and gas velocity using the approach presented in Chapter 2.

The concentrations are calculated in terms of mass fractions. The source term qφ in equation 6.1 has units kg/m3/s × [φ]. Therefore for component i the source term is specified in kg[i]/m3/s.

Transfer of mass and heat between catalyst particles and fluid

For packed bed chemical reactors, the transfer of heat and mass from the fluid to the particles of the bed is an important mass transfer step. For strongly endothermic or exothermic reactions, the temperature of the particle can be quite different from the temperature of the fluid in its vicinity. Likewise, for fast reactions, there can be a difference between the concentration of reactants and products near the particle surface and in the surrounding fluid. This difference is caused by the fact that the fluid velocity at the particle edge is zero, so convective transport of heat and mass is limited in the neighbourhood of the particle; this can be seen as a boundary layer around the particle where heat and mass transfer is by molecular transport (diffusion, conduction) only. The rate of conversion in the bed can be determined by this transport — depending on the reaction and mass transfer parameters — so it has to be taken into account in a fixed bed reactor model (as was shown explicitly by Wijngaarden and Westerterp, 1989 and 1993).

Heat and mass transfer between the particle and the fluid is governed by the particle Sherwood (for mass transfer) and Nusselt (for heat transfer) dimensionless numbers:

Shp,φ = kp,φdp

Dm,φ Nup =

kp,hdp

λf (6.64)

where kfp,φ is the heat transfer coefficient between the fluid and the particle for quantity φ, dp the particle diameter and Dm,φ the molecular diffusion coefficient of component φ. The Sherwood and Nusselt numbers depend on the flow conditions and are usually determined from empirical correlations as a function of the particle Reynolds and Schmidt respectively Prandtl numbers.

Since the flow and the boundary layer around a spherical particle is disturbed by neighbouring particles, the Sherwood number for a particle in a packed or fluidised bed differs from that of a single sphere in an infinite fluid. However, for high porosities (ε → 1), the packed bed relation should be equal to the single-sphere relation:

11 lim Shp = 2.0 + 0.66Re 2 Sc3 (6.65)ε→1

216

ε

( ) √

6.2 Models

For more realistic packed bed porosities, the value of Sherwood at low Reynolds numbers will go to a finite value that is not necessarily equal to 2.

lim Shp = f(ε) lim f(ε) = 2 (6.66) Re→0 ε→1

where f is a function of the bed packing. Commonly used functions for the particle Sherwood number (e.g. Dixon,

1979 and 1988; Papageorgiou and Froment, 1995) are of the form

0.255 Shp = Re2/3Sc1/3 Re > 100 (6.67)

This equation has the disadvantage that it is not valid for lower Reynolds numbers and also does not conform to equation (6.65).

Gunn (1978) derived a relationship for the Sherwood number in packed beds based on a statistical model of the bed. The result of this model is that

1 − ε lim Shp = 2.36 (6.68)Re↓0 ε

It is clear that this result does not follow the limiting behaviour for high porosities. This may be caused by the fact that the packed bed model used in the derivation is not valid for highly porous systems. As a relation that is valid for a wider Reynolds range, Gunn postulates:

Shp =(7 − 10ε + 5ε2)(1 + 0.7Re0.2Sc1/3)p (6.69)

+ (1.33 − 2.4ε + 1.2ε2)Re0p.7Sc1/3 Re < 105

The polynomial dependency on the porosity is somewhat arbitrary, and is chosen such that as Reynolds goes to zero, equation 6.68 is approximately followed for common bed porosities in the neighbourhood of 0.4, while for porosities of 1.0, the single-sphere value is found. The limiting relations are:

lim Shp = 7 − 10ε + 5ε2 (6.70)Re↓0

lim Shp = 2 + 1.4Re0.2Sc1/3 + 0.13Re0.7Sc1/3 (6.71)ε→1

The latter relation is not equal to equation (6.65), but is very similar for Re < 2000.

Gnielinski (1988) gives:

Shp = (1 + 1.5(1 − ε)) Sh2 turb (6.72)2 + lam + Sh2

217

√


with

Shlam = 0.644Re1/2Sc2/3 (6.73)

0.037Re0.8Sc Shturb = (6.74)

1 + 2.443Re−0.1(Sc2/3 − 1)

The limiting relations are:

lim Shp = 2 + 3(1 − ε) (6.75) Re↓0

lim Shp = 2 + Sh2 turb (6.76)lam + Sh2

ε→1

The latter is very close to equation (6.65); less than 10% difference for Re < 2000. However, equation (6.72) can only be used for Sc > 1 (which is often the case since molecular diffusion coefficients are usually low compared to the fluid viscosity).

For heat transfer, the same relations are used as those for mass transfer, with the dimensionless numbers replaced by their heat transfer analogies (Sh → Nu, Sc → Pr). This analogy is valid as long as the heat transfer mechanism is the same as the mass transfer mechanism (i.e., diffusion and conduction through the boundary layer). At higher temperatures, heat will also be transported by radiation. However, as high temperature applications are almost always gas-solid systems, heat transfer between the particle and the fluid by radiation will be less important; most radiative heat transfer will be from one particle to its neighbour. This effect is already taken into account in the axial and radial heat dispersion terms.

The different models are compared in figure 6.9. As can be seen, the results for the models of Gunn and Gnielinski are very similar (for Re > 10), so the choice between the two is somewhat arbitrary. The former more closely corresponds to the high-Reynolds model used by Dixon (1988). The model of Gunn is chosen here because of its somewhat simpler formulation.

To calculate the mass transfer between the solid particle and the bulk of the fluid, often the assumption is made that the transport term for different components φ are independent. For instance, Papageorgiou and Froment (1995) write

Φm,i = kpsv(ca,i − cb,i) = (1 − ε)Reff,i (6.77)

where Φm,i is the mass flux of component i (mol/m3/s), kp is the mass transfer coefficient (m/s), sv the specific particle surface area (m2/m3), cs,i the concentration of component i at the particle edge (mol/m3), cb,i the bulk concentration (mol/m3) and Ri,eff the effective reaction rate per volume catalyst

218

6.2 Models

1000

Gunn (1978) Gnielinski (1988) Dixon (1988) Single sphere 100

10

1

0.1

Re [-]

Figure 6.9: Different literature models for the particle Sherwood number (or Nusselt number for heat transfer) as a function of the particle Reynolds number for a packed bed porosity of 0.4 and Sc(= Pr) = 10.

material (mol/m3/s). However, this approach leads to problems for multicomponent reaction systems. As an example, consider the following simple isothermal oligomerisation reaction

νA → B

Let us assume that the fluid is a gas that consist solely of A, B and inert material N. Since the pressure gradient over the boundary layer is negligible, it follows for the partial pressures of A and B:

p − pN = pb,A + pb,B = ps,A + ps,B ⇒ cb,A − cs,A = cs,B − cb,B (6.78)

In steady state, the mass of A that is consumed and therefore transported to the particle is equal to the mass of B that is produced. Hence

Φm,A = −νΦm,B (6.79)

Sh

[-], Nu

[-]

0.1 1 10 100 1000 10000

219

∑

∑

∑


But also, according to 6.77

Φm,A = kp,Asv(cs,A − cb,A) Φm,B = kp,Bsv(cs,B − cb,B) (6.80)

Hence, the mass transfer coefficients for A and B must be related by:

kp,A = νkp,B (6.81)

This is unlikely as the mass transfer coefficients can be calculated independently from (for instance) equation 6.69. In reality, the stream of molecules A will cause a drift flux toward the particle that will hinder molecules B to diffuse outward. In this simple system, it is quite easy to correct for the drift flux, but for non-isothermal multi-component reaction systems, another approach must be taken.

A more fundamental way to approach the mass transfer problem is by the Maxwell-Stephan equation:

1 dxi vj − vi = (6.82)

xi dz j

xj Di,j

where xi is the mole fraction of component i, z is the distance into the boundary layer, vi is the velocity of component i (equal to the flux divided by the concentration) and Di,j the binary Maxwell-Stephan diffusivity of components i and j. This equation states that the driving force (the chemical potential gradient on the left hand side) is in equilibrium with the resistance that molecules i feel from all other components in the boundary layer. The equation is somewhat cumbersome to solve, and since it is only a small step in a complete packed bed reactor model, a simplified approach (Wesselingh and Krishna, 1990) is followed here. The differential equation (6.82) is approximated by its difference form:

Δxi vj − vi = (6.83)

xi j

xj ki,j

Here ki,j is the binary mass transfer coefficient for species i and j; the velocities vi and vj are calculated at the mean boundary layer concentration and temperature. This equation is accurate enough for most engineering applications, especially in the light of the uncertainties in mass transfer and other parameters in the model, while retaining the basic features of the full differential equation. Non-equimolar diffusion will also have an effect on the heat transfer through the boundary layer, causing a thermal drift:

Φ ” Φ ” Hi (6.84)h = −kp,hΔT + i i

220

6.3 Model verification and validation

¯where Hi is the enthalpy of species i. Equations (6.83) and (6.84) can be solved simultaneously with the diffusion-reaction equations inside the catalyst particles.


The code that is validated is a combination of the Dolfyn flow solver and the models for packed bed reactors developed in this work. Since the flow regime and geometry are relatively simple (low velocity, mostly cylindrical geometry, rectangular grid), the accuracy of the flow solution itself is quite sufficient (much higher than the accuracy of the catalyst bed model and the accuracy in which parameter like bed porosity are known). Therefore, simulations done with the complete CFD model (Dolfyn and packed bed code) can be used to verify or validate the packed bed code.

The verification of the model focuses on the correctness of the results in numerical sense, for flow in a simple cylindrical packed bed. The validation of the sub-models (porosity profiles, velocity profiles, dispersion of mass and heat) has been done in the previous chapters. Here this will repeated for the full 2-dimensional CFD model, comparing results to literature data for steam heated tubes (flow and heat transfer) and a laboratory-scale ethane oxidation reactor (flow, mass and heat transfer).

6.3.1 Flow in a packed tube

To establish a base case, the flow resistance in a cylindrical, isothermal tube was calculated using the CFD model as well as a hand calculation using the classical (differential) Ergun equation. The pressure at the outlet of the tube was taken as atmospheric and the inlet pressure was increased to set a given flow rate. A symmetrical boundary condition was applied at the wall (slip wall) and no radial particle distribution profiles were taken into account. Therefore, the geometry and boundary conditions closely approximate the assumptions of the hand calculation. The results are shown in Figure 6.10. It can be seen that, as expected, the CFD model and the hand calculation give very similar results in this case (the root mean square of the difference is about 1.6 %). The data deviates a little from a straight line in the log-log plot because of the effect of decreasing density with decreasing gas pressure.

For the one-dimensional case, the number of cells in the CFD model does not influence the results very strongly. There need to be enough cells in the axial direction to capture the pressure profile so that the density variation with pressure is accurately represented. Since there are no profiles in radial direction

221


Δp [

Pa]

10000

1000

100

10

1

0.1 1.0E+01 1.0E+02 1.0E+03 1.0E+04 1.0E+05

No wall Hand calculation

Re [-]

Figure 6.10: Comparison between a hand calculation based on the classical Ergun equation with variable density and a CFD model that is a close approximation of the assumptions of the hand calculation (see table 6.1)

Table 6.1: Calculation parameters for a one-dimensional isothermal tube number of cells in bed zone (x × y) number of iterations

70 × 30 2500

fluid wall condition porosity profile inlet condition

temperature N2

H2O NO NH3

outlet condition

ideal gas mixture symmetry none specified velocity and composition 20 ◦C 0.80 kg/kg 0.10 kg/kg 0.05 kg/kg 0.05 kg/kg absolute pressure 101325 Pa

temperature chemical reaction

constant none

222


Table 6.2: Calculation parameters for an isothermal tube number of cells in bed zone (x × y) number of iterations

70 × ny 2500

fluid ideal gas mixture wall condition no slip adiabatic porosity profile yes bed diameter 1 m

D/d 12.5 inlet condition specified velocity and composition

velocity 0.5 m/s temperature 20 ◦C N2 0.80 kg/kg H2O 0.10 kg/kg NO 0.05 kg/kg NH3 0.05 kg/kg

outlet condition absolute pressure 101325 Pa temperature chemical reaction

constant none

for any variable, the number of cells in radial direction does not influence the results.

When the wall effect is taken into account, the number of cells in radial direction becomes important because the radial profiles of porosity, tortuosity and specific area needs to be captured. Also, the velocity profile at the wall needs to be captured to get a good estimate of the wall friction. In order to increase the number of cells near the wall, a variable grid spacing is used according to:

[ ]f rj

= j

(6.85)R n

where ri is the radial position of division line j, R is the radius of the tube, n the number of divisions in radial direction and f the spacing factor. Here, a value of 1.25 is used for f .

A number of simulations were performed with different numbers of cells in radial direction for the conditions given in table 6.2. The radial velocity profile in the bed is shown as a function of the number of cells in the radial (y) direction is shown in figure 6.11, and the deviation from the last solution is shown in 6.12. It can be seen that in this case at least 40 cells are needed in radial direction to approach the final solution to 1-2 %.

223


0

0.2

0.4

0.6

0.8

1

1.2 v

[m/s

]

10 20 30 40 50 60

ny

0.2 0.25 0.3 0.35 0.4 0.45 0.5

r [m]

Figure 6.11: Radial velocity profiles in an isothermal packed tube for different radial mesh densities. Simulation parameters according to table 6.2.

The pressure drop over the bed is shown in figure 6.13 as a function of the number of radial cells. It can be concluded that with a radial cell count of 50, the radial profiles are well represented for this case (less than 1% difference in pressure drop). For higher tube to particle diameter ratio’s a higher resolution near the wall may be required to capture the radial profiles, but the relative effect of the wall zone on the flow in the tube becomes smaller for D/d > 12.5. Conversely, for lower ratios, the radial profiles are more easily captured in good resolution, but the effect of the wall zone on the flow in the tube is larger. In this chapter, we will look at beds with D/d < 12.5 and use ny = 50.

The calculations are performed using a first order upwind discretization scheme. The case with ny = 50 was recomputed using a second order central difference scheme. The velocity profile in the bed was nearly identical to the first order case (the root mean square of the difference was about 5e-5 m/s). The pressure drop over the bed was also very similar (183.5 versus 183.3 Pa). Therefore, it is shown that for macroscopic flow through packed beds like these, first order calculations can be used without significant loss of accuracy.

224


0

0.01

0.02

0.03

0.04

0 10 20 30 40 50 60

Dev

iati

on f

rom

fin

al s

olut

ion

[m/s

]

*

Number of radial cells

Figure 6.12: Mean root square of the deviation of the velocity profile and the final solution (60 radial cells) as a function of the number of radial cells. For the simulations with high ny (marked *), the number of axial cells (nx) in the bed was increased to 120 to improve cell shape.

The pressure drop over the bed described by table 6.2 with D/d = 12.5 and the same model with a larger particle size so that D/d = 5 is calculated at different inlet velocities. The results are compared with the earlier simulations without wall effect (which is the same as the result of the classical Ergun equation) in figure 6.14. Obviously, the pressure drop of the low bed-to-particle diameter ratio decreases drastically due to the bypassing of flow through the wall zone. In fact, the ratio between the pressure drop in a tube with fixed, finite D/d to that of an identical tube with no wall effect taken into account is constant (not a function of the flow rate), at least at higher flow rates.

The radial velocity profiles for the case of table 6.2 for different inlet velocities are shown in figure 6.15. The same profiles are compared to the 1dimensional simulations done in chapter 4 in figure 6.16, where they are normalised with the inlet velocity. It can be seen that for the lower flow cases (Re ≤ 100), the shape of the profiles changes with the flow rate; the maximum velocity peak becomes more pronounced. At higher flow rates (Re > 100),

225


*

100

120

140

160

180

200 Δp

[P

a]

-50

-40

-30

-20

-10

0

10

devi

atio

n fr

om f

inal

val

ue [

%]

0 20 40 60 80 100

ny [#]

Figure 6.13: Pressure drop over the packed bed as a function of the number of radial cells, simulations according to table 6.2. For the simulations with high ny (marked *), the number of axial cells (nx) in the bed was increased to 120 to improve cell shape.

the turbulent term is dominant and the shape of the profiles remains about the same. It can be seen that there is a good correspondence between the 1-dimensional model and the 2-dimensional CFD model. The CFD results are compared to the same literature points used for the 1-dimensional model in chapter 4 in figure 6.17. The correspondence of the model with the measured data is as good as may be expected; the oscillations in the velocity profile damp out at a somewhat smaller distance to the wall, which is inherited from the porosity profile used.

6.3.2 Heat transfer in a steam heated packed tube

In order to verify the transport of heat through dispersion, a steam heated packed tube is calculated. The geometry is basically the same as in the previous section, except that the wall is not taken as adiabatic but has a constant temperature of 100 ◦C. The temperature of the inlet stream is 20 ◦C. As

226


1000

800

600

Δp [

Pa]

400

200

0

Figure 6.14: Pressure drop as a function of inlet velocity for the isothermal packed bed of table 6.2 with different bed-to-particle diameter ratios.

Table 6.3: References and conditions for data points in figure (6.17).

D/d = 5

D/d = 12.5

D/d = μ

0 0.2 0.4 0.6 0.8 1 v0 [m/s]

symbol w0,m [m/s] dp [mm] Rep D/d note 1 2 3 4 5

0.5 0.25-1.5 0.25-1.5

0.5 1

8.0 268 4.5 76-450 9.8 165-987 7.0 235 9.8 658

6.3 11.1 5.1 7.1 5.1

a a a a a

6 7 8 9

0.004 0.076 0.102 0.526

8.6 4 8.6 77 8.6 103 8.6 532

9.30 9.30 9.30 9.30

b b b b

a) Bey and Eigenberger (1997). Fluid: air; velocity profiles measured downstream of a packed bed supported by a 3.5 mm long monolith to preserve the flow profile. b) Giese et al. (1998). Fluid: special liquid mixture (viscosity 8.5 × 10−6m2/s); velocity measured inside bed by Laser-Doppler velocimetry.

227


v [m

/s]

0.9

0

0.3

0.6

1.2

1.5

1.8

2.1

0.01 0.1

0.25 0.5

1.0

v0 [m/s]

0.2 0.25 0.3 0.35 0.4 0.45 0.5

r [m]

Figure 6.15: Velocity profiles for simulations based on table 6.2 with different inlet velocities.

the flow field is now no longer isothermal, the physical properties (density, viscosity, conductivity and specific heat) vary with position.

Grid density variation calculations were performed for beds with bed-toparticle diameter ratio of 5 and 12.5. The resulting temperature profiles are shown in figures 6.18 and 6.19. It can be seen that for a radial cell count of 50 or more, the temperature profiles are not influenced by the grid density.

To show the influence of the flow rate, calculations were performed at different inlet Reynolds numbers. The parameters for these calculations are shown in table 6.4.

As expected, it can be seen that the influence of the wall region increases with increasing Reynolds numbers. Figure (6.21) gives the radial thermal conductivity as a function of the distance to the wall and figure (6.22) gives the axial thermal conductivity. These figures show the transition from molecular conduction to conduction by convective mixing. It can be seen that for low Reynolds numbers, the solid conductivity (which is much higher than the conductivity of the stagnant fluid) is dominant, so the conductivity increases with

228

4 1D Re= 1000 1D Re= 10

2D Re= 4.0 2D Re= 39.6

3 2D Re= 395.9

v/v 0

[-]

2D Re= 989.9

2

1

0


Figure 6.16: Velocity profiles from 2-D CFD simulations compared to earlier 1-D simulations for different values of the Reynolds number

Table 6.4: Values of the parameters used for flow rate dependency calculations

0.2 0.25 0.3 0.35 0.4 0.45 0.5 r [m]

Quantity symbol value Bed diameter Bed length Particle diameter Bed-to-particle diameter ratio Bed bulk porosity Bed bulk tortuosity Axial wall tortuosity

D L dp

D/dp

εb

τb

τw

0.075 m 0.2 m 7.5 mm 10 0.38 1.41 1.1


λs

εR



M p Re0

air 0.028 kg/mol 105 Pa 0.1 - 1000

229


0 0 1 2 3 4 5 6 7 8 9

0.5

1

1.5.

2

2.5

3

3.5

4.0 v 0

/v m

[-]

1 2 3

4 5 6

7 8 9

Rep = 4

Rep = 39.6

Rep = 989.9

x/rp [-]

Figure 6.17: Velocity profiles from 2-D CFD simulations compared to literature data, see table 6.3.

decreasing porosity. For higher Reynolds numbers, the fluid conductivity (due to mixing) is dominant, so the conductivity increases with fluid velocity. As the velocity is (in general) higher where the porosity is higher, the conductivity plot is inverted: where there is a maximum conductivity for low flow (e.g., at 0.7 particle diameters from the wall), there is a minimum conductivity for high flow. Directly at the wall, radial mixing is absent, so the conductivity there is not influenced much by the Reynolds number.

To our knowledge, the radial profile of axial and radial conductivity has not been measured directly. This would require a large number of temperature measurements at a large number of radial positions in the packed bed. The temperature profiles that can be found in existing literature do not have a sufficient resolution to compute these profiles. It would be interesting to see if the profiles predicted by the packed bed model (and especially the predicted inversion of the peaks) could be reproduced experimentally.

230


100

ny

20

3080 50

100

60

40

20

Figure 6.18: Temperature profile for D/d=5, for different radial cell counts

T [

°C]

0 1 2 3 4 5 x/rp [-]

231


100

ny

20

3080

50

100

60

40

20

Figure 6.19: Temperature profile for D/d=12.5, for different radial cell counts

T [

°C]

0 1 2 3 4 5 6 7 8 9 10 11 12 13 x/rp [-]

232


96

97

98

99

100

T [

°C]

T [

°C]

20

40

60

80

100

Re=0.1 Re=1.0 Re=10 Re=100 Re=1000

0 1 2 3 4 5

x/rp [-]

Figure 6.20: Temperature profiles in a steam-heated tube for different inlet Reynolds numbers. Parameters for these calculations are given in table 6.4.

233


0

1

2

3

4

5

6

a R [

cm2 /

s]

a R [

cm2 /

s]

0

4

8

12

16

20

24 Re=0.1 Re=1.0 Re=10 Re=100 Re=1000

0 1 2 3 4 5 x/rp [-]

Figure 6.21: Radial thermal conductivity in a steam heated tube for different inlet Reynolds numbers. Parameters as given in table 6.4. Note that the lines for Re=0.1 and Re=1 almost coincide.

234


0

1

2

3

4

5

a A [

cm2 /

s]

a A [

cm2 /

s]0

100

200

300

400

500 Re=0.1 Re=1.0 Re=10 Re=100 Re=1000

0 1 2 3 4 5 x/rp [-]

Figure 6.22: Axial thermal conductivity in a steam heated tube for different inlet Reynolds numbers. Parameters as given in table 6.4.

235


To validate the results of the CFD model, it is compared to the so-called wall heat conduction (WHC) model developed in recent years by Winterberg et.al. (2000). The WHC model is presented as the new standard model to compute heat transfer between the wall and a packed bed, and is tuned to and validated with a broad range of literature data for different systems and boundary conditions in axisymmetric geometries (Winterberg and Tsotsas, 2000). The detailed dispersion model was already compared to the WHC model using a 2-dimensional, constant property model in Chapter 5. Here it is compared to the results of the full CFD solution. For this comparison the WHC model is also implemented in the CFD code.

Evolving from a lumped parameter model, the WHC model has some practical disadvantages when applied in a CFD code. For instance, the velocity in the centre of the bed is needed to calculate the effective radial conductivity in at any location in the cell. In a CFD code, this velocity usually not available at a cell level. More fundamentally, information that cannot be deduced from the state of a cell and its neighbours in a CFD simulation, but depends on cells that are far removed from that cell, can also in reality not have a direct influence on the cell itself. Therefore, all important parameters should depend only on local variable values. Therefore, the fact that in the WHC model thermal conductivity in each cell depends on the Peclet number in the centre of the bed cannot be physically correct. In fact, it may well be that the geometry of the bed is such that it is hard to determine where this ’centre’ is.

In the current implementation, as a workaround, the velocity in the centre of the bed is estimated in the CFD code at the start of each iteration by calculating the average velocity in the bed with the third power of the wall distance as weighing factor.

The detailed model gives similar results as the WHC model for steam heated tubes, as can be seen in figure 6.23, although for higher values of the Reynolds number the temperature in the centre of the bed seems to be underestimated somewhat by the detailed model. The behaviour of the CFD model is not different from the 2-dimensional finite difference model shown in the previous chapter. Again, it should be noted that the WHC model was tuned to the experimental data while the detailed model was derived and based on numerical simulation results for packed beds of identical spherical particles. Some differences could arise from the fact that the data set contains measurements for (somewhat) non-spherical and/or non-homogenous packings.

236


A comparison with a different set of experimental data is shown in figure 6.24. It can be seen that the fit in this case is quite good. The temperature jump in the near bed zone, which is caused by a combination of the flow profile near the wall and the radial effective conductivity profile, appears to correspond well to the measured profile. The temperature in the centre of the bed seems to be slightly underestimated.

20

30

40

50

60

70

80

90

100

WHC model

Detailed model

Measured values

T [

°C]

0 1 2 3 4 5 6 7 8

x/rp [-]

Figure 6.23: Comparison between the WHC model of Winterberg et.al. (2000) and the detailed model developed in this work for a steam heated tube, ceramic particles (λs = 1.2 W/m/K) and air (Re=480). The large symbols are measured values after Dixon (1988), as published by Winterberg et.al. (2000). Note that the WHC model was fitted to this data and the detailed model was not.

237


T [

°C]

120

100

80

60

40

20

0

Calculated

Measured

0 1 2 3 4 5 6 7 8 x/rp [-]

Figure 6.24: Results of the detailed model compared with experimental data given by Derckx and Dixon (1996) for a heated wall 2” tube, 1/4” ceramic spheres with air (Re=550, L=0.1175 m).

238

( ) ( )[

] ( )

{ ( )


6.3.3 Catalytic oxidation of ethane

The catalytic oxidation of ethane at intermediate temperatures is used by Vortmeyer and Haidegger (1991) to validate different approaches to heat transport calculations for wall-cooled reactions. The catalytic reaction has an acceptable rate at temperatures that can be achieved in laboratory conditions, and where the rate of the homogeneous reaction is negligible.

C2H6 + 3.5O2 → 2CO2 + 3H2O (6.86)

Vortmeyer and Haidegger give overall reaction rate equations for this reaction in a packed bed of 4 mm diameter, PdO coated, γ-alumina particles with a porosity ε0 of 40 %. The original article contains a number of printing errors, so the corrected equations are given here:

⎧ [ ] 9 n ⎪ −EA Tb1+ΔT1 y ⎪ kI exp fs T ≤ 597.36K ⎪ RT Tb1 yb ⎪ ⎪ ( )n ⎪ ⎨ kII exp (0.05407T ) y fs 597.36K > T > 626.03K rε0 = ( ) (

yb )⎪ 1000 1000 ⎪ kIII 3 − 3.62

2 ⎪ T T ⎪ ⎪ ( ) n ⎪ 3 ⎩ 1000 y+3.72 − 0.77 f T ≥ 626.03KT yb

s

(6.87)

where

−6.231 Tb2+ΔT2(0.024T − 13.63) T ≤ 658.75K

n = Tb2

1 T > 658.75K

ΔT1 = |Tb1 − T |ΔT2 = |Tb2 − T |Tb1 = 570.0 [K]

Tb2 = 581.3 [K]

kI = 3.0 × 106 [mol/m 3/s]

kII = 22.5 × 10−17 [mol/m 3/s]

kIII = 10.0 [mol/m 3/s]

EA = 94470 [J/mol]

yb = 0.002

fs = 1.067

239


The reaction rate at a given location in the bed with porosity ε is given by

1 − ε r(ε) = r(ε0) (6.88)

1 − ε0

The reactor consists of a 40 mm diameter tube, filled with 4 mm particles. The active length of the bed is 16 mm, but it is sandwiched in between two sections of inactive particles in order to get a well-established flow profile. The concentration of ethane at the inlet is 0.5 mass % in air. The physical properties of the packing were taken from the original publication. Two cases given by Vortmeyer and Haidegger were simulated in the detailed CFD model: at a mass flow rate of 0.2 and 0.25 kg/m2/s. This gives particle Reynolds numbers (at inlet conditions) of 25 and 32. The inlet temperature was set to 603 Kelvin and the wall temperature was 601 K. The calculated temperature field clearly shows the hot spot in the first part of the catalyst bed. The results of the calculations are compared to the measured data by Vortmeyer and Haidegger in figures 6.25 to 6.28.

580

600

620

640

660

680

700

720

740

T [

K]

Rep = 25 Measured (r=0)

Calculated (r=0)

Measured (r=R)

Calculated (r=R)

0 0.1 0.2 0.3 0.4 0.5 Axial coordinate [m]

Figure 6.25: Axial temperature profile in the centre and at the wall of a catalytic ethane oxidation reactor. CFD simulation results compared with measured data by Vortmeyer and Haidegger (1991) for a particle Reynolds number of 25.

It can be seen that the measured profiles are reproduced quite well; about as good as the model proposed by Vortmeyer and Haidegger and much better than the alternative models presented by them (notably the wall heat transfer model). The hot spot temperature seems to be slightly underestimated by our

240


0

1

2

3

4

5

6

0 0.1 0.2 0.3 0.4 0.5

Mas

s fr

acti

on E

than

e [‰

]

Measured (r=0)

Calculated (r=0)

Measured (r=R)

Calculated (r=R)

Rep = 25

Axial coordinate [m]

Figure 6.26: Axial ethane mass fraction profile in the centre and at the wall of a catalytic ethane oxidation reactor. CFD simulation results compared with measured data by Vortmeyer and Haidegger (1991) for a particle Reynolds number of 25.

580

600

620

640

660

680

700

720

740

T [

K]

Rep = 32 Measured (r=0)

Calculated (r=0)

Measured (r=R)

Calculated (r=R)


Figure 6.27: Axial temperature profile in the centre and at the wall of a catalytic ethane oxidation reactor. CFD simulation results compared with measured data by Vortmeyer and Haidegger (1991) for a particle Reynolds number of 32.

241


6

0

1

2

3

4

5

Mas

s fr

acti

on E

than

e [‰

] Rep = 32 Measured (r=0)

Calculated (r=0)

Measured (r=R)

Calculated (r=R)


Figure 6.28: Axial ethane mass fraction profile in the centre and at the wall of a catalytic ethane oxidation reactor. CFD simulation results compared with measured data by Vortmeyer and Haidegger (1991) for a particle Reynolds number of 32.

model. Especially for the higher flow case, the hot spot location is shifted one or two centimetres upstream compared to the measurement data. It might be that the axial dispersion is slightly overestimated by our model. However, it could also be that the difference is caused by a measurement error (for instance because of the influence of the measuring probe on the flow in the bed).

242

6.4 Demonstration

6.4 Demonstration

6.4.1 Reduction of NO in a packed tube

As a first model reaction, the selective catalytic reduction (SCR) of nitrogen oxide with ammonia is selected. This process is often used to remove NOx

from flue gas streams to reduce air pollution. A well-known catalyst for these reactions is vanadium oxide. The NO reduction reaction can be described as:

4NO + 4NH3 + O2 → 4N2 + 6H2O (6.89)

As ammonia is at least as harmful as nitric oxide, neither NH3 nor NO will be present in a large excess. Under these conditions, it is expected that the rate of reaction will depend both on the NO concentration and on the NH3

concentration. The following reaction rate equation is used:

−EA RT−rNO = k0e c n m (6.90)NOcNH3

with

3)1−n−m/sk0 0.5301 (mol/m

EA 12240 J/kg

n 0.5284 −

m 0.6354 −

In the flue gas of a gas engine, the NO concentration could be as high as 6000 ppm. For the model reaction a concentration of 0.5 mole% will be used, with a stoechiometric amount of ammonia, in air. The reaction takes place at temperatures of about 250 ◦C. Due to the relatively low reactant concentrations, the effect of the heat of reaction can be neglected. Therefore, the reactor will be isothermal (provided of course it is well insulated).

In general, the design of a tubular reactor (i.e., tube diameter, particle diameter) will be a trade-off between the pressure drop, the catalyst bed volume needed and heat transfer to or from the reactor. For an isothermal (or autothermal) tubular reactor, a low bed-to-particle diameter design will usually not be the optimal solution. If a high pressure drop is permitted, a small catalyst particle size will usually be chosen. On the other hand, if a low pressure drop is required, a large diameter bed with a short length becomes attractive. For very shallow beds, an equal distribution of the flow over the bed will become difficult and an alternative geometry (for instance the Radial Flow Reactor discussed below) may have advantages.

For demonstration purposes, we will design a (relatively) low pressure drop isothermal tubular packed bed reactor. The design of the model reactor is

243

√


based on the following considerations: the basic reactor design is a cylindrical fixed bed, packed with spherical catalyst particles. The flue gas flow is 525 Nm3/h, and the maximum pressure drop over the catalyst bed is 100 mbar. The amount of catalyst that is used to treat the flue gas is 70 kg and the catalyst particles have a density of 920 kg/m3 . The design that follows from these assumptions is summarised in table 6.5. The pressure drop requirement leads to an unusually large catalyst particle size, and therefore a low catalyst usage and low conversion. The model clearly demonstrates the influence of the wall on the packed bed performance in this case.

Due to the relatively simple design, the reactor performance can be predicted quite easily using traditional methods. We will do this here to compare the results with the CFD model. The pressure drop can be calculated from the Ergun equation, using the bulk porosity and gas properties at inlet conditions. This gives a pressure drop of about 5300 Pa. If we assume that the diffusion rates of nitric oxide and ammonia do not differ too much, we can simplify the rate equation to a simpler n-th order scheme:

(n+m)−rNO = kcNO (6.91)

For the simplified kinetics, the effectiveness factor can be estimated from the Thiele modulus, which can be calculated from (Fogler, 1986):

(n+m−1)dp kρpcNOφ = (6.92)2 Deff

As the catalyst particles are large, it can be expected the value of φ will also be ”large”, i.e., the reaction takes place in the diffusion limited region where the effectiveness factor is inversely proportional with the Thiele modulus., i.e.

3 η ≈ (6.93)

φ

Using this approach, the calculated conversion is 37.8 %.

For the CFD model, a two-dimensional axisymmetric geometry was used with 120 x 60 rectangular cells. The simulation was stopped when the residuals had decreased by four orders of scale, which was after about 2000 iterations. The results are compared to the results of the simple model in table 6.6.

In this case, the effectiveness factor is defined as the actual local rate of conversion of NO as a fraction of the rate of conversion that would be reached

244

6.4 Demonstration

Table 6.5: Values of the parameters used for deNOx reaction calculations Quantity symbol value Bed diameter Bed length Particle diameter Bed-to-particle diameter ratio Bed bulk porosity Bed bulk tortuosity Axial wall tortuosity

D L dp

D/dp

εb

τb

τw

0.3 m 1.1 m 30 mm 10 0.41 1.41 1.1

Fluid Molar mass Outlet pressure Inlet velocity

M p w0,in

air, 0.5 % NO, 0.5 % NH3

0.028 kg/mol 101325 Pa 4 m/s

Table 6.6: Overall results of isothermal tubular model reactor simulations Quantity ’simple’ model CFD model pressure drop conversion average effectiveness factor

5300 Pa 37.8 % 8.2 %

4500 Pa 34.7 % 8.3 %

if there were no (internal and external) mass transfer limitation, i.e., the concentration everywhere in the catalyst particle was equal to the local bulk gas concentration. This number depends on the local concentrations, flow rate, catalyst volume (porosity) and surface area,. Therefore, it depends on the location in the bed; both on the axial position and on the wall distance. The number given in the table above is a volume average over the entire bed. The effectiveness factor profiles 100 mm from the inlet side and 100 mm from the outlet side of the bed are shown in figure 6.29.

The concentrations of the components are also a function of both the axial and the radial position in the bed. Figure 6.30 shows the radial NO concentration profiles at different axial positions. The concentration increases slightly from the centre of the bed towards the wall; there is a steep jump at a distance of about a half particle radius from the wall. It shows that there is a quite well-defined wall zone where the velocity is relatively high and conversion low. The mass transfer between the wall zone and the bulk zone is limited because the low-porosity zone at one particle radius from the wall acts as a barrier. In this case, the distance over which a dramatic effect on the concentration can be seen (0.5rp) is much smaller than the distance over which the porosity profile is important (5 − 10rp). However, due to the cylindrical geometry, the 0.5rp thick wall zone still is over 10% of the bed volume. In addition, since the

245


velocity in the wall zone is high, about 15% of the air passes through the wall zone.

The local values of the NO concentration, NO conversion rate and effectiveness factor are shown in figure 6.31.

0

0.02

0.04

0.06

0.08

0.1

0.12

η [-

]

y=0.1

y=1.0

0 2 4 6 8 10 x/rp [-]

Figure 6.29: Radial profiles of the effectiveness factor for the deNOx reaction near the bed entrance and near the bed exit of at tubular reactor

As expected, the difference between the traditional approach and the CFD model is not very large in this case. The traditional approach was made as simple as possible in this case, which required a simplification of the reaction rate equation. Of course, the ’simple’ model could be extended, for instance to take into account the actual rate equation (e.g., by using the Aris number approach described in chapter 2 of this work) or external mass transfer limitations. In this approach it is up to the engineer to know when more details still need to be taken into account and when the model is complex enough to give results with a sufficient accuracy. In the CFD model, the majority of these effects is taken into account by default, whether they are important in a given case or not.

246

6.4 Demonstration

0.3

0.35

0.4

0.45

0.5

0.55

NO

con

cent

rati

on [

mas

s%]

y=0.1

y=0.4

y=0.7

y=1.0

0 1 2 3 4 5 6 7 8 9 10

x/rp [-]

Figure 6.30: Radial NO concentration profiles (mass fraction) in a tubular reactor at different axial positions in the catalyst bed

6.4.2 Reduction of NO in a radial flow reactor

With a CFD code, we are of course not limited to tubular geometries. As an example, figure 6.32 shows the geometry of a simple Radial Flow Reactor (RFR) design, using the same amount of the same catalyst particles as the tubular design. The flow enters axially through the inner channel, then flows radially outward through the packed bed that is fixed between two concentric gauze cylinders, and finally flows out in axial direction through the annulus between the outer gauze cylinder and the reactor wall. As a first guess, the RFR is designed so that the cross sectional area of the inner tube is equal to the cross sectional area of the annulus, so that the average inlet and outlet velocity are equal. The remaining parameters are the same as those given in table 6.5; the RFR contains the same amount of catalyst as the tubular design given above. The additional parameters for the RFR are given in table 6.7.

The resulting velocity field is shown in figure 6.33. The RFR design suffers from the fact that in the central channel as the velocity decreases, the pressure increases (approximately) according to Bernoulli’s law. In the annulus, the

247


5.30e-3 5.07e-3 4.84e-3 4.51e-3 4.38e-3 4.15e-3 3.92e-3 3.69e-3 3.46e-3 3.23e-3 3.00e-3

NO CONCENTRATION MASS FRACTION

EFFECTIVENESS

REACTION RATE [MOL/M3/S]

0.110 0.098 0.088 0.077 0.066 0.055 0.044 0.033 0.022 0.011 0.00

0.240 0.216 0.192 0.168 0.144 0.120 0.069 0.072 0.048 0.024 0.000

Flow

Figure 6.31: NO concentration, effectiveness and NO consumption rate in the catalyst bed in a tubular reactor. Axis of symmetry is on the lower side of each field, the inlet is on the left side, the outlet to the right.

Table 6.7: Additional radial flow reactor design parameters Quantity symbol value Inlet channel diameter Bed outer diameter Reactor diameter Bed height Inlet velocity

R0

R1

R2

H w0,in

0.15 m 0.25 m 0.29 m 0.6 m 4 m/s

pressure decreases with increasing velocity. As a result, the pressure across the bed increases in axial direction, causing a maldistribution of the flow over the catalyst bed (see figure 6.34). The remedy for this would be to change the

248

6.4 Demonstration

RFR design so that the inlet and outlet are on the same side (although this is somewhat more difficult to construct), or to design inserts in the central channel and annulus to create a more even pressure profile.

On a more detailed level, it is interesting to see that no boundary layer is built up along the gauze layer in the central channel. This is caused by the radial component of the flow, which pushes the boundary layer into the bed. In the annulus, the reverse is true: the radial flow streaming out of the bed causes an additional build-up of the boundary layer, forcing the main flow outward in the direction of the reactor wall. As the boundary layer does not contribute (much) to the axial flow, the effective annulus cross sectional area decreases and the maximum velocity increases. The geometry could be modified to correct for this.

The maldistribution is increased because the maximum value of the pressure drop across the bed, near the end of the central channel, is located at a wall in the bed, causing a relatively large velocity in the wall zone.

R0

R1

H

R2

Figure 6.32: Radial flow geometry for the deNOx reactor

Some overall performance parameters of the simple RFR design are given below.

249


Velocity [m/s]

6.0

5.0

4.0

3.0

2.0

1.0

0.0

NO [kg/kg] 5.0e-3

4.0e-3

3.0e-3

2.0e-3

1.0e-3

0.0e-0

Rate [mol/m3/s]

0.20

0.16

0.04

0.12

0.10

0.08

0.06

0.04

0.02

0.00

Effect. [-] 0.10

0.08

0.07

0.06

0.05

0.04

0.03

0.02

0.01

0.00

Figure 6.33: Velocity field, NO concentration, reaction rate and effectiveness factor for NO in the radial flow deNOx reactor

pressure drop 14.5 Pa conversion 31 % average effectiveness factor 8.6 %

If we compare these results to the packed tube design, it is clear that the RFR design shows only a small decrease in conversion (from 34.7 to 31 %), due to additional flow maldistribution, but at a fraction of the pressure drop required for the tube design (14.5 instead of 4500 Pa). Of course, the radial flow reactor is less compact then the tubular reactor; the volume of the RFR reactor is about twice the volume of the packed bed. However, if one were to construct a shallow packed bed with a larger diameter to decrease the pressure drop, a diffuser section would be required to distribute the flow evenly over the bed. In practice, a radial flow reactor is more compact than a shallow bed reactor with the same pressure drop. Figure 6.33 shows the field values for the NO concentration, effectiveness factor and source for the RFR design. Obviously, the design is far from optimal, because of the choice to use the same particles as in the tubular reactor of the previous section.

250

6.4 Demonstration

0

5

10

15

20

25

p [P

a]

central annulus

bed pressure drop

0.2 0.3 0.4 0.5 0.6 0.7 0.8

h [m]

Figure 6.34: Axial static pressure profile in the central channel and in the annulus of the RFR

Both the effectiveness factor and the pressure drop are low, so the particle diameter can be reduced to increase the conversion. A conical insert in the central channel can help improve the equal distribution of flow over the bed. As a third measure, the annular outlet channel can be enlarged to allow for the area taken by the additional boundary layer thickness. As an example, all three these modifications have been made to the model. The particle size was reduced to 1 mm; this increases the pressure drop to 960 Pa which is much higher but still low compared to the pressure drop of the tube reactor. The effectiveness factor increases to 90-95 %. The NO concentration field is given in figure 6.35. As can be seen, the conversion in the main part of the bed is about 90 %, but there is some slip of NO through the wall zone. The region in the outlet channel where the NO concentration is higher because of the slip through the bottom wall zone of the bed is relatively large, but the velocity in that region is low because of the (extended) boundary layer. Therefore the overall decrease in conversion because of the wall slip is only one percent point from 90 to 89 %.

Note that although the change in geometry of the axial channel and the

251


annulus certainly has influence on the flow for the original, low pressure drop bed, the impact is not so large for the smaller particle size bed as in that case the pressure drop over the bed is much larger than the pressure drop in these channels.

NO [kg/kg] 5.0e-3

4.0e-3

3.0e-3

2.0e-3

1.0e-3

0.0e-0

Figure 6.35: Concentration of NO in the modified design radial flow deNOx reactor

6.4.3 Non-isothermal chemical reaction in a packed tube: oxidation of SO2

The oxidation of SO2 is a well-known and industrially important process. The reaction of SO2 with oxygen is catalysed by vanadium oxide

V2O5

SO2 + 1/2O2 � SO3 (6.94)

The equilibrium reaction is relatively slow and exothermic. Therefore, in the design of a reactor, two counteracting mechanism are important: for a high reaction rate the temperature needs to be as high as possible, while for a complete conversion the temperature should be low. Below about 425 ◦C the

252

( ) [ ]

6.4 Demonstration

equilibrium lies almost completely toward the left side of equation 6.94. The conventional reactor design (e.g., Fogler, 1986) consists of three or four packed beds in series, with interstage cooling, see figure 6.36.

SO2

400

Temperature [°C]

550450 500 600

cooling

cooling

cooling

SO3

0 25 50 75 100 Conversion [%]

Figure 6.36: Typical layout, temperature and conversion profile of a conventional SO3 reactor (based on data given by Fogler, 1986)

The equilibrium constant for reaction (6.94) can be calculated from thermodynamic data (JANAF, 1971):

11810pSO3 = 105129/T −4.869Kp = 1/2

= exp − 11.21 (6.95) pSO2 pO2 eq

T

with T in K and Kp in bar−1/2 . The intrinsic reaction rate is given by Harris and Norman (1972) as

1/2 [ ]

pSO2 pO2 pSO3 rSO2 = ( )2 1 −

1/2 (6.96)

1/2 KppSO2 pA + BpO2 + CpSO2 + DpSO3 O2

253

[ ]

[ ]

[ ]

( ) [ ]


where Pi is the partial pressure locally at the active surface of a catalyst particle.

4960 A = exp − 6.8

T B = 0

7350 C = exp − + 10.32

T 6370

D = exp − 7.38 T

Although this is not usually done in the industrial practice, it would be possible to carry out the SO2 oxidation in a cooled tube reactor instead of the staged bed concept shown in figure 6.36. In order to remove the reaction heat efficiently, tubes with a relatively low diameter should be used. In addition, the diameter of the catalyst particles used is relatively large (8 mm is a common value). Therefore, the bed-over-particle diameter ratio will be quite small and the wall effect will be important.

The CFD approach developed in this work is compared with the conventional engineering method. The latter is taken from Fogler (1986), who uses the SO2 oxidation process as an example calculation in his text book. He uses an overall kinetic equation ascribed to Eklund, which gives the overall rate for a bed packed with cylindrical, vanadia loaded pumice particles with a diameter of 8 mm and a length of 8 mm. This overall kinetic equation includes effects of internal and external mass transfer limitation, and therefore can only be used for these specific particles and for a limited range of flow conditions. It was found that the overall kinetics could not be reproduced using the intrinsic kinetics give above. Apparently, the reaction rate for the vanadia catalyst used by Fogler depends on the temperature and concentrations in a different way than the vanadia catalyst described by Harris and Norman (1972).

To make an honest comparison between the CFD approach and the engineering approach, an overall kinetic equation was derived from the intrinsic kinetics given above for 8 mm spherical particles. The overall reaction rate is given as a function of conversion and particle edge temperature; this function was fitted to a large number of microbalance solutions. Although somewhat unconventional, the best fit is obtained using polynomial expressions. Since the gas composition is given in terms of the conversion, this model is only valid for the reactor inlet conditions chosen (see table 6.9).

The overal reaction rate is given by

a1ξ2 pSO3−rSO2 = + a2ξ + a3 1 −

1/2 (6.97)

KppSO2 pO2

254

6.4 Demonstration

where ξ is the conversion of SO2 and the coefficients ai depend on temperature through a third order polynomial (in K):

ai = bi,1T 3 + bi,2T 2 + bi,3T + ci,4 (6.98)

The values of the coefficients are given in table 6.8. With these coefficients, the overall kinetic equation describes the simulation results with a RMS error less than 1% for temperatures between 400 and 600 ◦C and for conversions from 0 to 90 %.

Table 6.8: Fitted coefficients bi,j for the overall kinetics for an 8 mm vanadium oxide particle (equation 6.97) according to the intrinsic kinetics given in equation (6.96).

j / i 1 2 3 1 2.316e-6 -1.715e-6 -0.4634e-6 2 -5.495e-3 4.385e-3 0.811e-3 3 4.293 3.666 -0.413 4 -1104 999.0 54.18

The overall kinetics equation (6.97) is substituted in the engineering model given by Fogler. The solution of this ’simpler’ model still involves the integration of three coupled differential equations, for the conversion, temperature and pressure along the length of the reactor tube, so some programming is required. As a one-dimensional model, the radial dispersion taken into account through an assumed value of the heat transfer coefficient between the fluid and the wall. This parameter should also include wall effects and therefore will depend on the tube to particle diameter ratio.

The CFD simulation takes into account the wall effect and calculates the internal diffusion limitation of the particles at local bed conditions. Therefore, the model can be used for other particle and reactor geometries. The parameters used for both simulations are shown in table 6.9.

The results of the CFD simulation are shown in figure (6.37) and compared to the results of the engineering model in figure (6.38). What can be seen is that the SO2 concentration drops steadily; the conversion at the end of the bed (l = 2.8m) is about 36 % for the CFD model and 40 % for the engineering model.

In the CFD results, the exothermic reaction leads to a hot spot in the tube, at 1 m from the entrance. The simple model predicts a – less well-defined – hot spot one meter further downstream. This is probably caused by the fact that the wall effect is not taken into account in the 1D model: due to the wall effect, the velocity in the centre of the bed decreases, so that the hot spot occurs at a more upstream location.

255


0.75

0.60

0.45

0.30

0.15

0.00

E [-]

1.60 1.44 1.28 1.12 0.96 0.80 0.64 0.48 0.32 0.16 0.00

S(SO2) [mol/m3/s]

0.200 0.192 0.185 0.178 0.170 0.162 0.155 0.148 0.140 0.132 0.125

SO2 [kg/kg]

476

458

441

424

407

390

T [°C]

Figure 6.37: Simulation results for the SO2 oxidation tubular reactor model. Fields for the SO2 mass fraction, the temperature, the reaction rate and catalyst effectiveness. The axial length is scaled by a factor 1/10 compared with the radial distance.

256

6.4 Demonstration

Table 6.9: Values of the parameters used for SO2 oxidation reaction calculations Quantity symbol value Inlet SO2 mole fraction Inlet SO3 mole fraction Inlet oxygen mole fraction

xSO2,0

xSO3,0

xO2,0

0.11 0.001 0.1

Bed diameter D 0.07 m Bed length L 2.8 m Particle diameter dp 8 mm Bed-to-particle diameter ratio D/dp 8.75 Bed bulk porosity εb 0.41 Bed bulk tortuosity τb 1.41 Axial wall tortuosity τw 1.1 Particle material ceramic Particle material density kg/m3 600 Particle conductivity λs 1.2 W/m/K Particle emissivity εR 0.85 Outlet pressure p 101325 Pa Inlet velocity w0,in 4 m/s Inlet temperature T0 393 ◦C Wall temperature Tw 400 ◦C

The maximum temperature in the bed is a useful value to know since it will be needed to predict e.g., the degradation of the catalyst material. The maximum temperature predicted by the 1-D model is close to the maximum average value from the CFD model (430 and 435 ◦C respectively). However, the maximum temperature at the hot spot in the bed is much higher than the average temperature: about 470 ◦C. It is obvious that a 1-dimensional model can not be used to predict the highest temperature in the bed with a high accuracy.

As Vortmeyer and Haidegger (1991) have shown, the location and magnitude of the hot spot depend strongly on the heat transfer model. The CFD model is a useful tool to estimate such parameters.

6.4.4 Oxidation of SO2: alternative reactor design

As an alternative design for the SO2 oxidation reactor, a tube-in-tube concept is used. This concept, a sketch of which is shown in figure 6.39, is well known especially for steam reformers. The reaction takes place in reaction tubes, that consist of an inner and an outer tube. In the reactor there are two tube

257


T1D TCFD average TCFD max. X1D XCFD

0 0.5 1 1.5 2 2.5 3

Length [m]

480

460

440

1

0.8

0.6

Con

vers

ion

[-]

T [

°C]

0.4420

0.2400

0380

Figure 6.38: Axial conversion (X) and temperature (T) profiles in a tubular, wall-cooled SO2oxidation reactor. CFD model results (maximum and flow-averaged values) compared to the results of the 1-dimensional model given by Fogler (1986) for the same geometry and flow.

plates. The inner tubes are connected to the top tube plate and the outer tubes are connected to the bottom tube plate. In this way, the SO2 rich feed that enters at a relatively low temperature at the top can flow into the centre tubes. The centre tubes are filled with a packed bed of inactive, but relatively good heat conducting pellets. At the bottom end of the tube the flow reverses and flows up through the annulus that is filled with catalyst pellets. Here the oxidation reaction takes place. The excess heat is taken up by the feed stream in the centre tubes, which is heated up to the desired reaction temperature. In order to get sufficient exchange of heat between the feed and the reaction zone, the reactor tube assemblies need to be small in diameter. Therefore, the wall effect will be important in the design of this reactor. To obtain the desired temperature profile in the reaction zone, that is needed to have both a sufficiently large reaction rate and a sufficient conversion at the outlet, the heat transfer to the feed flow needs to be engineered carefully. One of the advantages of such a design is that the thermal expansion of the reactor

258

6.4 Demonstration

tubes can be easily accommodated (only the differential axial expansion of the inner and outer tube needs to be allowed). If needed, additional cooling or heating could be applied through the space in between the tube assemblies (for instance at startup). Another advantage is that any heat taken up by the feed flow reduces both the need to preheat the feed and the need to cool down the reaction mixture as is done in the conventional process (figure 6.36). Of course, there are disadvantages (more expensive construction, possibly a higher pressure drop) and an technical/economic study should show whether this concept has a better overall performance over any other designs, but this is outside the scope of this work.

SO2

Temperature [°C] 400 450 500 550 600

SO3

0 25 50 75 100

Conversion [%]

Figure 6.39: Demonstration system reactor for SO2 oxidation with approximate desired conversion and temperature profiles.

For the demonstration system, a single reaction tube assembly is considered. The geometry and bed parameters for single tube assembly are summarised in table 6.10. The outer wall of the assembly is assumed to be thermally insulated. The inner tube wall is assumed to have a low heat resistance compared with the resistances of the bed-wall interfaces on both sides. The kinetics of the reaction are taken as before (equation 6.96). The computational

259


model is a 2-dimensional, axisymmetric section of the tube assembly. Due in part to the large difference between the radial and axial dimensions, the require numbers of computational cells is rather large. The coarse grid model consist of 295 × 44 cells, the fine grid model 530 × 90.

Table 6.10: Values of the parameters used for demonstration model calculations Quantity symbol value Inner tube Diameter Bed length Particle diameter Bed-to-particle diameter ratio Bed bulk porosity Particle material Particle conductivity

Di

L dp

D/dp

εb

λs

140 mm 5 m 2 mm 70 0.40 ceramic or stainless steel 20 W/m/K

Outer tube Diameter Annulus width Particle diameter Bed-to-particle diameter ratio Bed bulk porosity Particle material Particle conductivity

Do

da

dp

D/dp

εb

λs

100 mm 30 mm 2.5 mm 12 0.40 ceramic 3 W/m/K

Fluid (mass %) 79 % N2, 11 % O2, 10 % SO2

Mass flow Φm 0.462 kg/s Inlet temperature Tin 400 ◦C Outlet pressure p 101325 Pa

The pressure drop over the complete assembly is 2.5 bar, which is due to the fact that relatively small particles are used both in the central channel and in the catalyst bed. The pressure difference has an effect on the gas density (see figure 6.40), and through this on the velocities, so that even at the same temperature level, the inlet and outlet velocity will be different. Figure 6.42 shows the velocity profiles at different axial positions in the tube assembly.

The temperature field is shown in figure 6.41. It shows that the feed flow is heated up from 400 ◦C to 600 ◦C in the central tube. In the catalyst bed, the oxidation reaction rate at first is high (about 3 mol/m3/s), causing the temperature to rise to a maximum of 680 ◦C. As equilibrium is approached, the reaction rate decreases (to about 1 mol/m3/s) and also the temperature slowly decreases because heat is given off to the feed stream. The decrease in

260

6.4 Demonstration

catalyst

inert packing

DENSITY [kg/m3]

2.10 1.93 1.76 1.59 1.42 1.25 1.08 0.91 0.74 0.57 0.40

Figure 6.40: Sketch of the reaction tube assembly geometry (left) and the gas density profile (right). Note that the length of the tube has been scaled by a factor 1/10 for clarity.

temperature shifts the reaction towards the product side so the reaction rate does not drop off to zero and the conversion still increases.

Figure 6.41 shows the SO2 concentration in the bed, and figure 6.43 shows the conversion in the centre line of the catalyst bed. As can be seen, the conversion in this first attempt at a design is still too low (about 60 %). However, the conversion is still increasing almost linearly up to the end of the bed, so the conversion can be increased by increasing the bed length.

Also shown in figure 6.41 is the effectiveness factor for the oxidation reaction. This is quite high (nearly 100 % at the bottom half of the bed to about 60 % at the top), which indicates that it is possible to increase the particle diameter to reduce the pressure drop without having increase the total catalyst amount. Of course, this would also decrease the bed-to-particle diameter ratio and thereby increase the wall effect. For optimal conversion and pressure drop it may be advantageous to use large particles in the bottom part of the bed and (increasingly) smaller particles in the top part.

As a first optimisation step, the particle size is increased from 2.5 to 10 mm

261


TEMP SO3 RATE EFFECT. [C] [mol/m3] [mol/m3/s]

680 652 624 596 568 540 512 484 456 428 400

0.199 0.187 0.175 0.163 0.151 0.139 0.127 0.115 0.103 0.091 0.079

3.484 3.136 2.787 2.439 2.090 1.742 1.394 1.045 0.697 0.348 0.000

[-]

1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00

Figure 6.41: Profiles of the temperature, the SO2 concentration, reaction rate and the effectiveness factor for the tube assembly (see figure 6.40 for a sketch of the model geometry). Note that the length of the tube has been scaled by a factor 1/10 for clarity.

and the bed length is increased to 8.5 m. The amount of heat transferred to the unreacted gas in the inner tube will increase both because of the increase in length and because of the (desired) increase in conversion (i.e., heat released by the reaction). Therefore, the inlet temperature can be decreased; in this case from 400 to 250 ◦C. The resulting axial temperature and conversion profile is show in figure 6.44. The overall conversion has increased to nearly 70 %. Further simulations could be performed to find the optimal inlet temperature, tube and particle diameters but this is outside the scope of this demonstration.

262

6.4 Demonstration

6 x=1 x=24 x=3 x=42 x=6

0

-2

-4

-6

-8

-10

-12

Figure 6.42: Velocity profiles at different axial positions in the tube assembly for the coarse grid model. Positive velocity is downward.

Axi

al v

eloc

ity

[m/s

]

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 r [m]

263


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

SO2

conv

ersi

on [

-]

560

580

600

620

640

660

680

700

Tem

pera

ture

[°C

]

Conversion

Temperature

0 1 2 3 4 5 6 x [m]

Figure 6.43: Conversion of SO2 and the temperature in the centre line of the catalyst bed as a function of axial position.

264

6.4 Demonstration

SO2

conv

ersi

on [

-]

0.5

0.4 550

0.3

0.2 500

0.1

0 450 0 1 2 3 4 5 6 7 8 9

x [m]

0.8 650

0.7

0.6 600

Tem

pera

ture

[°C

]

Figure 6.44: Conversion of SO2 and the temperature in the centre line of the catalyst bed as a function of axial position, for the modified reactor.

265


6.5 Conclusions

An open source, unstructured, finite volume CFD code (Dolfyn) has been extended with the sub-models developed in previous chapters of this work so that it can be used to simulate packed bed reactors. This combined code has been verified, validated and its use to design and optimise packed bed chemical reactors has been demonstrated. Validation was based on experimental data known from literature, and therefore not very extensive and limited to cylindrical packed beds.

The models for the bed packing properties were developed in Chapter 3 of this work. Under the restriction that hexagonal, axis-aligned cells are used in the catalyst bed(s), the equations derived there are easily adapted to be implemented in a CFD code. In discretization of the profiles, care has been taken to ensure that the total mass of catalyst in the reactor does not depend on the chosen grid density.

Apart from the catalyst bed properties, it is also important to take into account the physical properties of the gas phase as a function of composition, temperature and pressure. Contrary to common CFD practice, the density cannot be taken independent of the pressure when packed bed reactors are simulated even though the velocities are well below the speed of sound. It has been shown that the (non-isentropic) pressure loss that takes place in packed beds can and should be taken into account when the density is calculated.

The specific heat of most gases is a relatively weak function of the temperature. Therefore, if the temperature differences in a model are not too large, for most gases it is allowed to approximate the enthalpy as cp(T − T0) instead of the integral over cpdT , provided T0 is chosen wisely. It is important to realise that the enthalpy balance differs from a general scalar balance since the enthalpy is the leading variable for convective transport while the temperature is the leading variable for diffusive transport (conduction). Failure to modify the balance for the temperature field in this sense can lead to solutions that violate the second law of thermodynamics.

The thermal conduction near the wall is influenced strongly by the wall effect. A striking feature of the effective thermal conductivity profiles is that with increasing Reynolds number, an inversion of the profiles is predicted. For laminar flow, the gas phase conductivity is (in general) lower than the solid phase conductivity, while for high Reynolds numbers, there is intense mixing in the gas phase and therefore it can have higher conductivity than the solid phase. Consequently, the situation can arise that at low Reynolds numbers the thermal conductivity is lowest for high porosity zones, while at high Reynolds numbers the conductivity is highest for high porosity zones.

266

6.5 Conclusions

The results of simulations of a steam-heated tube using the packed bed CFD model were compared to experimental data by Dixon (1988) as presented by Winterberg et.al. (2000) and of Derckx and Dixon (1996). The radial temperature profile was seen to correlate well with the measured data. Especially the temperature drop in the wall zone is accurately predicted by the packed bed model.

For further validation, a laboratory scale ethane oxidation reactor was modelled and the results were compared to measurements by Vortmeyer and Haidegger (1991). Again, the model results correspond very well to the measurements, without using any fitting parameters.

As a demonstration, a reactor for the catalytic reduction of NO by ammonia was modeled using the combined code. For a tubular reactor design, the model results are compared with hand calculations. For this specific design, the hand calculation underestimates the conversion by 10 % because it does not take into account hte wall effect.

The tubular design is compared to a radial flow reactor design using the same amount of the same particles. The results showed that the RFR has a much lower pressure drop (14.5 Pa instead of 4500 Pa) and only a little lower conversion (31 instead of 35 %). The CFD model shows that this decrease in conversion is mainly caused by flow maldistribution: the axial pressure gradient in the inlet and outlet channels is comparable to the pressure drop over the bed. It also shows that the effectiveness factor is low, so the particle size could be reduced to increase the conversion further, at the cost of some extra pressure drop.

A modified design for the RFR was made based on the results of the first simulation. An insert was placed in the inlet channel and the outlet channel was widened to make up for the boundary layer. The particle size was reduced to 1 mm. The new simulation shows a much improved flow pattern and an increase in conversion to nearly 90 %. Although the importance of the wall zone is decreased with the decreased particle size, some slip of reactants through the wall zone can still be observed.

In order to demonstrate the code for a industrially relevant and interesting case, an SO2 reactor was simulated. The SO2 oxidation takes place in a relatively slow equilibrium reaction, where at temperatures where the reaction is sufficiently fast, the equilibrium does not allow full conversion of SO2. First a cooled tube design was calculated and compared to results of a 1-dimensional model given by Fogler (1986). Due to the fact that the 1-dimensional model does not take into account the wall effect, the hot spot location is predicted further downstream by this model. It is believed that the CFD model in this case gives the more correct prediction. Furthermore, it is clear that a

267


1-dimensional model can not accurately predict the peak temperature in the reactor since there is a quite defined radial temperature profile.

As a more practical alternative, a tube-in-tube concept is simulated. The advantage of the tube-in-tube concept over conventional SO2 oxidation reactors would be that the reaction heat is removed by the feed flow, so both the need for cooling and for feed preheating is reduced. This type of reactor is difficult to design from scratch because of the strong two-way interaction between the preheating of the feed and the reaction. It is shown that a CFD code is a useful tool to model and optimise such a system.

It has been shown that a CFD tool has been developed which can be used successfully in the modelling for design and optimisation of packed bed chemical reactors with complex reactions, non-standard geometry and high interaction of flow and heat transfer. Several simpler cases have been simulated for which literature models were available and the accuracy of the code was found to be satisfactory. The data generated by the simulations is relatively detailed, giving average pressure, velocity, temperature and concentrations at every point of the grid. Therefore, it is quite a task to duplicate this resolution in experiments and measurements, and validation of the model on a detailed level in a complex simulation is not possible at this time. This is valid for this work in particular, but also more in general as the possibilities for detailed modelling rapidly improve. It is a challenge for future experimentalists to make similar improvements in measurement techniques for chemical reactors.

268

6.5 Conclusions

Nomenclature

Roman Symbol units Variable A Pa s/m2 Coefficient for ’laminar’ part of friction equation B Pa s2/m3 Coefficient for ’turbulent’ part of friction equation ci mol/m3 Concentration of component i cp J/kg/K Heat capacity (at constant pressure) cp J/kg/K Heat capacity (at constant volume) d, dp m Particle diameter di m Distance to wall i D m Bed diameter Da m2/s Axial dispersion coefficient DAB m2/s Binary diffusion coefficient of A in B Di,m m2/s Diffusion coefficient of compound i in a mixture Dr m2/s Radial dispersion coefficient EA J/mol Arrhenius activation energy h m Co-ordinate along length of bed Fd N Drag force H m Length H J/mol Enthalpy k0 ... Pre-exponential reaction rate constant Kp bar1/2 Equilibrium constant for SO2 oxidation reaction M − Mach number Mi kg/mol Molecular mass for component i �n m Normal vector p Pa Pressure pc Pa Critical pressure p0 Pa Total (stagnation) pressure pi bar Partial pressure of component i qφ [φ]kg/m3/s Source term for variable φ ri mol/m3/s Reaction rate for component i R J/kg/K Universal gas constant (8.3144 J/kg/K) sv m2/m3 Specific outer particle surface S m2 Surface

269


Symbol t T Tc

Tr

�v Vi

�x x

units s K K − m/s m3

m m

Variable Time Temperature Critical temperature Reduced temperature (= T/Tr) Velocity vector Molecular diffusion volume of compound i Position vector Distance from wall

Greek Symbol units Variable ε - Porosity (volume open to flow / total volume) εb - Bulk porosity, porosity far from any walls εR - Emissivity γ - Specific heat ratio Γ kg m/s Diffusivity coefficient φ ... Arbitrary intensive variable Φm kg/s Mass flow η Pa s Dynamic viscosity λ W/m/K Thermal conductivity λf W/m/K Thermal conductivity of fluid λk − Relative distance from cell centre to cell face k λs W/m/K Thermal conductivity of solid μ D Dipole moment ν m2/s Kinematic viscosity ρ kg/m3 Density ρ0 kg/m3 Stagnation density τ - Tortuosity (path length / distance) τa - Tortuosity for paths in axial direction τa, 2D - Tortuosity for paths in axial direction, radial compo

nent τb - Tortuosity of the bed at bulk conditions (far from

the wall) τr - Tortuosity of the bed for paths in radial direction

3Ω m Volume ξ - Conversion

270

6.5 Conclusions

Literature

Derckx, O.R., Dixon, A.G. (1996), Determination of the fixed bed wall heat transfer coefficient using computational fluid dynamics, Num. Heat Transfer part A: Applications, 29(8), 777-794

Dixon, A.G., Cresswell, D.L. (1979), Theoretical prediction of effective heat transfer parameters in packed beds, A.I.Ch.E. J. 25(4),663-676


Ferziger, J.H., Peric, M, Computational methods for fluid dynamics, 3rd Ed., Springer Verlag, 2002

Fogler, H.S., Elements of chemical reaction engineering, Prentice-Hall, 1986

Gnielinski, V. (1988), Warmeubertragung Partikel-Fluid in durchstromten Haufwerken, section Gh in VDI-Warmeatlas, 5. Auflage, VDI Verlag, Dusseldorf

Gunn, D.J. (1978), Transfer of heat or mass to particles in fixed and fluidised beds, Int. J. Heat Mass Transfer 21, 467-476

Harris, J.L., Norman, J.R., Ind. Eng. Chem. Process Des. Dev. 11 (1972), 564

JANAF Thermochemical tables, 2nd ed., D.R. Stull, H. Prophet, Project Directors, NSRDS-NBS 37, Washington D.C.: U.S. Government Printing Office, 1971), as quoted by Fogler (1986)

Papageourgiou, J.N., G.F. Froment (1995), Simulation models accounting for radial voidage profiles in fixed bed reactors, Chem. Eng. Sci. 50(19), 3043-3056

Pun, W.M., D.B. Spalding, A General Computer Program for Two-dimensional Elliptic Flows, HTS 76/2, CHAM Ltd. (1976), as described on http://www.simuserve.com/cfd-shop/hts76-2.htm

Reid, R.C. , Prausnitz, J.M., Poling, B.E., The properties of gases and liquids, McGraw-Hill, 1987

Shapiro, A.H. , The dynamics and thermodynamics of compressible fluid flow, The Ronald press company, new york, 1953

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Vortmeyer, D., Haidegger, E. (1991), Discrimination of three approaches to evaluate heat fluxes for wall-cooled fixed bed chemical reactors, Chem. Eng. Sci. 46(10), 2651-2660

Wesselingh, J.A., R. Krishna (1990), Mass Transfer, Ellis Horwood series in chemical engineering, Ellis Horwood, Chichester, UK

Wijngaarden, R.J., K.R. Westerterp (1989), Do the effective heat conductivity and the heat transfer coefficient at the wall inside a packed bed depend on a chemical reaction? Weaknesses and applicability of current models, Chem. Eng. Sci. 44(8), 1653-1663

Wijngaarden, R.J., K.R. Westerterp (1993), A heterogeneous model for heat transfer in packed beds, Chem.Eng.Sci. 48(7), 1273-1280

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Chapter 7

Conclusions

The central hypothesis of the present work stated that in the modelling of packed bed reactors, an alternative approach is possible that is a combination of the engineering and scientific approaches and that combines the advantages of both. Four underlying sub-hypotheses were given as necessary conditions for the main hypothesis. We will first consider each of the sub-hypotheses and then return to the main hypothesis.

The topics that have been covered, ranging in scale from a single catalyst particle to a complete reactor are:

• Reaction and diffusion in a catalyst particle

• Structure of a packed bed, especially near a wall

• Flow and pressure loss in a packed bed

• Dispersion on the bed scale

• CFD model of a packed bed reactor

In each of these topics, we made contributions to modelling methods known from literature.

In chapter 2, we built on methods to calculate and estimate diffusion-limited reaction inside porous catalyst particles. The mass and heat balances in this case lead to a set of differential equations: one for each component in the reaction mixture plus one for the temperature. It is known that the actual number of degrees of freedom is equal to the number of independent reactions. This is always less than or equal to the number of components. We presented a straightforward method to reduce the number of differential equations to the number of reactions, based on pseudo concentrations.

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7. Conclusions

The computation time needed to solve the differential equations depends strongly on three factors: the particle geometry, the form of the kinetic equation and the number of simultaneous reactions taking place. For simple particle geometries and single reactions with simple kinetics, analytic solutions are available for the concentration profiles. These can be evaluated with little computational effort. For single reaction systems with general kinetics and particle geometry, a numerical solution is required that is more computation-ally intensive. Westerterp and Wijngaarden (1992) gave an approximation method (the so-called Aris number approach) that greatly reduces this effort. However, their method can not be used for cases with multiple simultaneous reactions.

For multiple reactions with general kinetics, numerical integration of the set of differential equations was formerly the only approach. Especially for complex particle geometries, where the equations are partial differential equations, the numerical solution is computationally intensive. We presented an approximation method based on the Aris number approach for general particle geometry and general reaction network kinetics. This greatly reduces the computation effort involved in the solution of this type of problem. The method has been demonstrated for the Sohio process for the ammoxidation of propene, described as a network of six first order, non-isothermal reactions. For this process, the solution of the full system with a general PDE solver required more than 1500 nodes and 100 seconds. The approximation method for this system used 93 nodes and took just 7 seconds.

As a result, we can conclude that it is possible to develop a method to simplify the calculation of chemical reaction with diffusion limitation in a catalyst particle for general kinetics and particle shape so that it requires a limited amount of computation time and is accurate enough to use in a reactor model. Therefore, the first sub-hypothesis can be accepted.

Chapter 3 focused on the effect of a wall on the structure of a packed bed. It is known that near the wall, the distribution of the particles is no longer random. In literature, this wall effect is often described by a porosity profile near the wall. However, for the detailed modelling of flow and transfer of mass and heat near the wall, not only the porosity but also other bed-structure related parameters like the specific particle outer surface area and the tortuosity of the flow channels are important. Mariani et. al. (2002) and Delmas and Froment (1988) used a fitting procedure to derive a particle centre distribution that matches measured porosity profiles. From this distribution, the particle outer surface area distribution can be calculated. However, we have shown that there are many particle centre distributions that give (more or less) the same porosity profile; the resulting distribution depends strongly

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on the fitting procedure used. To overcome this, we used numerical techniques to simulate bed packings of mono-disperse spherical particles in cylindrical vessels. The particle centre distributions from these simulations have a physical basis and therefore they describe the actual bed structure. In contrast with existing literature, we found that there is no evidence, neither in the literature experimental data base nor in the bed simulation results, that the bed structure depends on the vessel diameter for D/d > 5.

Apart from the porosity and the particle outer surface area profiles, the most important factor that governs transport in a packed bed is the tortuosity. Starting with the simulated packings, tortuosity profiles were determined using a particle tracking procedure. It was shown that as a first approximation, the local tortuosity near a wall is inversely proportional to the local porosity.

We can conclude that the structure of a packed bed near a wall can be characterised so that the main parameters that influence transport processes in the bed (porosity, particle outer surface area and tortuosity) can be described as a function of the location in the bed. Therefore, the second sub-hypothesis can be accepted.

The flow resistance of packed beds is usually described in terms of the particle size and the bed porosity. However, equations such as the well-known Ergun equation that describe a large experimental data base (with a variation of 15 %) are valid for random packings in cylindrical tubes. Therefore, they cannot be used directly for detailed models near the wall.

In chapter 4, a model for the flow resistance in a packed bed was developed where the bed is described as a set of parallel twisted flow channels. The resistance of the flow channels was written as a function of local porosity, specific particle outer surface area and tortuosity, using engineering correlations for pressure loss in flow channels. It was shown that when the wall effect is ignored, this model reduces to the Ergun equation. The values of the fitted coefficients of the Ergun equation are reproduced when plausible estimates are used for physical properties of the flow channels (e.g., wall roughness).

When the model is used with the near-wall profiles from chapter 3, velocity profiles near the wall are obtained. These were compared with measured velocity profiles known from literature. The model results fitted the experimental profiles quite well, especially when considering that the model does not contain any fit parameters and the fact that measuring velocity profiles inside packed beds is notoriously difficult and error prone.

The existing engineering rules to calculate flow resistance of packed beds have been adapted and rewritten in terms of the main parameters (porosity, particle outer surface area and tortuosity) so they can be used on a local scale

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7. Conclusions

in a CFD-based packed bed model. In this way, the wall effect is taken into account in a natural way through the profiles of the main parameters near the wall. Therefore, we can accept the third sub-hypothesis.

Next to convection, dispersion is the most important transport mechanism in packed beds. In the last few decades, there has been a trend towards packed bed dispersion models that take into account the porosity profile near the wall. One of the most prominent of these is the Wall Heat Conduction (WHC) model described by Winterberg et.al. (2000). The WHC model can not be used directly in a CFD calculation because it uses global information like the Peclet number at the centre of the bed. In chapter 5 an alternative model was presented that takes into account the tortuosity as well as the porosity profile. It was shown that for heat transfer to packed tubes, the results of our model are very similar to the WHC model, even though, in contrast to the WHC model, our model was not fitted to this data. Also, as it uses only local information, our model can be implemented in a CFD code where a cylindrical (or even symmetrical) geometry can not be assumed.

It can be concluded that a dispersion model has been developed that describes heat and mass transport in a packed bed based on the main bed parameters in such a way that it can be used locally in a CFD-based packed bed model. Therefore, the last sub-hypothesis can be accepted.

In chapter 6, all submodels presented in previous chapters were combined into a single packed bed reactor model that is implemented in a CFD code. The packed bed model has been validated (in a limited way) by comparing results with literature data for cylindrical tubes. Both for a steam-heated tube (Dixon, 1988) and for a bench-scale ethane oxidation reaction (Vortmeyer and Heidegger, 1991), the model results were found to correspond very well to the measurements. It should be noted that these results were obtained without any fitting parameters.

The code was demonstrated by the simulation of two examples of reactors where the flow in the bed is important: a low pressure drop, radial flow reactor for the selective catalytic reduction of NOx by ammonia in flue gasses and a SO2 oxidation reactor where the exchange of heat through the reactor wall and hence through the wall zone is important.

As the four sub-hypotheses have been accepted, we will now discuss the main hypothesis. We will first determine if the packed bed model fulfils the main requirements.

The first requirement is that the model should have a higher accuracy than the engineering approach. We have shown that for standard (cylindrical) geometry the model gives comparable results as the engineering rules. This is

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correct behaviour because engineering rules are fitted to experimental data for this geometry. For many cases, for instance wall-cooled exothermic reactors, our model gives more accurate results than a hand calculation because it takes into account the wall effect. For non-standard geometry, engineering rules cannot be used directly. Therefore, in the engineering approach the non-standard geometry needs to be approximated by an equivalent standard geometry. For our model, complex geometries can be modelled directly without simplification. Therefore we state that for these cases our model will be more accurate than the engineering rules. However, due to the lack of experimental data available, this has not been proven yet.

The simulation time should be short enough for practical purposes. The computation time needed to solve a model depends strongly on the complexity of, for example, the bed geometry and the reaction kinetics. For a simple case like a deNOx reaction in a tubular reactor, the computation time on a standard personal computer was found to be in the order of 10-20 minutes. For more complex cases like the SO2 oxidation in concentric tubes, the computation time was in the order of a few hours. This is deemed acceptable for day-today use in engineering practice. In future the calculation time can decrease significantly because neither the CFD solver nor the packed bed model code have been optimised for computation speed.

When used in combination with the Comflow preprocessor, the level of CFD expertise needed to set-up a simulation is limited. Due to the limitation to 2-D geometry, the generation of a grid is easy and not time-consuming. Consequently, the time required to set-up a simulation is very short, typically less than one hour. The CFD solver and the packed bed model also support 3dimensional grids, but this requires more expertise in pre- and postprocessing.

A further requirement is that the packed bed model should be generic in the sense that it can handle a wide range of bed geometries, particle geometries and reaction kinetics. Any bed geometry that can be defined in the CFD grid can be used in the packed bed model. The porosity, tortuosity and particle outer surface area profiles are based on packings of uniform spherical particles. For different particle shapes, this results in an overestimation of the wall effect. In principle, any particle shape can be used if the particle centre distribution near a wall is known for that particle geometry. This would require a relatively small change in the code.

For the estimation of the reaction rate, any particle shape can be used provided the geometry factor Γ for that shape is known. This value needs to be calculated only once for a given particle shape. The kinetics can currently be given as networks of general power law equilibrium reactions with Arrhenius type temperature dependence. More complex kinetics can be added to the code with a small programming effort.

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7. Conclusions

The source code of the packed bed model will be released under the GNU Public Licence. Also, the solver (Dolfyn) and the pre- and postprocessor (Comflow) that are used in this work have been released to the public as open source software. Therefore, we have a complete CFD tool for the modeling of packed bed reactors that is available freely to chemical reactor engineers worldwide.

A CFD tool has been developed which can be used for the design and optimisation of packed bed chemical reactors with complex reactions, nonstandard geometry and high interaction of flow and heat transfer. Engineering rules have been adapted and are applied in this tool on a local scale, using a relatively simple CFD solver as the computational engine. We have shown that this model is accurate, fast, easy to use and generic. Therefore we can conclude that the main hypothesis can be accepted.

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Literature Dixon, A.G. (1988), Wall and particle-shape effects on heat transfer in packed beds, Chem. Eng. Commun. 71, 217-237

Vortmeyer, D., Haidegger, E. (1991), Discrimination of three approaches to evaluate heat fluxes for wall-cooled fixed bed chemical reactors, Chem. Eng. Sci. 46(10), 2651-2660

Westerterp, K.R., R.J. Wijngaarden (1992), Principles of Chemical Reactor Engineering, Ullmann’s Encyclopedia of Industrial Chemistry, Vol. B4, 5


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Samenvatting

In het overgrote deel van de industriele chemische processen spelen katalysatoren, stoffen die een reactie versnellen zonder zelf te worden omgezet, een belangrijke rol. Meestal is er sprake van heterogene katalyse; de katalysator is hierbij een vaste stof, terwijl de reactanten als vloeistof en/of gas door de reactor stromen. Een voorbeeld van heterogene katalyse is de katalysator die in de uitlaatsystemen van auto’s wordt toegepast. Hierbij is de katalysator aangebracht op een keramische honingraatstructuur (monoliet). Voor industriele chemische processen die op een veel grotere schaal plaatsvinden zijn gestructureerde pakkingen zoals deze monolieten relatief duur. In de meeste chemische reactoren wordt daarom gebruik gemaakt van gestorte pakkingen van poreuze keramische extrudaten.

Voor de werking van reactoren met heterogene katalyse speelt een aantal transportprocessen een rol. De uitwisseling van reactanten en produkten tussen de processtroom en de katalysator is uiteraard van cruciaal belang. In veel gevallen moet ook warmte worden toe- of afgevoerd om een snelle en zo volledig mogelijke omzetting van de uitgangsstoffen in produkt te bewerkstelligen. De uitwisseling van warmte tussen het gepakte bed en de reactorwand is dan ook belangrijk. De transportprocessen worden beınvloed door het macroscopische stromingsprofiel in het gepakte bed. De invloed van de wand op de stroming is hiervoor een belangrijke factor.

Om tot een optimaal ontwerp van een dergelijke reactor te komen is het nodig om een model op te stellen waarin al deze processen worden meegenomen. Hiervoor is een scala aan mogelijkheiden, die verschillen in complexiteit en benodigde rekentijd. Als uitersten zijn er ruwweg twee aanpakken te onderscheiden, de ’engineering’aanpak en de ’wetenschappelijke’ aanpak.

Bij de engineeringaanpak wordt gestreefd naar een zo eenvoudig mogelijke beschrijving van de reactor, op basis van empirische modellen. De nauwkeurigheid moet voldoende zijn om het ontwerp en de optimalisatie van de reactor te ondersteunen. Het resultaat is typisch een model dat inlaat- en uitlaatcondities aan elkaar relateert. Deze beschrijving is specifiek voor een

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bepaald reactorontwerp. Het voordeel is dat de rekentijd meestal beperkt is; een nadeel is dat het model alleen geldt voor geometrieen waarvoor empirische relaties beschikbaar zijn.

Bij de wetenschappelijke aanpak, wordt zoveel mogelijk uitgegaan van natuurwetten op microschaal. Voor een gepakt bed reactor kunnen de stroming en transportprocessen op deeltjesschaal bijvoorbeeld worden doorgerekend met Computational Fluid Dynamics (CFD) technieken. Dit levert generieke en gedetailleerde modellen op waaruit veel inzicht kan worden verkregen in de optredende processen. Deze zijn echter meestal niet geschikt als ontwerpgereedschap vanwege de hoge mate van expertise die nodig is om het model op te stellen en te gebruiken. Bovendien maakt de lange rekentijd bij grotere hoeveelheden deeltjes het toepassen van deze modellen voor reactoren op industriele schaal onmogelijk.

De centrale hypothese van dit werk is dat er tussen de engineeringaanpak en de wetenschappelijke aanpak een middenweg is, waarbij de nauwkeurigheid van de resultaten hoger is dan die van de engineeringaanpak, terwijl de rekentijd (en de inspanning die nodig is om het model op te stellen) voldoende klein is zodat het model kan worden gebruikt voor het ontwerpen van chemische reactoren. Het hiervoor ontwikkelde ontwerpgereedschap bestaat uit een bibliotheek met routines die gekoppeld kan worden aan een CFD code. Dit maakt het mogelijk om een gepakt bed reactor door te rekenen met behulp van engineeringrelaties maar binnen het CFD domein. Om dit mogelijk te maken, moesten er op een aantal gebieden modellen worden ontwikkeld.

De belangrijkste processen in een gepakt bed reactor zijn reactie, convectie (stroming en drukval) en dispersie (menging, stof- en warmteoverdracht).

De reactie vindt plaats op de schaal van de katalysatordeeltjes. Voor het berekenen van reactie en diffusie voor eenvoudige gevallen (enkele reactie, eenvoudige deeltjesgeometrie en/of eenvoudige reactiekinetiek) zijn berekeningsmethoden beschikbaar die accuraat genoeg zijn en niet teveel rekentijd kosten. In veel praktische gevallen spelen echter meerdere reacties een rol. Er was geen geschikte methode bekend om deze situaties te berekenen. Daarom is er een methode ontwikkeld om de zogenaamde Arisgetal benadering uit te breiden voor reactienetwerken met willekeurige kinetiek in poreuze katalysatordeeltjes met willekeurige geometrie.

Convectie en dispersie vinden plaats op de schaal van het gepakte bed. Deze processen worden bepaald door de structuur van de pakking. In de buurt van wanden is de structuur van een willekeurige pakking anders dan ver van elke wand. Hierdoor zal ook het convectie- en dispersiegedrag in de buurt van een wand anders zijn dan ver er vandaan. In een CFD model van een gepakt bed moeten naar onze mening de transporttermen worden berekend op basis van de

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lokale bedstructuur. De engineeringmodellen voor het berekenen van convectie en dispersie in gepakte bedden maken echter zonder uitzondering gebruik van gemiddelde waarden voor de karakteristieken van het bed. Daarom was het nodig nieuwe modellen te ontwikkelen voor stroming en menging in gepakte bedden.

De structuur van het gepakte bed kan worden gekarakteriseerd door lokale parameters: porositeit, tortuositeit en specifiek uitwendig deeltjesoppervlak. De lokale waarden van deze parameters in de buurt van een wand werden bepaald uit een groot aantal gesimuleerde willekeurige stapelingen van uniforme bolvormige deeltjes. Deze stapelingen zijn gegenereerd met een hiervoor geschreven computerprogramma.

De stroming in een gepakt bed kan worden gezien als stroming door een groot aantal parallele kanalen die telkens gesplitst en samengevoegd worden. Er is een model ontwikkeld dat de stroming door deze kanalen beschrijft als functie van de lokale bedparameters. Voor standaardwaarden van de parameters bleek dit model identieke resultaten te geven als de algemeen gebruikte engineeringrelaties. Dit is opmerkelijk omdat het hier ontwikkelde model niet gefit is aan experimentele drukvaldata voor gepakte bedden. Bovendien kan dit model, in tegenstelling tot de engineeringrelaties, worden gebruikt om de stroming in de buurt van wanden te beschrijven.

Het dispersiegedrag van gepakte bedden werd eveneens beschreven als functie van de lokale bedparameters (porositeit, tortuositeit en specifiek uitwendig deeltjesoppervlak). De engineeringrelaties gebruiken gefitte parameters (zoals de warmteoverdrachtscoefficient naar de wand) of profielen die niet overeenkomen met de structuur van het bed. De waarden van deze parameters zijn over het algemeen bepaald aan de hand van meetwaarden voor cylindrische buisreactoren. Hierdoor kunnen deze relaties strikt genomen alleen gebruikt worden voor cylindrische bedden. Het hier ontwikkelde model gaat direct uit van de structuur van het bed. Omdat het model geen fitparameters bevat is het algemener dan de literatuurmodellen. De resultaten komen overeen met de literatuurmodellen en meetwaarden voor eenvoudige (cylindrische) gevallen. In principe kan het model echter ook worden gebruikt voor niet-cylindrische reactoren.

Als laatste stap zijn de verschillende modellen samengevoegd en gecombineerd met een vrij beschikbare, open source CFD code. Zo’n CFD berekening levert als resultaat de snelheid, druk, temperatuur en concentraties op ieder punt in het katalysatorbed. Experimentele gegevens waarmee zulke gedetailleerde resultaten kunnen worden gevalideerd, zijn in de literatuur nauwelijks voor handen. Een beperkte validatie is uitgevoerd aan de hand van gepubliceerde metingen aan een reactor voor de oxydatie van etheen. De berekende

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temperatuur- en concentratieprofielen kwamen goed overeen met de gemeten waarden.

Om de bruikbaarheid van het model als ontwerpgereedschap te demonstreren is een tweetal voorbeeldsystemen doorgerekend: de katalytische reductie van NO met ammoniak, zoals wordt toegepast in rookgasreiniging, en de oxydatie van zwaveldioxide voor de produktie van zwavelzuur. Verschillende reactorontwerpen werden vergeleken. Hiermee werd aangetoond dat het CFD model geschikt is om dit soort systemen te simuleren en te optimaliseren.

In de praktijk wordt het gebruik van CFD voor het ontwerpen van chemische reactoren beperkt door de hoge kosten van CFD software en de expertise die nodig is om deze te kunnen bedienen. Het gepakt bed model dat in het kader van dit onderzoek ontwikkeld is, wordt gedistribueerd als onderdeel van het pakket Comflow. Met dit pakket kunnen ingenieurs met basiskennis op het gebied van stroming en transportprocessen maar zonder specialistische CFD expertise gepakt bed reactoren gedetailleerd doorrekenen. Comflow wordt als open source software gratis beschikbaar gesteld aan de reactor-engineering gemeenschap.

We kunnen concluderen dat er een simulatietool is ontwikkeld dat gebruikt kan worden als ontwerpgereedschap voor gepakt bed reactoren met complexe reacties, niet-standaard geometrie en een hoge mate van interactie tussen stroming, reactie en warmteoverdracht. Het model levert resultaten met meer detail dan de standaard engineeringaanpak, zodat het mogelijk is om alternatieve reactoren met betere prestaties te ontwerpen. De centrale hypothese van dit werk kan dus worden geaccepteerd.

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Curriculum vitae

Bouke Tuinstra was born on December 8th, 1969 in Cleveland, Ohio, United States of America. Before he was one year of age, his family repatriated to the Netherlands. He passed secondary education (V.W.O.) in 1988 at the Alfrink College in Zoetermeer. From September 1988 to April 1996 he studied Chemical Engineering (Scheikundige Technologie) at the Delft University of Technology where he graduated in the chemical reactor engineering group of prof. Van den Bleek.

The research described in this thesis was started in December 1996 as a part-time occupation. Bouke Tuinstra was employed by Randstad Polytechniek (later called Yacht) and stationed at Comprimo Computational Engineering b.v. in Schiedam. He divided his time equally between commercial CFD projects and his PhD research. After less than one year, Comprimo Computational Engineering was merged into Stork Comprimo Schiedam and then discontinued. Bouke was stationed at the Stork Comprimo headquarters in Amsterdam, working part-time as an engineer. In the years that followed, Stork Comprimo was reorganised, renamed Stork Engineers & Contractors b.v. and finally sold off to Jacobs Engineering at the end of the year 2000.

In December 2000, Bouke Tuinstra started working as an R&D engineer at Stork Product Engineering in Amsterdam, while continuing his PhD work for 2 days a week. In November 2005 Stork Product Engineering was dissolved into the Stork Fokker AESP organisation. Bouke Tuinstra moved to Stork Inoteq, working full-time as Systems Engineer. In January 2007, Stork Inoteq was taken over by Stork FDO to form Stork FDO Inoteq b.v. At this company Bouke is employed to date.

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Dankwoord

In de jaren die ik met dit onderzoek bezig ben geweest, hebben veel mensen op de een of andere manier bijgedragen aan dit werk. Een aantal mensen wil ik niet ongenoemd laten. Henri Roelofs, Cor Merks en Hans de Lathouder waren namens Comprimo, Kiwa en DSM betrokken bij de totstandkoming van het project en de financiering voor de eerste 4 jaar. Met Henri heb tot 2002 samengewerkt bij verschillende Stork bedrijven en tijdens de eerste vier jaar was hij een onmisbare schakel tussen mij en de Stork organisatie. Hans heeft zich eerst namens DSM en daarna op persoonlijke titel ingezet voor het behoud van het Comflow pakket in woelige jaren. Met Cor heb ik gewerkt aan het modelleren van korrelbedreactoren voor leidingwaterontharding. Sjoerd Romkes was vanuit de TU Delft betrokken bij dit interessante stuk onderzoek. Helaas had het te weinig raakvlakken met het huidige werk om in deze dissertatie te worden opgenomen. Stan Vermeulen heeft als afstudeerder metingen voor mij gedaan aan een industriele schaal Radial Flow deNOx reactor; dat de resultaten achteraf onbruikbaar bleken door een beschadiging aan het inwendige van de reactor doet niets af aan zijn werk. De toenmalige Proeffabriek staf en collega promovendi wil ik bedanken voor ondersteuning en de goede sfeer in de tijd dat ik daar (on-)regelmatig aanwezig was.

De leden van de vereniging INUDENT hebben zich steeds ingezet voor het voortbestaan van het programma Comflow. Dankzij deze inspanningen, die soms niet door puur bedrijfseconomische overwegingen waren ingegeven, heb ik het programma Comflow jarenlang kunnen onderhouden. Bij het opheffen van de vereniging hebben de leden Comflow als open source software beschikbaar gesteld. Mede hierdoor kon ook de in dit werk ontwikkelde code op deze wijze vrij gegeven worden.

Verder wil ik de stichting Dolfyn en met name Henk Krus bedanken voor het ontwikkelen en vrijgeven van de Dolfyn solver en de vruchtbare samenwerking.

Bijzondere dank ben ik verschuldig aan mijn promotor Cock van den Bleek, die mijn begeleiding jarenlang in zijn vrije tijd heeft voortgezet nadat hij was gestopt met werken. Zelfs tot in zijn huis in Frankrijk kon ik hem met mijn manuscript lastigvallen. Bijzondere dank ook voor mijn begeleider HP Calis.

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Al jaren geleden heeft zijn carriere buiten de TU Delft voortgezet. Desondanks was hij steeds beschikbaar om tijdens een vrije middag het concept van een hoofdstuk door te nemen.

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