Master in Chemical Engineering - Repositório Aberto · Master in Chemical Engineering Hydrodynamic...

Master in Chemical Engineering

Hydrodynamic modeling of adsorption -

Application to the separation of xylenes

Master Thesis

Miguel Domingos da Silva

Thesis performed in the framework of Dissertation at

Supervisor from IFP: Dr. Frédéric Augier

Supervisor from FEUP: Prof. José Carlos Lopes

Department of Chemical Engineering

July of 2014

Hydrodynamic modeling of adsorption – Application to the separation of xylenes

Acknowledgments

First, I want to thank my supervisor from IFPEN, Dr. Fréderic Augier, for every time spent

helping me and advising me during the internship, also for putting me back on track when

things seemed confusing and dispersed.

I thank Professor José Carlos Lopes for accepting to be my supervisor from FEUP.

I thank Dr. Aude Royon-Lebeaud for her help in every detail, every problem, for the interest

demonstrated about my work and for the help with some issues that I had with Linux and

Fluent.

I thank PhD Student Leonel Gomes for the help with every call and every question that I had

even before the first day I arrived at IFPEN.

To all my friends, I thank you all for the positive thoughts that you sent me, they helped me a

lot to surpass the “being far away from home” feeling and gave me more confidence conclude

the project.

Finally, to my family, I thank you for being so supportive and believe that at the end of this

little amount of time that was the course I had all the capacity and knowledge to finish it.

Thank you all!

Abstract

Xylenes production has been increasing over time and with this there has been a major

research in developing and upgrading the separation processes. The main separation process

is the Simulated Moving Bed which consists in a chromatographic separation performed in

multi fixed-bed adsorption columns.

Within these columns there are several dispersive phenomena that can be related to

hydrodynamics or the non-linearity of the adsorption equilibrium, that deviate the flow

regime from the desired plug flow. Therefore, it is needed to study the coupling between

these dispersive phenomena and adsorption, so that the separation can be accurately

modelled. Recently, these interactions were studied by Augier et al (2008).

The main objectives of this project are to study and model hydrodynamics of an adsorption

bed of the SMB, couple it with the adsorption equilibrium and study the impact of obstacles

(such as pipes and beams) placed inside the porous media, and empty chambers before and

after the fixed bed, in the process of separation.

Results from CFD simulations show that hydrodynamics have a huge role in the separation and

cannot be separated from the adsorption equilibrium, in the absence of mass transfer

limitations. When mass transfer limitations are considered results show that 1D ideal model

approximations can be made to predict the separation, in the presence or absence of

obstacles in the porous media.

Keywords: Hydrodynamics, Adsorption, Simulated Moving Bed,

Computational Fluid Dynamics, Chromatography.

Resumo

Ao longo dos ultimos anos a produção de xilenos tem vindo a aumentar, e com isto o aumento

do interesse na investigação para melhorar os processos de separação dos mesmos. O processo

de separação que mais tem vindo a ser utilizado é o Leito Móvel Simulado, que consiste numa

separação cromatográfica feita em colunas de adsorção constituídas de leitos fixos

sobrepostos.

No interior destas colunas vários fenómenos dispersivos estão presentes, estes podem estar

relacionados com a hidrodinâmica ou devido à não linearidade do fenómeno de adsorção,

estes efeitos causam um desvio no regime de escoamento que o faz sair do pretendido fluxo

pistão. Assim sendo, é necessário estudar esses efeitos para que a separação possa ser

modelisada com precisão. Este caso foi recentemente estudado por Augier et al (2008)

Este trabalho tem como objectivos principais o estudo e modelisação da hidrodinâmica de um

leito de adsorção do Leito Móvel Simulado, acoplar a esse estudo o fenómeno de adsorção e

estudar o impacto de obstáculos (como tubos e barras) colocados no meio poroso e de zonas

livres antes e depois do meio poroso, na separação.

Os resultados das simulações CFD demonstram que a hidrodinâmica tem um papel principal na

separação e a mesma não pode ser dissociada da adsorção, aquando da absência de

limitações à transferencia de massa. Quando essas limitações são tomadas em conta, os

resultados obtidos demonstram que apoximações feitas por modelos ideais podem ser feitas

para prever a separação, quer na ausência ou presença de obstaculos dentro do meio poroso.

Palavras Chave: Hidrodinâmica, Adsorção, Leito Móvel Simulado, Dinâmica

de Fluídos Computacional, Cromatografia.

Declaration

I declare, under honor commitment, that this work is original and every non-original

contribution was referenced with the source identification.

1 Introduction ........................................................................................... 1

2 Bibliographic study .................................................................................. 2

2.1 Xylenes ........................................................................................... 2

2.2 Theoretical description and background .................................................. 7

2.3 Flow characterization methods ........................................................... 12

3 Technical Description ............................................................................. 19

3.1 Experimental Setup .......................................................................... 19

3.2 CFD approach ................................................................................. 20

3.3 Ideal model approximations ............................................................... 21

4 Results and discussion ............................................................................ 23

4.1 Hydrodynamics ............................................................................... 23

4.2 Adsorption ..................................................................................... 31

4.3 Ideal model approximations ............................................................... 34

5 Conclusions ......................................................................................... 39

References ................................................................................................ 40

Appendix 1 Tables and figures ..................................................................... 43

Appendix 2 User defined functions ................................................................ 45

List of Figures

Figure 1 – Molecular structure of the mixed xylenes. ..............................................................2

Figure 2 – Main utilizations of the mixed xylenes [Fabri et al., 2000]. .........................................3

Figure 3 – Simplified scheme of a chromatography. ................................................................5

Figure 4 – Simplified scheme of the TMB with its concentration profiles [Ruthven and Ching, 1989]. ...6

Figure 5 – Simplified scheme of the Simulated Moving Bed [Ruthven and Ching, 1989]. ....................7

Figure 6 - Example of RTD. ............................................................................................ 13

Figure 7 – Example of three RTD with the same mean times but with distinct variances. ................ 14

Figure 8 - Completely segregated system and perfectly mixed system both with the same RTD. ....... 15

Figure 9 – Schematic view of the experimental installation, respective dimensions and sensor

positions. .................................................................................................................. 19

Figure 10 – 2D geometry representative of the experimental installation. .................................. 20

Figure 11 – Inlet profile injections for experimental and simulated RTD’s .................................. 25

Figure 12 – Velocity field obtained and zoom of the inlet and outlet. ........................................ 25

Figure 13 – User defined scalar contours for the RTD simulation after injecting the profile displayed in

Figure 11. ................................................................................................................. 26

Figure 14– RTD obtained for a 2D simulation using an inlet UDF injection (solid lines) and comparison

with the experimental result (points). .............................................................................. 27

Figure 15 – Local mean age and variance obtained through Liu and Tilton’s methodology (2010) ....... 29

Figure 16 – Mean age for the three geometries for a flow rate of 6 m3.h-1. ................................. 31

Figure 17 - Separation of the xylene mixture performed with the 2D geometry without obstacles .... 32

Figure 18 – Separation of the xylene mixture performed with the 2D geometry with the cylinder as

obstacle ................................................................................................................... 33

Figure 19 - Separation of the xylene mixture performed with the 2D geometry with the quadrangular

prism as obstacle ........................................................................................................ 33

Figure 20 – RTD comparison between the three geometries .................................................... 36

Figure 21– Adsorption separation, without mass transfer limitations, results comparison for the 2D

geometry in the absence of obstacle ................................................................................ 36

Figure 22 – Adsorption separation, without mass transfer limitations, results comparison for the 2D

geometry with a cylinder as obstacle ................................................................................ 37

Figure 23 – Adsorption separation, with mass transfer limitations, results comparison for the 2D

geometry with a cylinder as obstacle ................................................................................ 38

Figure 24 - Schematic view of the experimental installation with the obstacles (cylinder at left and

prism at right), with the respective dimensions. ................................................................. 43

Figure 25 - Adsorption separation, with no mass transfer limitations, results comparison for the 2D

geometry with a quadrangular prism as an obstacle ............................................................. 44

List of Tables

Table 1 – Physical properties of mixed xylenes [Fabri et al., 2000]. ............................................4

Table 2 – Hydrodynamic simulation parameters ................................................................... 23

Table 3 – First two RTD moments obtained with different turbulence models at the outlet of the

vessel. ..................................................................................................................... 24

Table 4 – Mean of the age distribution of the probes for the two turbulence models ..................... 24

Table 5 –Comparison of the mean ages between a CFD simulation made at a flow rate of 6.63 m3.h-1

and the experimental results at a flow rate of 6 m3/h .......................................................... 28

Table 6 – Comparison of internal age distribution and variance for the geometry/installation with a

cylinder as obstacle. .................................................................................................... 30

Table 7 - Comparison of internal age distribution and variance for the geometry/installation with a

quadrangular prism as obstacle ....................................................................................... 30

Table 8 – Table of moments of age comparison between hydrodynamics and adsorption in the three

cases of study ............................................................................................................ 34

Table 9 – Simulated results, with Liu and Tilton’s methodology (2010), for the top and bottom free

flow zones for the first and second moments, variance and equivalent number of CSTR’s ............... 35

Table 10 – Moments distribution for the same residence time and different Schmidt number .......... 43

Table 11 – Table of moments of age comparison between the already presented cases and their

equivalent beds coupled with two CSTR’s (one on top and other on the bottom free flow zone). ...... 44

Notation and Glossary

List of symbols

Terminology

Spatial mean age(s) (s)

Langmuir coefficient of the compound i (m3/mol)

Concentration of the compound i (mol/m3)

k- model parameter (-)

CoV Coefficient of variation (-)

Diffusivity coefficient (m2/s)

Flow’s length scale (m)

Particle’s diameter (m)

Turbulent diffusion (m2/s)

Exit age distribution (-)

F Liquid-solid friction force (N/m3)

Spalding’s spatial invariant term (mol/m3)

Turbulent kinetic energy (m2/s2)

Bed permeability (m2)

Overall mass transfer coefficient (s-1)

Outlet’s length (m)

Molar mass (kg/mol)

nth raw moment (sn)

Moment order (-)

Number of CSTR (-)

Pressure (Pa)

Peclet number (-)

Turbulent model equation (-)

Adsorbed concentration of the compound i (kg/m3)

Volumetric flow (m3/s)

Adsorbed concentration of the compound i in equilibrium (kg/m3)

Adsorbent mass capacity for the compound i (kg/kg)

Molar source term (mol/m3.s)

Reynolds number (-)

Schmidt number (-)

Time (s)

Switching time (s)

Fluid interstitial velocity (m/s)

Fluid average superficial velocity (m/s)

Vessel’s volume (m3)

Fluid superficial velocity (m/s)

Position (m)

Greek symbols

Age frequency function (-)

Molecule’s age (s)

Molecules’ mean age (s)

Molecule’s age in a point (s)

* Damping coefficient (-)

Non-Darcy coefficient (m-1)

i Bed porosity (-)

Turbulent kinetic energy dissipation rate (m2/s3)

Volume average age frequency function (-)

Fluid dynamic viscosity (Pa.s)

Fluid effective dynamic viscosity (Pa.s)

nth central moment (sn)

nth normalized central moment (-)

Turbulent dynamic viscosity (Pa.s)

Mean residence time (s)

Fluid density (kg/m3)

Particle density (kg/m3)

Variance (s2)

k- model parameter of k (-)

k- model parameter of (-)

Residence time (s)

ω Specific turbulence dissipation rate (s-1)

Indexes and exponents

Longitudinal

Outlet

Transversal

Volume averaged

Radial

Abbreviations

CFD Computational fluid dynamics

CSTR Continuous stirred-tank reactor

DPFR Dispersive plug flow reactor

EB Ethylbenzene

MX m-xylene

MOX m-xylene and o-xylene mixture

OX o-xylene

PDEB p-diethylbenzene

PET Polyethylene terephthalate

PX p-xylene

RTD Residence time distribution

SMB Simulated moving bed

TMB True moving bed

UDF User defined function

Introduction 1

1 Introduction

Para-xylene (PX) is the main raw material for the manufacture of polyethylene terephthalate

(PET), a polymer used for the production of beverage bottles and polyester fibers. The

constant growth of the PET consumption has boosted the production of PX.

Since PX is always produced in a mixture with its isomers, there has been a big interest in

developing and improving the processes of separation of this mixture. The most used process

to perform this separation is the Simulated Moving Bed (SMB).

The objective of the companies that are interested in studying this process is to maximize PX

production. To do this it is necessary to evaluate the impact of the different dispersive

phenomena such as the intra-particle mass transfer and axial dispersion of the flow in the

fixed bed. It is also important to describe hydrodynamics of the system and its interactions

with adsorption, when its isotherm is not linear.

Then, for this master thesis project the main objectives proposed are to:

Study and model hydrodynamics of an adsorption bed of a SMB through CFD

simulations and experimental tracer injections.

Identify the effect that obstacles provoke in hydrodynamics.

Couple the adsorption with the previous model and observe the effect of

hydrodynamics in the xylene separation.

Bibliographic study 2

2 Bibliographic study

2.1 Xylenes

The term xylene designates the dimethylbenzene, an aromatic ring with two methyl groups.

There are three different molecular configurations in which the xylenes can appear, ortho

(1,2-dimethylbenzene, OX), meta (1,3-dimethylbenzene, MX) and para (1,4-dimethylbenzene,

PX), depending on which carbon atoms the methyl groups are attached. The term mixed

xylenes also considers the ethylbenzene (EB) which instead of two methyl groups has one

group ethyl and also shares the same molecular formula and molar mass as the xylenes, C8H10

and 106.16 g /mol. The molecular configurations of these four molecules are shown in Figure

Figure 1 – Molecular structure of the mixed xylenes.

Despite the similar molecular configuration of these isomers, they do not share the same

industrial interest. The PX is by far the most important aromatic C8, followed by the OX. To

valorise the isomers of PX, they are sent to an isomerization unit so they can be converted

into PX.

2.1.1 Mixed Xylenes - production and use

There are four ways to obtain these isomers: the catalytic reforming (80%) [Chem Systems

Ltd. 1995], pyrolysis gasoline (11.1%), toluene disproportionation (7.6%) and coke-oven light

oil (1.3%). EB is not present when the xylenes are produced via toluene disproportionation

[Cannella, 2007].

During 2007-2012 the consumption of mixed xylenes increased by 15%, the capacity also

increased in a rate of 31%. So, in the next years, it is expected the operational rates to

increase. In terms of consumption and production, the Asian market is the one that has the

highest share. For the isomers of xylene, PX accounted for 79% of 2012 global mixed xylenes

demand, followed by the OX with 7% of the mixed xylenes consumption, after comes the MX

with 1.3% [Xylenes report, IHS Chemical, 2012].

Figure 2 shows in a schematic view the main uses for the mixed xylenes.

Figure 2 – Main utilizations of the mixed xylenes [Fabri et al., 2000].

From the mixed xylenes recovered, between 50 and 60% are destined to PX production, from

10 to 15% to OX production, 10 to 25% are sent to gasoline blending and merely 1% to MX

production [Cannella, 2007]. Around 98% of the PX is consumed in the polyester chain, mainly

to produce PET. For this, the PX must be separated from its isomers, with high level purity, so

it can be used in this valuable industrial chain.

2.1.2 Separation of Mixed Xylenes

The physical properties of the mixed xylenes are shown in Table 1.

Table 1 – Physical properties of mixed xylenes [Fabri et al., 2000].

o-xylene m-xylene p-xylene ethylbenzene

boiling point (°C) 144.4 139.1 138.4 136.2

melting point (°C) -25.2 -47.9 13.3 -95.0

density at 25°C (kg/m3) 876.0 859.9 856.7 862.4

To separate this mixture the process of distillation is not economically viable due to the close

boiling points of these isomers. If the EB and the OX were separated from the other two

isomers it would be necessary 300 theoretical trays [Cannella, 2007]. This process would have

high energy consumption and the cost of the equipment would be impracticable, also the MX

would still need to be removed from the resultant mixture.

Crystallization was the first industrial PX purification process. It consists in cooling the xylene

mixture below the PX's freezing point, so it can be extracted from the other isomers, still in

the liquid phase. Even if it is possible to achieve a purity of 99%, due to the eutectic point the

maximal recovery rate of PX is around 60-65%, resulting in a high volume of recycled xylenes

to the isomerization unit [Fabri et al., 2000].

Nowadays, around 60% [Minceva and Rodrigues, 2007] of the PX produced comes from

chromatographic processes. This kind of industrial processes consists in the exploitation of

the difference of affinities of a given adsorbent for the PX and its isomers, and can achieve a

recovery rate of around 97% per pass, much higher than the crystallization [Candella, 2007].

2.1.2.1 Chromatography

With this method, a small sample of a mixture (mobile phase) is injected in a column filled

with the adsorbent (stationary phase). To better understand this process a scheme of a

chromatographic separation is shown in Figure 3.

Figure 3 – Simplified scheme of a chromatography.

The difference of affinities of the adsorbent for the compounds within the mixture will result

in different residence times: compounds with respective higher affinity will remain longer

inside the chromatographic column, while those with lower affinity will be less affected by

the adsorbent and will have shorter residence times. These laboratory scale chromatographic

columns must have small diameters to minimize radial heterogeneities. For industrial scale

columns this problem cannot be avoided since their diameters can vary between 5 and 10

meters. To maximize the mass transfer between the mobile phase and the so called

“stationary phase”, these are put in counter current, analogously to the heat transfer in heat

exchangers.

2.1.2.2 True Moving Bed (TMB) process

The principle of the TMB consists in the motion of the solid phase (adsorbent) and the liquid

phase (xylenes) in countercurrent within a column. The PX is the more strongly adsorbed

species, and must be separated from its isomers, OX, MX and EB, that are less strongly

adsorbed. The desorbent used is the para-diethylbenzene (PDEB), for who the adsorbent has

an intermediate affinity, so the order of affinities is PX>PDEB>EB>OX/MX. The process

separates the original mixtures into two streams: the extract (PX and PDEB) and the raffinate

(OX, MX, EB and PDEB). These streams must then be purified through distillation.

A simplified diagram of the TMB is shown in Figure 4.

Figure 4 – Simplified scheme of the TMB with its concentration profiles [Ruthven and Ching,

1989].

The most used desorbent is the p-diethylbenzene (PDEB), which molecular structure is similar

to the PX, but instead of the two methyl groups (CH3), PDEB has two ethyl groups (C2H5) in

the para position. When the stationary state is achieved, two mixtures may be recovered

from the column. The role of the desorbent is to clean the solid phase from the mixed

xylenes. The boiling point of the PDEB is different from the mixed xylenes, so that a posterior

distillation can be easily performed.

The main technical difficulty of this process is the circulation of the solid. It is hard to control

its velocity, resulting in heterogeneous motion of the adsorbent and consequently a great loss

of efficiency of the mass transfer between the bulk and the solid. The movement of the solid

is expensive, and it causes its corrosion and backmixing, which highly decreases the

separation efficiency of the process [Ruthven and Ching, 1989].

2.1.2.3 Simulated Moving Bed process

In this process instead of circulating the solid, the countercurrent is simulated by periodically

switching the inlets and outlets of the column towards the fluid flow. This process is called

Simulated Moving Bed (SMB) and is schematically represented in Figure 5.

Figure 5 – Simplified scheme of the Simulated Moving Bed [Ruthven and Ching, 1989].

The adsorbent is kept in several fixed beds. TMB’s respective four sections are visible and

distinguishable. Solid arrows show the actual streams at time t and the dashed arrows show

the streams in use after the stream switch, at t+t* where t* is the switching time.

While the TMB reaches a stationary state, the SMB leans towards a cyclic steady state that is

reached after several cycles of the unit’s operation. When the cyclic steady state is achieved,

the concentration profiles change over time in each fixed bed of the adsorber, although,

these profiles are stationary in each section of the SMB.

This process is ideal for xylene separation due to its ability to work continuously and

consequently to treat industrial quantities of mixed xylenes.

2.2 Theoretical description and background

In this section the equations and models used to describe hydrodynamics in free flow and

porous media, the mass transfer and the adsorption thermodynamics equilibrium will be

presented. Since in the scope of the project is based in the separation of a mixture of

xylenes, and this separation is assumed to be isothermal, the heat transfer equations are not

presented, only the mass balance will be studied.

2.2.1 Hydrodynamics

In this work, hydrodynamics are studied while assuming the flow as steady and

incompressible. Different models were chosen for two different media: free flow and porous

media.

2.2.1.1 Free flow

For the free flow media, if the turbulence effects are non-existent, the Navier-Stokes

equations for an incompressible Newtonian fluid will be solved:

upuu 2 (2.1)

where is the fluid density (kg/m3), u the fluid velocity (m/s), p the pressure (Pa), the

fluid dynamic viscosity (Pa.s) and 2 is the Laplacian operator. Since the Reynolds number is

high, , the models chosen to characterize the flow are the standard k-ε

turbulence model proposed by [Launder and Spalding (1972)] and standard k-ω turbulence

model developed by [Wilcox (1998)].

For the k-ε model, the turbulent viscosity is related to the local values of turbulent kinetic

energy (m2/s2) and its dissipation rate T (m2/s3) as shown in equation (2.2).

is a k-ε model parameter, determined experimentally and equals to 0,09. This introduces

two new transport equations, (2.3) and (2.4).

) ] (2.3)

, , , are constant parameters of the model determined experimentally and is

the production term.

The standard k-ω model is an empirical based model with transport equations for turbulent

kinetic energy and its specific dissipation rate . This model has been modified several

times to improve its accuracy and as result the turbulent viscosity is defined using a damping

coefficient , defined and characterized by [Wilcox (1998)], as shown in equation (2.5).

The transport equations (2.6) and (2.7) are used in this model.

] (2.6)

] (2.7)

These two turbulence models were used in a comparative study in order to perceive which

one is more suitable for this study.

2.2.1.2 Porous Media

The Brinkman-Forchheimer model, is a modification of the laminar Navier-Stokes equations

where the diffusion of momentum due to viscosity effects and the inertia due to friction

between the fluid and particles are modeled. This model has been used to simulate

hydrodynamics in porous media, [Chan et al. (2000)]. The fluids are considered to be

incompressible since they are in liquid state, and the equations to be solved are:

for continuity, where represents the fluid interstitial velocity (m/s).

( ) (2.9)

for momentum, is the inter-particle porosity, is the pressure (Pa), is the apparent liquid

viscosity (Pa.s), is the liquid density (kg/m3) and is the liquid-solid friction force (N/m3)

and is defined by,

(2.10)

v is the superficial velocity (m/s), is the permeability of the porous media (m2) and is the

non-Darcy term (m-1). In the case of fixed bed the values of and are expressed by the

Ergun’s law [Ergun (1952)] and their values can be obtained through equation (2.11) and

(2.12).

(2.11)

(2.12)

These empirical expressions were originally developed for one–dimensional flows within

isotropic media, but when the first term of the equation (2.9) / viscous term, is dominant the

use of the friction term coupled in the Navier-Stokes equation is acceptable, [Zeng and Crigg

(2006)].

2.2.2 Mass Transfer

2.2.2.1 Inter-particle phase

The mass transfer equations that are presented in this section were coupled with the solution

previously obtained for hydrodynamics. In the free flow regions there is no adsorption, the

concentration of a compound is transported by diffusion and convection.

( ) (2.13)

If the regime is turbulent, the turbulent diffusion must be added to the molecular. This

turbulent diffusion , can be obtained by assuming a turbulent Schmidt number of 0.7 and

using the turbulent viscosity obtained by turbulent model in use, either k-ε or k-ω, as shown

in equation (2.14) .

(2.14)

This turbulent diffusion, in opposite of the molecular diffusion that is assumed constant,

suffers a variation in space, this happens because it is a function of the turbulent viscosity

and consequently of the local velocity.

In the porous media the transport of species can be described by equation (2.15),

( ) (2.15)

i is the bed porosity, is the particle density (kg/m3), the molar mass of the compound

(kg/mol) and u is the interstitial velocity (m/s). The diffusivity coefficient in the equation

(2.15) comprises the molecular diffusion and the mechanical dispersion. Contrarily to the

molecular and turbulent diffusion that are isotropic, the mechanical dispersion is described

by a diagonal tensor comprising the radial ( ) and axial dispersion ( ), relatively to the

flow direction. These two vectors are calculated through the definition of the Peclet number

that is the product of the Reynolds and Schmidt numbers and quantifies the ratio between

convective and diffusive transport.

(2.16)

The longitudinal ( ) and transversal dispersion ( ) coefficients can be calculated by using a

constant Peclet number.

(2.17)

(2.18)

Since they are dependent to the local fluid velocity, the tensor can be calculated for a two-

dimensional domain such as,

) (2.19)

) (2.20)

The longitudinal and transversal Peclet numbers are assumed to take the value of 2 and 11

respectively, following the results obtained by [Founemy et al (1992)].

2.2.2.2 Intra-particle phase

Two possible ways to define the adsorption term are, through direct equilibria

between both phases using partial derivative equations (PDE), equation (2.21), or through a

linear driving force (LDF), equation (2.22), to simulate the mass transfer resistance between

the bulk and the adsorbed phase.

(2.21)

) (2.22)

is the overall mas transfer coefficient for the fluid film resistance (s-1). The equation (2.20)

can be applied when molecules migrate between the bulk and the adsorbed phases almost

instantaneously, so the mass transfer limitations are not important. When the mass transfer

limitations are substantial, the adsorbent must be discretized and the mass balance must be

solved locally taking into account a linear driving force and by adjusting the overall mass

transfer coefficient.

2.2.3 Adsorption Equilibria

The adsorbed concentration is obtained through the Langmuir multicomponent adsorption

model [Ruthven (1984)].

(2.23)

is the adsorbent mass capacity (kg/kg), is the adsorbent affinity (m3/mol) for each

compound on a given system, is the concentration (mol/m3) of a compound in the mobile

phase. This model is generally used for gas/solid adsorption systems. However, Daems et al.

(2006) found a good agreement between this model and experimental results for liquid/solid

adsorption for alkanes, alkenes and aromatics mixtures.

2.3 Flow characterization methods

2.3.1 Residence Time Distribution (RTD)

A classical approach to provide some basic information of the macro-mixing state and

hydrodynamics of a given system is the Residence Time Distribution (RTD) theory. To

determine the RTD of a system normally transient experiments with inert tracers are made.

These experiments consist in injecting an inert tracer at the inlet of a vessel (reactor,

column), and the measurement of its concentration at the outlet. These injections must be

close to the ideal, such as perfect step or Dirac. The RTD is represented by an external

residence time distribution or an exit age distribution, E(t). The function E(t) has the units of

time-1 and one example shown in the Figure 6.

Figure 6 - Example of RTD.

This function is defined as shown in equations (2.25) and (2.26)

∫ ( )

(2.24)

( ) ∫ ( ) ( )

∫ ∫ ( )

(2.25)

is the outlet tracer concentration (mol/m3), and are the fluid superficial and its

average at the outlet and is the outlet’s length. Residence time distributions can be

described by using an infinite set of parameters known as moments.

( ) (2.26)

is the nth raw moment of the distribution. For a better comprehension of these moments,

they must be centered relatively to the previous moment(s), as shown in equation (2.27).

∫ ( )

( ) (2.27)

Where is the nth central moment. The first moment is the mean of the distribution or mean

residence time and can be obtained through equation (2.28).

( ) (2.28)

It is equal to the average residence time of a system and it is defined as the expected value

of a molecule’s age at the outlet of a system. The second central moment measures the

variance and it is the degree of dispersion around the mean, equation (2.29).

∫ ( )

( ) ∫

(2.29)

If the value of the variance is closer to zero, it means that all the values are close to the

mean age, and the system is not very dispersive. As this value tends to grow, the dispersion

also grows and the points tend to spread out from each other and from the mean age.

One example of this variation is demonstrated in the Figure 7.

Figure 7 – Example of three RTD with the same mean times but with distinct variances.

The main limitation of the RTD calculations is that they just give information about the

system hydrodynamics at its outlet, leaving its internal information unknown. With this

limitation the RTD theory can be insufficient when non-linear phenomena is present, with this

in mind it is possible to conclude that this theory is incomplete to characterize

hydrodynamics. It is then needed to adopt a supplementary method that can enable one to

access the internal information of a vessel.

2.3.2 Degree of mixing

In the design of the continuous-flow reactors the characterization of the system made by the

RTD could be, in most of the cases, insufficient. This happens because the RTD

characterization is focused in demonstrating the macro mixing of a vessel. This problem could

be demonstrated by comparing the RTD’s of a perfectly mixed system, described by a

continuous stirred-tank reactor, and a system composed by parallel plug flow reactors. These

systems and their RTD are presented in the Figure 8, and also is their RTD.

Figure 8 - Completely segregated system and perfectly mixed system both with the same

To better describe the internal information of these two distinct systems, Danckwerts (1958)

and Zweetering (1959) proposed a criterion that describes the quality of the mixing of a

vessel. Danckwerts uses the concept of “age of the fluid at a point”, which implies the age

averaged over a region, small compared to the whole system and even smaller compared to

the scale of segregation. If , the age of a molecule, is defined as the time passed since the

molecule entered the system, it is possible to calculate the variance of the ages of all the

molecules in the system (equation (2.30)).

( ) (2.30)

is the mean age of all molecules in the system at a given time and the upper bar indicates

that the values are averaged over all the molecules. For each “point” (Danckwerts concept)

the variance of the mean age of the molecules is given by the equation (2.31).

( ) (2.31)

Where is the mean age of the molecules at a given point, and the upper bar indicates that

the values are averaged over all the “points” of the vessel.

For the cases described previously, in the “well-mixed” stirred tank system, the at each

point is the same and equal to the mean age of every molecule in the system, which gives a

variance for all the points the value of 0. In the other system the values for and

are the same for every molecule in the system. Danckwerts define the degree of mixing, J, as

a ratio between the two variances (equation (2.32)).

( ) (2.32)

This quantity lies between 0, for the systems mixed at a molecular scale, and 1, for a

completely segregated system. The degree of mixing helps to study and characterize the

internal information of a system, the impact of hydrodynamics on a system that is at a degree

of mixing between 0 and 1.

Recently, Liu (2012), developed a CFD methodology to transport the moments of internal age

distribution that allows the computation of the degree of mixing.

2.3.3 Transport of moments of internal age distribution

As shown previously, the raw moments of the RTD can be calculated through equation (2.33).

( ) ∫

(2.33)

The previous equation can be applied at any point in a system, as Danckwerts (1958) has

shown. The resulting equation can be solved in function of the spatial position, and can be

used to obtain local measurements of the tracer concentration in every point of the system.

( ) ∫ ( )

∫ ( )

( ) (2.34)

( ) is the nth moment frequency function.

Then when a pulse injection of tracer is performed in a steady and incompressible flow, the

transport of molecules is made by convection and diffusion, and the variation of

concentration, as function of time and position, is given by equation (2.35).

( ) ( ) (2.35)

D comprises all the dispersive phenomena in the system (molecular diffusion, turbulent

mixing, mechanical dispersion, etc). If this expression is multiplied by tn and integrated over

time it results in equation (2.36).

∫ ( )

∫ ( ) ( )

(2.36)

To simplify the equation, the term on the left-hand side can be integrated by parts, resulting

in equation (2.37).

∫ ( )

( )| ∫ ( )

(2.37)

Adding to this, Spalding (1958) considered a steady and incompressible flow in a closed

system and inferred the quantity I, as shown in equation (2.38).

∫ ( )

(2.38)

I is spatially invariant. With these two equations, (2.36) and (2.37), and considering that in

the equation (2.36) the term ( )| will be zero when , because C tends to be zero

faster than t goes to infinity [Liu and Tilton (2010)]. With ( )| , replacing equation (36)

in equation (35) and dividing everything by I, results in equation (2.39)

∫ ( )

[ (∫ ( )

∫ ( )

) (∫ ( )

∫ ( )

)] (2.39)

Where the invariance of u and D, and spatial invariance of I have been used. The term on the

left-hand side corresponds to n times the nth-1 moment of the RTD and the transport equation

of the nth moment, for an incompressible flow, is then shown in equation (2.40).

(2.40)

With the aid of this equation it is understandable that all the RTD moments are diffused and

convected. The equation (2.39) is generally solved by CFD solvers and can be easily used to

provide the spatial distribution of ages by simply adding a source term in steady state

simulations.

2.3.4 Degree of mixing calculation using the internal age distribution

The concept of internal age distribution can be used to calculate the degree of mixing [Liu

(2012)]. By taking a case of a steady incompressible flow where the tracer concentration is

constant at the inlet, and the age of every molecule at the inlet is zero, if the age can be

identified at any point of the system, then the age at any point of the system can be sampled

with the equation (2.40).

( ) ( )

∫ ( )

(2.41)

With this equation, concentration stands as a function of the molecular age. Repeating all the

steps made by Liu and Tilton (2010) and doing the integrations in molecular age, an equation

similar to the equation (2.39) can be obtained. The age frequency function of all molecules in

a system can be obtained by integrating the equation of a determined point, equation (2.41),

for all the points of the system.

By using the definition of the average age of all the molecules, it is easy to obtain a

relation that equals this average to the volume averaged distribution moments.

∫ ( )

(2.42)

With this equation, and knowing the concept of variance, Liu (2012) showed that Zwietering’s

(1959) degree of mixing was easily computed using the volume averaged first and second raw

moments of distribution.

(2.43)

This methodology can only be applied for steady incompressible flows. In CFD simulations this

can be a pragmatic way to obtain results about de mixing state of a given system and

compare them with others that produce the same RTD but have unrelated hydrodynamics.

Technical Description 19

3 Technical Description

3.1 Experimental Setup

In order to validate the CFD models and the chosen assumptions, concerning hydrodynamic

models, boundary conditions and the Peclet and Schmidt numbers, tracer tests were

performed using an experimental installation, represented in Figure 9.

Figure 9 – Schematic view of the experimental installation, respective dimensions and

sensor positions.

This installation was filled with glass spheres of a diameter of 1 mm with a resulting average

porosity of 0.357. The inlet volumetric flow varied between 2, 4, 6 and 8 m3/h. When

hydrodynamics inside the installation were stable, a saline solution was injected in a pulse

shape. The conductivity was measured throughout time with the aid of 13 conductivity

sensors with a frequency of 8 Hz. These 13 localizations of the sensors are also used for the

CFD simulations as probes in order to compare the experimental results with the numerical

results. The obtained RTD and its moments are compared with those obtained through CFD

simulations.

3.2 CFD approach

CFD simulations were performed to characterize hydrodynamics of the SMB adsorbent beds.

With such simulations it was possible to obtain the local moments of the internal age

distribution and the RTD resulting from the injection of inert tracer or obtain the outlet curve

concentrations of mixed xylenes in the presence of adsorption. In view of this knowledge, a

criterion is defined to develop new simple models that remain coherent in the presence of

non-linear phenomena, such as adsorption. This need to simplification also surges because the

real time of CFD simulations for these non-linear phenomena can take up to 72 hours, which

is a huge amount of time regarding to what is normal in simpler RTD simulations (2/3 hours).

When one needs to optimize or to perform a parameter fitting of this process, simulations of

72 hours renders such task are infeasible. It is then interesting to study the simplification of

the processes, such as 1D models which can take to converge.

3.2.1 2D CFD

In order to compare the hydrodynamic results for the experimental setup, a two-dimensional

geometry was drawn as shown in Figure 10.

Figure 10 – 2D geometry representative of the experimental installation.

To this geometry some modifications were made, such as the addition of macro scale

obstacles within the porous media, or the displacement of the inlet and outlet of the vessel.

To perform all the simulations the software ANSYS Fluent 15.0 was used. The mesh for this

geometry was optimized and adapted to all the changes made in the geometry.

Hydrodynamics of this volume were characterized through the mapping of the local moments

of internal age distribution obtained with Liu’s methodology (2012) and through RTD resulting

from pulse injections of inert tracer. These dynamic simulations performed in order to obtain

the RTD of the 2D domain can take up to 3 hours. Thus, it is interesting to study the accuracy

of traditional and simple reactor models that will be introduced in the next section.

3.3 Ideal model approximations

Generally, to ease the modelling of industrial processes, these are assumed to behave as ideal

vessels. The two main simple models are the Continuous Stirred-tank Reactor and the Plug

Flow Reactor. The first assumes that the vessel is perfectly mixed and that the concentration

is the same in every point of the vessel at any given time.

3.3.1 Continuous Stirred Tank Reactor

In ideal Continuous stirred tank reactors, the assumption that the concentration is uniform

throughout the vessel is made. Thus, the outlet concentration is equal to the concentration

inside the reactor, and its temporal variation can be calculated by:

iiiniiiinii RCCRCC

Q is the volumetric flow (m3/s) fed to the vessel, Ci,in the inlet concentration (mol/m3), τ is

the vessel's residence time (s) and Ri the molar source term (mol/m3.s). It must be assumed

that the fluid's density is constant, which is a decent assumption since xylenes are in liquid

state when separated.

As said before, the degree of mixing of a CSTR is equal to zero. The central moments of the

RTD of a CSTR were deduced with the help of an equation similar to (2.39) knowing that the

inlet moment mn,in is null equation (3.2),

nmmnmmm nnninnn 11,

The resulting equation for the distribution central moments is equation (3.3),

n n !1 (3.3)

where (n-1)! is the factorial of the order of the moment minus 1. When the process deviates

from perfect mixing, a cascade of CSTR is generally used for the modelling. It consists in

several consecutive CSTR linked between themselves. The equations of the distribution

moments of an enchainment of CSTR can be deduced with the help of the equation (3.2) but

where mn,in is not null, (except for the first reactor of the enchainment) since the moments of

a given reactor i depend on the moments of the previous reactor (equation (3.4)).

innnnninnn mnmmnmmm ,11,

where mn is the distribution raw moment of order n of the reactor i, mn-1 is the moment of

order n-1 of the reactor i and mn,in is the moment of order n of the reactor i-1.

Using this equation, the distribution central moments of a cascade of NCSTR reactors can be

deduced, if they share their residence times, and is equal to equation (3.3) multiplied by

NCSTR:

CSTRN1 (3.5)

22 CSTRN (3.6)

32 CSTRN (3.7)

The fourth distribution moment does not follow the logic order of the previous ones, and is

equal to 3(2+NCSTR)τ4. The degree of mixing for a homogeneous (constant τ) cascade of CSTR

was deduced. It only depends on the number of consecutive reactors:

NJ (3.8)

By changing the two variables of a cascade of CSTR, NCSTR and τ, it is possible to fit two from

four parameters (three distribution moments and J) to a more complex case.

Results and discussion 23

4 Results and discussion

4.1 Hydrodynamics

With the geometry shown in the previous chapter it is possible to study hydrodynamics of the

system by CFD simulations. The set of parameters used for these simulations are shown in

Table 2.

Table 2 – Hydrodynamic simulation parameters

Value Units

μ 1E-3 Pa.s

ρ 998.2 kg/m3

dp 1E-3 m

Pet 11

The bed porosity εi was calculated experimentally and the average value for this parameter

was 0.357. The diffusivity of the species is given by an user defined function (Appendix 2) and

has two different calculations, one for the porous media and the other for the free flow zones.

The inlet flow rate of the simulations was defined as being 6 m3/h since it gives a superficial

velocity of the fixed bed close to the one passing through the SMB beds.

4.1.1 Comparison with experimental results

The goal of such simulations is to obtain the spatial distribution of ages at every point and the

RTD of the experimental installation, and validate de CFD results with the aid of

experimental ones.

4.1.1.1 Turbulent models comparison

With the geometry drawn, it was necessary to study which hydrodynamic turbulent model was

best suited for this study, and their results must be compared to experimental data. These

comparisons were made by converging hydrodynamics of the system in steady state and then

use the equation (2.39), to calculate the internal age distribution for the probes shown in

Figure 9. The moments of the resulting simulations are displayed in Table 3.

Table 3 – First two RTD moments obtained with different turbulence models at the outlet of

the vessel.

Turbulent Model μ1 (s) σ2 (s2)

Standard k-ε model 29.06 3.23

Standard k-ω model 29.04 3.10

This table only includes the first two moments in the outlet. The moments of the age

distribution at the internal probes are displayed in the Table 4.

Table 4 – Mean of the age distribution of the probes for the two turbulence models

Top probes Middle probes Bottom probes Outlet

(s) 4.85 1.44 0.99 16.33 14.68 13.98 13.80 13.94 27.00 27.70 29.06

(s) 4.63 1.56 1.04 16.25 14.78 14.09 13.78 14.03 27.07 27.80 29.04

Since the experimental residence time obtained was 27.2 seconds and the variance at the outlet was

equal to 2.5 s2 the best method to do the simulations and represent hydrodynamics is the Standard k-

ω model since the residence time and the variance at the outlet are almost the same, the

criteria to choose was based in the first layer of probes and taking it into account the k-ω

model was chosen to be the model used in all the simulations of this project.

4.1.1.2 RTD and Internal age distribution

After choosing the best turbulence model for the system it was possible to do RTD

simulations. The simulations were made in transient state and the input was an injection

profile developed to represent the experimental injections of tracer, the profile was then

compiled to the solver as an UDF (Appendix 2). The profile result and comparison with the

experimental injection is presented in the Figure 11. This injection profile has this shape

because it is difficult to represent experimentally the perfect Dirac injection, since it is

manually controlled. Thus, for better accuracy of the results comparison, an approximated

profile was developed.

Figure 11 – Inlet profile injections for experimental and simulated RTD’s

The experimental values (red squares) are mean values calculated with the information of all

the injections made experimentally. With this profile and the velocity field obtained for a

steady state simulation of hydrodynamics, (Figure 12), RTD simulations were made.

Figure 12 – Velocity field obtained and zoom of the inlet and outlet.

Snapshots at different times, of a RTD simulation, were taken and are shown in Figure 13.

0 1 2 3 4

Experimental

Simulated

Figure 13 – User defined scalar contours for the RTD simulation after injecting the profile

displayed in Figure 11.

Figure 13 shows the behavior of the tracer injected along the column. It is possible to see

that exist dispersion inside the bed since the area covered by the tracer increases. To study

the behavior of the tracer inside the geometry and also to compare it with the experimental

ones, the concentration curves were obtained for each probe of the experimental

installation, and compared to the curves obtained experimentally. This data is divided in 3

sections depending on the zone of the bed where the probes are, Figure 14.

Figure 14– RTD obtained for a 2D simulation using an inlet UDF injection (solid lines) and

comparison with the experimental result (points).

For better comprehension of the figure, the graphic (a) (top) presents the results of the

probes 2 (blue), 3 (green) and 5 (red) of the Figure 9, the graphic (b) (middle) the probes 6

(dark blue), 7 (red), 8 (green), 9 (violet) and 10 (light blue) and the graphic (c) (bottom and

outlet) the probes 11 (green), 12 (red) and 13 (blue).

As it has been demonstrated before the probes inside the column are distributed by three

zones. The results of the first zone, in the beginning of the porous media, are represented in

the graphic (a) of the Figure 14, the second zone, in the middle of the porous media and the

representation of results appears in the graphic (b) and the third zone is at the end of the

experimental set-up, graphic (c).After analyzing these graphics some conclusions could be

made. In the graphic (a) it is shown that the numeric tracer takes more time since it enters

the column in the opposite side, this would be a problem because it could cause the

concentration profile shown in the snapshot of the Figure 13, to solve this problem other

simulations were made doing some changes in the diffusive term for the free zones, by

decreasing the Schmidt number which increases the degree of mixing. This turned out to be

discarded after showing that even for a very small Schmidt number the solution would be

better for the point that is far from the inlet, but for the other two it would be worse (Table

(a) top

(b) middle

(c) bottom

and outlet

10 in Appendix 1). In the middle zone (graphic (b)) results show that the first half of the

installation in the experiments was more dispersive that in the simulations, one option of

solution was to decrease the axial Peclet ( ). After one simulation this option was dropped

because it influenced the dispersion at all zones and the variance at the outlet would

increase, turning it far away from the experimental value.

Observing the graphic (c) it was easy to conclude that the variance at the outlet was almost

the same but the mean residence time had a delay of 1.5 seconds. to solve this delay without

changing the dispersion of the result, the solution proposed was to make simulations

considering a bigger flow rate to equalize the mean residence time of the experiments. The

best result for this solution was at a flow rate of 6.63 m3/h and it is shown in the next table.

This delay could be related to the flow meter error of 5% to highest flow rate

(5%*12m3/h=0.6m3/h).

This representation of the results was made graphically in order to avoid the integration

errors provoked by the noise read by the experimental sensors, which can result in an

overestimation of the moments.

Table 5 –Comparison of the mean ages between a CFD simulation made at a flow rate of 6.63

m3.h-1 and the experimental results at a flow rate of 6 m3/h

Top probes Middle probes Bottom probes +Outlet

(6.63 m3/h) 4.58 2.30 1.96 15.38 14.23 13.64 13.31 13.75 25.53 26.03 27.00

(6 m3/h)

4.17 2.73 1.87 18.90 16.31 15.72 15.52 15.10 25.83 27.19 27.26

The result brought another point of study, since the mean age is almost equal at the outlet.

The increase of flow rate just shows that the porosity inside the experimental installation is

not constant. This conclusion could also be validated based in the difference of residence

times between the top half of the installation and the bottom half. For the first half this

difference is around 12 seconds but for the experiments it is about 13~14 seconds, this result

shows that the top half of the installation could have a higher value of porosity than the

medium value of 0.357. In the bottom half the opposite situation is verified, 11 seconds for

the simulations and 9~11 seconds for the experiments. After some calculations it was possible

to have some approximation for the porosity of these two situations, those values were 0.37

in the top part and 0.32 in the bottom. This conclusion can also explain the difference

between the mean residence time obtained experimentally and through CFD simulations for

equal flow rate, and showed that if the porosity in the bed would be described by a constant

value, there would be no possibilities to approximate the simulated values to the

experimental ones. The transport of the moments of the internal age distribution was also

simulated. The local mean age and variance are shown in Figure 15, for the case where no

obstacle is placed inside the porous media.

Figure 15 – Local mean age and variance obtained through Liu and Tilton’s methodology

(2010)

Figure 15 demonstrates the result of a simulation through Liu’s methodology (2012). The

result for the mean age shows that the flow at the left side of the column is higher, as

expected, also there are highly delayed regions in the bottom corners and finally the mean

residence time is near the one obtained by a RTD simulation, 26.80 seconds. In the case of

the variance, the left side of the column has higher values derived from the delay observed in

the mean age distribution. . The result at the outlet for the variance is also near the values

obtained before, 2.04 s2.

The next step was to simulate the flow perturbations provoked by the presence of obstacles

inside the porous media and evaluate the resulting dispersion. The obstacles chosen and

added to the experimental setup were a cylinder and a quadrangular prism with the same

volume, the representation of these geometries is displayed in Figure 24 of Appendix 1, with

their respective dimensions. The results of those simulations and the comparison with the

experimental ones are displayed in the next tables.

Table 6 – Comparison of internal age distribution and variance for the geometry/installation

with a cylinder as obstacle.

Cylinder Top probes Middle probes Bottom probes +

Outlet

(s) 4.65 2.49 1.66 12.79 12.08 20.17 11.16 10.51 25.56 23.77 27.03

(s2) 3.59 0.31 0.58 2.31 0.58 6.13 0.60 0.99 1.08 1.52 5.69

(s) 3.73 2.96 1.64 14.08 13.84 22.12 12.85 12.07 23.87 23.34 24.99

(s2) 0.27 0.25 0.07 1.78 1.78 6.78 1.72 1.46 2.85 1.51 3.84

Table 7 - Comparison of internal age distribution and variance for the geometry/installation

with a quadrangular prism as obstacle

Prism Top probes Middle probes Bottom probes

+ Outlet

(s) 4.23 2.56 1.70 12.06 11.66 27.69 10.60 10.37 24.95 23.39 26.88

1.98 0.27 0.65 0.64 0.43 18.25 0.40 1.11 0.68 1.73 9.26

(s) 3.74 3.21 2.05 13.77 13.65 26.25 12.38 11.84 23.18 23.11 24.71

0.34 0.32 0.14 1.78 1.67 10.43 1.62 1.62 1.51 3.21 4.79

After analyzing the results for the two tables it is possible to take some conclusions about the

influence of the obstacles in hydrodynamics of the system in the experiments and in the

simulations.

For the first topic, the presence of obstacles shows that the residence time inside the column

decreases, as expected, because the obstacles occupy a portion of the bed’s total volume. Its

presence increases the variance at the outlet which was also expected because with the

inclusion of obstacles causes some perturbations in the system (deviations of the

streamlines), the changes on the results are well represented by the middle probe of the

middle probes line, where, comparing to the case without obstacles, the difference of the

mean age was about 1.74 seconds between the probe 8 and the other four and in the

presence of the cylinder this difference raises to 9.66 seconds and it is even higher in the

presence of the prism, 17.32 seconds.

The transport of moments of the age distribution was simulated, and the mean age, for the

three geometries, is shown in the Figure 16.

Figure 16 – Mean age for the three geometries for a flow rate of 6 m3.h-1.

The effect of the obstacles in the internal age distribution is well verified by this figure where

it is demonstrated that the geometry with no obstacles tend to have high delay regions in the

bottom corners of the geometry and the below the obstacle which can influence the

adsorption separation that will be discussed in the next chapter.

4.2 Adsorption

After studying hydrodynamics with the aid of an inert tracer injection, the adsorption was

studied. The parameters used to describe the adsorption equilibria were inserted in a UDF

presented in the Appendix 2.

The procedure of simulating this equilibrium was the same as the one of a RTD simulation.

The compounds for these simulations were the desorbent (PDEB), the PX and the mixture of

MX and OX, or MOX (the abbreviation used in the simulations).

For these simulations the mass transfer limitations were considered negligible, so the values

given to the mass transfer coefficients were much higher than the real ones, with this it will

be possible to compare the simplified models with the CFD for the most unfavorable case. For

this case, the geometries simulated were the ones presented and studied in the previous

chapter, but some changes were made, the inlet was changed to the same abscissa point as

the outlet.

The simulations were performed with the k- turbulence model and a flow rate of 6 m3/h.

The outlet concentration profiles of PX and MOX for the three geometries are in the Figures

17, 18 and 19.

Figure 17 - Separation of the xylene mixture performed with the 2D geometry without

obstacles

30 35 40 45 50 55 60 65 70

Time (s)

Figure 18 – Separation of the xylene mixture performed with the 2D geometry with the

cylinder as obstacle

Figure 19 - Separation of the xylene mixture performed with the 2D geometry with the

quadrangular prism as obstacle

For the three cases the chromatographic separation is evident, although it is possible to see

the long tails of both curves (PX and MOX) leaning to the right, when the obstacles are placed

inside the porous media. The adsorbent has the highest affinity for the PX which is retained

longer and has higher values of residence time than the MOX mixture. Residence times

increase comparing to the result of hydrodynamics, this happens because there are mass

transfer interactions between liquid and solid phases. The differences between the

hydrodynamic and adsorption results in terms of moments are in the Table 8.

30 35 40 45 50 55 60 65 70

Time (s)

30 35 40 45 50 55 60 65 70

Time (s)

Table 8 – Table of moments of age comparison between hydrodynamics and adsorption in the

three cases of study

Hydrodynamics

Adsorption

PX MOX

Case 1 (s) 2 (s2) CoV 1 (s)

(s2) CoV 1 (s)

2 (s2) CoV

2D 28.44 2.16 0.052 51.64 6.03 0.048 45.60 5.25 0.050

2D + Cylinder 27.04 5.69 0.088 47.06 14.02 0.080 41.54 10.43 0.078

2D + Prism 26.89 9.25 0.113 46.75 29.34 0.110 41.44 26.35 0.124

The previous table demonstrates what has been said previously, when comparing the

moments of age distribution with the moments of the concentration curves in the presence of

adsorption, for the same case both increase. The rise is more pronounced for the PX since is

the component with more affinity.

To better compare the results it was used another term, the coefficient of variation that is

defined to be the square root of the normalized variance [Liu and Tilton, (2010)]. With this

coefficient it is possible to compare the normalized dispersion of the RTD and the

concentration curves, as presented in the Table 9.

The results show values of CoV are not affected when adsorption is taken into account in the

absence of obstacles, but when an obstacle is added to the porous media there are some

changes. When the cylinder is placed, it increases the normalized dispersion (CoV) but in the

presence of adsorption the peaks are compressed and the CoV decreases (from 0.88 to ~ 0.8).

When the prism is placed inside the porous media the CoV gets even more increased, and in

the presence of adsorption the curve of MOX gets even more dispersed (higher CoV value).

This conclusion shows that it is important and necessary to study the coupling between

hydrodynamics and adsorption, because there is no linearity in the results.

This is an important result because it validates that hydrodynamics and adsorption are not

independent and also that hydrodynamics could strongly impact the efficiency of the

adsorption, it increases the variance of the system turning it more dispersive.

4.3 Ideal model approximations

As it was said previously, to ease the modelling of processes ideal approximations must be

made. In this project it was made the assumption that the free flow zones of the bed could

be modeled as CSTR’s. This simplification, if successful, can simplify the modeling of the bed

and decrease the simulation times.

The first thing to do was to calculate the first and second moments of the internal age

distribution at the end of each free flow zone and then calculate the number of CSTR

equivalent to that distribution, Table 9.

Table 9 – Simulated results, with Liu and Tilton’s methodology (2010), for the top and

bottom free flow zones for the first and second moments, variance and equivalent number of

CSTR’s

Since the result for the number of CSTR’s is close to one, each zone was replaced by a CSTR

with the same residence time, and consequently almost the same variance.

Thus, the top free flow zone it was simulated as being on CSTR with a mean residence time of

0.993 seconds, and to do the input for each simulation it was necessary to create a UDF with

the corresponding profile of the CSTR (Appendix 2). The bottom free flow zone was simulated

by feeding the results of simulations with the previous CSTR profile plus the porous media to

a second CSTR. With the mean residence time of the bottom free flow zone and the result of

the simulations it was possible calculate the moments of the age distribution at the outlet of

the system and compare them with simulations that include the complete geometries. The

simulations were made for the cases of the 2D geometry without obstacles, with a cylinder

and with a quadrangular prism inside the geometry. The results and comparisons are

displayed in the Table 11 on Appendix 1. Notice that the comparisons were made in the

presence or absence of adsorption. This was made to get a general overview of the

differences in every case and every phenomenon.

To better understand the results of the comparisons s the results were plotted and are

presented in the Figures 20, 21, 22 and 23.

1,e (s) 2,e (s

2) 2 (s2) NCSTR

Top 0.991 2.115 1.133 0.866

Bottom 1.633 5.477 2.810 0.949

Figure 20 – RTD comparison between the three geometries

Figure 20 shows that the geometries with obstacles suffer the same effect when the free flow

zones are substituted, by decreasing their peaks and increasing the variation. In the other

way in the absence of obstacles, when the free media regions are replaced by the two CSTR,

and the bed simulated as a plug flow reactor the opposite happens.

Figure 21– Adsorption separation, without mass transfer limitations, results comparison for

the 2D geometry in the absence of obstacle

10 15 20 25 30 35 40 45

Time (s)

1D + CSTR

Prism + CSTR

Cylinder

Cylinder + CSTR

30 35 40 45 50 55 60 65 70

Time (s)

2D global PX

1D + CSTR PX

2D global MOX

1D + CSTR MOX

The Figure 21 represents the results comparison between the 2D geometry without obstacles

and an approximate model composed by a 1D plus two CSTR’s (one for each free flow zone).

Results showed that the model represents almost identically the 2D simulation. Then, the

next step was to simulate this phenomenon with the 2D geometries that contained the

obstacles and compare them with the respective models. The plots with the results of these

simulations are in Figures 22, for the cylinder, and Figure 25 of Appendix 1, for the

quadrangular prism.

Figure 22 – Adsorption separation, without mass transfer limitations, results comparison for

the 2D geometry with a cylinder as obstacle

As it is displayed in Figure 22, the fact of using the real free flow zones or model them by

CSTR changes the behavior of the outlet results for the separation. The simplified model

increases the variance, for both obstacles (prism results in Figure 25 of Appendix 1). These

differences can be explained by the already higher variances given by the simplified model

when studying the RTD (Figure 20).

The final step was to simulate the separation with the real mass transfer coefficient and

compare them with the model approximations, since the differences of the results were

higher for the cases with the obstacles, in terms of variance, it was made the comparison

with a cylinder as an obstacle, Figure 23.

30 35 40 45 50 55 60 65 70

Time (s)

Cylinder PX

Cylinder MOX

Cylinder + CSTR PX

Cylinder + CSTR MOX

Figure 23 – Adsorption separation, with mass transfer limitations, results comparison for the

2D geometry with a cylinder as obstacle

This final comparison shows that with the real mass transfer limitations the use of a simple

model to substitute the free flow zones is possible, even in the presence of obstacles in the

porous media. It also shows that separation is worse than in the absence of mass transfer

limitations, which was expected.

Therefore, it is possible to conclude that the 1D model approximation is satisfactory in the

absence of obstacles, for every scenario of mass transfer limitations. For the cases where the

obstacles are placed inside the porous media, this is not possible because they disturb the

streamlines which cannot be represented directly by just one plug flow reactor, this could be

solved by substituting the porous media by several parallel dispersive plug flow reactors with

different lengths.

Another option to simplify this case is to replace the top free flow zone by a horizontal PFR

with several outlets and the bottom zone by a PFR with several inlets, in order to reproduce

the correct radial delay.

Finally, the simpler model is able to predict the chromatographic separation in the presence

of obstacles and considerable mass transfer limitations, this happens because these

limitations provoke a strong dispersion of the concentration fronts, which tend to conceal the

effect of the obstacles.

0 10 20 30 40 50 60 70 80 90 100

Time (s)

Cylinder PX

Cylinder + CSTR PX

Cylinder MOX

Cylinder + CSTR MOX

Conclusions 39

5 Conclusions

In this work, hydrodynamic modelling of an adsorption bed was studied through the use of

CFD simulations. This modelling was made by coupling the RTD theory and the methodology

developed by Liu and Tilton (2010) to transport the moments of the age distribution. The

results obtained were compared to experimental ones, and it was concluded that there was a

delay in terms of mean age that was solved by increasing the simulated flow rate to match

the same residence time as the one from the experimental results. With these results the

coupling between the turbulence model k-and the porous media Brinkman-Forchheimer

model was validated.

Two dispersive agents that influence the separation were discussed, free flow zones and

presence of obstacles. For the effect of obstacles inside the porous media of the bed results

show that this presence cause perturbations by deviating the streamlines and increase the

variance of the outlet results.

Adsorption equilibrium was also simulated throughout CFD simulations, firstly in the absence

of mass transfer limitations. These simulations demonstrated that hydrodynamics have a

major impact on the chromatographic separation, and the characterization of the adsorption

phenomena is not possible without first having hydrodynamics well defined.

The final step was to define simpler (ideal) model approximations that could reproduce the

behavior of some zones of the bed, this approximations were made by replacing the free flow

zones of the bed by CSTR’s with the same residence time. The results show that these

approximations are acceptable in absence of obstacles or when in the presence of strong mass

transfer limitations.

As a final overview, the objectives proposed were accomplished, there were some limitations

caused by the lack of time available for the study since the subject matter is complex and has

a quite big amount of its development recently done. This lack of time brought limitations to

represent one SMB bed by simpler 1D models, plug flow reactors and CTSR in the presence of

obstacles and to easily simulate the chromatographic separation in an industrial case. These

points can also be done in future studies to efficiently upgrade the xylene separation

performed by the SMB.

References 40

References

Augier, F., Laroche, C., & Brehon, E. (2008). Application of computational fluid

dynamics to fixed bed adsorption calculations: effect of hydrodynamics at laboratory

and industrial scale. Separation and Purification Technology, 63(2), 466-474.

Cannella, William J. "Xylenes and ethylbenzene." Kirk-Othmer Encyclopedia of

Chemical Technology (2007).

Daems, I., Leflaive, P., Méthivier, A., Baron, G. V., & Denayer, J. F. (2006). Influence

of Si: Al-ratio of faujasites on the adsorption of alkanes, alkenes and aromatics.

Microporous and mesoporous materials, 96(1), 149-156.

Danckwerts, P. V. (1958). The effect of incomplete mixing on homogeneous reactions.

Chemical Engineering Science, 8(1), 93-102.

Delgado, J. M. P. Q. (2006). A critical review of dispersion in packed beds. Heat and

mass transfer, 42(4), 279-310.

Ergun, S. (1952). Fluid flow through packed columns. Chem. Eng. Prog., 48.

Fabri, J., Graeser, U. and Simo, T. A. 2000. Xylenes. Ullmann's Encyclopedia of

Industrial Chemistry.

Giese, M., Rottschäfer, K., & Vortmeyer, D. (1998). Measured and modeled superficial

flow profiles in packed beds with liquid flow. AIChE Journal, 44(2), 484-490.

Liu, M. (2012). Age distribution and the degree of mixing in continuous flow stirred

tank reactors. Chemical Engineering Science, 69(1), 382-393.

Liu, M., & Tilton, J. N. (2010). Spatial distributions of mean age and higher moments

in steady continuous flows. AIChE Journal, 56(10), 2561-2572.

Minceva, M. and Rodrigues, A. E., AIChE J., 53(1), 138–149 (2007). (Understanding and

Revamping of Industrial Scale SMB Units for p-Xylene Separation)

Minceva, M., & Rodrigues, A. E. (2002). Modeling and simulation of a simulated moving

bed for the separation of p-xylene. Industrial & engineering chemistry research,

41(14), 3454-3461.

Xylenes market report, IHS Chemical Week, November 2012

Ruthven, D. M. (1984). Principles of adsorption and adsorption processes.

References 41

Ruthven, D. M., & Ching, C. B. (1989). Counter-current and simulated counter-current

adsorption separation processes. Chemical Engineering Science, 44(5), 1011-1038.

Sandberg, M. (1981). What is ventilation efficiency?, Building and Environment, 16(2),

123-135.

Spalding, D. B. (1958). A note on mean residence-times in steady flows of arbitrary

complexity. Chemical Engineering Science, 9(1), 74-77.

Zeng, Z., & Grigg, R. (2006). A criterion for non-Darcy flow in porous media. Transport

in Porous Media, 63(1), 57-69.

Zwietering, T. N. (1959). The degree of mixing in continuous flow systems. Chemical

Engineering Science, 11(1), 1-15.

Appendix 1 Tables and figures

Table 10 – Moments distribution for the same residence time and different Schmidt number

Schmidt number

0.3 0.01

μ(s) σ²(s2)

μ(s) σ²(s2) μ(s) σ²(s2)

Top probes

3.91 2.59

3.94 2.54

3.95 2.43

3.96 2.28 3.00 1.32

1.54 0.07

1.55 0.07

1.55 0.07 1.71 0.52

1.21 0.85

1.22 0.80

1.22 0.75

1.20 0.66 1.05 0.15

Middle probes

14.91 0.60

14.93 0.72

14.94 0.80

14.95 0.88 14.55 1.46

13.74 0.33

13.74 0.34

13.74 0.34 13.82 1.02

13.14 0.35

13.14 0.35 13.34 0.72

12.81 0.37

12.82 0.36

12.82 0.36 13.12 0.61

13.28 1.52

13.26 1.38

13.23 1.29

13.21 1.17 12.88 0.45

Bottom probes

25.30 1.07

25.30 0.98

25.29 0.91

25.27 0.82 25.06 0.24

25.78 0.09

25.78 0.10 25.89 0.69

Outlet 26.80 2.04

26.80 1.99

26.80 1.96

26.80 1.92 26.80 1.45

Figure 24 - Schematic view of the experimental installation with the obstacles (cylinder at

left and prism at right), with the respective dimensions.

Table 11 – Table of moments of age comparison between the already presented cases and

their equivalent beds coupled with two CSTR’s (one on top and other on the bottom free

flow zone).

Hydrodynamics

Adsorption

PX MOX

Case 1 (s) 2 (s2) CoV 1 (s)

2 (s2) CoV 1 (s) 2 (s2) CoV

2D global 28.44 2.16 0.051 51.64 6.03 0.048 45.60 5.25 0.050

1D + CSTR 28.24 1.87 0.048 52.44 6.04 0.047 45.91 5.56 0.051

2D + Cylinder 27.04 5.69 0.088 47.06 14.02 0.080 41.54 10.43 0.078

2D + Cylinder + CSTR

27.16 7.93 0.104 47.15 20.59 0.096 41.52 17.67 0.101

2D + Prism 26.89 9.25 0.113 46.75 29.34 0.116 41.44 26.35 0.124

2D + Prism + CSTR

26.85 9.02 0.112 46.84 29.08 0.115 41.49 29.57 0.131

Note: Adsorption cases have no mass transfer limitations.

Figure 25 - Adsorption separation, with no mass transfer limitations, results comparison for

the 2D geometry with a quadrangular prism as an obstacle

30 35 40 45 50 55 60 65 70

Time (s)

Prism PX

Prism MOX

Prism + CSTR PX

Prism + CSTR MOX

Appendix 2 User defined functions

/******************************** **

UDF to define the anisotropic diffusivity

*******************************************/

#include "udf.h"

/**********************************

************************************/

#define sct 0.7

#define dp 0.001

#define eps 0.357

#define Peax 2

#define Pera 11

DEFINE_ANISOTROPIC_DIFFUSIVITY(udf_anis,c,t,i,dmatrix)

if(THREAD_VAR(t).fluid.porous)

dmatrix[0][0]=C_R(c,t)*eps*(1e-

8+dp*(fabs(C_V(c,t)/Pera)+fabs(C_U(c,t)/Peax)));

dmatrix[0][1]=0;

dmatrix[1][0]=0;

dmatrix[1][1]=C_R(c,t)*eps*(1e-

8+dp*(fabs(C_V(c,t)/Peax)+fabs(C_U(c,t)/Pera)));

dmatrix[0][0]=1e-8*C_R(c,t)+C_MU_T(c,t)/sct;

dmatrix[0][1]=0;

dmatrix[1][0]=0;

dmatrix[1][1]=1e-8*C_R(c,t)+C_MU_T(c,t)/sct;

/*********************************************************************

UDF to specify steady-state parabolic pressure profile boundary

**********************************************************************

#include "udf.h"

DEFINE_PROFILE(uds1_profile,t,i)

real x[ND_ND]; /* this will hold the position vector

real y;

face_t f;

real flow_time=RP_Get_Real("flow-time");

real tempszero;

real buf;

tempszero = 0;

begin_f_loop(f,t)

F_CENTROID(x,f,t);

y = x[1];

buf=1.05*(0.91-exp(-(flow_time-0.2-tempszero)/0.2))*exp(-

(flow_time-0.72-tempszero)/0.43);

if (buf>=0.00001)

F_PROFILE(f,t,i) = buf;

F_PROFILE(f,t,i) = 0;

end_f_loop(f,t)

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