Computational Modelling of Gas-liquid Flow in Stirred Tanks-Thesis-2005

8/18/2019 Computational Modelling of Gas-liquid Flow in Stirred Tanks-Thesis-2005

1/218

Computational Modelling of Gas-Liquid

Flow in Stirred Tanks

A Thesis Submitted for the Degree of

Doctor of Philosophy

by

Graeme Leslie Lane

BE (Chem, Hons)

The University of Newcastle

Submitted November 2005

Revised submission August 2006


2/218

I hereby certify that the work embodied in this thesis is the result of original research

and has not been submitted for a higher degree to any other University or Institution.

(Signed) _________________________________________


3/218

ACKNOWLEDGEMENTS

I would like to express my gratitude to my supervisor, Professor Geoffrey Evans, for his

guidance and advice during the course of this project. I would also like to thank

Dr Phillip Schwarz (CSIRO), Dr Peter Witt (CSIRO) and Dr Greg Rigby (formerly at

the University of Newcastle) for their valuable assistance with various aspects of the

project. I would also like to thank my employer, CSIRO Minerals, whose sponsorship

has made this thesis possible.


4/218

i

ABSTRACT

This thesis describes a study in which the aim was to develop an improved method for

computational fluid dynamics (CFD) modelling of gas-liquid flow in mechanically-

stirred tanks. Stirred tanks are commonly used in the process industries for carrying out

a wide range of mixing operations and chemical reactions, yet considerable

uncertainties remain in design and scale-up procedures. Computational modelling is of

interest since it may assist in investigating the detailed flow characteristics of stirred

tanks. However, as shown by a review of the literature, a range of limitations have been

evident in previously published modelling methods.

In the development of the modelling method, single-phase liquid flow was firstly

considered, as a basis for extension to multiphase flow. A finite volume method was

used to solve the equations for conservation of mass and momentum, in conjunction

with the k -ε turbulence model. Simulation results were compared with experimental

measurements for tanks stirred by a Rushton turbine and by a Lightnin A315 impeller.

Comparison was made between different methods which account for impeller motion.

Accuracy was assessed in terms of the prediction of velocities, power and flownumbers, the presence of trailing vortices, pressures around the impeller, and the

turbulent kinetic energy and dissipation rate. The effect of grid density was investigated.

For gas dispersion in a liquid, the modelling method employed the Eulerian-Eulerian

two-fluid equations, again in conjunction with the k -ε turbulence model. The correct

specification of the equations was firstly reviewed. Different forms of the turbulent

dispersion force were compared. For the drag force, it was found that existing

correlations did not properly account for the effect of turbulence in increasing the

bubble drag coefficient. By analysing literature data, a new equation was proposed to

account for this increase in drag. For the prediction of bubble size, a bubble number

density equation was introduced, which takes into account the effects of break-up and

coalescence. The modelling method also allows for gas cavity formation behind

impeller blades.

Simulations of gas-liquid flow were again carried out for tanks stirred by a Rushton

turbine and by a Lightnin A315 impeller. Again, the impeller geometry was included


5/218

ii

explicitly. A series of simulations were carried out to test the individual effects of

various alternative modelling options. With the final method, based on developments in

this study, simulation results show reasonable overall agreement in comparison with

experimental data for bubble size, gas volume fraction, overall gas holdup and gassed

power draw. In comparison to results based on previously published modelling

methods, a significant improvement has been demonstrated. However, a number of

limitations have been identified in the modelling method, which can be attributed either

to the practical limitations on computer resources, or to a lack of understanding of the

underlying physics. Recommendations have been made regarding investigations which

could assist with further improvement of the CFD modelling method.


6/218

iii

TABLE OF CONTENTS

Chapter 1. Introduction ...............................................................................................1

1.1 General background ............................................................................................1

1.2 Aim of the study..................................................................................................4

1.3 Scope of the study ...............................................................................................4

1.4 Organisation of the thesis....................................................................................5

Chapter 2. Design and Fluid Flow Characteristics of Gas-Sparged Stirred Tanks.....7

2.1 Introduction........................................................................................................7

2.2 Applications of gas-sparged stirred tanks ..........................................................7

2.3 Design of gas-sparged stirred reactors ..............................................................10

2.4 Characteristics of the flow in tanks stirred by a Rushton turbine .....................11

2.5 Dimensionless groups and correlations............................................................14

2.6 Alternative impeller designs .............................................................................18

2.7 Scale-up of stirred tank reactors........................................................................20

2.8 Advanced experimental methods .....................................................................21

2.9 Conclusions......................................................................................................23

Chapter 3. Review of Modelling Methods................................................................27

3.1 Introduction.......................................................................................................27

3.2 Basic principles of computational fluid dynamics............................................27

3.3 Extension of the equations to two-phase flow ..................................................34

3.4 Review of simulations of single-phase flow in stirred tanks ............................37

3.5 Issues identified relating to single-phase modelling .........................................49

3.6 Review of simulations of gas-liquid flow in stirred tanks ................................52

3.7 CFD simulations of other systems with gas-liquid flow...................................58

3.8 Simulations of solids suspension in stirred tanks..............................................61

3.9 Differencing schemes for two-phase flow ........................................................63

3.10 Issues identified relating to two-phase modelling...........................................64

Chapter 4. CFD Simulations of Single-Phase Flow..................................................69

4.1 Introduction.......................................................................................................69

4.2 Simulations of single-phase flow with the Rushton turbine .............................69

4.3 Additional simulations of a tank stirred by a Rushton turbine ........................784.4 Prediction of detailed flow around the impeller...............................................82


7/218

iv

4.5 Prediction of turbulence ...................................................................................88

4.6 Modelling of the Lightnin A315 impeller........................................................91

4.7 Conclusions.......................................................................................................95

Chapter 5. Modelling Equations for Flow in Gas-Liquid Dispersions ...................135

5.1 Introduction.....................................................................................................135

5.2 Approaches to modelling ................................................................................135

5.3 Averaging procedure for the two-fluid equations...........................................138

5.4 Closure method for the interfacial force .........................................................142

5.5 Comparison of models for the turbulent dispersion force ..............................151

5.6 Evaluation of models for the turbulent dispersion force................................154

5.7 Added mass and lift forces..............................................................................156

5.8 Turbulence in two-phase flow ........................................................................161

5.9 Conclusions.....................................................................................................165

Chapter 6. The Mean Drag Coefficient in Turbulent Flow.....................................171

6.1 Introduction.....................................................................................................171

6.2 Drag coefficient in stagnant flow....................................................................172

6.3 Previous studies of drag in turbulent flow......................................................176

6.4 Development of a correlation for use in CFD simulations .............................184

6.5 Additional considerations for the CFD model ................................................193

Chapter 7. Modelling of Bubble Break-Up and Coalescence..................................211

7.1 Introduction....................................................................................................211

7.2 The population balance equation ....................................................................212

7.3 Derivation of the bubble number density equation.........................................213

7.4 Previously published literature relating to modelling of bubble size.............216

7.5 Theory of bubble break-up..............................................................................2187.6 Expressions for the break-up rate ...................................................................222

7.7 Theories for bubble coalescence.....................................................................225

7.8 Efficiency term for coalescence......................................................................227

7.9 Modification of the coalescence efficiency expression ..................................230

7.10 Prediction of ventilated gas cavities .............................................................233

7.11 Modelling within the framework of CFX4 ..................................................236

Chapter 8. CFD Simulations of Gas-Liquid Flow ..................................................241

8.1 Introduction.....................................................................................................241

8.2 Data for validation of the model with the Rushton turbine ............................241


8/218

v

8.3 Data for validation of the model with the Lightnin A315 impeller ...............245

8.4 Approach to development and validation .......................................................245

8.5 Modelling method for gas-liquid flow in tank stirred by Rushton turbine .....248

8.6 Modelling method for gas-liquid flow in tank stirred by Lightnin A315 .......252

8.7 Description of the modelling options..............................................................252

8.8 Simulation results for the tank stirred by a Rushton turbine..........................254

8.9. Results for simulations with the A315 impeller.............................................265

8.10 Conclusions...................................................................................................268

Chapter 9. Conclusions and Recommendations......................................................349

9.1 Introduction....................................................................................................349

9.2 Findings from the single-phase modelling.....................................................349

9.3 Findings from the two-phase modelling ........................................................350

9.4 Evaluation ......................................................................................................352

9.5 Recommendations..........................................................................................357

Nomenclature ............................................................................................................359

References .................................................................................................................365

Relevant Papers Published by the Author.................................................................383

Appendix A: Summary of the Mathematical Model for Gas-Liquid Flow.............385

A.1 Introduction...................................................................................................385

A.2 Equations for conservation of mass and momentum ....................................385

A.2 Reynolds stresses ...........................................................................................386

A.3 Interfacial forces.............................................................................................387

A.4 Bubble size model .........................................................................................390

A.5 Gas cavity model ............................................................................................391


9/218

Chapter 1. Introduction

1


1.1 General background

Mechanically-stirred tanks are widely used in the process industries, including

applications in production of chemicals, pharmaceuticals, foods, paper, minerals and

metals. Typical operations carried out in mixing tanks include blending of liquids,

contacting of a liquid with a gas or second immiscible liquid, solids suspension, and

chemical reactions. Despite many years of research and accumulated experience in the

design of this important type of equipment, the fluid flow behaviour of stirred tanks

remains a subject of active investigation. The design of a stirred tank needs to be

carefully matched to the particular operation, and due to the complex flow patterns

encountered, many uncertainties remain in design and scale-up procedures.

Operations involving multiphase mixtures, e.g. contacting of a liquid with a gas, another

immiscible liquid, particulate solids, or some combination of these, form a large

proportion of stirred tank applications. For multiphase operations, there are substantial

additional complexities which need to be addressed, compared with single-phase liquid

flow. Many of the uncertainties in design are related to multiphase aspects, and

therefore, the focus of this thesis is on multiphase flow. More specifically, this study

considers the case of gas-liquid contacting, which takes place in a significant proportion

of industrial stirred tank reactors.

Many experimental studies have been undertaken over the years to investigate the

characteristics of fluid flow in stirred tanks. Often, these studies have resulted inempirical correlations which relate a global parameter, e.g. power draw, mixing time or

mass transfer rate, to the geometric configuration and operating conditions (Kresta &

Wood, 1991; Tatterson, 1991). These empirical correlations have been applied in the

design of mixing tanks, in combination with practical experience. For new processes,

the design is also generally optimised through studies of the process at the laboratory

scale. Then, for the full-scale reactor, a scale-up procedure is applied, which is generally

based on similarity criteria and various empirical scale-up rules (Bartels, 2002).


10/218


2

In such design approaches, empirical correlations may be limited in their applicability,

since extensive data are only available for standard tank configurations and common

impeller types. Also, design based on global quantities does not take into account the

non-uniform and complex three-dimensional flow in a stirred tank. Furthermore, due to

the approximate nature of scale-up procedures, the performance of the production-scale

equipment has often been found to be far from the optimum which was identified at the

laboratory level (Bartels, 2002). Therefore, for greater confidence in design, other

approaches are necessary, where better understanding of the fluid dynamics in stirred

tanks is obtained, including information about the internal flow structures and the

distributed properties of the multiphase dispersion (Bakker, 1992).

One approach to investigating the detailed internal flow is through experimental studies

at the laboratory scale, taking advantage of the considerable advances in techniques

which have been made in recent years. A range of visualisation and advanced

measurement methods, such as laser doppler velocimetry (e.g. Costes & Couderc, 1988;

Hockey, 1990; Petterson & Rasmuson, 1998), have been applied. However, while such

experimental methods provide valuable information, there are also various limitations.

For example, it is very difficult to apply experimental methods to full-scale industrial

tanks, and therefore the uncertainties of scale-up cannot be addressed. Experimental

methods also generally involve the use of model fluids (e.g. water and air) and cannot

be applied to real industrial processes, which potentially involve corrosive materials,

high temperatures and high pressure.

Hence, computer modelling offers an attractive alternative approach for investigating

stirred tanks. Computer modelling allows investigation of the detailed internal flow intanks of non-standard designs at actual process conditions. Compared to experimental

methods, computer modelling also offers advantages such as the ability to address scale-

up issues or model full-scale reactors, and there is the potential to obtain data at a lower

cost in a shorter time frame (Fletcher, 1991). Computer simulation may also generate

data that are very difficult to obtain experimentally, and it may reduce the cost of pilot

plant development.

Computer simulation of stirred tanks can be achieved through the methods of

computational fluid dynamics (CFD). This is a method for obtaining numerical


11/218


3

solutions to the equations governing the flow of fluids and dispersed solids. Being based

on fundamental principles, the CFD method can be applied to new designs and new

geometries within given classes of problems, provided that the simulation method has

been sufficiently validated against representative test cases. Simulations of

mechanically-stirred tanks have been reported in the literature since the late 1970s

(Daskopoulos & Harris, 1996). The capabilities of CFD models have improved over the

years, due to continual improvements in the speed and memory capacity of computers,

and the on-going development of improved modelling procedures. However, accurate

simulation of this type of system is particularly challenging for CFD, and there are

many issues which must be addressed. This is even more so for the case of multiphase

problems. Therefore, developments in modelling stirred tanks have required on-going

efforts to refine and improve the method, and validation against experimental

measurements has remained necessary. Most modelling studies reported in the literature

have addressed only single-phase liquid flow, while studies considering multiphase flow

have been considerably fewer in number.

Procedures for CFD modelling of gas-sparged stirred tanks, as reported in the literature,

have presented a range of limitations. Earlier published studies adopted a simplistic

two-dimensional approach to the problem. While some later models have adopted a

more realistic three-dimensional geometry, predictive capabilities have been limited in

many cases since the impeller was not modelled explicitly. Instead, the impeller was

treated as a ‘black box’, with the fluid motion generated by the impeller being specified

by reference to empirical data. Only a small number of published studies have attempted

to include the impeller in an explicit way. Other typical simplifications of published

modelling methods have included the use of a single, fixed bubble size, but this ignoresthe variations in bubble diameter due to bubble break-up and coalescence. Furthermore,

there has been a lack of agreement regarding the form of the equations governing the

two-phase flow. For example, various authors have applied different expressions for the

inter-phase forces on bubbles, and have adopted different approaches to calculating

turbulent dispersion of the gas. In many published studies, the accuracy and reliability

of results is limited, or else uncertain due to limited extent of validation against

experimental measurements. Due to these various limitations, there has been a clear

need for further development and validation of CFD modelling methods.


12/218


4

1.2 Aim of the study

The aim of this study was to investigate modelling methods for the improved numerical

simulation of gas dispersion in a liquid in mechanically-stirred tanks. In such an

improved modelling method, it was intended that the method should be as generalised

as possible, so as to be applicable to different tank and impeller designs, and offer

increased predictive capabilities compared to previously published methods. Therefore,

the model should include an explicit representation of the impeller geometry.

Furthermore, the modelling method should provide sufficient data for the design or

evaluation of a gas-sparged stirred tank. Therefore, outputs of the CFD model should

include velocities, flow patterns, gas volume fractions, bubble sizes, gas holdup and

power consumption. The accuracy of the modelling method should be assessed by

comparison with experimental data.

1.3 Scope of the study

With the aim of developing improved modelling methods, this study considers a range

of issues affecting modelling. However, for reasons of practicality in developing and

validating this modelling method, limitations must be set on the range of tank designs,fluid properties and flow regimes considered. Thus, for the most part, investigations

have been limited to a configuration consisting of a ‘standard’ design baffled tank

stirred by a Rushton turbine (i.e. a six-bladed disc turbine). This tank configuration is

the system which has been used most often in research. Therefore, modelling of this

configuration provides the greatest opportunity for assessing the accuracy of the

modelling method, since most of the available experimental information refers to a tank

with this type of impeller.

Likewise, development has been based on a turbulent air-water system, since this

corresponds to the model system for which laboratory data is available for validation.

Of course, real industrial systems involve other gases and liquids, and the liquid, in

particular, may exhibit different characteristics, such as a non-Newtonian viscosity. The

liquid may also be mixed with suspended solids. However, it is preferable firstly to

develop modelling for a simpler system before considering such complexities.


13/218


5

In industrial practice a wider range of impeller types are employed, since other impeller

designs are claimed to possess advantages such as improved energy efficiency and

flexibility in gas handling. This study also extends to modelling of one such impeller,

being the Lightnin A315, which is a wide-bladed hydrofoil suitable for gas dispersion.

The flow patterns generated by the Rushton turbine and the A315 are quite different,

and by including tanks stirred by both impeller types, a degree of generality can be

demonstrated in the model.

1.4 Organisation of the thesis

The structure of the remainder of this thesis is outlined as follows:

Chapter 2 provides a description of the applications and general design features of gas-

sparged stirred tanks. A description is also given of the characteristics of the flow in

these systems, and the use of dimensionless groups and correlations to account for their

behaviour. Problems with design and scale-up procedures are outlined, and it is argued

that these problems might be addressed by CFD modelling.

Chapter 3 firstly summarises the general principles of computational fluid dynamics as

applied to stirred tanks. The literature is then reviewed relating to previous efforts in

modelling stirred tanks, firstly for single-phase flow, and then for the more complicated

case of gas-liquid flow. A number of issues are identified relating to modelling

procedures.

Chapter 4 describes work carried out to develop CFD modelling of single-phase flow in

a stirred tank, which provides a basis for the modelling of two-phase gas-liquid flow.

Different methods are compared for modelling the impeller motion, in terms of their

accuracy and computational requirements. The accuracy of modelling methods is

assessed in relation to velocities, the presence of trailing vortices, pressures near

impeller blades, and turbulence levels. Grid sensitivity is investigated.

Chapter 5 discusses the governing equations for two-phase flow. The derivation of

these equations is outlined in order to clarify the appropriate forms for terms such as

drag force, added mass, lift and turbulent dispersion force. The approaches of a number


14/218


6

of authors are considered for the specification of turbulent dispersion. The modelling of

turbulence in dispersed two-phase flow is also discussed.

Chapter 6 considers the specification of the drag coefficient for bubbles in turbulent

flow. The published literature on this topic is reviewed, from which it is found that drag

tends to increase due to turbulence. A unifying approach is then identified for

correlating data from several sources. This leads to an equation describing the effect of

turbulence on drag coefficient. Arguments are presented for the form of the equation

when extended to conditions for which experimental data are not yet available.

Chapter 7 describes modelling of bubble sizes in the tank using a bubble number

density equation. Models for break-up and coalescence are discussed and terms are

defined to take into account the efficiency and rates of break-up and coalescence. Also,

a modelling approach is outlined to account for gas cavity formation on impeller blades.

Chapter 8 describes the development of the CFD model for gas-liquid dispersion. The

sources of experimental data for validation are firstly described. A number of modelling

options are defined, so that comparison can be made between options which have been

used previously in the literature, and proposed new approaches based on developments

in this study. Results are presented for simulations at a number of operating conditions

with a tank stirred by a Rushton turbine and another tank stirred by a Lightnin A315

impeller. Simulation results are compared against experimental data available in the

literature, and it is shown that the preferred modelling options lead to significantly

improved agreement with data.

Chapter 9 summarises the main findings of this thesis, discusses some of the unsolved

problems and provides some recommendations for further work.


15/218

Chapter 2. Design and Fluid Flow Characteristics of Gas-Sparged Stirred Tanks

7

Chapter 2. Design and Fluid Flow Characteristics of Gas-

Sparged Stirred Tanks

2.1 Introduction

This chapter firstly illustrates some of the applications of gas-sparged stirred tanks in

industry. The design of such tanks is described in terms of typical configurations and

impeller types, and the flow characteristics of such tanks are discussed. The use of

global parameters and empirical correlations in design of stirred tanks is described, and

the issue of scale-up is outlined. It is seen that such approaches have a range of

limitations, which indicates the need for development of CFD modelling as a means of

assisting with the design of stirred tank reactors.

2.2 Applications of gas-sparged stirred tanks

As indicated in Chapter 1, mechanically-agitated tanks represent a very common and

important process operation across a wide range of process industries, including bulkand fine chemicals production, food and beverages, pharmaceuticals, and minerals

processing and metals production. Mixing tanks are employed for a range of duties,

ranging from simple blending of liquids to complex chemical reactions. Mixing

operations may be single-phase or multiphase (e.g. mixtures of liquids and solids, gas,

or a second immiscible phase). In multiphase operations the mixing vessel must meet

requirements such as suspension of solids, or break-up and dispersion of gas or liquid

phases as bubbles or droplets.

Indicating the importance of stirred tank reactors, it has been estimated (Butcher &

Eagles, 2002) that 50% of all chemical production takes place in batch stirred vessels,

representing an annual sales turnover value of US$1290 billion worldwide. Poor initial

design can lead to problems such as commissioning failures, production rates lower than

expected, and increased downstream processing costs, and these problems were

estimated as costing 0.5–3% of total turnover. Mixing problems also lead to additional

on-going maintenance and down-time costs. Furthermore, there may be costs due to

unnecessary overmixing (Tatterson, 1994). That is, because of uncertainty in design


16/218


8

procedures, it is necessary to overspecify the agitator power input or batch mixing time,

in order to ensure that the mixing is sufficient.

Gas-liquid contacting is an important industrial operation, since it has been estimated

(Tatterson, 1994) that about 25% of industrial reactions occur between a gas and a

liquid. A mechanically-stirred tank is often chosen for this purpose, although other

contacting methods are possible (Lee & Tsui, 1999). Examples of other contacting

methods include bubble columns, tray columns, and static mixers. The choice of

equipment is determined by factors such as the required residence times of gas and

liquid, the degree of conversion of reactants and selectivity for desired products, and

safety and flexibility of operation. A stirred tank may be preferred (Tatterson, 1994)

because it is possible to have a large inventory of liquid with a high degree of flexibility

over liquid residence time, and it may offer advantages such as a well-mixed

environment with relatively uniform reagent concentrations, pH and temperature.

Compared with equipment such as a bubble column, a stirred vessel can offer better

control over bubble size and spatial dispersion of the gas. Agitation also increases the

gas-liquid mass transfer rate. In addition to gas and liquid, a third solid phase is present

in some processing operations. In such a case, especially in minerals processing, a

stirred vessel is often used since agitation is also effective for keeping the solids in

suspension.

Gas-sparged stirred tanks are used for a variety of processes which may involve

reactions such as oxidation, hydrogenation, or chlorination. Some examples of such

industrial processes are as follows:

• Aerobic Fermentation: Stirred tank fermenters are widely used in the food and

pharmaceutical industries, where micro-organisms are exploited to produce a variety

of products such as yeast, antibiotics, enzymes, amino acids, vitamins, flavour

enhancers, and thickening agents (Sengha, 1994; Benz, 2003). The use of a stirred

tank allows for suspension of the micro-organisms and facilitates uniform conditions

of pH, temperature, and nutrient and substrate concentrations, while sparging of air

provides for the oxygen requirements of the micro-organisms.


17/218


9

• Hydrogenation: Industrially important hydrogenation reactions include the reaction

of hydrogen with vegetable oil for the production of margarine and related products

(Hasenhuettl, 1994). Mechanical agitation is required to suspend nickel catalyst

particles and disperse the hydrogen which is sparged into the bottom of the tank.

Typically, this takes place in a pressurised, high aspect ratio tank with multiple

impellers and internal heat transfer coils to control the exothermic reaction.

• Pressure oxidation: Pressure oxidation processes are used in the minerals industry

to treat sulfidic ores, to liberate metals into solution for subsequent recovery by

solvent extraction and electrowinning. A typical application for pressure leaching is

treatment of refractory gold ores, where gold is bound up in the grains of pyrite or

arsenopyrite minerals (Thomas et al., 2002). In this case, reactions take place at high

temperature and pressure in an autoclave, which is typically a horizontal elongated

pressure vessel internally divided into a number of compartments separated by

weirs. Each compartment is fitted with its own agitator and oxygen sparger.

• Bioleaching: This is another process used to extract metals from sulfide ores such as

sulfidic gold, copper and cobalt (Brierley & Briggs, 2002). Micro-organisms such as

bacteria and archaea species are introduced to the process, and these oxidise ferrous

ions or reduced sulfur species to obtain their metabolic energy, and in doing so

provide a pathway for leaching reactions. Air or CO2-enriched air is sparged into the

tank to meet the requirements of the microorganisms.

• Mineral flotation: This process involves physical separation rather than a chemical

reaction, based on the exploitation of wettability differences of particles (Yarar,1994). Applications are mainly in the minerals industry, but also include water

treatment and other applications. Air is introduced into a tank fitted with a radial-

style impeller, and particles are selectively attached to bubbles, which then rise to

the surface to form a froth, which is skimmed off.

It can be seen from these examples that many of the relevant industrial processes are

actually three-phase, since they involve gas, solids, and liquid. However, it is generallythe case that the most demanding requirement on the agitator (in terms of power


18/218


19/218


11

strength given to the impeller by the disc compared, for example, to an open-bladed

paddle, and the disc also prevents short-circuiting of gas along the shaft (Smith, 1985).

A ‘standard’ tank configuration has also evolved (Smith, 1985), as shown in Figure 2.2.

This is often used in experimental studies at the laboratory scale, and likewise in

numerical studies, thus providing a basis for comparing the results of different workers.

The standard tank configuration consists of a flat bottomed cylindrical tank filled to a

depth, H , equal to the tank diameter, T , with four full length baffles of width B = 0.1T .

Where the tank is fitted with a Rushton turbine, the impeller diameter D is normally

0.33T and is centrally mounted at a clearance, C , of 0.33 H . The Rushton turbine has a

disc with diameter D2 = 0.75 D, and the blade proportions are a length L B = 0.25 D and a

width W B = 0.2 D. This configuration has been used in the single-phase modelling

development in this thesis, and a very similar configuration has been used for the two-

phase modelling, but with an impeller clearance of 0.25T , in accordance with the

arrangement used by Barigou and Greaves (1992, 1996), whose data provided the main

basis for validation of the model.

In industrial practice, major deviations from the standard design are common. Examples

of such designs have been mentioned in Section 2.2, e.g. the use of high aspect ratio

tanks with multiple impellers for hydrogenation of vegetable oil, or the use of elongated

horizontal autoclaves for pressure oxidation. In such designs, the tank bottom is dished

rather than flat. Another variation is the use of a conical bottom. Also, in an effort to

overcome some of the perceived shortcomings of the Rushton turbine, alternative

impeller designs have been developed, and some of these have had commercial

acceptance. These include the Smith impeller, the Scaba impeller and the Lightnin A315(see Section 2.6). Nevertheless, given the large amount of published data relating to

standard design tanks stirred by a Rushton turbine and the scarcity of data relating to

other impellers, a tank with a Rushton turbine has been chosen in this study as the basis

for most of the modelling development.

2.4 Characteristics of the flow in tanks stirred by a Rushton turbine

The flow characteristics of the Rushton turbine in the turbulent flow regime have been

studied extensively. As described by various authors (e.g. Nouri, 1988; Hockey, 1990),


20/218


12

the Rushton turbine generates a strong radial jet which emanates from the impeller and

impinges on the tank wall, and then divides into two wall jets, leading to the formation

of two recirculating ring vortices in the upper and lower parts of the tank (Hockey,

1990). Liquid returns to the impeller from above and below in the central region near

the impeller shaft. Measurements of mean velocities and turbulent fluctuating velocities

throughout the tank have been reported by a number of authors, e.g. Hockey (1990) and

Mavros et al. (1996), using laser doppler velocimetry. Measurements such as these

show that the flow pattern is also characterised by a wide variation in turbulence levels

in different parts of the tank, with turbulent kinetic energy being highest near the

impeller and in the impeller discharge stream. Further analysis of the turbulent energy

dissipation and other characteristics of the turbulence, such as the integral length scale,

has been reported by a number of authors, e.g. Wu and Patterson (1989), and these

quantities were also found to vary widely. Wu and Patterson estimated that the local

energy dissipation rate near the impeller tip was more than 20 times the average energy

dissipation rate, and the energy dissipation in the impeller region and discharge stream

accounted for 60% of the total.

An important feature of the flow in the immediate vicinity of the impeller blades is the

presence of trailing vortices, which are strongly swirling structures produced on the

trailing sides of impeller blades due to flow separation. Van’t Riet & Smith (1975)

reported detailed measurements of velocities, pressure distributions, and the spatial

location of the trailing vortices formed behind blades of a Rushton turbine. They

observed pairs of vortices about one fourth of the blade height emanating from the top

and bottom edges of each impeller blade, with a strong reduction in pressure at the core

of the vortex. As well as radial flow impellers like the Rushton turbine, trailing vorticesare found in the flow produced by axial flow impellers (Bakker, 1992). For axial flow

impellers, a single tip vortex is produced (Smith, 1985).

Ranade and Joshi (1990) concluded that the trailing vortices play a central part in the

mechanism of energy dissipation of impellers, and are particularly important in

multiphase flows. For gas-liquid flow, it is found that due to the strong centrifugal

action and the low pressure at the vortex core, bubbles are drawn into the trailing

vortices near the blades, and then dispersed into the tank along the line of the vortex, as

it merges with the bulk flow in the tank (Smith, 1985).


21/218


13

While the basic flow pattern in a stirred tank has been described in many studies on the

basis of single phase flow, it is important to consider that the introduction of gas into a

stirred tank can have a major effect on the hydrodynamics, mainly because of the

manner in which the gas is drawn into the impeller (Smith, 1985). The interaction of the

gas with the impeller plays a crucial role in determining the flow characteristics in the

tank as a whole, especially in determining power consumption and circulation flow. Gas

introduced to an impeller tends to be drawn into the low pressure regions at the trailing

side of the blades, and it is found that at sufficiently high gas flow rates, ventilated gas

cavities are formed. It is found that an impeller operating in a gas-liquid mixture will

generally have a reduced power draw and pumping rate, due partly to the reduced

density of the mixture, and also due to the streamlining effect of the ventilated cavities

(Smith, 1985).

The pattern of gas dispersion will generally depend on a balance between the buoyant

energy of the gas and the power input of the impeller. Considering a Rushton turbine at

constant gas flow rate, at low impeller speed the buoyancy of the gas dominates the

flow, and the impeller is said to be flooded. At somewhat higher speeds, the impeller is

able to produce a radial dispersion action, and the impeller is said to be ‘loaded’.

Further increase in impeller speed leads to velocities in parts of the tank which are

sufficient to prevent bubbles rising, and then the ‘recirculating’ regime is obtained

(Tatterson, 1991).

Several regimes have also been defined for the interaction of the gas with the Rushton

turbine and the formation of gas cavities on the blades. At low gas flow rates, the vortexcavity regime is found, where the bubbles are drawn into the vortices and have the

appearance of a foam, while the single-phase structure of the vortices is maintained. As

the gas flow rate is increased, coalescence of the bubbles leads to growth of the cavities

and reduction in the volume of spinning liquid behind the blade. Eventually the gas

extends right up to the blade, and clinging cavities are formed. At higher gas flow rates

again, so-called large cavities are formed. A pattern of large and clinging cavities form

on alternate blades, known as the 3-3 configuration (Smith, 1985). An even higher gas

flowrates, large cavities form on all blades, and eventually the flooding point is reached,

where the flow of gas is so high that the impeller is ineffective for pumping liquid. Flow


22/218


14

maps have been produced to predict the gas cavity regime as a function of gas flow rate

and impeller speed, e.g. Warmoeskerken & Smith (1985). The reduction in power due

to gassing has been correlated with regard to the cavity regime (Tatterson, 1991).

2.5 Dimensionless groups and correlations

In design procedures for mechanically stirred tanks, there are a number of global

parameters or dimensionless groups which have been found useful in characterising

performance. Many experimental studies have aimed at determining the values of these

dimensionless groups and how they vary with the geometric configuration and operating

conditions. The most important basic characteristic of a stirred tank is usuallyconsidered to be the power consumption, and this is often given in terms of a

dimensionless power number (sometimes called the Newton number), according to

(Tatterson, 1991):

53 N

D N

P

l

P ρ

= , (2.1)

where P is the power consumption, ρ l is the liquid density, N is the impeller speed (in

Hz or s

-1

) and D is the impeller diameter.

For single-phase flow, it has been found for stirred tanks that in general, N P is a

function of the impeller Reynolds number, Re, given by:

l

l ND

μ

ρ 2Re = , (2.2)

where μ l is the liquid viscosity. For agitators operating in the laminar flow regime (at

low Reynolds number), the power number is found to decrease with increasingReynolds number, while in a fully turbulent system (approximately Re > 104), the

power number becomes fairly constant for a given tank and impeller geometry. For

example, the power number of a ‘standard’ design Rushton turbine in the turbulent

regime is about 5.0 (Tatterson, 1991).

Another useful characteristic is the primary flow rate of liquid produced by an impeller

at a given speed. This is expressed in terms of a dimensionless impeller flow number,

NQ, given by (Tatterson, 1991):


23/218


15

3 N

ND

QlQ = , (2.3)

where Ql is the liquid flow rate through the discharge area swept by the impeller.

Many studies have been carried out to investigate how the power and flow numbers of

various impellers vary as functions of parameters such as impeller diameter, clearance

of the impeller from the tank bottom, impeller blade width, number of blades, angle of

blades etc. (Tatterson, 1991). By analysing the ratio of power to flow number, the

energy efficiency of different impeller designs for circulation of the liquid can be

compared.

When gas is introduced to a stirred tank (usually through a sparger located below the

impeller), a reduction in power is observed at constant impeller speed, with the effect

generally increasing with increasing gas flow rate. For a Rushton turbine, the loss of

power may be as much as 60%, before the flooding point is reached (Middleton, 1997).

A wide range of correlations have been proposed to account for the effect of gas on

power draw (Tatterson, 1991).

Two dimensionless groups which are commonly used in characterising the gas-liquid

dispersion are the gas flow number, Flg, and the Froude number, Fr. The gas flow

number (or aeration number) is given by:

3Fl

ND

Qgg = , (2.4)

where Qg is the gas flow rate to the vessel, while the Froude number is given by:

g D N

2

Fr = . (2.5)

where g is the acceleration due to gravity. These dimensionless groups have been used

by some workers to correlate the gassed power draw of the impeller (e.g. Bakker et al.,

1994). The aeration and Froude numbers have also been used to develop maps of the

flow regimes of a number of radial and axially pumping impellers (Middleton, 1997),

which indicate the formation of various types of cavity or where the flooding point is

reached.


24/218


16

Besides the power draw of impellers, the most important design parameters for a gas-

sparged tank relate to the interfacial area created, and the consequent gas-liquid mass

transfer rate. Various correlations have been developed to predict the interfacial area, or

alternatively, separate correlations have been proposed for the gas holdup and average

bubble size, which together account for the interfacial area (Tatterson, 1991). Such

correlations are based on laboratory scale measurements for a limited number of

specific impellers and tank configurations, such as a Rushton turbine in a standard

design tank.

Most published correlations for the gas holdup, φ g, have taken the form:

Bsg

A

l

ghg v

V

PC ⎟

⎟ ⎠

⎞⎜⎜⎝

⎛ =φ , (2.6)

where Pg is the gassed power, V l is the tank liquid volume, and vsg is the superficial gas

velocity. For example, Bakker et al. (1994) recommended an equation of this form, with

values for water-air systems being C h = 0.16 ± 0.04, A = 0.33 and B = 0.67.

Various authors (e.g. Calderbank, 1958; Lee & Meyrick, 1970; Bouaifi et al., 2001)

have proposed correlations for the average bubble size in stirred tanks. This is usually

represented by the Sauter mean diameter, d 32, which is the bubble size which has the

same ratio of area to volume as the complete distribution. The Sauter mean diameter is

given as the ratio of the third and second moments of the bubble size distribution

function f (d ), according to:

∫

∫=)()(

)()(

2

3

32d d d f d

d d d f d d . (2.7)

This is the most useful measure of average size since it is directly related to gas holdup

and interfacial area, a, according to (Barigou & Greaves, 1996):

32

6

d a

gφ = . (2.8)

An example of a correlation for bubble size is that proposed by Calderbank (1958) for

coalescing systems:


25/218


17

0009.015.4 5.0

2.0

4.0

6.0

+

⎟⎟ ⎠

⎞⎜⎜⎝

⎛ = g

ll

g

V

Pd φ

ρ

σ , (2.9)

where σ is the surface tension coefficient.

Correlations have also been proposed for the overall interfacial area, which results from

the combination of holdup and bubble size distribution. An example of such a

correlation, applicable to disc turbines, is that according to Hughmark (1980):

187.0

32

42592.0

32

4231

21

38.1⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎟⎟

⎠

⎞⎜⎜

⎝

⎛ ⎟

⎠

⎞⎜

⎝

⎛ =

lll

gl

V

DdN

gWV

D N

NV

Qga

σ σ

ρ (2.10)

where W is the impeller blade width.

Rather than calculate the interfacial area, correlations have also been proposed for the

combined mass transfer coefficient and interfacial area term, k la, as a function of tank

operating conditions. Hence according to Bakker et al. (1994):

b

sg

a

l

g

klal v

V

PC ak

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = (2.11)

where for air-water systems, the constants are given as C kla = 0.015 ± 0.005, a = 0.6 and

b = 0.6. The overall mass transfer rate can then be calculated according to:

( )llll C C ak dt

dC −= * (2.12)

where C l is the average concentration of gas dissolved in the liquid, and*

lC is the

saturation concentration.

As reviewed by Tatterson (1991), there are many other published correlations of this

type. These may cover different types of impeller, or in some cases, correlations have

been extended to cover a range of temperatures and pressures (e.g. Sridhar & Potter,

1980). However, correlations according to different authors may be functions of

different variables, and sometimes give conflicting results. Such correlation methods

may not work well in tanks of non-standard design, and are very difficult to generalise

to different liquids and gases. In particular, the bubble size is very sensitive to small

concentrations of species in the liquid such as electrolytes, surfactants, alcohols, oils


26/218


18

etc. (Middleton, 1997), making it very difficult to generalise these correlations for

different chemical systems. Sometimes, separate correlations are given for ‘coalescing’

and ‘non-coalescing’ systems, but this approach is hardly likely to cover all possible

chemical systems.

Clearly, there are many situations where there will be no suitable correlation for a given

stirred tank reactor. On the other hand, these correlations may provide a useful guide to

how the behaviour of the gas-liquid dispersion will be affected by changing operating

conditions in an existing reactor. For example, according to equation 2.6, holdup will

increase with the power input as P0.33, and mean bubble diameter will decrease with

power input as P-0.4 according to equation 2.9.

2.6 Alternative impeller designs

Understanding of the flow produced by a gassed Rushton turbine has led to the

development of various alternative impellers for gas dispersion, which seek to improve

on the design of the Rushton turbine. The Rushton turbine has been criticised for two

main reasons. The first reason relates to the sharp drop-off of gassed power with

increasing gas flow rate, such that it is not uncommon in industrial operations to have a

power draw of 50% or less of the ungassed power (Nienow, 1990). It may be necessary

in the design to specify a motor large enough to mix the liquid in the case of zero gas

flow, even though most of the power capacity is not used under normal operating

conditions. Secondly, the Rushton turbine is relatively inefficient in terms of the liquid

flow produced for a given power input (Fraser et al., 1993). This is particularly a

problem where the mixing operation must also achieve some other duty such as solids

suspension or heat transfer, which are flow-controlled operations.

Other impeller types include those based on a disc turbine, but with curved, concave

blades (Bakker et al., 1994), which aim at reducing the strength of the trailing vortices

and therefore reducing cavity formation. Impellers of this type, such as the Scaba SRGT

hollow-blade type with parabolic blades (Nienow, 1990), have been shown to give

much flatter curves for power number as a function of gas flow number.


27/218


19

Gas-dispersion impellers of the downward axially-pumping hydrofoil type have also

been developed. Such impellers employ a high solidity ratio, where wide blades are

used to avoid gas shortcircuiting through the impeller. The hydrofoil shape gives

improved power to flow ratios, leading to smaller power input requirements. Designs of

this type have included the Prochem Maxflo T (Nienow, 1990), Lightnin A315 (Fraser

et al., 1993), Mixel TT, and the Narcissus impeller (Vlaev et al., 2002). Also, it has

recently been proposed to use upward-pumping wide-bladed hydrofoils. The use of such

impellers would be expected to lead to more stable behaviour, since the flows produced

by the impeller and by the buoyancy of the gas are working together in the same

direction, rather than being in opposed directions, which has been found to lead to flow

and torque instabilities (Nienow & Bujalski, 2004).

Of all these options, modelling of just one such impeller design has been included in

this study, being the Lightnin A315 impeller. This is a wide-bladed hydrofoil (see

Figure 2.3) designed to produce downward axial flow. Other common axial flow

impellers, e.g. the pitched bladed impeller or thin-bladed hydrofoils such as the Lightnin

A310 are only suited to dispersing relatively small amounts of gas. However, due to its

high solidity ratio (Bakker, 1992), the Lightnin A315 is able to disperse gas flow rates

of similar magnitude to a Rushton turbine without becoming flooded.

The Lightnin 315 is generally placed at a similar clearance to a Rushton turbine (about

1/3 of tank height) and pumps downwards towards the tank bottom, producing a single

recirculating flow pattern, as distinct from the two circulation loops of a Rushton

turbine. Compared with the Rushton turbine, the manufacturers (Lally, 1987; Fraser et

al., 1993) claim several advantages including the following:

• Due to its hydrofoil shape, the Lightnin A315 has a much lower power number

(~0.7 compared with ~5.0 for a Rushton turbine), and this leads to a greater

hydraulic efficiency, so the impeller is able to recirculate gas bubbles more easily.

The lower power number also implies reduced torque, which leads to reduced cost in

the type of agitator motor required.

• The Lightnin A315 maintains a relatively flat gassed power curve, so that, up to

about a gas flow number of 0.4, the power draw remains fairly constant, whereas for


28/218


20

the Rushton turbine the power drops off very rapidly. This has advantages since the

motor does not need to be overdesigned to cope with larger power requirement of

ungassed mixing (e.g. at start-up or in a plant upset), of which only a fraction is used

in normal operation.

• The Lightnin A315 produces lower levels of shear, which is advantageous in

biochemical applications, where high shear rates may be destructive to

microorganisms.

• Since the flow produced is directed toward the bottom, and since the impeller

discharge flow rate is higher, this impeller is more suited to solids suspension.

• Laboratory measurements have indicated that the Lightnin A315 can produce mass

transfer rates at least 10–15% higher for the same power input.

Since advantages such as these are claimed for the Lightnin A315, and since such

impellers are relevant to industrial practice, it has been thought worthwhile to include

modelling of the Lightnin A315 in this study. It was beyond the scope of this study to

assess claims of the manufacturer such as higher mass transfer. Rather, the modelling

has aimed at demonstrating the applicability of the method to a tank with this type of

impeller, so as to indicate a degree of generality in the modelling method.

2.7 Scale-up of stirred tank reactors

In the development or improvement of chemical processes, it is normal to carry out tests

at laboratory scale to determine, for example, reaction kinetics, residence times, product

yield, etc. A scale-up procedure is then required, to ensure that the reaction rate and

product yield achieved in a small tank (e.g. 1–10 litres) translate to economic results on

a much larger tank (possibly thousands of cubic metres in volume). Likewise, to study

an existing full-scale process, it is useful to be able to scale down to laboratory scale.

For a new process, an intermediate-size pilot plant is often built in order to reduce risk.

Poor scale-up procedures can have serious economic implications in terms of excessive

energy costs or less than expected production rates (Wernersson & Trägårdh, 1999).


29/218


21

The essential difficulty in scale-up (Oldshue et al., 1992) is that it is not possible, by any

scale-up technique, to maintain all of the properties of a stirred tank constant as the

physical scale is increased. For example, stirred tanks are often characterised in terms of

dimensionless groups such as the Reynolds number, Re, the Froude number, Fr, and the

Weber number, We. However, these are proportional to ND2, N

2 D and N

2 D

3

respectively, so scaling while keeping any one of these groups constant means that the

other groups cannot be kept constant (Oldshue et al., 1992). An often-used scale-up

procedure is to maintain constant power per unit volume, but with this approach, as the

physical scale is increased, the distribution of fluid shear rates will change, since the

maximum shear rate will increase while the average shear rate will decrease. Also,

applying the principle of constant power per unit volume, mixing and circulation times

increase with scale.

Hence, scale-up is an uncertain process. According to Oldshue et al. (1992) the general

procedure should be to determine the feature of the process which is considered to be

controlling, and to scale up while keeping that characteristic constant. Other aspects of

the process will inevitably be different. For gas-liquid operations in stirred tanks, there

appears to be little agreement in the literature as to the best scale-up procedure.

Tatterson (1991) summarised some recommended procedures. For example, Westerterp

et al. (1963) recommended scale-up based on equal tip speed and D/T ratio. However,

Bourne (1964) recommended a basis of equal power per unit volume. Nishikawa et al.

(1981) recommended maintaining equal vvm (volumetric flow of gas per unit liquid

volume). According to several authors (Chandrasekharan & Calderbank, 1981; Oldshue,

1994), the usual assumption of maintaining geometric similarity is not necessarily the

best approach.

2.8 Advanced experimental methods

Earlier studies of stirred tanks were limited to measurements of a global nature, e.g.

power draw (e.g. using a torque meter) or overall interfacial area or k la (usually inferred

through measurement of some well-defined chemical reaction). Brief mention is made

here of more advanced experimental methods, since these provide one approach towards

gaining more detailed knowledge of the internal, three-dimensional flow in a stirred

tank.


30/218


22

For transparent, single-phase flow, advanced non-intrusive measurement techniques

such as particle image velocimetry (PIV) (e.g. Sharp & Adrian, 2001) and laser doppler

velocimetry (LDV) (e.g. Yianneskis et al., 1987; Costes & Couderc, 1988; Wu &

Patterson, 1989; Hockey, 1990; Mavros et al., 1996; Lee & Yianneskis, 1998) are

available, and these have been applied to obtain detailed measurements of liquid

velocities, flow patterns and turbulence parameters in stirred tanks.

For multiphase mixtures, there are limitations on the applicability of methods such as

LDV and PIV due to opacity of the mixture. However, one technique for two-phase

mixtures is phase doppler particle anemometry (PDPA), although this has only been

applied to very dilute two-phase mixtures, e.g. in PDPA measurements of a solids

suspension by Petterson and Rasmuson (1998), solids concentration was limited to

0.06%. More recently, PIV has been successfully applied at reasonable gas

concentrations. Aubin et al. (2004a) applied PIV to determine the mean velocities and

turbulent quantities in aerated vessels stirred by downward and upward pumping

pitched blade turbines. The dimensionless aeration number was 0.01 and the total gas

holdup was in the range 3.7–5. 8%.

There are other methods are available for investigating dense multiphase flows,

although these generally involve the use of an intrusive probe. For example, probes

have been developed to measure local gas volume fraction (e.g. Bakker, 1992; Barigou

& Greaves, 1996; Bombač et al., 1997) and local bubble size (e.g. Barigou & Greaves,

1992).

Another experimental method, which is non-intrusive and can be applied to opaque

liquids and multiphase flows, is electrical resistance tomography (ERT) (Mann et al.,

1996). This is based on using an array of electrodes positioned around the periphery of a

vessel to map out internal differences in electrical conductivity or resistivity over

different cross-sections in a tank. Therefore, it can be applied to various applications

where there are differences in conductivity. Mann et al. (1996) reported its use for

imaging mixing of strong brine solution into a lower concentration solution, imaging of

a vortex air-core inside a stirred vessel, and measuring the gas-voidage distribution in a

gas-sparged mechanically-stirred vessel.


31/218


23

All such experimental techniques are in general limited to laboratory or pilot-scale

tanks, since carrying out measurements at full-scale is generally too expensive or

impractical. In addition, the requirements of transparent liquids and ambient

temperatures and pressures limit most experimental investigations to model fluids (e.g.

air and water). While there is some potential for ERT to overcome this limitation,

measurements seem mainly limited to concentration fields, and issues remain relating to

spatial resolution and quantitative interpretation of the measured electrical signals.

Hence, in terms of experimental methods, it is difficult to investigate the effects of

scale-up or to investigate the behaviour of real, reacting systems.

2.9 Conclusions

In this chapter, typical applications of gas-sparged stirred tanks have been outlined, and

the design features and flow characteristics of these tanks have been discussed. It has

been shown that how the global or overall characteristics can be predicted using various

dimensionless parameters, empirical correlations and scale-up rules. However, such

approaches present a range of difficulties. One consideration is that the designs of

industrial tanks often differ from the laboratory configurations where correlations have

been developed. Also, correlations and scale-up rules only provide estimates of global

parameters, without any insight into the details of the fluid flow. The flow is very non-

uniform in many ways, such as with respect to velocities, turbulence, and phase

distributions. In addition, transient behaviour may need to be described, e.g. with the

addition of a reagent to a stirred reactor which undergoes a fast chemical reaction, the

details of the mixing process may be important, rather than merely an estimate of the

mixing time. Scale-up procedures also present another problem, since it is often unclear

as to which parameter should be kept constant, e.g. superficial velocity or power per

unit volume, and it is not possible to keep all aspects of a process similar with changing

physical scale.

Hence, as emphasised by Bakker (1992), for proper understanding of the fluid dynamics

and reliable design and scale-up, it is necessary to investigate internal flow structures

and phase distributions. Experimental methods, such as LDV, PIV, ERT and various

probes, provide one approach towards obtaining the required information about internal


32/218


24

flow structures. However, such methods may be time-consuming and costly, they may

be limited to transparent liquids and model fluids, and are generally limited to the

laboratory scale. Hence, there is a clear need for reliable computational modelling

methods.


33/218


25

Figure 2.1. Six-bladed Rushton turbine (Mixing Equipment Co.).

D2

D

WB

H

C

LB

T

B

Liquid level

Drive shaft

6 bladed disc

turbineBaffle

D2

D

WB

H

C

LB

T

B

Liquid level

Drive shaft

6 bladed disc

turbineBaffle

Figure 2.2. Layout of a standard configuration tank with a Rushton turbine (notation and relative

dimensions given in the text).


34/218


26

Figure 2.3. Lightnin A315 impeller (from Post Mixing website:

http://www.postmixing.com/mixing%20forum/impellers/impellers.htm).


35/218

Chapter 3. Review of Modelling Methods

27


3.1 Introduction

This chapter reviews previous efforts at CFD modelling of stirred tanks. Before

proceeding with this review, the basic principles of CFD modelling have been

summarised, to provide some background to the modelling issues. Then, the published

literature has been reviewed, considering both single-phase flow and gas-liquid

dispersion in stirred tanks. Also, some mention is made of relevant simulations in other

related situations, such as solids suspension in stirred tanks or gas-liquid flow in bubble

columns. By reviewing the literature, the prior state of development of relevant

modelling methods has been assessed. Several principle issues were identified relating

to shortcomings of previously published modelling methods, or the need for further

clarification where different authors have proposed different approaches.

3.2 Basic principles of computational fluid dynamics

Computational fluid dynamics is a method for simulating the flow of liquids, gases and

particulate solid particles, either separately or in some multiphase combination, along

with other associated phenomena such as chemical reactions and mass and heat transfer.

This is done through the numerical solution of the basic equations governing

conservation of mass, momentum and enthalpy in a fluid, where the fluid is assumed to

be a continuum without regard to the details of its molecular structure.

For an isothermal flow, the governing fundamental equations are the equations of

conservation of mass and momentum, which are given in vector notation by (Bird et al.,

1960):

0)( =•∇+∂∂

u ρ ρ

t (3.1)

( )( ) guuuuu ρ μ ρ ρ +∇+∇•∇+−∇=•∇+∂

∂ T)()(

L pt

(3.2)

where ρ is fluid density, u is the velocity vector, p is the dynamic pressure, t is time, and

μ L is the laminar viscosity.


36/218


28

Equation 3.2 is often referred to as the Navier-Stokes equation. The application of

equations 3.1 and 3.2 is most straightforward when applied to laminar flow of a single-

phase fluid. However, practical flow problems generally involve a range of

complexities, such as the presence of turbulence or the dispersion of one phase in

another. The same equations still apply, and application to such complex flows without

further modification is referred to as direct numerical solution (DNS). However, the

DNS approach is generally impractical due to the huge computer resources required.

Therefore, averaged forms of these equations are usually solved. In such cases where

the equations have been averaged, empirical closure relations must be applied, and the

appropriate forms of these closures remain an active subject of investigation.

Whether using the exact form or an averaged form of the conservation equations, it is

usually not possible to obtain an analytical solution to these equations, except for a

limited number of simple geometries, and therefore a numerical routine is required to

find a solution (Fletcher, 1991). The method involves representation of the governing

partial differential equations by a set of algebraic equations which can be solved on a

computer (Fletcher, 1991). The geometry of interest is discretised into a number of

nodes or cells and the equations are solved over these cells using a range of iterative

techniques. Methods available for spatial discretisation include spectral, finite element

and finite volume methods (Fletcher, 1991). Finite volume meshes are most commonly

used. Within this category, meshes may be regular or body fitted, and structured or

unstructured. The simulations carried out for this study are based on body-fitted, block-

structured finite volume meshes, which is typically the case in previously published

work related to CFD simulation of stirred tanks.

Numerical solution of the Navier-Stokes equations involves a great deal of complexity,

and therefore, to assist in carrying out such CFD simulations, generalised software

packages have been developed and commercialised, and these are commonly used as a

basis for setting up a model, which may then be customised within the limits allowed by

the software supplier, e.g. through user-supplied subroutines. Commonly used

commercial software packages include Fluent, CFX4, and Star-CD. Many of the

modelling studies mentioned here in reviewing the literature were carried out using one

of these codes as a basis, although custom-developed codes were used in other cases,

and sometimes the particular code is not mentioned. The modelling work carried out for


37/218


29

this thesis was performed within the framework of CFX4, using additional user-written

subroutines as required.

The conservation equations, equations 3.1 and 3.2, are applied most readily to laminar

flow. However, the majority of practical flow problems involve turbulent flow, meaning

that the flow exhibits unsteady time fluctuations and related turbulent eddy structures.

In principle, direct numerical simulation may be attempted, wherein all temporal and

spatial scales of the flow are resolved. However, this approach tends to lead to an

enormous demand on computer memory and computation time, beyond the practical

capabilities of most computers. Hence, for practical simulations it is normal to use

approximate methods based on averaged forms of the equations, leading to the

introduction of a turbulence model. An intermediate approach known as Large Eddy

Simulation (LES) involves averaging over the smaller turbulence structures while still

calculating the large turbulent structures. However, in most cases, the equations

governing fluid flow are time averaged following a procedure called Reynolds

averaging (Fletcher, 1991; CFX4 Solver Manual, 2002).

In this approach, the instantaneous values of variables such as velocity u are expressed

as the sum of a mean and a turbulent fluctuating component, U and u′, according to:

uUu ′+= . (3.3)

These expressions are substituted in equations 3.1 and 3.2, and the equations are then

averaged. This leads to the following form of the equations:

0)( =•∇+∂∂

U ρ ρ

t , (3.4)

guuUUUUU

ρ ρ μ ρ ρ

+′′−∇+∇•∇+−∇=•∇+∂∂

))((()()(

T LPt

, (3.5)

where each capitalised variable represents an averaged mean quantity. It should be

emphasised that since the numerical method solves for mean values of variables, the

resulting flow field will look considerably smoother compared to an actual “snapshot”

of the flow for a given instant in time, as would be obtained for example by

experimental methods such as particle image velocimetry (PIV).

It is seen that the Reynolds-averaged equation of momentum conservation contains an

additional term, )uu ′′•∇− ρ , called the Reynolds stress, which arises because the


38/218


30

product of fluctuating velocity components is non-zero. The Reynolds stress tensor

represents the additional momentum transport due to the cascade of turbulent eddies.

The averaged equations cannot be solved unless a closure expression is specified for the

Reynolds stress, and this requires a turbulence model.

Turbulence modelling is a large field in itself and remains a major challenge for the

science of fluid mechanics. Turbulence models vary in their level of complexity, and

accordingly in the level of detail at which they attempt to model the phenomena of

turbulence. Simpler models include the Prandtl mixing length model, which however

requires a priori knowledge of the mixing length. The most popular models are the two-

equation models, such as the k – ε model.

A first step in turbulence models such as k – ε is to invoke the Boussinesq eddy viscosity

concept (Bakker, 1992; CFX4 Solver Manual, 2002), where the Reynolds stress is

assumed to be given in terms of a turbulent viscosity, μ T , and the mean flow velocity

gradients according to:

( ) IUUuu k T ρ μ ρ 3

2T −∇+∇=′′− , (3.6)

where k is the turbulent kinetic energy per unit mass, which can be expressed in terms

of the fluctuating velocity component, u′, as:

2

21 u′=k , (3.7)

and I is the identity tensor.

The turbulent viscosity, μ T , is calculated according to:

ε ρ μ μ

2k

C T = , (3.8)

where ε is the rate of dissipation of turbulent kinetic energy per unit mass and C μ is a

constant whose value is usually given as 0.09 (CFX4 Solver Manual, 2002).

The variables k and ε in the turbulence model are calculated through conservation

equations according to (CFX4 Solver Manual, 2002):

ρε σ

μ

μ ρ

ρ

−+⎟⎟ ⎠

⎞

⎜⎜⎝

⎛

∇⎟⎟ ⎠

⎞

⎜⎜⎝

⎛

+•∇=•∇+∂

∂ Π k k t

k

k

T

L)(

)(

U , (3.9)


39/218


31

k c Π

k c

t

T L

2

21)()( ε

ρ ε

ε σ

μ μ ε ρ

ε ρ

ε

−+⎟⎟ ⎠

⎞⎜⎜⎝

⎛ ∇⎟⎟

⎠

⎞⎜⎜⎝

⎛ +•∇=•∇+

∂∂

U , (3.10)

where c1, c2, σ k and σ ε are model constants (values for these are given in Table 8.3), and

Π is the production of turbulent kinetic energy by mean velocity gradients given by:

( ) ( ) ( )( )k Π T LT L ρ μ μ μ μ +•∇+•∇−∇+∇•∇+= UUUUU3

2T . (3.11)

In applying boundary conditions at the walls in turbulent flow calculations with the k -ε

model, it is usual to use wall functions to model the boundary layer, since otherwise a

highly refined grid is necessary to resolve the steep gradients of velocity and other

variables in the boundary layer. The wall function approach is formulated using the

concept of a universal law of the wall (Launder and Spalding, 1974; Lathouwers, 1999;

CFX4 Solver Manual, 2002). This concept has been supported by experimental

evidence for a range of flows and assumes that there is constant stress in the near-wall

region and the eddy length scale is proportional to the distance from the wall. These

assumptions lead to a logarithmic velocity profile near the wall. A relationship is

obtained between the shear stress at the wall, τ w, and the velocity component parallel to

the wall in the adjacent node, adjt u , , according to:

adjt

adj

adjw u

Ey

k C

,

21

41

)ln( +=

κ ρ τ

μ , (3.12)

where κ is the von Karman constant, E is the roughness constant, C μ is a constant with

the usual value of 0.09, k adj is the turbulent kinetic energy in the cell adjacent to the

wall, and +adj y is the scaled distance to the wall, which is given by:

adjw

adj yk C

yμ

ρ μ 2

14

1

=+ , (3.13)

where yadj is the distance from the adjacent cell centre to the wall. The equation for the

wall stress provides a boundary condition for the tangential velocity component. The

value of the turbulent kinetic energy in the cell adjacent to the wall is obtained by

solving the transport equation for k , but using a special treatment for the production

term (Lathouwers, 1999; CFX4 Solver Manual, 2002). The transport equation for


40/218


32

turbulent energy dissipation rate is not solved adjacent to walls, but instead, its value is

obtained through the relation (Lathouwers, 1999):

adj

adj

yk C

κ ε μ

23

43

= . (3.14)

For the wall function approach to be valid, it is necessary that the first cell centre

adjacent to the wall should lie within the part of the boundary layer described by the

logarithmic velocity profile. Therefore, for the wall function approach to work properly,

it is required that the non-dimensionalised distance to the wall of each neighbouring cell

should fall in the range 30011


41/218


33

More complicated turbulence mo

Computational Modelling of Gas-liquid Flow in Stirred Tanks-Thesis-2005

Documents

Transcript of Computational Modelling of Gas-liquid Flow in Stirred Tanks-Thesis-2005