FLUENT 6 - SHARCNET · PDF fileto describe the changes in the particle population, in addition...

FLUENT 6.3

Population Balance Module Manual

September 2006

Copyright c© 2006 by Fluent Inc.All rights reserved. No part of this document may be reproduced or otherwise used in

any form without express written permission from Fluent Inc.

Airpak, FIDAP, FLUENT, FLUENT for CATIA V5, FloWizard, GAMBIT, Icemax, Icepak,Icepro, Icewave, Icechip, MixSim, and POLYFLOW are registered trademarks of FluentInc. All other products or name brands are trademarks of their respective holders.

CHEMKIN is a registered trademark of Reaction Design Inc.

Portions of this program include material copyrighted by PathScale Corporation2003-2004.

Fluent Inc.Centerra Resource Park

10 Cavendish CourtLebanon, NH 03766

Contents

Preface UTM-1

1 Introduction 1-1

1.1 The Discrete Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-1

1.2 The Standard Method of Moments . . . . . . . . . . . . . . . . . . . . . 1-2

1.3 The Quadrature Method of Moments . . . . . . . . . . . . . . . . . . . . 1-2

2 Population Balance Model Theory 2-1

2.1 The Particle State Vector . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1

2.2 The Population Balance Equation (PBE) . . . . . . . . . . . . . . . . . 2-2

2.2.1 Particle Growth and Dissolution . . . . . . . . . . . . . . . . . . 2-2

2.2.2 Particle Birth and Death Due to Breakage and Aggregation . . . 2-3

2.2.3 Particle Birth by Nucleation . . . . . . . . . . . . . . . . . . . . 2-5

2.3 Solution Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-5

2.3.1 The Discrete Method . . . . . . . . . . . . . . . . . . . . . . . . 2-5

2.3.2 The Standard Method of Moments (SMM) . . . . . . . . . . . . 2-8

2.3.3 The Quadrature Method of Moments (QMOM) . . . . . . . . . . 2-10

2.4 Reconstructing the Particle Size Distribution from Moments . . . . . . . 2-11

3 Using the FLUENT Population Balance Model 3-1

3.1 Population Balance Module Installation . . . . . . . . . . . . . . . . . . 3-1

3.2 Loading the Population Balance Module . . . . . . . . . . . . . . . . . . 3-1

c© Fluent Inc. July 19, 2006 i

CONTENTS

3.3 Population Balance Model Setup . . . . . . . . . . . . . . . . . . . . . . 3-2

3.3.1 Enabling the Population Balance Model . . . . . . . . . . . . . . 3-2

3.3.2 Defining Population Balance Boundary Conditions . . . . . . . . 3-6

3.3.3 Specifying Population Balance Solution Controls . . . . . . . . . 3-7

3.3.4 Coupling With Fluid Dynamics . . . . . . . . . . . . . . . . . . . 3-8

3.3.5 Specifying Interphase Mass Transfer Due to Nucleation and Growth 3-9

4 Postprocessing for the Population Balance Model 4-1

4.1 Population Balance Solution Variables . . . . . . . . . . . . . . . . . . . 4-1

4.2 Reporting Derived Population Balance Variables . . . . . . . . . . . . . 4-2

4.2.1 Computing Moments . . . . . . . . . . . . . . . . . . . . . . . . 4-2

4.2.2 Displaying a Number Density Function . . . . . . . . . . . . . . 4-3

5 UDFs for Population Balance Modeling 5-1

5.1 Population Balance Variables . . . . . . . . . . . . . . . . . . . . . . . . 5-1

5.2 Population Balance DEFINE Macros . . . . . . . . . . . . . . . . . . . . . 5-1

5.2.1 DEFINE PB BREAK UP RATE . . . . . . . . . . . . . . . . . . . . . . 5-2

5.2.2 DEFINE PB COALESCENCE RATE . . . . . . . . . . . . . . . . . . . 5-4

5.2.3 DEFINE PB NUCLEATION RATE . . . . . . . . . . . . . . . . . . . . 5-5

5.2.4 DEFINE PB GROWTH RATE . . . . . . . . . . . . . . . . . . . . . . . 5-7

5.3 Hooking a Population Balance UDF to FLUENT . . . . . . . . . . . . . . 5-9

A DEFINE HET RXN RATE Macro A-1

A.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-1

A.2 Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-2

A.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-3

A.4 Hooking a Heterogeneous Reaction Rate UDF to FLUENT . . . . . . . . A-4

ii c© Fluent Inc. July 19, 2006

Using This Manual

The Contents of This Manual

The FLUENT Population Balance Model Manual tells you what you need to know tomodel population balance with FLUENT. In this manual, you will find background infor-mation pertaining to the model, a theoretical discussion of the model used in FLUENT,and a description of using the model for your CFD simulations.

Typographical Conventions

Several typographical conventions are used in this manual’s text to facilitate your learningprocess.

• An informational icon ( i ) marks an important note.

• An warning icon ( ! ) marks a warning.

• Different type styles are used to indicate graphical user interface menu items andtext interface menu items (e.g., Iso-Surface panel, surface/iso-surface com-mand).

• The text interface type style is also used when illustrating exactly what appears onthe screen or exactly what you need to type into a field in a panel. The informationdisplayed on the screen is enclosed in a large box to distinguish it from the narrativetext, and user inputs are often enclosed in smaller boxes.

• A mini flow chart is used to indicate the menu selections that lead you to a specificcommand or panel. For example,

Define −→Boundary Conditions...

indicates that the Boundary Conditions... menu item can be selected from the Definepull-down menu, and

display −→grid

indicates that the grid command is available in the display text menu.

c© Fluent Inc. July 19, 2006 UTM-1

Using This Manual

The words before the arrows invoke menus (or submenus) and the arrows pointfrom a specific menu toward the item you should select from that menu. In thismanual, mini flow charts usually precede a description of a panel or command, ora screen illustration showing how to use the panel or command. They allow youto look up information about a command or panel and quickly determine how toaccess it without having to search the preceding material.

• The menu selections that will lead you to a particular panel are also indicated(usually within a paragraph) using a “/”. For example, Define/Materials... tellsyou to choose the Materials... menu item from the Define pull-down menu.

Mathematical Conventions• Where possible, vector quantities are displayed with a raised arrow (e.g., ~a, ~A).

Boldfaced characters are reserved for vectors and matrices as they apply to linearalgebra (e.g., the identity matrix, I).

• The operator ∇, referred to as grad, nabla, or del, represents the partial derivativeof a quantity with respect to all directions in the chosen coordinate system. InCartesian coordinates, ∇ is defined to be

∂

∂x~ı+

∂

∂y~+

∂

∂z~k

∇ appears in several ways:

– The gradient of a scalar quantity is the vector whose components are thepartial derivatives; for example,

∇p =∂p

∂x~ı+

∂p

∂y~+

∂p

∂z~k

– The gradient of a vector quantity is a second-order tensor; for example, inCartesian coordinates,

∇(~v) =

(∂

∂x~ı+

∂

∂y~+

∂

∂z~k

)(vx~ı+ vy~+ vz~k

)This tensor is usually written as

∂vx∂x

∂vx∂y

∂vx∂z

∂vy∂x

∂vy∂y

∂vy∂z

∂vz∂x

∂vz∂y

∂vz∂z

UTM-2 c© Fluent Inc. July 19, 2006

Using This Manual

– The divergence of a vector quantity, which is the inner product between ∇and a vector; for example,

∇ · ~v =∂vx∂x

+∂vy∂y

+∂vz∂z

– The operator ∇ · ∇, which is usually written as ∇2 and is known as theLaplacian; for example,

∇2T =∂2T

∂x2+∂2T

∂y2+∂2T

∂z2

∇2T is different from the expression (∇T )2, which is defined as

(∇T )2 =

(∂T

∂x

)2

+

(∂T

∂y

)2

+

(∂T

∂z

)2

Technical Support

If you encounter difficulties while using FLUENT, please first refer to the section(s) ofthe manual containing information on the commands you are trying to use or the typeof problem you are trying to solve. The product documentation is available from theonline help on the documentation CD, or from the Fluent Inc. User Services Center(www.fluentusers.com).

If you encounter an error, please write down the exact error message that appearedand note as much information as you can about what you were doing in FLUENT.Then refer to the following resources available on the Fluent Inc. User Services Cen-ter (www.fluentusers.com):

• Installation and System FAQs - link available from the main page on the UserServices Center. The FAQs can be searched by word or phrase, and are availablefor general installation questions as well as for products.

• Known defects for FLUENT - link available from the product page. The defects canbe searched by word or phrase, and are listed by categories.

• Online Technical Support - link available from the main page on the User ServicesCenter. From the Online Technical Support Portal page, there is a link to theSearch Solutions & Request Support page, where the solutions can be searched byword or phrase.

The User Services Center also provides online forums, where you can discuss topics ofmutual interest and share ideas and information with other Fluent users, and the abilityto sign up for e-mail notifications on our latest product releases.

c© Fluent Inc. July 19, 2006 UTM-3

Using This Manual

Contacting Technical Support

If none of the resources available on the User Services Center help in resolving the prob-lem, or you have complex modeling projects, we invite you to call your support engineerfor assistance. However, there are a few things that we encourage you to do before calling:

• Note what you are trying to accomplish with FLUENT.

• Note what you were doing when the problem or error occurred.

• Save a journal or transcript file of the FLUENT session in which the problem oc-curred. This is the best source that we can use to reproduce the problem andthereby help to identify the cause.

UTM-4 c© Fluent Inc. July 19, 2006

Chapter 1. Introduction

In FLUENT the population balance model is provided as an addon module with thestandard FLUENT licensed software. A special license is required to use the populationbalance module.

Several industrial fluid flow applications involve a secondary phase with a size distribu-tion. The size distribution of particles, including solid particles, bubbles, or droplets,can evolve in conjunction with transport and chemical reaction in a multiphase system.The evolutionary processes can be a combination of different phenomena like nucle-ation, growth, dispersion, dissolution, aggregation, and breakage producing the disper-sion. Thus in multiphase flows involving a size distribution, a balance equation is requiredto describe the changes in the particle population, in addition to momentum, mass, andenergy balances. This balance is generally referred to as the population balance. Casesin which a population balance could apply include crystallization, precipitative reactionsfrom a gas or liquid phase, bubble columns, gas sparging, sprays, fluidized bed polymer-ization, granulation, liquid-liquid emulsion and separation, and aerosol flows.

To make use of this modeling concept, a number density function is introduced to accountfor the particle population. With the aid of particle properties (e.g., particle size, porosity,composition, etc.), different particles in the population can be distinguished and theirbehavior can be described.

FLUENT offers three solution methods to the population balance equation: discretizedpopulation balance, standard method of moments, and quadrature method of moments.

• Section 1.1: The Discrete Method

• Section 1.2: The Standard Method of Moments

• Section 1.3: The Quadrature Method of Moments

1.1 The Discrete Method

In the discrete method, the particle population is discretized into a finite number of sizeintervals. This approach has the advantage of computing the particle size distribution(PSD) directly. This approach is also particularly useful when the range of particle sizesis known a priori and does not span more than two or three orders of magnitude. In thiscase, the population can be discretized with a relatively small number of size intervalsand the size distribution that is coupled with fluid dynamics can be computed. Thedisadvantage of the discrete method is that it is computationally expensive if a largenumber of intervals is needed.

c© Fluent Inc. July 19, 2006 1-1

Introduction

1.2 The Standard Method of Moments

The standard method of moments (SMM) is an efficient alternative to the discrete popula-tion balance approach. In this approach, the population balance equation is transformedinto a set of transport equations for moments of the distribution. The ith moment isdefined by integrating the number density throughout the particle space weighted withthe particle property raised to its ith power. It is generally sufficient to solve only a fewmoment equations, typically three to six. This may provide a significant reduction in thenumber of equations to be solved compared with the discretized approach. Apart fromthe computational advantage, the SMM approach is useful when the entire distributionis not needed and certain average and total quantities are sufficient to represent the par-ticle distribution. Typically, the zeroth moment represents the total number density, thesecond moment represents the total surface area per unit volume, and the third momentrepresents the total mass density.

In the SMM approach, no assumptions are made about the size distribution, and the mo-ment equations are formulated in a closed form involving only functions of the momentsthemselves. However, this exact closure requirement poses a serious limitation, as aggre-gation (with the exception of the constant aggregation kernel) and breakage phenomenacannot be written as functions of moments.

1.3 The Quadrature Method of Moments

The quadrature method of moments (QMOM) has a similar advantage as the SMM interms of computational costs, but replaces the exact closure needed by SMM with anapproximate closure. This allows application of QMOM to a broad range of applicationswithout any limitations.

1-2 c© Fluent Inc. July 19, 2006

Chapter 2. Population Balance Model Theory

This chapter presents an overview of the theory and the governing equations for themethods used in FLUENT to predict particle growth and nucleation.

• Section 2.1: The Particle State Vector

• Section 2.2: The Population Balance Equation (PBE)

• Section 2.3: Solution Methods

• Section 2.4: Reconstructing the Particle Size Distribution from Moments

2.1 The Particle State Vector

The particle state vector is characterized by a set of “external coordinates” (~x), whichdenote the spatial position of the particle, and “internal coordinates” (φ), which couldinclude particle size, composition, and temperature. From these coordinates, a numberdensity function n(~x, φ, t) can be postulated where φ ∈ Ωφ, ~x ∈ Ω~x. Therefore, theaverage number of particles in the infinitesimal volume dV~xdVφ is n(~x, φ, t)dV~xdVφ. In

contrast, the continuous phase state vector is given by ~Y ≡ [Y1(~x, t), Y2(~x, t), . . . , Yc(~x, t)]

The total number of particles in the entire system is then

∫Ωφ

∫Ω~x

n dV~xdVφ (2.1-1)

The local average number density in physical space (i.e., the total number of particlesper unit volume) is given by

N(~x, t) =∫

Ωφ

n dVφ (2.1-2)

The total volume fraction of all particles is given by

α(~x, t) =∫

Ωφ

n V (φ)dVφ (2.1-3)

where V (φ) is the volume of a particle in state φ.


Population Balance Model Theory

2.2 The Population Balance Equation (PBE)

Assuming that φ is the particle volume, the transport equation for the number densityfunction is given as

∂

∂t[n(V, t)] +∇ · [~un(V, t)] +∇v · [Gvn(V, t)]︸︷︷︸

Growth term

=

1

2

∫ V

0a(V − V ′, V ′)n(V − V ′, t)n(V ′, t)dV ′︸︷︷︸

Birth due to Aggregation

−∫ ∞

0a(V, V ′)n(V, t)n(V ′, t)dV ′︸︷︷︸

Death due to Aggregation

+∫

Ωv

νg(V ′)β(V | V ′)n(V ′, t)dV ′︸︷︷︸Birth due to Breakage

− g(V )n(V, t)︸︷︷︸Death due to Breakage

(2.2-1)

The boundary and initial conditions are given by

n(V, t = 0) = nv; n(V = 0, t)Gv = n0 (2.2-2)

where n0 is the nucleation rate in particles/m3-s.

2.2.1 Particle Growth and Dissolution

The growth rate based on particle volume, Gv, (m3/s) is defined as

Gv =∂V

∂t(2.2-3)

The growth rate based on particle diameter (or length) is defined as

G =∂L

∂t(2.2-4)

The volume of a single particle V is defined as KvL3, and therefore the relationship

between Gv and G is

Gv = 3KvL2G (2.2-5)

The surface area of a single particle, A, is defined as KaL2. Thus for a cube or a sphere,

Ka = 6Kv.

i Dissolution of particles can be represented as negative growth.


2.2 The Population Balance Equation (PBE)

2.2.2 Particle Birth and Death Due to Breakage and Aggregation

The birth and death of particles occur due to breakage and aggregation processes. Exam-ples of breakage processes include crystal fracture in crystallizers and bubble breakagedue to liquid turbulence in a bubble column. Similarly, aggregation can occur due toparticle agglomeration in crystallizers and bubble coalescence in bubble column reactors.

Breakage

The breakage rate expression, or kernel [7], is expressed as

g(V ′)β(V | V ′)

where

g(V ′) = breakage frequency; i.e., the fraction of particles of volume V ′ breakingper unit time (m−3s−1)

β(V | V ′) = probability density function (PDF) of particles breaking from volumeV ′ to a particle of volume V

The birth rate of particles of volume V due to breakage is given by

Bbr =∫

Ωv

νg(V ′)β(V | V ′)n(V ′)dV ′ (2.2-6)

where g(V ′)n(V ′)dV ′ particles of volume V ′ break per unit time, producing νg(V ′)n(V ′)dV ′

particles, of which a fraction β(V | V ′)dV represents particles of volume V . ν is the num-ber of child particles produced per parent (e.g., two particles for binary breakage).

The death rate of particles of volume V due to breakage is given by

Dbr = g(V )n(V ) (2.2-7)

The PDF β(V | V ′) is also known as the particle fragmentation distribution function,or daughter size distribution. Several functional forms of the fragmentation distributionfunction have been proposed, though the following physical constraints must be met:the masses of the fragments must sum to the original particle mass, and the number offragments formed has to be correctly represented.



Mathematically, these constraints can be written as follows:

• For conservation of mass

∫ V ′

0β(V | V ′)dV = 1 (2.2-8)

• For binary breakage, β is symmetric about V/V ′ = 0.5; i.e.,

β(V ′ − V | V ′) = β(V | V ′) (2.2-9)

Aggregation

The aggregation kernel [7] is expressed as

a(V, V ′)

The aggregation kernel has units of m3/s, and is sometimes defined as a product of twoquantities:

• the frequency of collisions between particles of volume V and particles of volumeV ′

• the “efficiency of aggregation” (i.e., the probability of particles of volume V coa-lescing with particles of volume V ′).

The birth rate of particles of volume V due to aggregation is given by

Bag =1

2

∫ V

0a(V − V ′, V ′)n(V − V ′)n(V ′)dV ′ (2.2-10)

where particles of volume V −V ′ aggregate with particles of volume V ′ to form particlesof volume V . The factor 1/2 is included to avoid accounting for each collision eventtwice.

The death rate of particles of volume V due to aggregation is given by

Dag =∫ ∞

0a(V, V ′)n(V )n(V ′)dV ′ (2.2-11)

i The breakage and aggregation kernels depend on the nature of the physicalapplication. For example, in gas-liquid dispersion, the kernels are functionsof the local liquid-phase turbulent dissipation.


2.3 Solution Methods

2.2.3 Particle Birth by Nucleation

Depending on the application, spontaneous nucleation of particles can occur due to thetransfer of molecules from the primary phase. For example, in crystallization from so-lution, the first step is the phase separation or “birth” of new crystals. In boiling ap-plications, the creation of the first vapor bubbles is a nucleation process referred to asnucleate boiling.

The nucleation rate is defined through a boundary condition as shown in Equation 2.2-2.


As discussed in Chapter 1: Introduction, the population balance equation can be solvedby three different methods in FLUENT: the discrete method, the standard method ofmoments (SMM), and the quadrature method of moments (QMOM). For each method,the FLUENT implementation is limited to a single internal coordinate correspondingto particle size. The following subsections describe the theoretical background of eachmethod and list their advantages and disadvantages.

2.3.1 The Discrete Method

The discrete method (also known as the classes or sectional method) was developed byHounslow [5], Litster [6], and Ramkrishna [12]. It is based on representing the continuousparticle size distribution (PSD) in terms of a set of discrete size classes or bins, asillustrated in Figure 2.3.1. The advantages of this method are its robust numerics andthat it gives the PSD directly. The disadvantages are that the bins must be defined apriori and that a large number of classes may be required.



i+1i

n(V)

V

Figure 2.3.1: A Particle Size Distribution as Represented by the DiscreteMethod

Numerical Method

In FLUENT, the PBE is written in terms of volume fraction of particle size i:

∂

∂t(ρsαi) +∇ · (ρsuiαi) +

∂

∂V

(GvρsαiV

)= ρsVi(Bag,i −Dag,i +Bbr,i −Dbr,i) + 0iρsV0n0

(2.3-1)

where ρs is the density of the secondary phase and αi is the volume fraction of particlesize i, defined as

αi = NiVi i = 0, 1, · · · , N − 1 (2.3-2)

where

Ni(t) =∫ Vi+1

Vin(V, t)dV (2.3-3)

and Vi is the volume of the particle size i. In FLUENT, a fraction of α, called fi, isintroduced as the solution variable. This fraction is defined as

fi =αiα

(2.3-4)

where α is the total volume fraction of the secondary phase.



The nucleation rate n0 appears in the discretized equation for the volume fraction of thesmallest size V0. The notation 0i signifies that this particular term, in this case ρsV0n0,appears in Equation 2.3-1 only in the case of the smallest particle size.

The growth rate in Equation 2.3-1 is discretized as follows [5]:

∂

∂V

(GvρsαiV

)= ρsVi

[(Gv,i−1Ni−1

Vi − Vi−1

)−(

Gv,iNi

Vi+1 − Vi

)](2.3-5)

The volume coordinate is discretized as [5] Vi+1/Vi = 2q where q = 1, 2, . . . and is referredto as the “ratio factor”.

The particle birth and death rates are defined as follows:

Bag,i =N∑k=1

N∑j=1

akjNkNjxkjξkj (2.3-6)

Dag,i =N∑j=1

aijNiNj (2.3-7)

Bbr,i =N∑

j=i+1

αjNjβ(Vi | Vj) (2.3-8)

Dbr,i = αiNi (2.3-9)

where aij = a(Vi, Vj) and

ξkj =

1 for Vi < Vag < Vi+1, where i ≤ N − 1

0 otherwise(2.3-10)

Vag is the particle volume resulting from the aggregation of particles k and j, and isdefined as

Vag = [xkjVi + (1− xkj)Vi+1] (2.3-11)

where

xkj =Vag − Vi+1

Vi − Vi+1

(2.3-12)



If Vag is greater than or equal to the largest particle size VN , then the contribution toclass N − 1 is

xkj =Vag

VN(2.3-13)

i Note that there is no breakage for the smallest particle class.

2.3.2 The Standard Method of Moments (SMM)

The SMM, proposed by Randolph and Larson [13] is an alternative method for solvingthe PBE. Its advantages are that it reduces the dimensionality of the problem and thatit is relatively simple to solve transport equations for lower-order moments. The disad-vantages are that exact closure of the right-hand side is possible only in cases of constantaggregation and size-independent growth, and that breakage modeling is not possible.The closure constraint is overcome, however, through QMOM (see Section 2.3.3: TheQuadrature Method of Moments (QMOM)).

Numerical Method

The SMM approach is based on taking moments of the PBE with respect to the internalcoordinate (in this case, the particle size L).

Defining the kth moment as

mk(~x, t) =∫ ∞

0n(L; ~x, t)LkdL k = 0, 1, · · · , N − 1 (2.3-14)

and assuming constant particle growth, its transport equation can be written as

∂

∂t(ρmk) +∇ · (ρ~umk) = ρ(Bag,k −Dag,k +Bbr,k −Dbr,k) + 0kn0 + Growth (2.3-15)

where

Bag,k =1

2

∫ ∞0

n(λ)∫ ∞

0a(u, λ)(u, λ)(u3 + λ3)k/3n(u)dudλ (2.3-16)

Dag,k =∫ ∞

0Lkn(L)

∫ ∞0

a(L, λ)n(λ)dλdL (2.3-17)

Bbr,k =∫ ∞

0Lk∫ ∞

0g(λ)β(L | λ)n(λ)dλdL (2.3-18)

Dbr,k =∫ k

0Lkg(L)n(L)dL (2.3-19)



N is the specified number of moments and n0 is the nucleation rate. The growth term isdefined as

Growth ≡∫ ∞

0kLk−1G(L)n(L, t)dL (2.3-20)

and for constant growth is represented as

kGmk−1 (2.3-21)

Equation 2.3-16 can be derived by using

u3 = L3 − λ3; dL =u2

L2du

and reversing the order of integration. From these moments, the parameters describingthe gross properties of particle population can be derived as

Ntotal = m0 (2.3-22)

Ltotal = m1 (2.3-23)

Atotal = Kam2 (2.3-24)

Vtotal = Kvm3 (2.3-25)

d32 =m3

m2

(2.3-26)

These properties are related to the total number, length, area, and volume of solidparticles per unit volume of mixture suspension. The Sauter mean diameter, d32, isusually used as the mean particle size.

To close Equation 2.3-15, the quantities represented in Equations 2.3-16–2.3-19 need tobe expressed in terms of the moments being solved. To do this, one approach is to assumesize-independent kernels for breakage and aggregation, in addition to other simplificationssuch as the Taylor series expansion of the term (u3 +λ3)k/3. Alternatively, a profile of thePSD could be assumed so that Equations 2.3-16–2.3-19 can be integrated and expressedin terms of the moments being solved.

In FLUENT, an exact closure is implemented by restricting the application of the SMMto cases with size-independent growth and a constant aggregation kernel.



2.3.3 The Quadrature Method of Moments (QMOM)

The quadrature method of moments (QMOM) was first proposed by McGraw [9] formodeling aerosol evolution and coagulation problems. Its applications by Marchisio etal. [8] have shown that the method requires a relatively small number of scalar equationsto track the moments of population with small errors.

The QMOM provides an attractive alternative to the discrete method when aggregationquantities, rather than an exact PSD, are desired. Its advantages are fewer variables(typically only six or eight moments) and a dynamic calculation of the size bins. Thedisadvantages are that the number of abscissas may not be adequate to describe the PSDand that solving the Product-Difference algorithm may be time consuming.

Numerical Method

The quadrature approximation is based on determining a sequence of polynomials or-thogonal to n(L) (i.e., the particle size distribution). If the abscissas of the quadratureapproximation are the nodes of the polynomial of order N , then the quadrature approx-imation

∫ ∞0

f(L)n(L)dL ≈N∑i=1

f(Li)wi, (2.3-27)

is exact if f(L) is a polynomial of order N or smaller [2]. In all other cases, the closerf(L) is to a polynomial, the more accurate the approximation.

A direct way to calculate the quadrature approximation is by means of its definitionthrough the moments:

mk =N∑i=1

wiLki . (2.3-28)

The quadrature approximation of order N is defined by its N weights wi and N abscissasLi and can be calculated by its first 2N moments m0, . . . ,m2N−1 by writing the recursiverelationship for the polynomials in terms of the moments mk. Once this relationship iswritten in matrix form, it is easy to show that the roots of the polynomials correspondto the eigenvalues of the Jacobi matrix [11]. This procedure is known as the Product-Difference algorithm [3]. Once the weights and abscissas are known, the source termsdue to coalescence and breakage can be calculated and therefore the transport equationsfor the moments can be solved.


2.4 Reconstructing the Particle Size Distribution from Moments

Applying Equations 2.3-27 and 2.3-28, the birth and death terms in Equation 2.3-15 canbe rewritten as

Bag,k =1

2

N∑i=1

wiN∑j=1

wj(L3i + L3

j)k/3a(Li, Lj) (2.3-29)

Dag,k =N∑i=1

LkiwiN∑j=1

wja(Li, Lj) (2.3-30)

Bbr,k =N∑i=1

wi

∫ ∞0

Lkg(Li)β(L | Li)dL (2.3-31)

Dbr,k =N∑i=1

wiLki g(Li) (2.3-32)

Theoretically, there is no limitation on the expression of breakage and aggregation kernelswhen using QMOM.

The nucleation rate is defined in the same way as for the SMM. The growth rate forQMOM is defined by Equation 2.3-20 and represented as

N∑i=1

wiLk−1i G(Li) (2.3-33)

to allow for a size-dependent growth rate.

2.4 Reconstructing the Particle Size Distribution from Moments

Given a set of moments, the most likely PSD can be obtained based on the “statisticallymost probable” distribution for turbulent flames [10], which was adapted for crystalliza-tion problems by Baldyga and Orciuch [1].

The number density function n(L) is expressed as

n(L) = exp

(N−1∑i=0

AiLi

)(2.4-1)

The equation for the kth moment is now written as

mk =∫ ∞

0Lk exp

(N−1∑i=0

AiLi

)dL k = 0, 1, · · · , N − 1 (2.4-2)

Given N moments, the coefficients Ai can be found by a globally convergent Newton-Raphson method to reconstruct the particle size distribution (e.g., Figure 2.4.1).



0 0.0005 0.001 0.0015Length(m)

0

2e+13

4e+13

6e+13

8e+13

Num

ber

Den

sity

(#/

m3 −

m)

Original PSDReconstructed PSD

Figure 2.4.1: Reconstruction of a Particle Size Distribution


Chapter 3. Using the FLUENT Population Balance Model

This chapter provides basic instructions to install the population balance model and solvepopulation balance problems in FLUENT. It assumes that you are already familiar withstandard FLUENT features, including the user-defined function procedures described inthe FLUENT UDF Manual. This chapter describes the following:

• Section 3.1: Population Balance Module Installation

• Section 3.2: Loading the Population Balance Module

• Section 3.3: Population Balance Model Setup

3.1 Population Balance Module Installation

The population balance module is provided as an addon module with the standard FLU-ENT licensed software. A special license is required to use the population balance module.

3.2 Loading the Population Balance Module

The population balance module is loaded into FLUENT through the text user interface(TUI). The module can only be loaded when a valid FLUENT case file has been set orread. The text command to load the module is

define −→ models −→addon-module.

A list of FLUENT addon modules is displayed:

FLUENT Addon Modules:

0. none

1. MHD Model

2. Fiber Model

3. PEM Fuel Cell Model

4. SOFC Fuel Cell Model

5. Population Balance Model

Enter Module Number: [0] 5


Using the FLUENT Population Balance Model

Select the Population Balance Model by entering the module number 5. During theloading process a scheme library containing the graphical and text user interface, and audf library containing a set of user defined functions are loaded into FLUENT. A messageAddon Module: popbal1.1...loaded! is displayed at the end of the loading process.

The population balance module setup is saved with the FLUENT case file. The moduleis loaded automatically when the case file is subsequently read into FLUENT. Note thatin the saved case file, the population balance module is saved with the absolute path.Therefore, if the locations of the population balance module installation or the savedcase file are changed, FLUENT will not be able to load the module when the case file issubsequently read.

3.3 Population Balance Model Setup

Following the loading of the population balance module, enable either the mixture orEulerian multiphase model. This will allow you to activate the population balance model,where you will specify the appropriate parameters, and supply multiphase boundaryconditions. These inputs are described in this chapter. Using the double-precision versionof FLUENT when solving population balance problems is highly recommended.

i A limitation of the population balance model is that it can be used only onone secondary phase, even if your problem includes additional secondaryphases.Note that a three-phase gas-liquid-solid case can be modeled, where thepopulation balance model is used for the gas phase and the solid phaseacts as a catalyst.

3.3.1 Enabling the Population Balance Model

The procedure for setting up a population balance problem is described below. (Notethat this procedure includes only those steps necessary for the population balance modelitself; you will need to set up other models, boundary conditions, etc. as usual. See theFLUENT User’s Guide for details.)

1. Start the double-precision version of FLUENT.

2. To enable the population balance model, follow the instructions in Section 3.2: Load-ing the Population Balance Module.

Remember to enable the mixture or Eulerian multiphase model.

3. Open the Population Balance Model panel (Figure 3.3.1).

Define −→ Models −→Population Balance...

4. Specify the population balance method under Method.



Figure 3.3.1: The Population Balance Model Panel

• If you select Discrete, you will need to specify the following parameters:

Definition can be specified as a Geometric Ratio or as a File. If GeometricRatio is selected, then the Ratio Exponent must be specified. If File isselected, you will click the Load File... button and select the bin size filethat you want loaded.

You can input the diameter through the text file, with each diameterlisted on a separate line, starting from the largest to the smallest diameter.Hence, you are not limited by the choices specified in the panel.

Number specifies the number of particle size bins used in the calculation.

Ratio Exponent specifies the exponent q used in the discretization of thegrowth term volume coordinate (see Section 2.3.1: Numerical Method).

Min specifies the minimum bin size L0 ≡ (V0/Kv)1/3.

Max displays the maximum bin size, which is calculated internally.

Kv specifies the value for the particle volume coefficient (Kv in Section 2.3.1: Nu-merical Method). By default, this coefficient has a value of π/6.



To display a list of the bin sizes in the console window, click Print Bins. Thebin sizes will be listed in order of size, from the largest to the smallest. Thisoption is only available when the Geometric Ratio Definition is selected.

• If you select Standard Moment under Method, you will specify the number ofMoments under Parameters.

• If you selected Quadrature Moment under Method, you will set the number ofmoments to either 4, 6 or 8 under Parameters.

5. Select the secondary phase from the Phase drop-down list for which you want toapply the population balance model parameters.

6. For all population balance methods, you can enable the following under Phenomena:

Nucleation Rate allows you to specify the nucleation rate (particles/m3-s). Youcan select constant or user-defined from the drop-down list. If you select con-stant, specify a value in the adjacent field. If you have a user-defined function(UDF) that you want to use to model the nucleation rate, you can choose theuser-defined option and specify the appropriate UDF.

Growth Rate allows you to specify the particle growth rate (m/s). You can selectconstant or user-defined from the drop-down list. If you select constant, specifya value in the adjacent field. If you have a user-defined function (UDF) thatyou want to use to model the growth rate, you can choose the user-definedoption and specify the appropriate UDF.

Aggregation Kernel allows you to specify the aggregation kernel (m3/s). Youcan select constant, luo-odel, or user-defined from the drop-down list. If youselect constant, specify a value in the adjacent field. If you select luo-model, theaggregation rate will be calculated based on Luo’s aggregation kernel describedin Section 2.2.2: Particle Birth and Death Due to Breakage and Aggregation.If you have a user-defined function (UDF) that you want to use to modelthe aggregation rate, you can choose the user-defined option and specify theappropriate UDF.

Breakage Kernel allows you to specify the particle breakage rate (particles/m3-s). You can select constant, luo-model, or user-defined from the drop-downlist. If you select constant, specify a value in the adjacent field. If you selectluo-model, the breakage rate will be calculated based on Luo’s breakage kerneldescribed in Section 2.2.2: Particle Birth and Death Due to Breakage andAggregation. If you have a user-defined function (UDF) that you want to useto model the breakage rate, you can choose the user-defined option and specifythe appropriate UDF.

See Chapter 5: UDFs for Population Balance Modeling for details about UDFs forthe population balance model.



7. Specify the boundary conditions for the solution variables.


See Section 3.3.2: Defining Population Balance Boundary Conditions below.

8. Specify the initial guess for the solution variables.

Solve −→ Initialize −→Initialize...

9. Solve the problem and perform relevant postprocessing functions.

Solve −→Iterate...

See Chapter 4: Postprocessing for the Population Balance Model for details aboutpostprocessing.



3.3.2 Defining Population Balance Boundary Conditions

To define boundary conditions specific to the population balance model, use the followingprocedure:

1. In the Boundary Conditions panel, select the secondary phase(s) in the Phase drop-down list and then open the appropriate boundary condition panel (e.g., Fig-ure 3.3.2).


Figure 3.3.2: Specifying Inlet Boundary Conditions for the Population Bal-ance Model

2. In the Multiphase tab, under Boundary Condition, select the type of boundary con-dition for each bin (for the discrete method) or moment (for SMM and QMOM) aseither Specified Value or Specified Flux.

Note that the boundary condition variables (e.g., phase-2 Bin-0) are labeled accord-ing to the following:

phase bin/moment-ith bin/moment

where the ith bin/moment can range from 0 (the first bin or moment) to N − 1,where N is the number of bins/moments that you entered in the Population BalanceModel panel.



3. Under Population Balance Boundary Value, enter a value or a flux as appropriate.

• If you selected Specified Value for the selected boundary variable, enter a valuein the field adjacent to the variable name. This value will correspond to thevariable fi in Equation 2.3-4 (for the discrete method) ormk in Equation 2.3-15(for SMM or QMOM).

• If you selected Specified Flux for the selected boundary variable, enter a valuein the field adjacent to the variable name. This value will be the spatialparticle volume flux dV/dxi.

3.3.3 Specifying Population Balance Solution Controls

In the Solution Controls panel (Figure 3.3.3, equations for each bin (e.g., air Bin-0 fraction)will appear in the Equations list. The default value under Under-Relaxation Factors for thepopulation balance equations is 0.5, and the default Discretization scheme is First OrderUpwind.

Figure 3.3.3: The Solution Controls Panel



3.3.4 Coupling With Fluid Dynamics

To couple population balance modeling of the secondary phase(s) with the overall problemfluid dynamics, a Sauter mean diameter (d32 in Equation 2.3-26) may be used to representthe particle diameter of the secondary phase. The Sauter mean diameter is defined asthe ratio of the third moment to the second moment for the SMM and QMOM. For thediscrete method, it is defined as

d32 =

∑NiL

3i∑

NiL2i

(3.3-1)

To specify the Sauter mean diameter as the secondary phase particle diameter, openthe Phases panel (Define/Phases...). Select the secondary phase and click Set. In theresulting Secondary Phase panel (e.g., Figure 3.3.4), select sauter-mean from the Diameterdrop-down list under Properties. Note that a constant diameter or user-defined functionmay also be used.

Figure 3.3.4: The Secondary Phase Panel for Hydrodynamic Coupling



3.3.5 Specifying Interphase Mass Transfer Due to Nucleation and Growth

In applications that involve the creation, dissolution, or growth of particles (e.g., crys-tallization), the total volume fraction equation for the particulate phase will have sourceterms due to these phenomena. The momentum equation for the particulate phasewill also have source terms due to the added mass. In FLUENT, the mass sourceterm can be specified using the UDF hook DEFINE HET RXN RATE, as described in Ap-pendix A: DEFINE HET RXN RATE Macro, or using the Phase Interaction panel, describedbelow.

As an example, in crystallization, particles are created by means of nucleation (n0), anda growth rate (G) can also be specified. The mass transfer rate of formation (in kg/m3-s)of particles of all sizes is then

m = 3ρKv

∫ ∞0

L2Gn(L)dL

=1

2ρKa

∫ ∞0

L2Gn(L)dL

(3.3-2)

For the discrete method, the mass transfer rate due to growth can be written as

m = ρ∫ ∞

0Gvn(L)dL

= ρ∫ ∞

0Gvn(V )dV

= ρ∑i

Gv,iNi

(3.3-3)

If the nucleation rate is included in the total mass transfer, then the mass transferbecomes

m = ρV0n0 +∑i

ρGv,iNi (3.3-4)

i For the discrete method, the sources to the population balance equationsmust sum to the total mass transfer rate. To access the sources, you canuse the macro C PB DISCI PS (cell, thread, i).

See Chapter 5: UDFs for Population Balance Modeling for more information aboutmacros for population balance variables.



For the SMM, only a size-independent growth rate is available. Hence, the mass transferrate can be written as

m =1

2ρKaGm2 (3.3-5)

For the QMOM, the mass transfer rate can be written as

m =1

2ρKa

∑i

L2iwiG(Li) (3.3-6)

For both the SMM and QMOM, mass transfer due to nucleation is negligible, and is nottaken into account.

i Note that for crystallization, the primary phase is comprised of multiplecomponents. At the very least, there is a solute and a solvent. To definethe multicomponent multiphase system, you will need to activate SpeciesTransport in the Species Model panel for the primary phase after activatingthe multiphase model. The rest of the procedure for setting up a speciestransport problem is identical to setting up species in single phase. Theheterogeneous reaction is defined as:

Solute (liquid) −→ Crystal (secondary phase)

When the population balance model is activated, mass transfer between phases for non-reacting species (such as boiling) and heterogeneous reactions (such as crystallization)can be done automatically, in lieu of hooking a UDF.

For simple unidirectional mass transfer between primary and secondary phases due tonucleation and growth phenomena of non-reacting species, go to the Phases panel (Fig-ure 3.3.5) and click the Interaction... button. This will open the Phase Interaction panel(Figure 3.3.6). Click the Mass tab to specify the Mass Transfer of species between thephases. Specify the Number of Mass Transfer Mechanisms involved in your case. From thedrop-down list under From Phase, select the phase that you want to transfer mass from.In the To Phase drop-down list, select the phase that you want to transfer mass to.



You have a choice of four mechanisms used to transfer mass. Under Mechanism selectfrom the drop-down list

none if you do not want any mass transfer between the phases.

constant-rate for a fixed, user-specified rate.

user-defined if you hooked a UDF describing the mass transfer mechanism.

population-balance for an automated method of mass transfer, not involving a UDF.

Click OK to save the settings.

For heterogeneous reactions, the Species Transport model has to be activated for the pri-mary phase. In the Phases panel (Figure 3.3.7), click the Interaction... button. This willopen the Phase Interaction panel (Figure 3.3.8). Click the Reactions tab to specify the stoi-chiometry for the reactant and the product. At the bottom of the Phase Interaction panel,select population-balance as the Reaction Rate Function. Click OK to save the settings. Ei-ther this method or the use of the UDF, described in Appendix A: DEFINE HET RXN RATE

Macro, will produce the same results.

Figure 3.3.5: The Phase Interaction Panel for Non-reacting Species



Figure 3.3.6: The Phase Interaction Panel for Non-reacting Species

Figure 3.3.7: The Phases Panel



Figure 3.3.8: The Phase Interaction Panel for a Heterogeneous Reaction


Chapter 4. Postprocessing for the Population BalanceModel

FLUENT provides postprocessing options for displaying, plotting, and reporting variousparticle quantities, which include the main solution variables and other auxiliary quan-tities.

• Section 4.1: Population Balance Solution Variables

• Section 4.2: Reporting Derived Population Balance Variables

4.1 Population Balance Solution Variables

Solution variables that can be reported for the population balance model are:

• Bin-i fraction (discrete method only), where i is N-1 bins/moments.

• Number density of Bin-i fraction (discrete method only)

• Diffusion Coef. of Bin-i fraction/Moment-i

• Sources of Bin-i fraction/Moment-i

• Moment-i (SMM and QMOM only)

• Abscissa-i (QMOM method only)

• Weight-i (QMOM method only)

Bin-i fraction is the fraction (fi) of the volume fraction for the ith size bin when usingthe discrete method. Number density of Bin-i fraction is the number density (Ni) inparticles/m3 for the ith size bin. Moment-i is the ith moment of the distribution whenusing the standard method of moments or the quadrature method of moments.

i Though the diffusion coefficients of the population variables are available(e.g., Diffusion Coef. of Bin-i fraction/Moment-i), they are set to zero be-cause the diffusion term is not present in the population balance equations.


Postprocessing for the Population Balance Model

4.2 Reporting Derived Population Balance Variables

Two options are available in the Report menu that allow you to report compute momentsand number density on selected surfaces or cell zones of the domain.

4.2.1 Computing Moments

You can compute moments for the population balance model using the Moments forPopulation Balance panel (Figure 4.2.1).

Report −→ Population Balance −→Moments...

Figure 4.2.1: The Moments for Population Balance Panel

The steps for computing moments are as follows:

1. For the discrete method, specify the Number of Moments. For the SMM andQMOM, the number of moments is set equal to the number of moments that weresolved, and thus cannot be changed.

2. For a surface average, select the surface(s) on which to calculate the moments inthe Surfaces list.

3. For a volume average, select the volume(s) in which to calculate the moments inthe Cell Zones list.

4. Click Print to display the moment values in the console window.

5. To save the moment calculations to a file, click Save... and enter the appropriateinformation in the resulting Select File dialog box. The file extension should be.pb.


4.2 Reporting Derived Population Balance Variables

4.2.2 Displaying a Number Density Function

You can display the number density function for the population balance model using theNumber Density Function panel (Figure 4.2.2).

Report −→ Population Balance −→Number Density...

Figure 4.2.2: The Number Density Function Panel

The steps for displaying the number density function are as follows:

1. Specify the Report Type as either a Surface Average or a Volume Average.

2. Under Plot Type, specify how you would like to display the number density functiondata.

Histogram displays a histogram of the discrete number density (Ni). The numberof divisions in the histogram is equal to the number of bins specified in thePopulation Balance Model panel. This option is available only with the discretemethod.

Curve displays a smooth curve of the number density function.

Write To File allows you to write the number density function data to a file (seebelow).

3. In the Fields list, select the data to be plotted.

Discrete Number Density (Ni) is the number of particles per unit volume of phys-ical space in the ith size bin plotted against particle diameter size i. Thisoption is available only with the discrete method.

Length Number Density Function (n(L)) is the number of particles per unit vol-ume of physical space per unit particle length plotted against particle diameter.


Postprocessing for the Population Balance Model

Volume Number Density Function (n(V )) is the number of particles per unit vol-ume of physical space per unit particle volume plotted against particle volume.

4. Choose the surface(s) on which to plot the number density function data in theSurfaces list.

5. Click Plot... to display the data.

6. (optional) Click Print to display the number density function data in the consolewindow.

7. (optional) To save the number density function data to a file, select the Write ToFile option. The Plot... button will change to the Save... button. Clicking on theSave... button will open a Select File dialog box, in which you can specify a nameand save a file containing the plot data. The file extension should be .pbd.


Chapter 5. UDFs for Population Balance Modeling

5.1 Population Balance Variables

The macros listed in Table 5.1.1 can be used to return real variables associated withthe population balance model. The variables are available in both the pressure-basedand density-based solvers. The macros are defined in the sg pb.h header file, which isincluded in udf.h.

Table 5.1.1: Macros for Population Balance Variables Defined in sg pb.h

Macro Argument Types ReturnsC PB DISCI cell t c, Thread *t, int i fraction (fi) of the total volume

fraction for the ith size binC PB SMMI cell t c, Thread *t, int i ith momentC PB QMOMI cell t c, Thread *t, int i ith moment, where i = 0, 1, 2, 3, 4, 5C PB QMOMI L cell t c, Thread *t, int i abscissa Li, where i = 0, 1, 2C PB QMOMI W cell t c, Thread *t, int i weight wi, where i = 0, 1, 2C PB DISCI PS cell t c, Thread *t, int i net source term to ith size binC PB SMMI PS cell t c, Thread *t, int i net source term to ith momentC PB QMOMI PS cell t c, Thread *t, int i net source term to ith moment

5.2 Population Balance DEFINE Macros

This section contains descriptions of DEFINE macros for the population balance model.Definitions of each DEFINE macro are contained in the udf.h header file.

• DEFINE PB BREAK UP RATE

• DEFINE PB COALESCENCE RATE

• DEFINE PB NUCLEATION RATE

• DEFINE PB GROWTH RATE


UDFs for Population Balance Modeling

5.2.1 DEFINE PB BREAK UP RATE

You can use the DEFINE PB BREAK UP RATE macro if you want to define your own particlebreakage kernel. The function is executed at the beginning of every time step.

Usage

DEFINE PB BREAK UP RATE(name, cell, thread, d 1, d 2, freq, pdf)

Argument Type Descriptionchar name UDF namecell t cell Cell indexThread *thread Pointer to the secondary phase threadreal d 1 Parent particle diameter or lengthreal d 2 Diameter of one of the daughter particles after breakage;

the second daughter particle diameter is calculated byconservation of particle volume

real *freq Pointer to the breakage frequencyreal *pdf Pointer to the breakage distribution function (PDF)

Function returnsreal

There are seven arguments to DEFINE PB BREAK UP RATE: name, cell, thread, d 1, d 2,freq, and pdf. You will supply name, the name of the UDF. cell, thread, d 1, d 2,freq, and pdf are variables that are passed by the FLUENT solver to your UDF.

Example

Included below is a example UDF for a parabolic breakage kernel (see Section 2.2.2: Par-ticle Birth and Death Due to Breakage and Aggregation) that has been formulated interms of a numeric constant C [4]. In this example, the breakage frequency g(V ) = 1and the breakage PDF is defined as

β(L | λ) =3CL2

λ3+(

1− C

2

)(72L8

λ9− 72L5

λ6+

18L2

λ3

)

The PDF respects the constraints in Section 2.2.2: Particle Birth and Death Due toBreakage and Aggregation. Depending on the value of C, different behaviors are observed.For example, if C = 2, the particle breakage has a uniform distribution. If 0 < C < 2,a concave parabola is obtained, meaning that it is more likely to obtain unequally sizedfragments than equally sized fragments. The opposite of this is true for 2 < C < 3.



Values outside of the range of 0 to 3 are not allowed because the PDF cannot have anegative value.

i In the UDF example below, the breakage PDF has been converted fromthe length-based kernel to the volume-based kernel as follows:

β(V | V ′) =β(L | λ)

3L2

where V = KvL3 and V ′ = Kvλ

3.

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

UDF that computes the particle breakage rate

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

#include "udf.h"

#include "sg_pb.h"

#include "sg_mphase.h"

DEFINE_PB_BREAK_UP_RATE(break_up_kernel, cell, thread, d_1, d_2, freq, pdf)

/*d_1 is the particle breaking into d_2 and its complementary diameter

d’_2 such that d_13 = d_23 + d’_23*/

/*parabolic kernel is symmetric about v = 0.5*/

real C=1., f_1, f_2, f_3, f_4, func;

real lambda = d_1;

real v = pow(d_2,3.)/pow(d_1,3.);

f_1 = C/(pow(lambda,3.));

f_2 = 24.*pow(v,2.);

f_3 = -24.*v;

f_4 = 6.;

func = f_1 + ((1.-C/2.)/pow(lambda,3.))*(f_2 + f_3 + f_4);

/*freq returns the number of d_1’s breaking up*/

*freq = 1; /*units are 1/s*/

/*pdf returns the daughter distribution*/

*pdf = func; /*units are i/m3*/



5.2.2 DEFINE PB COALESCENCE RATE

You can use the DEFINE PB COALESCENCE RATE macro if you want to define your ownparticle aggregation kernel. The function is executed at the beginning of every time step.

Usage

DEFINE PB COALESCENCE RATE(name, cell, thread, d 1, d 2)

Argument Type Descriptionchar name UDF namecell t cell Cell indexThread *thread Pointer to the secondary phase threadreal d 1, d 2 Diameters of the two colliding particles


There are five arguments to DEFINE PB COALESCENCE RATE: name, cell, thread, d 1,and d 2. You will supply name, the name of the UDF. cell, thread, d 1, and d 2 arevariables that are passed by the FLUENT solver to your UDF. Your UDF will need toreturn the real value of the aggregation rate.

Example

Included below is a example UDF for a Brownian aggregation kernel. In this example,the aggregation rate is defined as

a(L, λ) = a(V, V ′) = β0(L+ λ)2

Lλ

where β0 = 1× 10−17 m3/s.



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

UDF that computes the particle aggregation rate

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

#include "udf.h"

#include "sg_pb.h"


DEFINE_PB_COALESCENCE_RATE(aggregation_kernel,cell,thread,d_1,d_2)

real agg_kernel;

real beta_0 = 1.0e-17 /* aggregation rate constant */

agg_kernel = beta_0*pow((d_1+d_2),2.0)/(d_1*d_2);

return agg_kernel;

5.2.3 DEFINE PB NUCLEATION RATE

You can use the DEFINE PB NUCLEATION RATE macro if you want to define your ownparticle nucleation rate. The function is executed at the beginning of every time step.

Usage

DEFINE PB NUCLEATION RATE(name, cell, thread)

Argument Type Descriptionchar name UDF namecell t cell Cell indexThread *thread Pointer to the secondary phase thread


There are three arguments to DEFINE PB NUCLEATION RATE: name, cell, and thread.You will supply name, the name of the UDF. cell and thread are variables that arepassed by the FLUENT solver to your UDF. Your UDF will need to return the real

value of the nucleation rate.



Example

Potassium chloride can be crystallized from water by cooling. Its solubility decreaseslinearly with temperature. Assuming power-law kinetics for the nucleation rate,

n0 = Kn(S − 1)Nn

where Kn = 4× 1010 particles/m3-s and Nn = 2.77.

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

UDF that computes the particle nucleation rate

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

#include "udf.h"

#include "sg_pb.h"


DEFINE_PB_NUCLEATION_RATE(nuc_rate, cell, thread)

real J, S;

real Kn = 4.0e10; /* nucleation rate constant */

real Nn = 2.77; /* nucleation law power index */

real T,solute_mass_frac,solvent_mass_frac, solute_mol_frac,solubility;

real solute_mol_wt, solvent_mol_wt;

Thread *tc = THREAD_SUPER_THREAD(thread); /*obtain mixture thread */

Thread **pt = THREAD_SUB_THREADS(tc); /* pointer to sub_threads */

Thread *tp = pt[P_PHASE]; /* primary phase thread */

solute_mol_wt = 74.55; /* molecular weight of potassium chloride */

solvent_mol_wt = 18.; /* molecular weight of water */

solute_mass_frac = C_YI(cell,tp,0);

/* mass fraction of solute in primary phase (solvent) */

solvent_mass_frac = 1.0 - solute_mass_frac;

solute_mol_frac = (solute_mass_frac/solute_mol_wt)/

((solute_mass_frac/solute_mol_wt)+(solvent_mass_frac/solvent_mol_wt));

T = C_T(cell,tp); /* Temperature of primary phase in Kelvin */

solubility = 0.0005*T-0.0794;

/* Solubility Law relating equilibrium solute mole fraction to Temperature*/



S = solute_mol_frac/solubility; /* Definition of Supersaturation */

if (S <= 1.)

J = 0.;

else

J = Kn*pow((S-1),Nn);

return J;

i Note that the solubility and the chemistry could be defined in a separateroutine and simply called from the above function.

5.2.4 DEFINE PB GROWTH RATE

You can use the DEFINE PB GROWTH RATE macro if you want to define your own particlegrowth rate. The function is executed at the beginning of every time step.

Usage

DEFINE PB GROWTH RATE(name, cell, thread,d i)

Argument Type Descriptionchar name UDF namecell t cell Cell indexThread *thread Pointer to the secondary phase threadreal d i Particle diameter or length


There are four arguments to DEFINE PB GROWTH RATE: name, cell, thread, and d i. Youwill supply name, the name of the UDF. cell, thread, and d i are variables that arepassed by the FLUENT solver to your UDF. Your UDF will need to return the real valueof the growth rate.



Example

Potassium chloride can be crystallized from water by cooling. Its solubility decreaseslinearly with temperature. Assuming power-law kinetics for the growth rate,

G = Kg(S − 1)Ng

where Kg = 2.8× 10−8 m/s and Ng = 1.

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

UDF that computes the particle growth rate

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

#include "udf.h"

#include "sg_pb.h"


DEFINE_PB_GROWTH_RATE(growth_rate, cell, thread,d_1)

/* d_1 can be used if size-dependent growth is needed */

/* When using SMM, only size-independent or linear growth is allowed */

real G, S;

real Kg = 2.8e-8; /* growth constant */

real Ng = 1.; /* growth law power index */

real T,solute_mass_frac,solvent_mass_frac, solute_mol_frac,solubility;

real solute_mol_wt, solvent_mol_wt;

Thread *tc = THREAD_SUPER_THREAD(thread); /*obtain mixture thread */

Thread **pt = THREAD_SUB_THREADS(tc); /* pointer to sub_threads */

Thread *tp = pt[P_PHASE]; /* primary phase thread */

solute_mol_wt = 74.55; /* molecular weight of potassium chloride */

solvent_mol_wt = 18.; /* molecular weight of water */

solute_mass_frac = C_YI(cell,tp,0);

/* mass fraction of solute in primary phase (solvent) */

solvent_mass_frac = 1.0 - solute_mass_frac;

solute_mol_frac = (solute_mass_frac/solute_mol_wt)/

((solute_mass_frac/solute_mol_wt)+(solvent_mass_frac/solvent_mol_wt));

T = C_T(cell,tp); /* Temperature of primary phase in Kelvin */


5.3 Hooking a Population Balance UDF to FLUENT

solubility = 0.0005*T-0.0794;

/* Solubility Law relating equilibrium solute mole fraction to Temperature*/

S = solute_mol_frac/solubility; /* Definition of Supersaturation */

if (S <= 1.)

G = 0.;

else

G = Kg*pow((S-1),Ng);

return G;

i Note that the solubility and the chemistry could be defined in a separateroutine and simply called from the above function.

5.3 Hooking a Population Balance UDF to FLUENT

After the UDF that you have defined using DEFINE PB BREAK UP RATE, DEFINE PB

COALESCENCE RATE, DEFINE PB NUCLEATION RATE, or DEFINE PB GROWTH RATE is inter-preted or compiled, the name that you specified in the DEFINE macro argument (e.g.,agg kernel) will become visible and selectable in the appropriate drop-down list underPhenomena in the Population Balance Model panel (Figure 3.3.1).


Appendix A. DEFINE HET RXN RATE Macro

A.1 Description

You need to use DEFINE HET RXN RATE to specify reaction rates for heterogeneous reac-tions. A heterogeneous reaction is one that involves reactants and products from morethan one phase. Unlike DEFINE VR RATE, a DEFINE HET RXN RATE UDF can be specifieddifferently for different heterogeneous reactions.

During FLUENT execution, the DEFINE HET RXN RATE UDF for each heterogeneous re-action that is defined is called in every fluid cell. FLUENT will use the reaction ratespecified by the UDF to compute production/destruction of the species participating inthe reaction, as well as heat and momentum transfer across phases due to the reaction.

A heterogeneous reaction is typically used to define reactions involving species of differ-ent phases. The bulk phase can participate in the reaction if the phase does not haveany species (i.e. phase has fluid material instead of mixture material). Heterogeneousreactions are defined in the Phase Interaction panel.

c© Fluent Inc. July 19, 2006 A-1

DEFINE HET RXN RATE Macro

A.2 Usage

DEFINE HET RXN RATE(name,c,t,r,mw,yi,rr,rr t)

Argument Type Descriptionchar name UDF name.cell t c Cell index.Thread *t Cell thread (mixture level) on which

heterogeneous reaction rate is to be applied.Hetero Reaction *r Pointer to data structure that represents

the current heterogeneous reaction(see sg mphase.h).

real mw[MAX PHASES][MAX SPE EQNS] Matrix of species molecular weights.mw[i][j] will give molecular weight ofspecies with ID j in phase with index i.For phase which has fluid material, themolecular weight can be accessed asmw[i][0].

real yi[MAX PHASES][MAX SPE EQNS] Matrix of species mass fractions.yi[i][j] will give molecular weight ofspecies with ID j in phase with index i.For phase which has fluid material,yi[i][0] will be 1.

real *rr Pointer to laminar reaction rate.real *rr t Currently not used. Provided for future use.

Function returnsvoid

There are eight arguments to DEFINE HET RXN RATE: name, c, t, r, mw, yi, rr, and rr t.You will supply name, the name of the UDF. c, t, r, mw, yi, rr, and rr t are variablesthat are passed by the FLUENT solver to your UDF. Your UDF will need to set the valuesreferenced by the real pointer rr.

A-2 c© Fluent Inc. July 19, 2006

A.3 Example

A.3 Example

The following compiled UDF, named arrh, defines an Arrhenius-type reaction rate. Therate exponents are assumed to be same as the stoichiometric coefficients.

#include "udf.h"

static const real Arrhenius = 1.e15;

static const real E_Activation = 1.e6;

#define SMALL_S 1.e-29

DEFINE_HET_RXN_RATE(arrh,c,t,hr,mw,yi,rr,rr_t)

Domain **domain_reactant = hr->domain_reactant;

real *stoich_reactant = hr->stoich_reactant;

int *reactant = hr->reactant;

int i;

int sp_id;

int dindex;

Thread *t_reactant;

real ci;

real T = 1200.; /* should obtain from cell */

/* instead of compute rr directly, compute log(rr) and then

take exp */

*rr = 0;

for (i=0; i < hr->n_reactants; i++)

sp_id = reactant[i]; /* species ID to access mw and yi */

if (sp_id == -1) sp_id = 0; /* if phase does not have species,

mw, etc. will be stored at index 0 */

dindex = DOMAIN_INDEX(domain_reactant[i]);

/* domain index to access mw & yi */

t_reactant = THREAD_SUB_THREAD(t,dindex);

/* get conc. */

ci = yi[dindex][sp_id]*C_R(c,t_reactant)/mw[dindex][sp_id];

ci = MAX(ci,SMALL_S);

c© Fluent Inc. July 19, 2006 A-3

DEFINE HET RXN RATE Macro

*rr += stoich_reactant[i]*log(ci);

*rr += log(Arrhenius + SMALL_S) -

E_Activation/(UNIVERSAL_GAS_CONSTANT*T);

/* 1.e-40 < rr < 1.e40 */

*rr = MAX(*rr,-40);

*rr = MIN(*rr,40);

*rr = exp(*rr);

A.4 Hooking a Heterogeneous Reaction Rate UDF to FLUENT

After the UDF that you have defined using DEFINE HET RXN RATE is interpreted or com-piled (see the Fluent UDF Manual for details), the name that you specified in the DEFINEmacro argument (e.g., arrh) will become visible and selectable under Reaction Rate Func-tion in the Reactions tab of the Phase Interaction panel. (Note you will first need to specifythe Total Number of Reactions greater than 0.)

A-4 c© Fluent Inc. July 19, 2006

Bibliography

[1] J. Baldyga and W. Orciuch. Barium Sulfate Precipatation in a Pipe – An Exper-imental Study and CFD Modeling. Chemical Engineering Science, 56:2435–2444,2001.

[2] H. Dette and W. J. Studden. The Theory of Canonical Moments with Applicationsin Statistics, Probability, and Analysis. John Wiley & Sons, New York, NY, 1997.

[3] R. G. Gordon. Error Bounds in Equilibrium Statistical Mechanics. Journal ofMathematical Physics, 56:655–633, 1968.

[4] P. J. Hill and K. M. Ng. New Discretization Procedure for the Breakage Equation.AIChE Journal, 41:1204–1216, 1995.

[5] M. J. Hounslow, R. L. Ryall, and V. R. Marshall. A Discretized Population Balancefor Nucleation, Growth, and Aggregation. AIChE Journal, 34(11):1821–1832, 1988.

[6] J. D. Litster, D. J. Smit, and M. J. Hounslow. Adjustable Discretization PopulationBalance for Growth and Aggregation. AIChE Journal, 41(3):591–603, 1995.

[7] H. Luo and H. F. Svendsen. Theoretical Model for Drop and Bubble Breakup inTurbulent Dispersions. AIChE Journal, 42(5):1225–1233, 1996.

[8] D. L. Marchisio, R. D. Virgil, and R. O. Fox. Quadrature Method of Moments forAggregation-Breakage Processes. Journal of Colloid and Interface Science, 258:322–334, 2003.

[9] R. McGraw. Description of Aerosol Dynamics by the Quadrature Method of Mo-ments. Aerosol Science and Technology, 27:255–265, 1997.

[10] S. B. Pope. Probablity Distrubutions of Scalars in Turbulent Shear Flow. In Tur-bulent Shear Flows, volume 2, pages 7–16, 1979.

[11] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery. NumericalRecipes. Cambridge University Press, Cambridge, England, 1992.

[12] D. Ramkrishna. Population Balances: Theory and Applications to Particulate Sys-tems in Engineering. Academic Press, San Diego, CA, 2000.

[13] A. D. Randolph and M. A. Larson. Theory of Particulate Processes: Analysis andTechniques of Continuous Crystallization. Academic Press, San Diego, CA, 1971.

c© Fluent Inc. July 19, 2006 Bib-1

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