IMPERIAL COLLEGE, LONDON

A study of the mixing efficiency in a lab scale bioreactor using CFD

By

David Docherty and

Thomas Jones

Supervised by Dr Kontoravdi, Prof Shah and Prof Xu,

[16/12/2011]

1

1. Abstract

The world’s energy demand is ever increasing and with current resources under threat, it is

essential to look to renewable energy for the future. Liquid biofuels have potential to be a

leading form of renewable energy, especially due to the advances that have been made in

the conversion of cellulosic biomass into bioethanol. This report details a study into the

mixing efficiency of a lab scale bioreactor that does precisely this; using pre-treated

miscanthus, a high yield biofuel feedstock, as the source of cellulose. The mixing efficiency is

analysed with the use of Computational Fluid Dynamics (CFD). A geometrically accurate

model of the reactor is created using AutoCAD, ‘meshed’ using ANSYS ICEM and then

imported into ANSYS FLUENT where CFD simulations are performed. Simulations are run to

investigate how particle size, solids loading and impeller speed affected the mixing

efficiency. Results are compared both quantitatively and qualitatively from the output of

FLUENT. The best case is found to have a solids loading of around 15%, particle diameter as

small as practicable and an impeller speed of 400-600 rpm. A compromise is required as

cases that show the best mixing have a greater power requirement, which ultimately may

reduce the efficiency of the process. Limitations in the model are observed and reasonable

caution is urged over accuracy of the results. However it is concluded this study forms a

strong base on which further investigation into the process can be performed.

Keywords: Mixing, CFD, homogeneity, Particle size, Solids loading, Impeller speed

2

Acknowledgments Many thanks to Prof Xu for her suggestions, guidance and help throughout the project, Wenbo Zhan for his willingness to assist in developing an effective ANSYS FLUENT model, Jeremy Bartosiak-Jentys for providing information on the processes going on in the reactor and access to the reactor itself and Ryo Torii for his assistance in using ANSYS ICEM.

3

2. Nomenclature

Latin Alphabet

Symbol Meaning Units

A Area m2

Lift Coefficient N/A

Virtual Mass coefficient N/A

Drag Coefficient N/A

model constant N/A

model constant N/A

model constant N/A

model constant N/A

model constant N/A

Particle diameter m

Impeller diameter m

Force due to phase interaction acting on phase k Kg.m-2.s-2

Drag force Kg.m-2.s-2

Virtual mass force Kg.m-2.s-2

Lift force Kg.m-2.s-2

Gravitational acceleration m.s-2

Turbulence production term m2.s-2

Turbulent kinetic energy m2.s-2

Mixing length m

Impeller rotational speed rps

“Just-suspension” speed rps

Power number N/A

Flow number N/A

Number of particles per unit volume m-3

Unit vector at a normal to the surface N/A

4

Number of planes N/A

Power W

Pressure Kg.m-1.s-2

Impeller discharge flow m3.s-1

Reynolds Number N/A

Turbulence source term N/A

Impeller/tank geometry constant N/A

Time s

T Torque N.m

TD Tank Diameter m

Velocity (one-dimensional) m.s-1

Velocity vector m.s-1

Averaged velocity vector m.s-1

Average of the averaged velocity vector m.s-1

Velocity Fluctuation m.s-1

Average of the velocity fluctuation m.s-1

Reynolds stresses m2.s-2

Velocity vector m.s-1

Settling velocity m.s-1

Velocity of particle relative to fluid m.s-1

Volume fraction N/A

Average volume fraction N/A

Greek Alphabet

Under relaxation factor N/A

Turbulent kinetic energy dissipation rate m2.s-2

Generic transport variable N/A

Dynamic viscosity Kg.m-1.s-1

Kinematic viscosity m2.s-1

5

Density Kg.m-3

Turbulent viscosity of the fluid Kg.m-1.s-1

Concentration of dispersed phase (v/v) N/A

Effective diffusivity m2s-1

Subscripts

phase N/A

Cartesian coordinate N/A

Cartesian co ordinate N/A

Dispersed phase N/A

Fluid phase N/A

particle N/A

6

Table of Contents 1. Abstract .................................................................................................................... 1

2. Nomenclature ........................................................................................................... 3

3. Introduction .............................................................................................................. 9

3.1. Aims ........................................................................................................................... 12

4. Background and Theory .......................................................................................... 14

4.1. Mixing ........................................................................................................................ 14

4.1.1. Mixing in stirred tanks ....................................................................................... 14

4.1.2. Mixing Correlations ............................................................................................ 18

4.2. Expected findings ...................................................................................................... 20

4.3. Previous work on the CFD modelling of stirred tanks .............................................. 20

4.4. Mathematical Modelling ........................................................................................... 21

4.4.1. Basic fluid flow modelling .................................................................................. 21

4.4.2. CFD application to multiphase systems ............................................................. 22

4.4.3. Turbulence .......................................................................................................... 27

4.5. CFD solutions ............................................................................................................. 33

4.5.1. Geometry ........................................................................................................... 33

4.5.2. Meshing .............................................................................................................. 33

4.5.3. Boundary conditions .......................................................................................... 33

4.5.4. Solution procedure ............................................................................................. 34

4.5.5. Modelling stirred tanks in CFD ........................................................................... 37

4.6. Bioreactor set up ....................................................................................................... 38

5. Computational Methods ......................................................................................... 40

5.1. Geometry generation ................................................................................................ 40

5.2. Mesh creation ........................................................................................................... 41

5.3. Simulation ................................................................................................................. 42

5.3.1. Theoretical models ............................................................................................. 42

5.3.2. Solution Methods ............................................................................................... 44

5.4. Assumptions .............................................................................................................. 44

5.4.1. Particle size ........................................................................................................ 44

5.4.2. Density ............................................................................................................... 45

6. Results and Discussion ............................................................................................ 46

6.1. Effectiveness of the model ........................................................................................ 46

6.2. Base Case ................................................................................................................... 47

6.3. Impeller Speed .......................................................................................................... 48

6.4. Particle size ................................................................................................................ 54

6.5. Solids loading............................................................................................................. 56

6.6. Baffles ........................................................................................................................ 59

6.7. Limitations of the results .......................................................................................... 60

6.7.1. Geometry ........................................................................................................... 60

7

6.7.2. MRF model ......................................................................................................... 61

6.7.3. Theoretical models ............................................................................................. 61

6.7.4. Reaction effects .................................................................................................. 61

6.7.5. Initial loading conditions .................................................................................... 62

6.7.6. Assumptions ....................................................................................................... 62

6.8. Recommendations for further work ......................................................................... 62

6.8.1. Transient effects ................................................................................................. 63

6.8.2. Reaction ............................................................................................................. 63

6.8.3. Models used ....................................................................................................... 63

6.8.4. Geometry and scale up ...................................................................................... 64

7. Conclusions ............................................................................................................. 65

8. Bibliography ........................................................................................................... 66

9. Appendices ............................................................................................................. 71

8

List of Figures Figure 1: Typical Rushton disc turbine ..................................................................................... 14

Figure 2: Typical flow pattern (Nagata, 1975). ........................................................................ 15

Figure 3: Graph showing S against T/D with T/C as a parameter for disc turbine impellers .. 19

Figure 4: Tank schematic ......................................................................................................... 38

Figure 5: Model of the flow domain produced ........................................................................ 40

Figure 6: Mesh produced ......................................................................................................... 42

Figure 7: Base case flow field generated ................................................................................. 46

Figure 8: Base Case results....................................................................................................... 48

Figure 9: Results obtained at impeller speeds of 100 and 150 rpm ....................................... 49

Figure 10: Results obtained at impeller speeds of 300 and 400 rpm ...................................... 50

Figure 11: Results obtained at impeller speeds of 600 and 800 rpm ...................................... 50

Figure 12: Results obtained at an impeller speed of 1000 rpm .............................................. 51

Figure 13: showing Impeller Speed against power requirement ............................................ 53

Figure 14: Showing the effect of impeller speed against homogeneity .................................. 53

Figure 15: Results obtained with particle diameters of 1.5 and 2 mm .................................. 55

Figure 16: Results obtained with particle diameters of 0.5 and 1 mm ................................... 55

Figure 18: graph illustrating the change in homogeneity with volume fraction ..................... 57

Figure 19: Results obtained with volume fractions of 0.3 and 0.4 .......................................... 58

Figure 20: Results obtained with volume fractions 0.1, 0.15 and 0.2 ..................................... 59

Figure 21: Results obtained when baffles are introduced....................................................... 60

Figure 22: Graph showing S against T/D with T/C as a parameter for disc turbine impellers 71

List of Tables Table 1: Information about various biomass feedstock (Boersma et al, 2011) ...................... 11

Table 2: Key geometric parameters of the mixing tank .......................................................... 20

Table 3: Commonly used correlations for the drag coefficient ............................................... 26

Table 4: A summary of the main turbulence models .............................................................. 28

Table 5: Impeller and Tank dimensions ................................................................................... 39

Table 6: Software used ............................................................................................................ 40

Table 7: Constants required by the realizable model .............................................................. 43

Table 8: Discretisation used in simulations ............................................................................. 44

Table 9: Variables 'base case' values ....................................................................................... 47

Table 10: Base Case parameters, used to calculate just suspension speed ............................ 72

Table 11: Power requirements for varying impeller speed ..................................................... 73

Table 12: Homogeneity calculations for the base case ........................................................... 74

9

3. Introduction

Managing the world’s energy resources and requirements is one of the greatest challenges

facing mankind. How this is dealt with will have a huge effect on generations to come. With

the world’s population reaching 7 billion and continuing to rise, resources will be continually

stretched and the need to find alternative energy sources will become even more

paramount. The demand for energy is also going to be greatly affected by the economic

growth in developing countries especially the BRIC countries.

Currently around 80% of the world’s energy supply comes from fossil fuels, with

combustible renewables contributing approximately 10% (of which liquid biofuel provides

only a small fraction). The other proportion is made up from nuclear, hydropower and other

green sources such as wind and solar (IEA, 2010). The continued reliance on fossil fuels to

supply most of the world’s energy is having a great impact on the climate due to the release

of greenhouse gases. It is essential that the release of these gases, namely carbon dioxide, is

reduced to limit the devastating effects that are sure to ensue. This impact can be reduced if

more low carbon emission technology and green energy sources are introduced.

The demand for oil is over 86 million barrels per day, equating to over 30 billion barrels per

year. (EIA, 2011) This is a massive demand and with concern over the long term future of oil,

opens up a large potential market for alternative fuels. Liquid biofuel is the most promising

form of alternative fuel to take advantage of the high dependence on oil as it is easily

incorporated into current technology. Plant biomass is the only feasible feedstock for the

sustainable production of biofuels, due to its high energy content and ability to be grown

quickly and cheaply (la Grange et al, 2010).

Biomass can be used in many ways to generate energy, the easiest and most common

method is just to burn it and generate energy from the heat given off. The method that is of

interest for this report is conversion of biomass to liquid biofuels. The biomass first must be

broken down into simpler sugar molecules, and then fermented to produce alcohol. This is

then distilled to produce a purer product, with higher fuel content. This practice is nothing

10

new; alcohol has been produced for human consumption amongst other uses for tens of

thousands of years.

The most common biofuel is bioethanol which is produced through the fermentation of

sugars, starch and cellulose. Currently maize, wheat and sugar cane are the main forms of

biomass used to produce ethanol fuel. Due to the nature in which maize and sugar are

farmed and then converted into bioethanol the overall process does very little to help

reduce carbon dioxide emissions or reduce the need for fossil fuels. The production of

biofuels has risen significantly in recent years, this increase in production has been linked

with increased food prices as a large amount of arable land is required for the crops

(Bounds, 2007).

Even though there are concerns with the production of biofuels, it remains one of the most

promising renewable energy sources due to its large-scale availability and low cost, in fact

ethanol fuels are currently used widely in some countries already, namely USA and Brazil.

The ethanol is blended with petrol and used as a transport fuel. In Brazil it is mandatory to

blend ethanol with petrol; the current legal requirement is around 25%. This has been

implemented to reduce the countries’ dependence on oil (de Brito Cruz, 2008).

There has been a shift away from the use of sugar and starch crops as the source for

biofuels due to the problems mentioned, and advancement in technology to convert

cellulose into ethanol. This is extremely promising due to the fact that cellulose is the largest

component in plant cell walls and is therefore abundant. The introduction of cellulosic

ethanol will allow ethanol fuels to play a much larger role in renewable energy (World

Energy Outlook, 2006).

Miscanthus is a type of grass with great potential for ethanol production. It is rich in

cellulose and has advantages over traditional biofuel crops as it is not currently used as a

food source. It is able to produce more ethanol per hectare and needs little attention to

grow and develop. Miscanthus is compared to other biofuel crops in Table 1, it can be seen

that miscanthus has the potential to be a very useful biomass source. Advances in the

11

conversion of cellulose to ethanol mean that using miscanthus to produce ethanol is now a

potentially economically viable alternative. This study focuses on a lab scale reactor that

converts pre-treated miscanthus into ethanol (Balan et al, 2007).

Table 1: Information about various biomass feedstocks (Boersma et al, 2011)

Feedstock

Harvestable

Biomass

(Tons/acre)

Ethanol(gal/acre)

Million acres

needed for 35

billion gallons

of ethanol

% 2006

harvested U.S.

cropland

Corn grain 4.5 456 12.6 24.4

Corn stover 3.3 300 19.1 37.2

Corn total 7.8 756 7.6 14.8

Switchgrass 4.6 421 13.6 26.5

Miscanthus 13.2 1198 4.8 9.3

The process for bioethanol production from lignocellulosic biomass consists of

delignification, saccharification, fermentation, recovery and purification of ethanol.

(Boonchild et al, 2010) For the commercial and economically viable production of cellulosic

ethanol the total energy balance must be considered. The main reason that biofuel cannot

be generated from lignocellulosic biomass at an industrial scale is the absence of low-cost

technology enabling the process (Lynd et al, 2005).

The main barrier to the production of ethanol from cellulosic biomass is the recalcitrance of

the biomass materials. The sugars that have potential to be fermented are locked in a

complex polymer composite created to resist biological and chemical degradation. This

recalcitrance is due to the crosslinking between the polysaccharides and the lignin via ester

and ether linkages. To make use of the sugar, the cellulose must be disconnected from the

lignin. Once disconnected, the cellulose can be broken down into simple monosaccharides

which can easily be fermented (U.S. Dept. of Energy, 2006).

This report focuses on understanding the fluid dynamics of a lab-scale bioreactor that

incorporates the production of saccharolytic enzymes, the hydrolysis of carbohydrate

components to sugar, the fermentation of hexose sugars and the fermentation of pentose

sugars. These all occur in the same step in the process configuration called consolidated

bioprocess (CBP). The CBP strategy is applicable to any fermentation product. Even though

12

ethanol is currently the most popular many other fuels are of potential interest, including

butanol, dimethylfuran, and gamma-Valerolactone (Lynd et al, 2005) (Tokay, 2002).

Mixing operations are central to many industrial processes; therefore having a good

understanding can lead to drastic improvements in product quality and efficiency of

operations. Gaining a good understanding of the fluid dynamics of the lab scale bioreactor in

question, could lead to improvement of efficiency and conversion. It would also help with

the design process when generating a scaled up industrial size version of the reactor.

In order to understand the mixing and fluid flow in the reactor Computational Fluid

Dynamics (CFD) will be used. CFD is a computer modelling tool that has found increasing use

in engineering design (e.g. automotive and aerospace industries) and atmospheric modelling

among other applications. It solves the Navier-Stokes equations that govern the fluid flow

along with empirical correlations and semi-empirical models to obtain spatially- and

temporally-resolved flow fields which can be further processed to produce graphical

representations of the process being studied. CFD is a very useful tool and has the scope to

simulate a wide range of complex flow situations; including stirred tank reactors.

3.1. Aims

The aim of this study is to use CFD to determine the optimal particle size and stirrer speed

for different loading conditions of miscanthus in order to achieve the best mixing efficiency

in an exact model of a prototype reactor. This reactor is currently being used to study the

production of ethanol from pre-treated miscanthus and a good understanding of the

optimum conditions to promote mixing is necessary. This information can aid the

experimental study and it is hoped the reactor can be used as a basis for the scaling up of

the operation for more efficient production of biofuels in the future. This is quite a general

study that can be used as a starting point for future investigation into the reactor system

both experimentally and computationally.

Producing an appropriate computational model requires the use of engineering drawing

software to create a geometrically accurate representation of the tank, meshing software

13

and a CFD code to solve the flow field inside the tank.

By varying the volume fraction of the dispersed phase (i.e. miscanthus particles), the

rotation speed of the impeller and particle size, numerical solutions at different loading and

operating conditions can be obtained, allowing for the mixing efficiency to be assessed.

14

Figure 1: Typical Rushton disc

turbine

4. Background and Theory

4.1. Mixing

Mixing is the process of manipulating a heterogeneous physical system with the intent of

making it more homogeneous. Solid-liquid mixing is nothing new in industry; it is one of the

key operations in many processes such as pharmaceuticals, food and paper production. This

has led to many different studies into the interactions that occur during mixing operations.

Mixing can be defined as an operation that causes reduction of inhomogeneity in order to

achieve a desired process result. The inhomogeneity can be one of concentration, phase or

temperature.(Atiemo-Obeng et al, 2004) This report deals with the process of achieving a

homogenous phase after bringing together a solid and a liquid, the homogenous phase

being created though suspension of the solid particles in the liquid. Mixing can be described

at different scales, depending on the area of interest. Micromixing looks at mixing on a

molecular scale, macromixing is on a larger scale and looks at the bulk distribution, which is

the focus of this report, and mesomixing is in between the two and is mainly related to

concentration gradients (Baldyga et al, 1999).

4.1.1. Mixing in stirred tanks

To create a suspension of solids in the liquid phase, it is

necessary to mechanically agitate the tank. This is usually

achieved with the rotation of an impeller within the tank.

The tank being modelled in this report uses a Rushton

disc turbine impeller, (a typical one is shown in Figure 1,

Kaufmann, 2011). The type of impeller that is used will

depend on the particular function that is required, as

each impeller will generate a different flow pattern. It is

worth choosing the correct impeller for the situation, as this will greatly affect mixing.

Not only is it worth choosing the correct impeller for the specific type of mixing required, it

is also important to take the impeller diameter, and clearance into account as it will have an

15

impact on the flow pattern and the effectiveness of mixing. The effect of impeller clearance

can be seen in Figure 2 (Nagata, 1975). As seen, if the impeller clearance is too large, there

is a risk that solids may build up in the area directly under the impeller. If the clearance is

too small then there is a risk that particles will build up in the bottom corners of the tank.

Figure 2 also shows the circulation loops created in the flow pattern, the impeller in the

figure is a pitched blade turbine. This type of impeller is good for axial mixing, because of

the slanted blades. The Rushton disc turbine that is used in the tank under investigation is

especially good for radial mixing.

Figure 2: Typical flow pattern (Nagata, 1975).

The impeller performance can be described by its power and flow numbers. The power

number, Np, is used to calculate the impeller power requirements and the flow number, Nqd,

indicates the pumping capacity of the impeller for a given tank geometry (Chudacek, 1985).

The Reynolds number, Re, indicates whether the fluid flow regime is laminar or turbulent.

The mixing ability can be greatly affected by which regime the flow is in. When Re≤10, the

flow is completely laminar, when 10≤Re≤1000, the flow is partly laminar or transitional,

When Re ≥ 1000 the flow is completely turbulent. (Nagata, 1975) To generate good solid-

liquid mixing, it is likely that a turbulent flow regime will be required. The dimensionless

numbers are defined by the following equations (4.1), (4.2), (4.3) (Nagata, 1975):

16

(4.1), (4.2), (4.3)

Where P is the power input, ρ is the density, N is the stirrer speed, D is the impeller

diameter, qd is the impeller discharge flow, and µ is the fluid viscosity.

There are many ways to improve the efficiency of mixing ability of a tank; e.g. varying the

number of impellers (the reactor that is investigated during this study makes use of two of

impellers on a single shaft). It is also possible to introduce baffles, which aim to limit the

tangential flow and convert it into an axial vertical flow. This will reduce the possibility of a

vortex occurring, vortexing is detrimental to solid-liquid mixing because the vortex is likely

to cause solids to build up at the bottom. Baffles are more effective when used with lower

viscosity liquids, as vortexing is less of a problem in high viscosity liquids. (Edwards et al,

1985) The reactor in question has the option to add baffles, so the effect will be investigated

as part of this study.

4.1.1.1 Settling velocity

Settling is the process whereby particles immersed in a fluid settle to the bottom of the

tank. This is due to gravity pulling them downwards. Several variables have an effect on the

ease with which a particle will settle, these include relative particle-fluid densities and

particle diameter. An estimation of the how easily a particle will settle is obtained from in

equation (4.4) (Perry et al, 1984).

√( ( )

) (4.4)

As can be seen from equation (4.4) an increase in both particle diameter and relative

density will increase the speed of settling and thus the difficulty in obtaining a

homogeneous mixture. Therefore it is expected, when investigating the effect of particle

diameters in this study, larger particles will be more difficult to mix and produce a less

17

homogeneous mixture.

4.1.1.2 Homogeneity

The homogeneity of a solid-liquid mixture can be quantified by measuring the average

volume concentration over the entire tank at different planes and with the use of equation

(4.5) (Hosseini, Patel et al. 2010)

√∑ ( )

(4.5)

Where n, Xv and are the number of planes, solid volume concentration, and the average

solid volume concentration respectively. The more planes (n) that are considered, the more

the result from the equation will be representative the whole system. Although this is a

good tool to compare the relative homogeneity of different cases, it is not an absolute

measure of mixing.

4.1.1.3 Packing Limit

The packing limit is defined as the largest volume fraction that can be occupied by the solid

granular phase. Monodispersed spheres can have quite a low packing limit, as this there will

be a fair amount of space between the particles that other particle will not be able to fit.

However if the particles are all different sizes, smaller ones will be able to fill the small gaps

between larger ones, meaning the packing limit will be increased. For a polydispersed

system the packing limit will reach a value close to 1.

4.1.1.4 Mixing time

Mixing time or blend time is defined as the length of time that it takes for a mixture to reach

a specified level of homogeneity. This is important to take into account when assessing the

mixing efficiency, if the mixing time is long; it has the potential to reduce the efficiency

dramatically. Many factors can affect the mixing time such as impeller type, location and

speed (Montante, Moštěk et al.). Although transient calculations will not been carried out as

18

part of this report, it is something that needs to be taken into account when deciding on the

best operating conditions.

4.1.2. Mixing Correlations

4.1.2.1 Zwietering correlation

Settling is one of the main issues that has to be overcome when mixing solid particles into a

liquid phase. The minimum speed at which the impeller needs to rotate to stop settling

occurring is called the just suspension speed; Njs. “just suspension” means that no solid

particle should remain static on the bottom of the tank for longer than two seconds. This

was defined by Zwietering in 1958, by introducing the following correlation (Zwietering,

1958):

(

)

(4.6)

Where S is the dimensionless number accounting for the effect of geometry, ν is kinematic

viscosity, dp is the particle diameter, g is gravitational acceleration, Δρ is the difference in

density, ρ is the liquid density, Xv is the solid volume fraction, and D is the impeller diameter.

The Zwietering correlation is very general, so is good for making initial calculations. However

more studies indicate that it shouldn’t be used when special geometries are involved, or if

the solids’ loading is above 15% by volume (Nagata, 1975) (Tokay, 2002).

Figure 3 shows experimental correlations to calculate S from TD/D and TD/C, where TD, D and

C are the tank diameter, Impeller diameter and clearance below the impeller respectively.

Once S has been found it is possible to determine the NJS value (Nagata, 1975).

19

Figure 3: Graph showing S against TD/D with TD/C as a parameter for disc turbine impellers

The various geometric parameters for the tank under investigation are given in Table 2,

using these values and the correlations shown in Figure 3; S is calculated to be 4. This can

then be used in equation (4.6) to find the just suspension speed. If the base case conditions

are assumed, which is shown in Table 10 in Appendix A, the just suspension speed is

calculated to be 150 rpm. This gives a good first approximation in the absence of detailed

experimental and computational analyses (detailed calculations in Appendix A).

The Zwietering correlation, equation (4.6), shows that there are many conditions and

parameters that will cause the mixing ability to vary. The investigation of these parameters

is the main purpose of this study.

20

Table 2: Key geometric parameters of the mixing tank

Parameter Value (mm)

Tank height, H 156

Tank diameter, TD 105

Impeller diameter, D 50

Impeller clearance, C 20

4.2. Expected findings

From the mixing theory outlined the expected outcomes from the study are:

An increase in the particle diameter will be detrimental to mixing efficiency. As

settling velocity is proportional to particle diameter the settling velocity will increase

and thus the ease with which homogeneity is achieved will decrease.

Increasing impeller rotational speed will improve the mixing. This will increase the

velocity of the fluid and particles which will increase the Reynolds number and hence

the amount of turbulence. A more turbulent flow regime should be more conducive

to mixing.

Higher solids loading will decrease the mixing efficiency. As there are more particles

present more agitation will be necessary to produce a similar level of homogeneity.

4.3. Previous work on the CFD modelling of stirred tanks

There are numerous examples of the application of CFD to the modelling of stirred tanks in

the literature. The majority of these focus upon systems involving a homogeneous fluid and

the flow generated by an impeller (Mavros et al 2001). Multiphase mixing has been studied,

although often to model gas-liquid systems (Lane et al 2005) or in some cases liquid-liquid

systems (Hernandez et al 2011). The studying of biological processes has been performed,

for example the study of cell population growth (Lapin et al 2006).

There is a lack of work in the literature on the modelling of general solid-liquid mixing

21

parameters. While studies have been done that focus on improving correlations for specific

situations (Ochieng and Onyango 2007) or measuring just suspension speed there is not a

large body of work related to studying the effect of impeller rotational speed or solids

loading on the steady state mixing profile; particularly for the study of solids such as

miscanthus.

4.4. Mathematical Modelling

The process of simulating flow fields in CFD is not trivial. Below is a description of the

process and theory behind the CFD modelling.

4.4.1. Basic fluid flow modelling

The governing equations for general fluid flow are the ‘Navier-Stokes’ equations. Developed

in the 19th Century using the principles of conservation of mass and momentum these

equations form the basis of any numerical or analytical solution of fluid flow. The standard

Navier-Stokes equations for an incompressible Newtonian fluid are given in vector form by

equation (4.7) and equation (4.8).

(

) (4.7)

(4.8)

Where ρ is the density, u the velocity vector, t time, g acceleration due to gravity (or any

other body force acting upon the control volume), p the pressure and µ the viscosity.

The Navier-Stokes equations have a wide range of practical uses and can describe situations

as diverse as fluid flow in pipes to modelling the weather. They are partial differential

equations which cannot be solved analytically except for the simplest cases. Therefore,

numerical algorithms have been developed to solve these equations, and these are

generally referred to as CFD.

22

4.4.2. CFD application to multiphase systems

There are several different approaches to modelling this type of system. In each case the

Navier-Stokes equations are solved for the continuous phase. The dispersed phase can be

treated in two different ways, the ‘Lagrangian’ or the ‘Euler’ approach. The Lagrangian

approach treats the secondary phase as a set of discrete particles dispersed in the fluid and

hence utilises Newton’s laws of motion when resolving particle movement. The Euler

approach treats the secondary phase as a separate, immiscible, continuous phase and must

involve several particle interaction terms.

4.4.2.1 Lagrangian approach

Two types of Lagrangian approach are presented here, the ‘Euler-Lagrange’ and ‘direct

numerical simulation’.

Euler-Lagrange- This approach treats the secondary phase as individual discrete

entities (e.g. particle or droplet). This requires Newton’s laws of motion to be

applied to the particle when resolving its motion and effect on the continuous flow

field. This involves estimations for the particle-particle interaction, particle-wall

interaction and the drag force. The dispersed phase droplets are taken to have a

fixed size and shape. The continuous phase is solved using the Navier-Stokes

equations. This approach requires large computational effort as the number of

particles increases.

Direct numerical simulation- This approach reduces the grid size used for solving the

equations down to a scale where individual particles are ‘wrapped’ by the grid and

thus only one relevant velocity acts upon the particle. Turbulence can therefore be

resolved directly. Clearly this approach requires large computational demand and is

not often used. A variation on this approach only resolves larger turbulent eddies

directly and smaller eddies are modelled by closures. This ‘Large Eddy Simulation’

reduces computational demand (Biovin et al., 2000).

Another similar model is the Volume-of-fluid approach. This deals with situations where the

particles are deformable (gas-liquid, liquid-liquid systems). This method also requires high

23

computational demand and is not particularly applicable to the system being modelled and

hence is not considered for this project.

Given the computational demand of the Lagrangian approaches described above they are

not particularly suitable for industrial processes where particle loading is usually high and

many computations are regularly performed. Therefore this project focuses on the Euler-

Euler approach.

4.4.2.2 Euler Approach

In this approach the Navier-Stokes equations are solved simultaneously for each phase

present. However several phase interaction terms must be included to account for particle-

particle, particle-fluid and particle-boundary interactions. Compared to the Lagrangian

approach less computational effort is required, although it does necessitate the use of

closures for certain terms left undecided by the averaged momentum equations. It is not

clear however, whether the physical effects corresponding to these terms are properly

represented by the discrete particle models, since these replace fluid flow field by its locally

averaged form (Jackson, 2000). Hence accuracy is not severely compromised.

The Euler approach uses momentum and mass balances that incorporate a term

representing force per unit volume and momentum transfer at the phase interfaces to

numerically solve the Navier-Stokes equations. The mass and momentum balances for each

phase (neglecting inter-phase mass transfer) are shown in Cartesian form in equations 4.9

and 4.10 respectively.

( )

( ) (4.9)

( )

( )

[ (

)] (4.10)

24

Where the subscript k denotes the phase and represents the forces acting upon the

control volume due to phase interaction. This quantity can be further split into several

different variables that represent the different interactions. In order to account for

turbulent effects the value of the continuous phase viscosity is modified, this is further

explored in section 4.4.3.

Phase interaction models

To model the exchange of momentum between interfaces can be split into 3 terms. ,

and that represent forces on each particle (lift, drag and virtual mass). The

significance of each contribution varies depending upon situation; in general no one term is

usually insignificant (Jakobsen et al 1997).

(4.11)

Lift force

The lift force represents the effect of the liquid phase shearing motion on the dispersed

phase and its movement. Zhang and Prosperetti (1994) derived the following definition.

[( ) ( )] (4.12)

Where ϕ is the volume fraction of the dispersed phase and the lift coefficient. This is

generally accepted to have a value of 0.5 for spherical particles however Auton et al (1988)

have reported a value of 2 for the case of cylindrical particles. There is some uncertainty in

the literature as to the algebraic form of the lift force and its accuracy. However for the

purposes of this study, equation 4.12 is assumed to be valid.

25

Virtual mass force

The force required to accelerate the apparent mass of continuous phase directly in the

locale of the dispersed phase is modelled by the virtual mass force ( ). Again Zhang and

Prosperetti (1994) derived equation 4.13 to describe the force.

(

( )

) (4.13)

Where is the virtual mass coefficient (equal to 0.5 for low values of ). Given that both

and can take the value 0.5, in this case it can be shown that is frame

independent. However this depends upon and particle shape hence is not a universal

result.

Drag force

The drag force is an important part of the phase interactions; it describes the force

experienced by the particle as it moves through the fluid. It is often represented as

| | (4.14)

With w being the velocity of the fluid relative to the particle, A the projected area of the

particle and the drag coefficient. The drag coefficient is a function of the Reynolds

number and hence not a universal constant. Several correlations for the drag coefficient

exist; Table 3 (Ricard 2005) illustrates some of the more commonly used relations.

26

Table 3: Commonly used correlations for the drag coefficient

Correlation Author(s)

( ) Ishii and Zuber (1979)

( ) Schiller and Nauman (1933)

Ihme et al. (1973)

These correlations are valid for spherical particles in the range . With the

particle Reynolds number ( ) given by

| |

(4.15)

Where is the diameter of the particle. For a Reynolds number above 1000 a of 0.44 is

commonly used. Other more extensive correlations exist, e.g. the Syamlal-O’Brien model,

this defines the drag coefficient as:

(

√

)

(4.16)

With the relative Reynolds number and a correlation for the terminal settling

velocity (see Appendix B for full listing of these parameters).

For the general case of dispersed particles in a fluid medium, the expression for the drag

force can be generalised to the drag force per unit volume. The number of particles per unit

volume is given by

27

(4.17)

Where is the volume of the particle. Multiplying this by the expression for the drag force

on a single particle yields.

| | (4.18)

This is the drag force per unit volume used in equation (4.12).

4.4.3. Turbulence

4.4.3.1 Turbulence and CFD

The modelling of turbulence is crucial not only to the analysis of complex flow but also in

the application of CFD. The many eddies and vortices have complex effects impacting on the

accuracy of analytical and numerical models. There are several approaches that have been

proposed over the years to dealing with turbulence. The most common is ‘Reynolds-

Averaging’.

The velocity of the fluid at any point will fluctuate between many values. Rather than trying

to determine the instantaneous velocity it can be split into the ‘ensemble average velocity’

( ) and a fluctuation ( ).

(4.19)

Substituting this result into the Navier-Stokes equations and assuming

The average of fluctuations is zero ( )

Averaging an average makes no difference ( )

28

Average of derivative is equivalent to derivative of average (

)

Yields, in Cartesian form, the Reynolds-averaged Navier stokes equation (RANS)

(

)

(4.20)

The

terms in the RANS equations are referred to as Reynolds stresses. They are an

important part of CFD turbulence modelling and are modelled through turbulence closure

models. A good summary of the common models has been published by Wilcox (1998) and

is summarised in Table 4 (Ricard 2005).

Table 4: A summary of the main turbulence models

Reynolds stresses

Boussinesq’s eddy viscosity

hypothesis

Algebraic models that

don’t include transport

equations for turbulence properties

Prandtl’s mixing length hypothesis

Constant eddy viscosity

Models including transport

equations for turbulence properties

One equation models

Two equation models

model

model

Direct turbulent stress models

Algebraic stress models

Reynolds stress models

Boussinesq’s eddy viscosity relies on the assumption that the turbulent stresses are

proportional to the mean velocity gradient. This is important in the modelling of Reynolds

stresses. Boussinesq’s eddy viscosity hypothesis introduces a turbulent viscosity term. This

term is not a fluid property, being dependant on the local turbulence rate. In the simpler

models this property is assumed to be constant over the whole flow domain; however more

complex models incorporate the transport of turbulence properties.

29

Non- transport of turbulence

These models can be split into two main types

Prandtl’s mixing length hypothesis- This assumes that the eddy viscosity is

proportional to the velocity gradient and a constant, the ‘mixing length’( ). This

model proposed in the 20th century has found wide use, however it does have

limitations. As

approaches zero parameters such as heat diffusivity and eddy

viscosity also do. This is often not a good approximation of the physical situation and

can cause large errors.

Constant eddy viscosity- This is a relatively self-explanatory model that assumes

eddy viscosity to be constant. While being suitable for simple calculations this model

is not appropriate for more complex situations.

Transport of turbulence

Transport models can be split into two main types:

One-equation models that use the local kinetic energy of the turbulent motion ( ) to

determine the local velocity gradient and subsequently approximate the turbulent

viscosity

Two equation models that include another transport term turbulent dissipation ( )

or turbulent frequency ( ) in a second transport equation that determines the

lengthscale governing the size of the larger eddies

model

One of these models, the model, in particular has been used widely and is recognised

generally as a good model (Wilcox, 1998). This model introduces two equations, one for the

estimation of turbulent dissipation ( ) and one that approximates turbulent kinetic energy

( ) shown in Cartesian form in equations 4.21 and 4.22

30

( )

( )

(

)

( ) (4.21)

( )

( )

(

) ( ) (4.22)

Where G and S represent turbulence production and source terms and and

constants. is the turbulent viscosity, this is found from

(4.23)

Where is another constant.

Improvements to the model

There have been modifications to the model to account for particular effects. The

RNG model uses Re-normalisation groups method (RNG) developed by Yakhot et al

(1987) to renormalize the N-S equations to account for the effects of smaller scales of

motion. This modifies the equation 4.22 by differing the source term to account for these

different scales of motion.

The realizable model satisfies certain mathematical constraints for the Reynolds

stresses consistent with the physics of turbulent flows. (Sharcnet, 2010) It modifies equation

4.21 and uses slightly different constants in equation 4.22. It is believed to give a better

approximation for rotational effects.

Turbulence models not relying on Boussinesq’s hypothesis

Despite it’s wide use and acceptance, the model and other models using Boussinesq’s

eddy viscosity hypothesis have the limitation of not capturing the effects of rotational flow

31

or anisotropy. Other models that follow a different approach determine individual

components of the Reynolds stresses. Reynolds stress models use direct transport equations

for turbulent stresses. However these models have been found to produce no better results,

with an additional large computational demand compared to the model (Erdal et al

2000). Algebraic Stress models replace the partial differential equations of the Reynolds

stress models with algebraic forms. These have produced good results for a limited number

of applications.

Due in part to a lack of a better alternative, it was determined the model was the best

available model for use in this project.

4.4.3.2 Walls and turbulence modelling

The presence of walls in a stirred tank affects the turbulence. While a zero velocity is applied

close to the wall, the fluid has a high velocity not far away from this position. Therefore

there is a strong velocity gradient near the wall which has an effect on the turbulence.

Consequently close to the wall turbulence requires special treatment. There are several

approaches to modelling the effects which are summarised below.

Standard wall functions

This approach uses semi-empirical correlations to connect the wall with the fully turbulent

region. This allows the turbulent model to remain unchanged and the viscous sub layer close

to the wall is left unresolved. This means a significant reduction in computational demand.

Near wall model

In this approach, the near wall region is fully meshed and solved. This is a more accurate

approach but has the disadvantage of higher computational demand.

32

Enhanced wall treatment

Enhanced wall treatment attempts to reconcile the accuracy of the near wall method with

the reduced computational demand of standard wall functions. It combines the accuracy of

a two-layer model with fine wall meshes without reducing the accuracy of wall functions.

4.4.3.3 Multiphase

The turbulent considerations so far have generally only considered the turbulence produced

in single phase flow. In the case of multiphase flow turbulence caused by the movement of

particles can be significant. There are several approaches to modelling these multiphase

effects, described below.

Turbulence dispersion- Introducing terms that deal with turbulence production (e.g.

the S ‘source term’ in the model) can describe diffusional motion due to

turbulence

Sato’s eddy viscosity model (Sato 1975)- This model incorporates a fluctuation

velocity into equation 4.19 to account for particle induced turbulence. The

expression for the turbulent viscosity subsequently reads

| | (4.24)

The turbulence production caused by particles and shearing is known as G (equation 4.25). It

is generally regarded as being proportional to the interphase drag force. Boisson and Malin

(1996) define G as

| | (4.25)

Where is the shear stress and the fraction of the work done by the particles that is

transferred to the large eddies. This typically varies from 0.2 to 0.75.

33

4.5. CFD solutions

CFD utilises the theory outlined using numerical methods and algorithms to solve the fluid

flow problem. Following is a description of the different elements required to both produce

a general fluid dynamic model and one specific to this study.

4.5.1. Geometry

Before any computation is done the geometry of the model must be formed in order to

define the problem. This involves using software to produce a 3-D model of the flow domain

in question which can subsequently be used as a starting point for the CFD solving process.

In the case of this project an exact model of the stirred tank bioreactor must be made.

4.5.2. Meshing

Once the geometry is formed the meshing process can begin. Meshing is the process of

splitting the flow domain into a series of smaller regularly shaped control volumes, where

the N-S equations are solved. The control volumes together form a ‘mesh’ of the entire

domain. The mesh size must be sufficiently small such that the solution is mesh

independent. Meshing is necessary as the partial differential equations solved in CFD must

be solved numerically (except in the most basic of cases where an analytical solution is

possible). There are several different ways of ‘meshing’ including unstructured meshes,

hybrid meshes, meshes with embedded refinement and meshes with arbitrary interfacing.

Gosman (1998) has published comprehensive reviews of the different types available.

4.5.3. Boundary conditions

Boundary conditions must be applied to the walls and interfaces in order to accurate model

the situation. For example no-slip at the wall is one condition that is often applied. In this

project enhanced wall functions have been applied at the wall.

34

4.5.4. Solution procedure

The procedure to solve the N-S equations uses a discretisation scheme. There are several

schemes available including finite element, finite volume and discontinuous galerkin

(Hubbard, 2011). In this case a finite volume scheme is adopted. The principle steps in this

scheme are described below

1. Integration of the flow equations over all control volumes (making use of Gauss’

divergence theorem)

2. Substitution of finite-type approximations into the integrated equations the

discretisation is achieved as this converts the integral equations into a set of

algebraic ones.

3. Solution of the algebraic equations is then found using either a segregated or

iterative solver (segregated solver uses a series of separate matrices that are solved

consecutively to reach a solution while an iterative solver fits the equations to one

matrix which is then solved iteratively).

An example of discretisation using a generic variable, ζ, is shown below. The N-S equation

( )

( ) ( ) (4.26)

Can be applied to a control volume yielding

∫ ( )

∫ ( )

∫ ( )

∫

(4.27)

By applying Gauss’s divergence theorem this becomes

∫ ( )

∫ ( )

∫ ( )

∫

(4.28)

35

This solution is then discretised using a differencing scheme. Spatial Discretisation is

necessary as the value of ζ in the centre of each cell has been determined, but for the

advection equations ζ at the interface of the cell is required. This is found by interpolating

between cells.

4.5.4.1 Differencing schemes

There is a range of differencing schemes available to achieve the discretisation:

First order: This assumes the centre cell value of ζ represents an average cell value

and is thus used as the face value.

Power Law: This scheme interpolates the face value of ζ by using the solution to a

one-dimensional diffusion-conduction equation.

Second order: Second order differencing uses a Taylor series expansion of the cell

centred solution around the cell centroid. This is known as a multidimensional linear

reconstruction approach. This produces higher accuracy results than first order.

Central differencing scheme: Central differencing is a second order accurate

scheme that is appropriate for Large eddy Simulation models, providing high

accuracy for this type of model. It assumes a linear change between cell centre and

face value.

QUICK: This type of scheme uses a weighted average of the second order and central

interpolations value. This generally provides a higher order of accuracy than other

schemes.

4.5.4.2 Temporal discretisation

For the case of time dependant phenomena along with the described spatial discretisation

temporal discretisation is also required. For the variable ζ a general expression for its time

dependence is given by.

( ) (4.29)

36

If this is discretized using the first order backwarding scheme this becomes

( )

(4.30)

Due to the non-linearity of equation 4.30 so called ‘under-relaxation’ factors must be used.

This defines the variation of ζ at each iteration.

(4.31)

Where is the under relaxation factor.

4.5.4.3 Pressure velocity coupling

To solve the discretised continuity and momentum equations a scheme is required that

introduces a pressure term to the momentum equation for an incompressible fluid and thus

allows estimation of the velocity field. The various schemes are:

SIMPLE: Semi-Implicit Method for Pressure-Linked Equations (Patankar and Spalding

1971) uses a guessed pressure p and a subsequent correction factor, p’, required to

satisfy the continuity equation. The resulting equation is then solved using an AMG

algorithm.

SIMPLEC: this approach is similar to SIMPLE however uses a modified correction

factor (a change in the flux equation causes this). This approach can improve

convergence.

PISO: Pressure Implicit with Splitting of Operators follows a similar method to

SIMPLE but includes more correction factors, namely skewness and neighbour

correction. It also solves the pressure correction twice; requiring additional

computational demand.

37

4.5.5. Modelling stirred tanks in CFD

There are several approaches to modelling the rotation of an impeller inside a tank. These

are known as Multiple Reference Frame (MRF), Imposed boundary conditions (IBC) and

Sliding mesh.

4.5.5.1 Imposed boundary conditions

This approach uses a fixed reference frame and relies upon boundary conditions imposed in

the impeller region. This is the model used in much early work (often only for the study of a

homogeneous liquid) e.g. Ranade and Joshi (1990). Values of the velocity and turbulent

kinetic energy are applied to boundary of the impeller and shaft and used to simulate the

rotation. This model has the drawback of relying on empirical correlations or experimental

data to set the boundary conditions. In many cases these are not available for different

impeller geometries or for multiphase systems.

4.5.5.2 Multiple reference frames

The MRF model (developed by Luo et al 1994) uses separate concentric zones rotating at

different speeds to each other. An inner zone (in the region of the impeller) that employs a

rotating reference frame and an outer zone (representing the rest of the tank) that remains

stationary. The fluxes at the interface between zones are matched which allows transfer of,

among other things, momentum to the remainder of the tank. This approach gives a good

indication of steady state flow field for an impeller of arbitrary geometry.

4.5.5.3 Sliding mesh

The sliding mesh method is very similar to MRF except that rather than the reference frame

rotating, the inner zone’s mesh rotates. This means the equations of motion are solved

relative to a rotating grid. This solution is applicable to transient situations and has been

found to produce good results for this situation.

38

Comparison of the different models has found that MRF and sliding mesh provide good

agreement with experimental data while IBC was very sensitive to the input boundary

conditions (Daskopoulos and Harris (1996)). Sliding mesh in particular was found to be most

accurate.

4.6. Bioreactor set up

The stirred tank being modelled in this project is a 2 L jacketed reactor with a working

volume of 1.7 L (main dimensions listed in Table 5, schematic Figure 4). The impeller type is

a double Rushton disc turbine (Figure 1) with dimensions also described in Table 5.

Figure 4: Tank schematic

39

Table 5: Impeller and Tank dimensions

The tank is loaded with pre-treated miscanthus particles and the reaction media. The

particles are generally cylindrical in shape, varying in length from 2-5 mm and have an

average diameter of 0.5 mm. The reaction takes place in the aqueous phase and such the

reaction media is a fluid with similar properties to pure water.

Dimension Size (mm)

Blade width 12

Blade height 12

Disc diameter 20

Shaft diameter 8

Tank diameter 105

Liquid height 156

Impeller clearance 20

Impeller shaft length 136

40

5. Computational Methods

To produce an accurate model of the tank, three tasks must be achieved:

Create an accurate geometry of the tank

‘Mesh’ the geometry

Use CFD to simulate the mixing

The software chosen for each stage is outlined in Table 6. The different types of software

were chosen due to both availability and appropriate compatibility.

Table 6: Software used

Function Software

Creating Geometry AutoCAD

Developing Mesh ANSYS ICEM CFD version 13.0

Running Simulations ANSYS FLUENT version 13.0

5.1. Geometry generation

After using the drawing software a model of the flow domain illustrated in Figure 5 was

produced for meshing. This was exported into ICEM in an STL file format.

Figure 5: Model of the flow domain produced

41

5.2. Mesh creation

The geometry was imported from the STL file generated in AutoCAD. Before generating the

mesh it is important to carry out a few operations to strengthen the model. As the model is

initially imported as one continuous surface it was necessary to split this into the individual

surfaces. Curves were then generated from the surfaces, using the extract curve tool, and

the curves subsequently segmented. Points were then added to the geometry, by inserting

them at the end of the curves. It was then possible to split the geometry into individual

parts, making allowing definition of the different zones in FLUENT.

It was then necessary to create “bodies” in the different geometry zones. Three bodies were

created, one for the main volume of the tank and two more for the sections surrounding

each Impeller. The bodies are created to indicate the volumes where fluid is present (and

conversely areas where no fluid is present i.e. inside solid objects such as the impeller and

shaft.). To achieve this ICEM uses ‘flood fill’ to ‘fill’ any volumes where a body is assigned

with fluid. These fluid zones are then the areas ‘meshed’. Different bodies allow definition

of different zones in FLUENT.

The mesh is then ready to be created, due to the complexity of the geometry, in particular

the areas around the impeller, it was not possible to use blocks to generate a precise and

more even mesh. Therefore the mesh was generated using the “unstructured” method,

which uses tetrahedral shapes. The mesh was improved by introducing prism layers on the

surface of the walls and impeller. This produces a more refined mesh closer to the wall to

resolve the boundary layer. The mesh size was chosen to give accurate results, but with a

compromise on computational demand. The mesh generated can be seen in Figure 6.

42

Figure 6: Mesh produced

5.3. Simulation

After selecting ANSYS FLUENT as the simulation software there were several decisions to be

made concerning both the theoretical models and numerical methods to be used.

5.3.1. Theoretical models

Different models, described in section 4, exist to model the phase interaction, impeller

rotation and other aspects necessary to produce an accurate simulation. The models chosen

are described below.

5.3.1.1 Multiphase

As presented in section 4.3.2 there are two types of approach to dealing with a multiphase

system, Euler approach and the Lagrangian approach. Due in part to the extra

computational demand required in the Lagrangian approach it was decided to use the Euler-

Euler model. This model treats the secondary phase a separate continuous phase and thus

requires phase interaction models.

43

5.3.1.2 Phase interaction models

The drag force, lift force and virtual mass force are used to model phase interactions. After

initial simulations it was found that inclusion of the lift force and virtual mass force had a

negligible effect upon results (this is to be expected when phase densities are not too

dissimilar). For simplicities sake they were omitted from the simulation (as is the FLUENT

default).

It was decided to use the inbuilt correlations in FLUENT rather than to explicitly state the

drag coefficient. The Syamlal-O’Brien drag model was selected for this study.

5.3.1.3 Turbulence models

Due to its wide use and general accuracy the model was selected for this study.

Recent improvements to this model (RNG and realizable) were also considered and it was

decided the realizable model would be used. Due to potentially strong particle-wall

effects enhanced wall treatment was used as a good compromise between accuracy and

computational demand. Section 4.3.3 outlines the theory behind this model and some

constants involved. Table 7 shows the chosen values of constants required by the selected

models.

Table 7: Constants required by the realizable model

Constant Value

1.9

1.3

5.3.1.4 Impeller rotation model

As described in section 4.4.5 the most appropriate rotational model for a steady state

system is the MRF model, this is the one chosen for this study. This model was implemented

By defining separate regions that enclose the impeller blades to have a ‘rotating reference

frame’ relative to the remainder of the tank. As part of this model the boundary conditions

must be specified. Each wall that adjoins a rotating region must also have a rotating

44

reference frame; however this frame is defined relative to the adjacent cell zone and set to

zero. This means in the absolute frame these walls will be rotating.

5.3.2. Solution Methods

A pressure based solver was used and the pressure-velocity coupling scheme SIMPLE

employed. This scheme has proven to be effective in most cases and was adjudged suitable.

The following algorithms were chosen for the spatial discretisation.

Table 8: Discretisation used in simulations

Discretisation parameter Algorithm

Gradient Least Squares Cell Based

Momentum QUICK

Volume Fraction First Order Upwind

Turbulent Kinetic Energy QUICK

Turbulent Dissipation Rate QUICK

5.4. Assumptions

5.4.1. Particle size

The particles are of varying size as a result of the pre-treatment process. Consequently it

was assumed that the solution is polydispersed, i.e. the packing limit is arbitrarily close to 1

(as opposed to a monodispersed phase of perfect spheres that would tend to the FLUENT

default of 0.63). The shape of the particles is also varies, for the sake of simplicity in defining

the particle size an average ‘particle diameter’ was computed by taking the average volume

of the range of particle sizes and working out an ‘effective diameter’. From the given

estimates of particle size they were taken as generally cylindrical, with length 2-5 mm and

diameter 0.5 mm. This yields and average diameter 1mm. this was used as the base case

parameter (full calculations can be found in Appendix C).

45

5.4.2. Density

Miscanthus has a dry weight density of 150 . However for the purpose of this study

it is assumed that its actual density, when fully saturated with fluid, will in fact be higher

than that of the reaction media. This is because the grass will absorb water in the tank

increasing its overall density. At the point of full saturation air gaps in the miscanthus

structure will be mostly filled with water to a point where the overall miscanthus density is

higher than that of the surrounding fluid. An assumed density of 1100 was used.

46

6. Results and Discussion

6.1. Effectiveness of the model

Before gathering and interpreting results from the simulations carried out on FLUENT, it is

important to ensure that the model generated is effective. This can be done by comparing

the velocity profile in the simulation to the expected flow pattern in the reactor. All of the

cases carried out showed a similar flow pattern, thus to demonstrate the models

effectiveness shows the velocity field from the base case is shown Figure 7. The flow

pattern generated from the simulations provides a good comparison to the literature

(Nagata, 1975) shown in Figure 2. This indicates that the MRF model generated provides a

reasonable representation of the fluid dynamics within the bioreactor.

Figure 7: Base case flow field generated

47

6.2. Base Case

The base case conditions are shown in Table 9.

Table 9: Variables 'base case' values

Variable Base case value

Solid Loading 15% (v/v)

Impeller speed 400 rpm Particle Diameter 1 mm

A fixed value for each variable was chosen as a standard to be used during the investigation

of another variable. These standard values formed the ‘base case’ Table 9. Figure 8 shows

the results when this case was run. As can be seen there is reasonable homogeneity within

the tank. It was therefore decided that the base case values were suitable choices as they

did not produce particularly efficient or poor mixing and thus will not dominate in the

contribution to mixing efficiency.

It was chosen to measure the homogeneity by visualising the volume fraction of the

secondary phase on both a horizontal and vertical plane. The vertical plane is taken along

the x axis and the horizontal plane at a region in between the two impellers. This allows a

good estimation of the general mixing profile within the tank and provides clear evidence of

homogeneity to compare between cases.

48

Figure 8: Base Case results

6.3. Impeller Speed

The “just suspension” speed was estimated previously, with the use of the Zweitering

correlation (equation 4.6), to be approximately 150rpm. To check how the model generated

compared to this estimation, simulations were carried out at 100 and 150 rpm. The results

can be seen below in Figure 9. The 100 rpm case indicates that it is below the just

suspension speed as there are zones along the bottom of the tank showing a high volume

fraction of miscanthus. At 150 rpm the zones of high miscanthus volume fractions at the

base of the tank are no longer present; this suggests that 150 rpm is above the just-

suspension speed. This shows that the just-suspension speed is in between 100 and 150

rpm, which is lower, but none the less a good match to the estimation generated from the

Zweitering correlation. The just suspension may be lower than estimated due to a number

of reasons. Chief among these that the correlation is modelled for single impeller systems.

Therefore it is not surprising that the just-suspension speed is lower.

49

It can be seen that neither of the cases show good homogenous mixing. This would suggest

that just because the impeller speed is above the just-suspension speed, it does not

guarantee good homogenous mixing. The cases both have areas of high miscanthus volume

fraction. The results do not show complete symmetry, even steady state was reached.

Simulations were carried out at higher impeller speed to further investigate the effect of

changing impeller rotational speed.

Figure 9: Results obtained at impeller speeds of 100 and 150 rpm

50

Figure 10: Results obtained at impeller speeds of 300 and 400 rpm

Figure 11: Results obtained at impeller speeds of 600 and 800 rpm

51

Figure 12: Results obtained at an impeller speed of 1000 rpm

Figure 10 and Figure 11 show a progressive increase in impeller speed, from 300 to 800 rpm

(all other variables kept at their respective base case value). The figures show that

increasing the impeller speed causes the homogeneity of the suspension to increase, the

800 rpm case shows almost compete homogenous mixing, this being backed up by Figure 14

(the homogeneity calculated from equation 4.5). This is an unsurprising result as the

increasing impeller speed causes the flow in the tank to become increasingly turbulent (as

the Reynolds number increases with increasing velocity) and provides more energy for

agitation. However, as shown in Figure 12, increasing the speed will only improve mixing up

to a certain point, after which increasing the impeller speed will have a detrimental effect

on the mixing. The reason for this is at higher speeds, the centrifugal force in the circulation

loops begins to dominate and cause a reduction in homogeneity.

The improved mixing observed at higher rotational speeds is desirable, however this comes

at the cost of higher energy and power requirement. The power requirements at each

impeller speed were calculated using the torque evaluated in FLUENT and equation (6.1)

(table and results in appendix D), the results are shown in Figure 13.

52

(6.1)

Where P is the power and T the torque. This shows that increasing the impeller speed has

little effect on the power requirement up until around 400 rpm, where the power

requirements significantly increase. This is something that needs to be taken into account

when choosing the impeller speed for the bioreactor. Even though an impeller speed at 800

rpm will cause better homogenous mixing, the power requirement is a lot greater which

could make the overall process less efficient, it is essential to balance the power

requirements against the effectiveness of mixing when choosing the operating impeller

speed.

The impeller speed has a large effect on the homogeneity and should also be between the

“just suspension speed” and the speed for maximum homogeneity. From the results shown,

it would be recommended to operate nearer the maximum speed as this shows a dramatic

improvement in homogeneity. However the high power requirement at 800 rpm means this

speed, while producing the best mixing, may not be the optimum when considering the

process as a whole. The processes’ eventual aim is to produce fuel for use an energy source

and such as low an energy input as possible is desired. Therefore it is recommended to

operate between 400 and 600 rpm as this combines good mixing with low power

requirements.

53

Figure 13: showing Impeller Speed against power requirement

Figure 14: Showing the effect of impeller speed against homogeneity

0

2

4

6

8

10

12

14

16

0 200 400 600 800 1000

Po

we

r R

eq

uir

em

en

t (W

)

Impeller Speed (rpm)

0.85

0.87

0.89

0.91

0.93

0.95

0.97

0.99

0 200 400 600 800 1000

Ho

mo

gen

ity

Impeller speed (rpm)

54

6.4. Particle size

Simulations were performed using a particle diameters ranging from of 0.25 mm to 2 mm.

Figure 15 and Figure 16 show the distribution of the secondary dispersed phase and its

volume fraction at different sizes.

It can be seen from Figure 15 and Figure 16 that larger particle sizes are detrimental to the

overall mixing efficiency. With a particle size of 2 mm there are large regions where particles

collect in high concentrations, particularly around the edge of the tank. It is noted there are

large regions around the impeller where particles also gather. It can be seen that the smaller

particle size of 1.5 mm produces better overall homogenity although it still contains

significant regions of high secondary phase concentrations.

As the particle size reduces the homogenity of the mixture drastically improves. There are

much fewer and smaller regions of high secondary phase concentration and there are large

regions of relative homogenity. It is interesting to note that at lower particle sizes there is

no longer agglomeration at the wall, in fact the opposite becomes true with small areas near

the wall of the tank becoming devoid of any particles. The effect of reducing particle

diameter on mixing efficeincy is less pronounced at smaller sizes; however reduction in

particle accumulation around the impeller region is noted.

55

Figure 15: Results obtained with particle diameters of 1.5 and 2 mm

Figure 16: Results obtained with particle diameters of 0.5 and 1 mm

0.50

0.45

0.40

0.35

0.30

0.25

0.20

0.15

0.10

0.05

0.00

Volume

Fraction

Particle size

1.5 mm 2 mm

0.50

0.45

0.40

0.35

0.30

0.25

0.20

0.15

0.10

0.05

0.00

Volume

Fraction

Particle size

0.5 mm 1 mm

56

It is to be expected that a reduction in particle size will improve the mixing. An estimate of

the free settling velocity of a particle is described by equation 4.4.

As can be seen an increase in the particle diameter will increase the settling velocity and

thus make a solution containing larger particles more difficult to mix. For example a particle

of diameter 2 mm will have a settling velocity of 0.078 ms-1 while a particle of diameter 0.5

mm has a settling velocity 0.039 ms-1 of (full calculations in Appendix E). The correlation for

the ‘just suspension’ adds further weight to this conclusion. As can be seen from equation

4.4 an increase in particle diameter will also increase the speed required to keep the

particles in suspension, this demonstrates that mixing is more difficult with larger particles.

An increase in this speed means an increase in the impeller torque and thus energy input

into the system to fluidize the particles which is representative of the difficulty in producing

good mixing. This explains the large difference in homogeneity noted between these two

particle sizes; a result consistent with the literature (Hosseini et al, 2010).

6.5. Solids loading

Simulations were conducted that tested the effect of increasing the solid loading in the tank.

It was difficult to find an effective method of qualitatively comparing the mixing. This owing

to the fact that different cases had different volume fractions and thus regions of what

appeared to be particularly high or low volume fractions. However as there is a different

overall volume fraction in the tank in each case, a direct comparison is not always effective.

It was therefore decided to look for general homogeneity throughout the tank and not pay

too much attention to the colours observed. Using equation 4.5 it was also possible to

quantitatively compare results, displayed in Figure 17.

57

Figure 17: graph illustrating the change in homogeneity with volume fraction

It was found that as volume fraction increased, the homogeneity decreased. The

homogeneity is useful tool to compare between cases, but is not an absolute measure of

mixing (e.g. a homogeneity of 0.9 does not mean the tank is 90% perfectly mixed, just better

mixed than a case with a homogeneity of 0.8). Therefore qualitative analysis was also

required.

For volume fractions higher than 0.2 (Figure 18) large regions of high secondary phase

concentration are observed collecting around the impeller and wall regions however areas

in-between have a relatively low particle concentration. This mixing is clearly much poorer

than volume fractions between 0.1 and 0.2 (Figure 19).

0.88

0.89

0.9

0.91

0.92

0.93

0.94

0.95

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

Ho

mo

gen

ity

Volume fraction

58

Figure 18: Results obtained with volume fractions of 0.3 and 0.4

This again is an unsurprising result. It stands to reason that as the number of particles

increases the ease with which they are mixed to a desired level of homogeneity. With more

particles present more energy must be provided in the form of agitation to produce a similar

level of homogeneity. From equation 4.6 it can be seen that an increase in volume fraction

increases the necessary impeller speed (and hence energy input) required to produce

complete just suspension. Therefore to produce a similar level of mixing more energy will

also be required as the volume fraction increases.

From the results above it is recommended that the tank be operated with a volume fraction

between 15 and 20%. This of course depends on the trade-off between mixing efficiency

and production requirements.

59

Figure 19: Results obtained with volume fractions 0.1, 0.15 and 0.2

6.6. Baffles

The introduction of baffles into the tank was investigated. Baffles are designed to improve

the mixing in the tank by promoting more axial flow. As can be seen from Figure 20 the

mixing is improved compared to the overall base case. It is noted that the secondary phase

collects in the top region of the tank. This is in part due to the increased axial flow pushing

more particles to the top of the tank. Geometry inaccuracies may also contribute to this,

namely that the top of the tank is not modelled as a free surface (explained further in

section 6.7.1). This could encourage particles to collect near the top of the tank. There are

also regions where it appears the particles are gathering around the baffles. This effect may

be detrimental to the overall mixing performance however it will be a small effect and

outweighed by the beneficial effects of baffles.

Despite this it is believed that the regions of high particle concentration are small, and

overall the baffles are an improvement on the mixing performance.

60

Figure 20: Results obtained when baffles are introduced

6.7. Limitations of the results

6.7.1. Geometry

It is also noted that in some cases, particularly when baffles are present, there is a collection

of particles at the top of the tank. There is a flaw in the model used in that the top surface

of the tank is modelled as a solid wall. in reality it is a free surface open to the air in the

tank. This inaccuracy in the model is most likely responsible for this effect but also may have

implications on the predicted mixing in the remainder of the tank.

The model produced is also not a perfect representation of the tank geometry. The bottom

of the tank is taken as a perfectly flat surface when in reality it is slightly conical. This will

mean the mixing observed is not necessarily representative of the true mixing to be found.

However this particular effect will mean the simulations will most likely give a conservative

estimation of the mixing efficiency, as a flat bottomed tank is not as conducive to

homogeneity as slightly rounded one. The geometry did not take into account the presence

of temperature probes and other equipment protruding into the mixture. This means the

61

true mixing observed in the tank may not be fully represented by the simulations

undertaken.

6.7.2. MRF model

The sections of the impeller shaft that were not inside the moving zones were modelled as

stationary, due to the boundary condition stipulating walls cannot move relative to the

adjacent cell zone. This is not a representation of the true situation inside the tank as in

reality these sections would be rotating at the same speed as the moving zone. This

inaccuracy can go some way to explaining why poor mixing was often observed in between

the impeller zones. To overcome this problem a separate number of thin rotating zones

would have to be specified that allowed shaft rotation to be taken into account.

Implementing this approach will improve accuracy of results but will add to the

computational demand.

6.7.3. Theoretical models

The theoretical models used are by no means 100% accurate. There are known limitations

with the model, particularly with respect to capturing rotational or anisotropic flow.

The Euler-Euler multiphase model may not be entirely suitable for this case given the size of

the particles used. As the secondary phase is treated as a concentration field rather than

discrete particles this model is not suited to handling large particle sizes.

6.7.4. Reaction effects

The purpose of studying the mixing within the reactor is to improve the overall efficiency of

the process. The main process taking place in the reactor is the conversion of cellulose to

biofuel. This study did not take into account the effects this reaction would have on the

chemical and physical properties of the species inside the reactor which would have an

influence on the mixing performance. However as the reaction progresses the miscanthus

will be broken down and the mixing efficiency is likely to improve due to the smaller particle

sizes present.

62

6.7.5. Initial loading conditions

The model was initialised with the secondary phase already perfectly dispersed throughout

the tank. This is not representative of the true initial conditions which may have caused

initial overestimation of the ease with which the tank reached steady state and the overall

mixing performance.

6.7.6. Assumptions

As outlined in section 5.4 several assumptions were made regarding the particle size and

miscanthus density.

6.7.6.1 Particle size

This assumption, while necessary, is not representative of the true system. Both the variety

of different particle sizes and the generally cylindrical shape will mean the true systems

secondary phase may well behave differently in its fluid and particle interactions. This

behaviour will therefore impact on the mixing performance noted and could influence

results.

6.7.6.2 Density

After performing sensitivities on different density sizes it was found that the results were

not particularly sensitive to a change in the chosen density (in the region of 1000 Kgm-3).

However the assumption of full saturation may not be entirely correct. It may well be the

case that the density will evolve over time as the miscanthus absorbs more water and thus

the behaviour of the dispersed may change as a result.

6.8. Recommendations for further work

Given the general nature of this study it is hoped the results can inform future investigation

in this area. From the limitations noted there are several areas that can be investigated.

63

6.8.1. Transient effects

As noted in section 6.6 no transient simulations were carried out. The following are areas

highlighted as potential aspects to investigate.

6.8.1.1 Mixing time

The mixing time is the time taken for the multiphase system to reach a defined level of

homogeneity. The mixing time has no effect on the steady state mixing, however if it is

found to be particularly large at certain conditions this can impact the recommended

operating conditions. By studying the effect of changing rotational speed and other

variables on mixing time, a clearer picture of the optimal conditions to promote mixing and

reaction in the tank will be formed. In order to achieve a transient simulation the ‘sliding

mesh’ model outlined in section 4.4.5.3 must be employed.

6.8.1.2 Change of properties with time

As noted certain properties of the fluid medium and particles may change over time. By

incorporating an estimate of the change in these properties a more accurate picture of the

processes going on in the tank can be built.

6.8.2. Reaction

The aim of this study was to visualise the effect certain parameters had on the mixing

efficiency with the knowledge efficient mixing will improve the reaction process. The next

logical step is to include the reaction in the model gaining a fuller appreciation of the

process as a whole. This inclusion of the reaction process could start with examining the

mass transfer of species (e.g. Glucan) between phases and enzymatic reactions taking place.

6.8.3. Models used

There are limitations to the models used in this study. Many models used are semi or

completely empirical and thus not necessarily specific to the process. By investigating

64

different types of models for multiphase or turbulence a more accurate picture of the

system can be built.

In particular investigation of the different drag models available may be beneficial. As the

system has a wide range of particle size and shapes standard correlations may not

effectively describe the particle-fluid interactions which are known to be different in

spherical and cylindrical particles. This will improve the accuracy of phase interaction

models and thus provide a better account of the mixing process

6.8.4. Geometry and scale up

As noted in section 6.6.1 the geometry is not perfectly accurate. By improving the areas

noted a better model of the tank will be built. In particular the free surface at the top of the

tank may have important implications for the mixing process. Once a more accurate

geometry is settled upon it may be possible to investigate the effects of scaling this lab-scale

reactor to a larger process. This is obviously the eventual aim of studying this bioreactor; to

test the feasibility of this process on a large scale.

65

7. Conclusions

From this study it can be concluded:

A larger particle size is detrimental to the mixing performance. It is recommended to

operate at as low a particles size as reasonably practicable.

The mixing efficiency is improved as the impeller rotational speed is increased.

However approaching 1000 rpm the influence of the centrifugal force begins to

dominate causing detrimental effects on the mixing efficiency. When taking power

considerations into account it is recommended to operate between 400 and 600 rpm

as this combines lower power requirement with good mixing.

Solids loading of above 20% is detrimental to the overall mixing performance and

thus it is recommended to operate below this level.

These results are consistent with the expected outcomes.

Due to general nature of this study several limitations have been noted and areas for

improvement and further work that include incorporating reaction and transient effects into

the model are suggested. Reasonable caution is urged over the accuracy of the results

(given limitations noted) however the general trends observed should not change with

increased accuracy of the model.

The aims of this project were to get an estimate of the mixing efficiency in the reactor and

the effect of changing certain variables. These aims have been achieved. A good model of

the tank has been produced and a rigorous investigation of certain variables conducted with

clear results and recommendations. This study should be able to form a strong base on

which further investigation into the process can be done.

66

8. Bibliography

Atiemo-Obeng, V.,A., Kresta, S.,M., Paul, E.,L.(2004) Handbook of Industrial Mixing – Science

and Practice, John Wiley & Sons.

Auton, T.R., Hunt, J.C.R., & Prud’homme, M. (1988) The force exerted on a body in inviscid,

unsteady, non-uniform rotational flow. J. Fluid Mech. 197, 241-257

Balan, V., Bals, B., Chundawat, S. P. S., da Costa Sousa, L., Dale, B. E., Murnen, H.K. (2007)

Optimization of Ammonia Fiber Expansion (AFEX) Pretreatment and Enzymatic Hydrolysis of

Miscanthus x giganteus to Fermentable Sugars, Biotechnol Prog.

Baldyga, J. & Bourne, J. R., (1999), Turbulent Mixing and Chemical Reactions, John Wiley &

Sons, Chichester, UK

Biosson, N., Malin, M.R (1996), The Numerical prediction of two-phase flow in bubble

columns, J. Num. Meth. Fluid 23, 1289-1320

Biovin, M., Simonin, O., Squires, K. (2000) On the prediction of gas-solids flow with two way

coupling using large eddy simulation, Physics of Fluids 12 (8), 2080-2090.

Boersma, N., Caverny, J.D., Dohleman, F.G., Heaton, E.A., Voigt, T.B. (2011) Miscanthus for

Biofuel Production

Boonchird, C., Moukamnerd, C., Kino-oka, M., Sugiyama, M., Kaneko, Y., Harashima, C.S.,

Noda, H., Ninomiya, K., Shioya, S., Katakura, Y. (2010) Ethanol production from biomass by

repetitive solid-state fed-batch fermentation with continuous recovery of ethanol.

Bounds, A. (2007-09-10). "OECD warns against biofuels subsidies". Financial Times.

http://www.ft.com/cms/s/0/e780d216-5fd5-11dc-b0fe-0000779fd2ac.html#axzz1dzYwBr6U

accessed on 1/12/2011

67

De Brito Cruz, CH (2008) Bioethanol in Brazil,

http://www.biofuels.apec.org/pdfs/apec_200810_brito-cruz.pdf accessed on 14/11/2011

Chudacek, M. W. (1985). "Impeller power numbers and impeller flow numbers in profiled

bottom tanks." Industrial & Engineering Chemistry Process Design and Development 24(3):

858-867.

Daskopoulos, Ph., harris, C.K. (1996) Three-dimensional CFD simulations of turbulent flow in

baffled stirred tanks: an assessment of the current position, IChemE Symposium Series 140 ,

1-13.

Edwards, M.F., Harnby, N., Nienow, A.W. (1985) Mixing in the Process Industries,

Butterworths

EIA, Energy Information Administration, (2011) web page:

http://www.eia.gov/forecasts/ieo/liquid_fuels.cfm accessed on 07/11/11

Erdal, F.M., Shirazi, S.A., Mantilla, I., Shoham, O. (2000) Computational Fluid Dynamics (CFD)

study of bubbles carry-under in gas-liquid cylindrical cyclone separators, Society of

Petroleum engineers production and facilities 15 (4), 217-222.

Gosman, A.D. (1998). "Developments in Industrial Computational Fluid Dynamics." Chemical

Engineering Research and Design 76(2): 153-161.

la Grange DC, Haan R, van Zyl WH (2010) Engineering cellulolytic ability into bioprocessing

organisms, Springer-Verlag

Hernandez, M., Quijano, G., Muñoz, R., Bordel, S. (2011) Modeling of VOC mass transfer in

two-liquid phase stirred tank, biotrickling filter and airlift reactors, Chemical Engineering

Journal. 172 (2-3), 961-969.

Hosseini, S., D. Patel, et al. (2010). Study of solid–liquid mixing in agitated tanks through

68

electrical resistance tomography. Chemical Engineering Science 65(4): 1374-1384.

Hosseini, S., D. Patel, et al. (2010). Study of Solid−Liquid Mixing in Agitated Tanks through

Computational Fluid Dynamics Modeling, Industrial & Engineering Chemistry Research 49(9):

4426-4435.

Hubbard, M., University of Leeds, Finite Volume Schemes: A Tutorial

http://www.comp.leeds.ac.uk/meh/Talks/FVTutorial.pdf accessed on 01/12/11

IEA, 2010, web page: http://www.iea.org/textbase/nppdf/free/2010/key_stats_2010.pdf

accessed on 1/12/2011.

Ishii, M., Zuber, N. (1979) Drag Coefficients and relative velocity in bubbly, droplet or

particulate flows, AIChE J 25, 843-855

Ihme, F., Schmidt-Traub, H., Brauer, H. (1972) Theoretische unterschung uber die

Umstromung und den Stoffubergang an Kuglen, Chem. Ing. Tech. 44, 306

Jackson R., (2000) The Dynamics of Fluidized Particles, Cambridge University Press

Jakobsen, H.A., Sannaes, B.H., Grevskott, S., Svendsen, H.F. (1997) Modelling of vertical

bubble-driven flows, Ind. Eng. Chem. Res. 36, 4052-4074

Kaufmann- http://www.ewkaufmann.com/product.asp?specific=jqqqcqn8 accessed on

26/11/2011

Lane, G.L., Schwarz, M.P., Evans, G.M. (2005) Numerical modelling of gas–liquid flow in

stirred tanks, Chemical engineering science, 60 (8-9), 2203-2214

Lynd LR, van Zyl WH, McBride JE, Laser M (2005) Consolidated bioprocessing of cellulosic

biomass: an update.

Lynd LR, Germgrss TU, Wyman CE (1999) Biocommodity engineering. Biotechnol Prog

69

Lapin, A., Schmid, J., Reuss, M. (2006) Modeling the dynamics of E. coli populations in the

three-dimensional turbulent field of a stirred-tank bioreactor—A structured–segregated

approach, Chemical Engineering science, 61 (14), 4783-4797

Luo, J.Y., Issa, R.I., Gosman, A.D. (1994) Prediction of impeller-induced flow using multiple

frames of reference, IChemE Symp. Series 136, 549-556

Mavros, P., Mann, R., Vlaev, D., Bertrand, J. (2001) Experimental visualization and CFD

simulation of flow patterns induced by a novel energy-saving dual-configuration impeller in

stirred vessels, TransIChemE 79 (A), 857-866

Montante, G., M. Moštěk, et al. CFD simulations and experimental validation of

homogenisation curves and mixing time in stirred Newtonian and pseudoplastic liquids.

Chemical Engineering Science 60(8-9): 2427-2437.

Nagata, S. (1975) Mixing Principles and applications, Wiley

Ochieng, A., Onyango, M.S. (2008) Drag models, solids concentration and velocity

distribution in a stirred tank, powder technology, 181 (1), 1-8

Patankar, S.V., Spalding, D.B (1972) A calculation procedure for heat, mass and momentum

transfer in three-dimensional parabolic flows, int. Journal of heat and mass transfer 15,

1787-1800

Perry, R. H., Green, D. (1984) Perry’s Chemical Engineers’ Handbook; McGraw-Hill: New York,

Ranade, V.V., Joshi, J.B (1990) Flow generated by a disc turbine. Part II- mathematical

modelling and comparison with experimental data, Chem. Eng. Res. Dev. 68A, 34-50

Ricard, F.X. (2005) Application of electrical resistance tomography to pharmaceutical mixing

processes, Department of Chemical engineering and chemical technology, Imperial College

of science, technology and medicine

70

Sato, Y., Segokuchi, K. (1975) Liquid velocity distribution in two phase bubble flow,

international journal of multiphase flow, 2, 79

Schiller, L., Nauman, A. (1933) A drag coefficient correlation, VDI Zeits. 77, 318-320

Sharcnet-

https://www.sharcnet.ca/Software/Fluent13/help/flu_th/flu_th_sec_turb_realizke.html

Accessed on 04/12/11

Tokay, B.,A. (2002) "Biomass Chemicals" in Ullmann's Encyclopedia of Industrial Chemistry,

Wiley-VCH, Weinheim

U.S. Department of Energy, Breaking the biological barriers to cellulosic ethanol, 2006,

http://genomicscience.energy.gov/biofuels/2005workshop/b2blowres63006.pdf accessed

on 26/09/2011.

Wilcox, D.C (1998) Turbulence modelling for CFD, 2nd edition DCW Industries, Inc.

World Energy outlook 2006 http://www.iea.org/textbase/nppdf/free/2006/weo2006.pdf

accessed on 29/11/2011

Yakhot, V.,. Orszag, S.A. (1987) Renormalization group and local order in strong turbulence

Nuclear Physics B - Proceedings Supplements, 2, 417-440

Zhang, D.Z & Prosperetti, A. (1994) averaged equations for inviscid disperse two-phase flows

and their closure for dilute suspensions. Int. J. Multiphase Flow. 23, 425-453

Zwietering, TH.N. (1958) Chem. Eng. Sci. 8, 244-253

71

9. Appendices

A

Figure 21: Graph showing S against T/D with T/C as a parameter for disc turbine impellers

To calculate NJS, TD/D and TD/C are found first:

Figure 21 was then used to find an S value of 4. The just speed is calculated using the

Zwietering equation (4.6):

(

)

72

Table 10: Base Case parameters, used to calculate just suspension speed

Parameter Value

Particle size 1mm

Density of fluid 1000 Kg/m3

Density of Miscanthus 1100 Kg/m3

Volume fraction of Miscanthus 0.15

Liquid Viscosity 1E-06

B

Syamlal-O’Brien drag coefficient correlation

( √( ) ( ) )

C

Particle average diameter size calculation. Taking an average length of 3.5 mm this yields an

volume of

Approximating the particles to spheres this yields an average diameter of

73

√

For the sake of simplicity and given it is only an approximation this was reduced to 1 mm.

D

Table 11: Power requirements for varying impeller speed

Impeller rotational speed (rpm) N (rps) T (N.m) P (W)

100 1.666667 0.020969 0.219585

200 3.333333 0.071881 1.505467

300 5 0.058596 1.840858

400 6.666667 0.080012 3.351552

600 10 0.123316 7.748202

700 11.66667 0.144029 10.55786

800 13.33333 0.160159 13.41747

E

√( ( )

)

For a particle size of 2mm (assuming drag coefficient = 0.44)

√( ( )

)

√( ( )

)

74

F

Sample calculation for the homogeneity

√∑ ( )

Table 12: Homogeneity calculations for the base case

400 rpm

Xv: 0.15

Plane av Xv (xv-avxv)^2

z-005 0.18411863 0.001164081

z-025 0.15859292 7.38383E-05

z-050 0.19066076 0.001653297

z-075 0.22129293 0.005082682

z-100 0.24864234 0.009730311

z-125 0.21136649 0.003765846

z-150 0.23097607 0.006557124

0.936723758