ansys fluent 12 - بیت دانلودs2.bitdownload.ir/engineering/all.ansys/ansys 12...

ANSYS FLUENT 12.0

Population Balance

Module Manual

April 2009

Copyright c© 2009 by ANSYS, Inc.All Rights Reserved. No part of this document may be reproduced or otherwise used in

any form without express written permission from ANSYS, Inc.

Airpak, Mechanical APDL, Workbench, AUTODYN, CFX, FIDAP, FloWizard, FLUENT,GAMBIT, Iceboard, Icechip, Icemax, Icepak, Icepro, Icewave, MixSim, POLYFLOW, TGrid,

and any and all ANSYS, Inc. brand, product, service and feature names, logos andslogans are registered trademarks or trademarks of ANSYS, Inc. or its subsidiaries

located in the United States or other countries. All other brand, product, service andfeature names or trademarks are the property of their respective owners.

CATIA V5 is a registered trademark of Dassault Systemes. CHEMKIN is a registeredtrademark of Reaction Design Inc.

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

ANSYS, Inc. is certified to ISO 9001:2008

See the on-line documentation for the complete Legal Notices for ANSYS proprietarysoftware and third-party software. If you are unable to access the Legal Notice, contact

ANSYS, Inc.

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 . . . . . . . . . . . . . . . . . . . . 2-13

2.3 Solution Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-13

2.3.1 The Discrete Method . . . . . . . . . . . . . . . . . . . . . . . . 2-13

2.3.2 The Standard Method of Moments (SMM) . . . . . . . . . . . . 2-17

2.3.3 The Quadrature Method of Moments (QMOM) . . . . . . . . . . 2-19

2.4 Reconstructing the Particle Size Distribution from Moments . . . . . . . 2-20

3 Using the ANSYS FLUENT Population Balance Model 3-1

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

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

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 . . . . . . . . 3-9

3.3.3 Specifying Population Balance Solution Controls . . . . . . . . . 3-10

Release 12.0 c© ANSYS, Inc. January 12, 2009 i

CONTENTS

3.3.4 Coupling With Fluid Dynamics . . . . . . . . . . . . . . . . . . . 3-11

3.3.5 Specifying Interphase Mass Transfer Due to Nucleationand Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-12

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 BREAKUP RATE FREQ . . . . . . . . . . . . . . . . . . . 5-2

5.2.2 DEFINE PB BREAKUP RATE PDF . . . . . . . . . . . . . . . . . . . . 5-3

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

5.2.4 DEFINE PB NUCLEATION RATE . . . . . . . . . . . . . . . . . . . . 5-6

5.2.5 DEFINE PB GROWTH RATE . . . . . . . . . . . . . . . . . . . . . . . 5-8

5.3 Hooking a Population Balance UDF to ANSYS FLUENT . . . . . . . . . 5-10

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 ANSYS FLUENT . . . . A-4

ii Release 12.0 c© ANSYS, Inc. January 12, 2009

Using This Manual

The Contents of This Manual

The ANSYS FLUENT Population Balance Model Manual tells you what you need toknow to model population balance with ANSYS FLUENT. In this manual, you will findbackground information pertaining to the model, a theoretical discussion of the modelused in ANSYS 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.

• A 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 dialog box, surface/iso-surface com-mand).

• The text interface type style is also used when illustrating exactly what appearson the screen or exactly what you need to type into a field in a dialog box. Theinformation displayed on the screen is enclosed in a large box to distinguish it fromthe narrative text, and user inputs are often enclosed in smaller boxes.

• A mini flow chart is used to guide you through the navigation pane, which leadsyou to a specific task page or dialog box. For example,

Models −→ Multiphase −→ Edit...

indicates that Models is selected in the navigation pane, which then opens thecorresponding task page. In the Models task page, Multiphase is selected from thelist. Clicking the Edit... button opens the Multiphase dialog box.

Release 12.0 c© ANSYS, Inc. January 12, 2009 UTM-1

Using This Manual

Also, a mini flow chart is used to indicate the menu selections that lead you to aspecific command or dialog box. For example,

Define −→Injections...

indicates that the Injections... menu item can be selected from the Define pull-downmenu, and

display −→mesh

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

In this manual, mini flow charts usually precede a description of a dialog box orcommand, or a screen illustration showing how to use the dialog box or command.They allow you to look up information about a command or dialog box and quicklydetermine how to access it without having to search the preceding material.

• The menu selections that will lead you to a particular dialog box or task pageare also indicated (usually within a paragraph) using a “/”. For example, De-fine/Materials... tells you to choose the Materials... menu item from the Definepull-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

UTM-2 Release 12.0 c© ANSYS, Inc. January 12, 2009

Using This Manual

– 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

– 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

Release 12.0 c© ANSYS, Inc. January 12, 2009 UTM-3

Using This Manual

Technical Support

If you encounter difficulties while using ANSYS FLUENT, please first refer to the section(s)of the 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, or from the User Services Center (www.fluentusers.com).

If you encounter an error, please write down the exact error message that appeared andnote as much information as you can about what you were doing in ANSYS FLUENT. Thenrefer to the following resources available on the User Services Center (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 ANSYS FLUENT - link available from the product page. Thedefects can be 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.

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 log a technical supportrequest (www.fluentusers.com) to obtain further assistance. However, there are a fewthings that we encourage you to do before logging a request:

• Note what you are trying to accomplish with ANSYS FLUENT.

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

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

UTM-4 Release 12.0 c© ANSYS, Inc. January 12, 2009

Chapter 1. Introduction

In ANSYS FLUENT the population balance model is provided as an addon module withthe standard ANSYS FLUENT licensed software.

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.

ANSYS FLUENT offers three solution methods to the population balance equation: dis-cretized population balance, standard method of moments, and quadrature method ofmoments.

• 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.

Release 12.0 c© ANSYS, Inc. January 12, 2009 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 Release 12.0 c© ANSYS, Inc. January 12, 2009

Chapter 2. Population Balance Model Theory

This chapter presents an overview of the theory and the governing equations for themethods used in ANSYS 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

pg(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.



2.2.2 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 [18], 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

pg(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 pg(V ′)n(V ′)dV ′

particles, of which a fraction β(V | V ′)dV represents particles of volume V . p 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: thenormalized number of breaking particles must sum to unity, the masses of the fragmentsmust sum to the original particle mass, and the number of fragments formed has to becorrectly represented.



Mathematically, these constraints can be written as follows:

• For the normalization condition:

∫ V ′

0β(V | V ′)dV = 1 (2.2-8)

• For conservation of mass

p∫ V ′

0m (V ) β(V | V ′)dV = m (V ′) (2.2-9)

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

β(V ′ − V | V ′) = β(V | V ′) (2.2-10)

The following is a list of models available in ANSYS FLUENT to calculate the breakagefrequency:

• constant value

• Luo model

• Lehr model

• Ghadiri model

• user-defined model

ANSYS FLUENT provides the following models for calculating the breakage PDF:

• parabolic PDF

• generalized PDF for multiple breakage fragments


The breakage frequency models and the parabolic and generalized PDFs are describedin detail in the sections that follow.



Luo and Lehr Breakage Kernels

The Luo and Lehr models are integrated kernels, encompassing both the breakage fre-quency and the PDF of breaking particles. The general breakage rate per unit volume isusually written [15] as

Ωbr (V, V ′) = ΩB (V ′) η (V | V ′) [1/m3/sec] (2.2-11)

where the original particle has a volume V ′ and the daughter particle has a volumeV . In the previous expression, ΩB (V ′) is the breakage frequency, and η (V | V ′) is thenormalized daughter particle distribution function. For binary breakage, the breakagekernel must be symmetrical with respect to V

V ′= 0.5.

The general form is the integral over the size of eddies λ hitting the particle with diameterd (and volume V ). The integral is taken over the dimensionless eddy size ξ = λ/d. Thegeneral form is

Ωbr(V, V ′) = K∫ 1

ξmin

(1 + ξ)2

ξnexp (−bξm) dξ (2.2-12)

where the parameters are as shown in Table 2.2.1:

Table 2.2.1: Luo and Lehr Model Parameters

K [1/m3/sec] n b m

Luo 0.9238ε1/3d−2/3α 11/3 12(f 2/3 + (1− f)2/3 − 1

)σρ−1ε−2/3d−5/3 −11/3

Lehr 1.19ε−1/3d−7/3σρ−1f−1/3 13/3 2σρ−1ε−2/3d−5/3f−1/3 −2/3



Ghadiri Breakage Kernels

The Ghadiri model [7, 22], in contrast to the Luo and Lehr models, is used to model onlythe breakage frequency of the solid particles. You will have to specify the PDF model todefine the daughter distribution.

The breakage frequency f is related to the material properties and impact conditions:

f =ρsE

2/3

Γ5/3v2L5/3 = Kbv

2L5/3 (2.2-13)

where ρs is the particle density, E is the elastic modulus of the granule, and Γ is theinterface energy. v is the impact velocity and L is the particle diameter prior to breaking.Kb is the breakage constant and is defined as

Kb =ρsE

2/3

Γ5/3(2.2-14)

Parabolic PDF

The breakage PDF function contains information on the probability of fragments formedby a breakage event. It provides the number of particles and the possible size distributionfrom the breakage. The parabolic form of the PDF implemented in ANSYS FLUENTallows you to define the breakage PDF such that

β (V | V ′) = 0.5

[C

V ′ +1− C/2

V ′

24(

V

V ′

)2

− 24(

V

V ′

)+ 6

](2.2-15)

where V and V ′ are the daughter and parent particle volumes, respectively. Dependingon the value of the shape factor C, different behaviors will be observed in the shape ofthe particle breakage distribution function. For example, if C = 2, the particle breakagehas a uniform distribution. If 0 < C < 2, a concave parabola is obtained, meaning thatit is more likely to obtain unequally-sized fragments than equally-sized fragments. Theopposite of this is true for 2 < C < 3. Values outside of the range of 0 to 3 are notallowed, because the PDF cannot have a negative value.

Note that the PDF defined in Equation 2.2-15 is symmetric about V/V ′ = 0.5.



Generalized PDF

The generalized form of the PDF implemented in ANSYS FLUENT allows you to simulatemultiple breakage fragments (> 2) and to specify the form of the daughter distribution(e.g., uniform, equisized, attrition, power law, parabolic, binary beta). The model itselfcan be applied to both the discrete method and the QMOM.

Considering the self-similar formulation [25] where the similarity z is the ratio of daughter-to-parent size (i.e., z ≡ V

V ′), then the generalized PDF is given by

pβ(V |V ′) =θ(z)

V ′ (2.2-16)

The kth moment of θ(z) (bk) is

bk =∫ 1

0zkθ(z) dz =

Bk(V′)

V ′k(2.2-17)

where

Bk(V′) =

∫ V ′

0V kpβ(V |V ′)dV (2.2-18)

The conditions of number and mass conservation can then be expressed as

b0 =∫ 1

0θ(z)dz = p (2.2-19)

b1 =∫ 1

0zθ(z)dz = 1 (2.2-20)

The generalized form of θ(z)[6] can be expressed as

θ(z) =∑

i

wipizqi−1(1− z)ri−1

β(qi, ri)(2.2-21)

where i can be 0 or 1, which represents θ(z) as consisting of 1 or 2 terms, respectively.For each term, wi is the weighting factor, pi is the averaged number of daughter particles,qi and ri are the exponents, and β(qi, ri) is the beta function. The following constraintsare imposed on the parameters in Equation 2.2-21:

∑i

wi = 1 (2.2-22)



∑i

wipi = p (2.2-23)

∑i

wi(piqi

qi + ri

) = 1 (2.2-24)

In order to demonstrate how to transform the generalized PDF to represent an appro-priate daughter distribution, consider the expressions shown in Table 2.2.2:

Table 2.2.2: Daughter Distributions

Type θ(z) p Constraints

Equisized[12] pδ(z − 1p) p p ≥ 2

Attrition[12] δ(z − (1− ε)) + δ(z − ε) 2 ε 1Power Law[29] (ν + 1) zν−1 ν+1

ν0 < ν ≤ 1

Parabolic -a[29] (ν + 2)(ν + 1)zν−1(1− z) ν+2ν

0 < ν ≤ 2Austin[2] w (ν1 + 1) zν1−1 w(1 + 1

ν1) ν1, ν2 > 0

+(1− w) (ν2 + 1) zν2−1 +(1− w)(1 + 1ν2

) 1 ≥ w ≥ ν1(ν2−1ν2−ν1

)

Binary Beta -a[11] 60z2(1− z)2 2 N/ABinary Beta -b[20] 2

β(ν,ν)zν−1(1− z)ν−1 2 ν > 0

Uniform[29] p(p− 1)(1− z)p−2 p p ≥ 2

In Table 2.2.2, δ is the Dirac delta function, w is a weighting coefficient, and ε, ν, ν1,and ν2 are user-defined parameters.

The generalized form can represent the daughter distributions in Table 2.2.2 by usingthe values shown in Table 2.2.3.



Table 2.2.3: Values for Daughter Distributions in General Form

Type w0 p0 q0 r0 w1 p1 q1 r1 Constraints

Equisized 1 p ∞∗ ∞∗ N/A N/A N/A N/A p ≥ 2Attrition 0.5 2 ε 1 0.5 2 1 ε ε 1

Power Law 1 ν+1ν

ν 1 N/A N/A N/A N/A 0 < ν ≤ 1Parabolic 1 ν+2

νν 2 N/A N/A N/A N/A 0 < ν ≤ 2

Austin w ν1+1ν1

ν1 1 1− w ν2+1ν2

ν2 1 ν1, ν2 > 0

1 ≥ w ≥ ν1(ν2−1ν2−ν1

)

Binary Beta** 1 2 ν ν N/A N/A N/A N/A ν > 0Uniform 1 p 1 p− 1 N/A N/A N/A N/A p ≥ 2

(*)You can approximate ∞ by using a very large number, such as 1e10.(**)Binary Beta -a is a special case of Binary Beta -b when ν = 3.

i Note that for the ANSYS FLUENT implementation of the generalized formof the PDF, you will only enter values for w0, p0, q0, r0, and q1, and theremaining values (w1, p1, and r1) will be calculated automatically.

Aggregation

The aggregation kernel [18] 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 ′ (2.2-25)

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 ′ (2.2-26)

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.

The following is a list of aggregation functions available in ANSYS FLUENT:

• constant

• Luo model

• Free molecular model

• Turbulent model


The Luo, free molecular, and turbulent aggregation functions are described in detail inthe sections that follow.

Luo Aggregation Kernel

For the Luo model [17], the general aggregation kernel is defined as the rate of particlevolume formation as a result of binary collisions of particles with volumes Vi and Vj:

Ωag (Vi, Vj) = ωag (Vi, Vj) Pag (Vi, Vj) [m3/sec] (2.2-27)

where ωag (Vi, Vj) [m3/sec] is the frequency of collision and Pag (Vi, Vj) is the probabilitythat the collision results in coalescence. The frequency is defined as follows:

ωag (Vi, Vj) =π

4

(di

2 + dj2)ninjuij (2.2-28)

where uij is the characteristic velocity of collision of two particles with diameters di anddj and number densities ni and nj. Two physical mechanisms are behind the calculationof this velocity. The first mechanism is turbulent mixing. Assuming that the particles’size lies in the inertial range of turbulence, and the turbulence is isotropic, the mixingvelocity ut

ij of the two particles can be expressed as

utij =

(u2

i + u2j

)1/2(2.2-29)



where

ui = 1.43 (εdi)1/3 (2.2-30)

The expression for the probability of aggregation is

Pag = exp

−c1

[0.75

(1 + x2

ij

) (1 + x3

ij

)]1/2

(ρ2/ρ1 + 0.5)1/2(1 + xij)3

We1/2ij

(2.2-31)

where c1 is a constant of order unity, xij = di/dj, ρ1 and ρ2 are the densities of theprimary and secondary phases, respectively, and the Weber number is defined as

Weij =ρldi(uij

t)2

σ(2.2-32)

Free Molecular Aggregation Kernel

Real particles aggregate and break with frequencies (or kernels) characterized by complexdependencies over particle internal coordinates [28]. In particular, very small particles(say up to 1µm) aggregate because of collisions due to Brownian motions. In this case, thefrequency of collision is size-dependent and usually the following kernel is implemented:

a(Li, Lj) =2kBT

3µ

(Li + Lj)2

LiLj

(2.2-33)

where kB is the Boltzmann constant, T is the absolute temperature, µ is the viscosity ofthe suspending fluid. This kernel is also known as the Brownian kernel or the perikinetickernel.

Turbulent Aggregation Kernel

During mixing processes, mechanical energy is supplied to the fluid. This energy createsturbulence within the fluid. The turbulence creates eddies, which in turn help dissipatethe energy. The energy is transferred from the largest eddies to the smallest eddies inwhich it is dissipated through viscous interactions. The size of the smallest eddies is theKolmogorov microscale, η, which is expressed as a function of the kinematic viscosityand the turbulent energy dissipation rate:

η =

(v3

ε

)1/4

(2.2-34)

In the turbulent flow field, aggregation can occur by two mechanisms:



• viscous subrange mechanism: this is applied when particles are smaller than theKolmogorov microscale, ν

• inertial subrange mechanism: this is applied when particles are bigger than theKolmogorov microscale. In this case, particles assume independent velocities.

For the viscous subrange, particle collisions are influenced by the local shear within theeddy. Based on work by Saffman and Turner [27], the collision rate is expressed as,

a(Li, Lj) = ςT

√8π

15γ

(Li + Lj)3

8(2.2-35)

where ςT is a pre-factor that takes into account the capture efficiency coefficient of tur-bulent collision, and γ is the shear rate:

γ =ε

v

0.5

(2.2-36)

For the inertial subrange, particles are bigger than the smallest eddy, therefore they aredragged by velocity fluctuations in the flow field. In this case, the aggregation rate isexpressed using Abrahamson’s model [1],

a(Li, Lj) = ςT 23/2√

π(Li + Lj)

2

4

√(U2

i + U2j ) (2.2-37)

where U2i is the mean squared velocity for particle i.

The empirical capture efficiency coefficient of turbulent collision describes the hydro-dynamic and attractive interaction between colliding particles. Higashitani et al. [9]proposed the following relationship:

ςT = 0.732(

5

NT

)0.242

; NT ≥ 5 (2.2-38)

where NT is the ratio between the viscous force and the Van der Waals force,

NT =6πµ(Li + Lj)

3λ

8H(2.2-39)

Where H is the Hamaker constant, a function of the particle material, and λ is thedeformation rate,

λ =(

4ε

15πv

)0.5

(2.2-40)


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 ANSYS FLUENT: the discrete method, the standard methodof moments (SMM), and the quadrature method of moments (QMOM). For each method,the ANSYS FLUENT implementation is limited to a single internal coordinate correspond-ing to 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 developedby Hounslow [10], Litster [16], and Ramkrishna [25]. It is based on representing thecontinuous particle size distribution (PSD) in terms of a set of discrete size classes orbins, as illustrated in Figure 2.3.1. The advantages of this method are its robust numericsand that it gives the PSD directly. The disadvantages are that the bins must be defineda priori 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 ANSYS FLUENT, the PBE is written in terms of volume fraction of particle size i:

∂

∂t(ρsαi) +∇ · (ρsuiαi) +

∂

∂V

(Gvρsαi

V

)= 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

Vi

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

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

fi =αi

α(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 [10]:

∂

∂V

(Gvρsαi

V

)= ρsVi

[(Gv,i−1Ni−1

Vi − Vi−1

)−(

Gv,iNi

Vi+1 − Vi

)](2.3-5)

The volume coordinate is discretized as [10] Vi+1/Vi = 2q where q = 1, 2, . . . and isreferred to 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

g(Vj)Njβ(Vi | Vj) (2.3-8)

Dbr,i = g(Vi)Ni (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.

Breakage Formulations for the Discrete Method

The default breakage formulation for the discrete method in ANSYS FLUENT is based onthe Hagesather method [14]. In this method, the breakage sources are distributed to therespective size bins, preserving mass and number density. For the case when the ratiobetween successive bin sizes can be expressed as 2n where n = 1, 2, . . ., the source in bini, (i = 1, . . . , N) can be expressed as

Bb(i) =N∑

k=i+1,i6=N

Ωb(vk, vi) +i∑

k=i,i6=N

xi+1,kΩb(vi+1, vk) +i−1∑

k=1,i6=1

(1− xi,k)g(vi+1Ωb(vi, vk)

(2.3-14)

Here

Ωb(vk, vi) = Nkg(vk)β(vk, vi) (2.3-15)

A more mathematically rigourous formulation is given by Ramakrishna [13], where thebreakage rate is expressed as

Bb(i) =N∑i

ni,kg(vk)Nk (2.3-16)

where

ni,k =∫

vi

vi+1vi+1 − v

vi+1 − vi

β(vk, v)dv +∫ vi

vi−1

v − vi−1

vi − vi− 1β(vk, v)dv (2.3-17)

The Ramakrishna formulation can be slow due to the large number of integration pointsrequired. However, for simple forms of β, the integrations can be performed relativelyeasily. The Hagesather formulation requires fewer integration points and the differencein accuracy with the Ramakrishna formulation can be corrected by a suitable choice ofbin sizes.



i To keep the computing time reasonable, a volume averaged value is used forthe turbulent eddy dissipation when the Luo model is used in conjunctionwith the Ramakrishna formulation.

2.3.2 The Standard Method of Moments (SMM)

The SMM, proposed by Randolph and Larson [26] 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 (2.3-18)

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 (2.3-19)

where

Bag,k =1

2

∫ ∞

0n(λ)

∫ ∞

0a(u, λ)(u, λ)(u3 + λ3)k/3n(u)dudλ (2.3-20)

Dag,k =∫ ∞

0Lkn(L)

∫ ∞

0a(L, λ)n(λ)dλdL (2.3-21)

Bbr,k =∫ ∞

0Lk∫ ∞

0g(λ)β(L | λ)n(λ)dλdL (2.3-22)

Dbr,k =∫ k

0Lkg(L)n(L)dL (2.3-23)

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 (2.3-24)



and for constant growth is represented as

kGmk−1 (2.3-25)

Equation 2.3-20 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 (2.3-26)

Ltotal = m1 (2.3-27)

Atotal = Kam2 (2.3-28)

Vtotal = Kvm3 (2.3-29)

d32 =m3

m2

(2.3-30)

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 2.3-19, the quantities represented in Equations 2.3-20–2.3-23 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 2.3-20–2.3-23 can be integrated and expressedin terms of the moments being solved.

In ANSYS FLUENT, an exact closure is implemented by restricting the application of theSMM to 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 [21] formodeling aerosol evolution and coagulation problems. Its applications by Marchisio etal. [19] 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

∫ ∞

0f(L)n(L)dL ≈

N∑i=1

f(Li)wi, (2.3-31)

is exact if f(L) is a polynomial of order N or smaller [5]. 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 . (2.3-32)

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 [24]. This procedure is known as the Product-Difference algorithm [8]. 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.



Applying Equations 2.3-31 and 2.3-32, the birth and death terms in Equation 2.3-19 canbe rewritten as

Bag,k =1

2

N∑i=1

wi

N∑j=1

wj(L3i + L3

j)k/3a(Li, Lj) (2.3-33)

Dag,k =N∑

i=1

Lki wi

N∑j=1

wja(Li, Lj) (2.3-34)

Bbr,k =N∑

i=1

wi

∫ ∞

0Lkg(Li)β(L | Li)dL (2.3-35)

Dbr,k =N∑

i=1

wiLki g(Li) (2.3-36)

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 2.3-24 and represented as

N∑i=1

wiLk−1i G(Li) (2.3-37)

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 [23], which was adapted for crystalliza-tion problems by Baldyga and Orciuch [3].

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).


2.4 Reconstructing the Particle Size Distribution from Moments

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 ANSYS FLUENT PopulationBalance Model

This chapter provides basic instructions to install the population balance model and solvepopulation balance problems in ANSYS FLUENT. It assumes that you are already familiarwith standard ANSYS FLUENT features, including the user-defined function proceduresdescribed in the ANSYS 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 ANSYSFLUENT licensed software.

3.2 Loading the Population Balance Module

The population balance module is loaded into ANSYS FLUENT through the text userinterface (TUI). The module can only be loaded when a valid ANSYS FLUENT case filehas been set or read. The text command to load the module is

define −→ models −→addon-module.

A list of ANSYS FLUENT addon modules is displayed:

FLUENT Addon Modules:

0. none

1. MHD Model

2. Fiber Model

3. Fuel Cell and Electrolysis Model

4. SOFC Model with Unresolved Electrolyte

5. Population Balance Model

Enter Module Number: [0] 5


Using the ANSYS 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 ANSYS FLUENT.A message Addon Module: popbal...loaded! is displayed at the end of the loading process.

The population balance module setup is saved with the ANSYS FLUENT case file. Themodule is loaded automatically when the case file is subsequently read into ANSYS FLU-ENT. Note that in the saved case file, the population balance module is saved with theabsolute path. Therefore, if the locations of the population balance module installationor the saved case file are changed, ANSYS FLUENT will not be able to load the modulewhen the case file is subsequently 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 ANSYS 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 theANSYS FLUENT User’s Guide for details.)

1. Start the double-precision version of ANSYS 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 dialog box (Figure 3.3.1).

Models −→ Population Balance −→ Edit...

4. Specify the population balance method under Method.



Figure 3.3.1: The Population Balance Model Dialog Box

• 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 dialog box.

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 (as describedin Section 2.2.1: Particle Growth and Dissolution). By default, this coef-ficient 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-model, free-molecular-model, turbulent-model, or user-defined from the drop-down list:

• If you select constant, specify a value in the adjacent field.

• If you select luo-model, the Surface Tension for Population Balance dialogbox will open automatically to allow you to specify the surface tension (seeFigure 3.3.2). The aggregation rate for the model will then be calculatedbased on Luo’s aggregation kernel (as described in Section 2.2.2: ParticleBirth and Death Due to Breakage and Aggregation).

• If you select free-molecular-model, then Equation 2.2-33 is applied.

• If you select turbulent-model, the Hamaker Constant for Population Balancedialog box will open automatically to allow you to specify the Hamakerconstant (see Figure 3.3.3). More information about this model is avail-able in Section 2.2.2: Turbulent Aggregation Kernel.



Figure 3.3.2: The Surface Tension for Population Balance Dialog Box

Figure 3.3.3: The Hamaker Constant for Population Balance Dialog Box

• 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 specifythe appropriate UDF.

Breakage Kernel allows you to specify the particle breakage frequency (particles/m3-s).You can select constant, luo-model, lehr-model, ghadiri-model, or user-definedfrom the Frequency drop-down list:

• If you select constant, specify a value in the adjacent field.

• If you select luo-model, the Surface Tension for Population Balance dialogbox will open automatically to allow you to specify the surface tension (seeFigure 3.3.2). The frequency used in the breakage rate will then be calcu-lated based on Luo’s breakage kernel (as described in Section 2.2.2: Par-ticle Birth and Death Due to Breakage and Aggregation).

• If you select lehr-model, the Surface Tension and Weber Number dialogbox will open automatically to allow you to specify the surface tensionand critical Weber number (see Figure 3.3.4). The frequency used in thebreakage rate will then be calculated based on Lehr’s breakage kernel (asdescribed in Section 2.2.2: Particle Birth and Death Due to Breakage andAggregation).

• If you select ghadiri-model, the Ghadiri Breakage Constant for PopulationBalance dialog box will open automatically to allow you to specify thebreakage constant (see Figure 3.3.5). The frequency will then be calcu-lated based on Ghadiri’s breakage kernel (as described in Section 2.2.2: Par-ticle Birth and Death Due to Breakage and Aggregation).

• If you have a user-defined function (UDF) that you want to use to model



Figure 3.3.4: The Surface Tension and Weber Number Dialog Box

Figure 3.3.5: The Ghadiri Breakage Constant for Population Balance DialogBox

the frequency for the breakage rate, you can choose the user-defined optionand specify the appropriate UDF.



If you selected constant, ghadiri-model, or user-defined for Frequency, then youcan specify the probability density function used to calculate the breakage rateby making a selection in the PDF drop-down list. You can select parabolic,generalized, or user-defined:

• If you select parabolic, the Shape Factor for Parabolic PDF dialog box willopen automatically to allow you to specify the shape factor C (see Fig-ure 3.3.6). The PDF used in the breakage rate will then be calculatedaccording to Equation 2.2-15 (as described in Section 2.2.2: Particle Birthand Death Due to Breakage and Aggregation).

Figure 3.3.6: The Shape Factor for Parabolic PDF Dialog Box

• If you select generalized, the Generalized pdf for multiple breakage dialogbox will open automatically (Figure 3.3.7).

Figure 3.3.7: The Generalized pdf for multiple breakage Dialog Box



Perform the following steps in the Generalized pdf for multiple breakagedialog box:

(a) Select either One Term or Two Term from the Options list. Your selec-tion will determine whether i in Equation 2.2-21 is 0 or 1, respectively.

(b) Enter a value for the averaged Number of Daughters. It can be anyreal number (including non-integers, e.g., 2.5), as long as it is not lessthan 2.

(c) Define the parameter(s) for Equation 2.2-21 in the Input Parametersgroup box. When One Term is selected from the Options list, youmust enter a value for qi0. When Two Term is selected from theOptions list, you must enter values for wi0, pi0, qi0, ri0, and qi1. Forinformation about appropriate values for these parameters to resultin the daughter distributions shown in Table 2.2.2, see Table 2.2.3.

(d) Click the Validate/Apply button to save the settings. The text boxesin the All Parameters group box will be updated, using the values youentered in the Input Parameters group box, as well as values derivedfrom the constraints shown in Equations 2.2-22–2.2-24.

(e) Verify that the values in the All Parameters group box represent yourintended PDF before clicking Close.

• If you have a user-defined function (UDF) that you want to use to modelthe PDF for the breakage rate, you can choose the user-defined option andspecify the appropriate UDF. See Chapter 5: UDFs for Population BalanceModeling for details about UDFs for the population balance model.

Choose between the default Hagesather formulation and the Ramakrishna for-mulation. Detailed information about these two methods can be found inSection 2.3.1: Breakage Formulations for the Discrete Method.

7. Specify the boundary conditions for the solution variables.

Boundary Conditions

See Section 3.3.2: Defining Population Balance Boundary Conditions below.

8. Specify the initial guess for the solution variables.

Solution Initialization

9. Solve the problem and perform relevant postprocessing functions.

Run Calculation

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 task page, select the secondary phase(s) in the Phasedrop-down list and then open the appropriate boundary condition dialog box (e.g.,Figure 3.3.8).

Boundary Conditions

Figure 3.3.8: 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., Bin-0) are labeled according tothe following:

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 dialog box.



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) or mk in Equation 2.3-19(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 Equations dialog box (Figure 3.3.9), equations for each bin (e.g., phase-2 Bin) willappear in the Equations list.

Solution Controls −→ Equations...

The default value under Under-Relaxation Factors (in the Solution Controls task page) forthe population balance equations is 0.5, and the default Discretization scheme (in theSolution Methods task page) is First Order Upwind.

Figure 3.3.9: The Equations Dialog Box



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 2.3-30) 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, open theSecondary Phase dialog box.

Phases −→ Secondary Phase −→ Edit...

In the Secondary Phase dialog box (e.g., Figure 3.3.10), select sauter-mean from the Di-ameter drop-down list under Properties. Note that a constant diameter or user-definedfunction may also be used.

Figure 3.3.10: The Secondary Phase Dialog box 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 phase willalso have source terms due to the added mass. In ANSYS 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 dialog box, de-scribed below.

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

∫ ∞

0L2Gn(L)dL

=1

2ρKa

∫ ∞

0L2Gn(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

L2i wiG(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 dialog box for the primary phase after ac-tivating the multiphase model. The rest of the procedure for setting up aspecies transport problem is identical to setting up species in single phase.The heterogeneous 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 task pageand click the Interaction... button. This will open the Phase Interaction dialog box(Figure 3.3.11). 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 theprimary phase. In the Phases task page, click the Interaction... button. This willopen the Phase Interaction dialog box (Figure 3.3.12). Click the Reactions tab to spec-ify the stoichiometry for the reactant and the product. At the bottom of the PhaseInteraction dialog box, select population-balance as the Reaction Rate Function. ClickOK to save the settings. Either this method or the use of the UDF, described in Ap-pendix A: DEFINE HET RXN RATE Macro, will produce the same results.

Figure 3.3.11: The Phase Interaction Dialog Box for Non-reacting Species



Figure 3.3.12: The Phase Interaction Dialog Box for a Heterogeneous Reac-tion


Chapter 4. Postprocessing for the Population BalanceModel

ANSYS FLUENT provides postprocessing options for displaying, plotting, and reportingvarious particle quantities, which include the main solution variables and other auxiliaryquantities.

• 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 computed 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 Population BalanceMoments dialog box (Figure 4.2.1).

Report −→ Population Balance −→Moments...

Figure 4.2.1: The Population Balance Moments Dialog Box

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 Write... 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 dialog box (Figure 4.2.2).

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

Figure 4.2.2: The Number Density Function Dialog Box

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 dialog box. This option is available only with thediscrete method.

Curve displays a smooth curve of the number density function.

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.

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.


Postprocessing for the Population Balance Model

4. Choose the cell zones on which to plot the number density function data in the CellZones list.

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

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

7. Click Write to save the number density function data to a file. The Select File dialogbox will open, where you can specify a name and save a file containing the plotdata. 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 volumefraction 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 BREAKUP RATE FREQ

• DEFINE PB BREAKUP RATE PDF

• DEFINE PB COALESCENCE RATE

• DEFINE PB NUCLEATION RATE

• DEFINE PB GROWTH RATE


UDFs for Population Balance Modeling

5.2.1 DEFINE PB BREAKUP RATE FREQ

You can use the DEFINE PB BREAKUP RATE FREQ macro if you want to define the breakagefrequency using a UDF. The function is executed at the beginning of every time step.

Usage

DEFINE PB BREAKUP RATE FREQ(name, cell, thread, d 1)

Argument Type Descriptionchar name UDF namecell t cell Cell indexThread *thread Pointer to the secondary phase threadreal d 1 Parent particle diameter or length

Function returnsreal

There are four arguments to DEFINE PB BREAKUP RATE FREQ: name, cell, thread, andd 1. You will supply name, the name of the UDF. cell, thread, and d 1 are variablesthat are passed by the ANSYS FLUENT solver to your UDF.

Example

Included below is an example of a UDF that defines a breakage frequency (see Sec-tion 2.2.2: Particle Birth and Death Due to Breakage and Aggregation) that is based onthe work of Tavlarides [4], such that

g(V ′) = C1ε1/3

(1 + α)d2/3exp

(−C2

σ(1 + α)2

ρlε2/3d5/3

)(5.2-1)

where C1 and C2 are constants, ε is the dissipation rate, d is the parent diameter, σ isthe surface tension, α is the volume fraction of the dispersed phase, and ρl is the densityof the primary phase.



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

UDF that computes the particle breakage frequency

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

#include "udf.h"

#include "sg_pb.h"

#include "sg_mphase.h"

DEFINE_PB_BREAKUP_RATE_FREQ(break_up_freq_tav, cell, thread, d_1)

real epsi, alpha, f1, f2, rho_d;

real C1 = 0.00481, C2 = 0.08, sigma = 0.07;

Thread *tm = THREAD_SUPER_THREAD(thread);/*passed thread is phase*/

epsi = C_D(cell, tm);

alpha = C_VOF(cell, thread);

rho_d = C_R(cell, thread);

f1 = pow(epsi, 1./3.)/((1.+epsi)*pow(d_1, 2./3.));

f2 = -(C2*sigma*(1.+epsi)*(1.+epsi))/(rho_d*pow(epsi,2./3.)*pow(d_1, 5./3.));

return C1*f1*exp(f2);

5.2.2 DEFINE PB BREAKUP RATE PDF

You can use the DEFINE PB BREAKUP RATE PDF macro if you want to define the breakagePDF using a UDF. The function is executed at the beginning of every time step.

Usage

DEFINE PB BREAKUP RATE PDF(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 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




There are five arguments to DEFINE PB BREAKUP RATE FREQ: 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 ANSYS FLUENT solver to your UDF.

Example

Included below is an example of a UDF that defines a breakage PDF (see Section 2.2.2: Par-ticle Birth and Death Due to Breakage and Aggregation) that is parabolic, as defined inEquation 2.2-15.

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

UDF that computes the particle breakage PDF

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

#include "udf.h"

#include "sg_pb.h"


DEFINE_PB_BREAKUP_RATE_PDF(break_up_pdf_par, cell, thread, d_1, d_2)

real pdf;

real kv = M_PI/6.;

real C = 1.0;

real f_2, f_3, f_4;

real V_prime = kv*pow(d_1,3.);

real V = kv*pow(d_2,3.);

f_2 = 24.*pow(V/V_prime,2.);

f_3 = -24.*(V/V_prime);

f_4 = 6.;

pdf = (C/V_prime) + ((1.-C/2.)/V_prime)*(f_2 + f_3 + f_4);

return 0.5*pdf;

5.2.3 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 ANSYS FLUENT solver to your UDF. Your UDF willneed to return 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;

5.2.4 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 ANSYS FLUENT solver to your UDF. Your UDF will need to return thereal 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.

5.2.5 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 ANSYS FLUENT solver to your UDF. Your UDF will need to return thereal value of 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 */



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 ANSYS FLUENT

After the UDF that you have defined using DEFINE PB BREAKUP RATE FREQ, DEFINE PB

BREAKUP RATE PDF, DEFINE PB COALESCENCE RATE, DEFINE PB NUCLEATION RATE, orDEFINE PB GROWTH RATE is interpreted or compiled, the name that you specified in theDEFINE macro argument (e.g., agg kernel) will become visible and selectable in the ap-propriate drop-down list under Phenomena in the Population Balance Model dialog box(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 ANSYS FLUENT execution, the DEFINE HET RXN RATE UDF for each heteroge-neous reaction that is defined is called in every fluid cell. ANSYS FLUENT will use thereaction rate specified by the UDF to compute production/destruction of the speciesparticipating in the reaction, as well as heat and momentum transfer across phases dueto 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 dialog box.

Release 12.0 c© ANSYS, Inc. January 12, 2009 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 ANSYS FLUENT solver to your UDF. Your UDF will need to setthe values referenced by the real pointer rr.

A-2 Release 12.0 c© ANSYS, Inc. January 12, 2009

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);

Release 12.0 c© ANSYS, Inc. January 12, 2009 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 ANSYS 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 dialog box. (Note you will first need tospecify the Total Number of Reactions greater than 0.)

A-4 Release 12.0 c© ANSYS, Inc. January 12, 2009

Bibliography

[1] J. Abrahamson. Collision Rates of Small Particles in a Vigourously Turbulent Fluid.Chemical Engineering Science, 30:1371–1379, 1975.

[2] L. Austin, K. Shoji, V. Bhatia, V. Jindal, K. Savage, and R. Klimpel. Some resultson the description of size distribution as a rate process in various mills. IndustrialEngineering and Chemical Process Design Devices, 15(1):187–196, 1976.

[3] 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.

[4] C. A. Coulaloglou and L. L. Tavlarides. Description of interaction processes inagitated liquid-liquid dispersions. Chem. Eng. Sci., 32:1289–1297, 1977.

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

[6] R. B. Diemer and J. H. Olson. A moment methodology for coagulation and breakageproblems: Part 3—generalized daughter distribution functions. Chemical Engineer-ing Science, 57(19):4187–4198, 2002.

[7] M. Ghadiri and Z. Zhang. Impact Attrition of Particulate Solids: Part 1. A Theo-retical Model of Chipping. Chemical Engineering Science, 57:3659–3669, 2002.

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

[9] K. Higashitani, K. Yamauchi, Y. Matsuno, and G. Hosokawa. Turbulent Coagulationof Particles Dispersed in a Viscous Fluid. Chemical Engineering Journal Japan,16(4):299–304, 1983.

[10] 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.

[11] M. A. Hsia and L. L. Tavlarides. Simulation analysis of drop breakage, coales-cence and micro-mixing in liquid–liquid stirred tanks. Chemical Engineering Jour-nal, 26(3):189–199, 1983.

[12] M. Kostoglou, S. Dovas, and A. J. Karabelas. On the steady-state size distribution ofdispersions in breakage processes. Chemical Engineering Science, 52(8):1285–1299,1997.

Release 12.0 c© ANSYS, Inc. January 12, 2009 Bib-1

BIBLIOGRAPHY

[13] S. Kumar and D. Ramkrishna. On the Solution of Population Balance Equationsby Discretization - I., A Fixed Pivot Technique. Chemical Engineering Science,51(8):1311–1332, 1996.

[14] Hagesaether L., Jakobsen H.A.1, and Svendsen H.F. A model for Turbulent BinaryBreakup of Dispersed Fluid Particles. Chemical Engineering Science, 57(16):3251–3267, 2002.

[15] F. Lehr, M. Millies, and D. Mewes. Bubble-Size distributions and flow fields inbubble columns. AIChE Journal, 48(11):2426–2443, 2002.

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

[17] H. Luo. Coalescence, breakup and liquid circulation in bubble column reactors. PhDthesis, Norwegian Institute of Technology, Trondheim, Norway, 1993.

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

[19] 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.

[20] B. J. McCoy and M. Wang. Continuous-mixture fragmentation kinetics: particle sizereduction and molecular cracking. Chemical Engineering Science, 49(22):3773–3785,1994.

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

[22] R. Moreno-Atanasio and M. Ghadiri. Mechanistic Analysis and Computer Sim-ulation of Impact Breakage of Allomerates: Effect of Surface Energy. ChemicalEngineering Science, 61(8):2476–2481, 2006.

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

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

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

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

Bib-2 Release 12.0 c© ANSYS, Inc. January 12, 2009

BIBLIOGRAPHY

[27] P.G. Saffman and J.S. Turner. On the Collision of Droplets in Turbulent Clouds.Journal of Fluid Mechanics, 1:16–30, 1956.

[28] M. Schmoluchowski. Versuch Einen Mathematischen Theorie der Koagulationstech-nik Kolloider Losungen. Zeitschrift fur Physikalische Chemie, 92:129–168, 1917.

[29] R. D. Vigil and R. M. Ziff. On the stability of coagulation–fragmentation populationbalances. Journal of Colloids and Interface Science, 133(1):257–264, 1989.

Release 12.0 c© ANSYS, Inc. January 12, 2009 Bib-3

ansys fluent 12 - بیت دانلودs2.bitdownload.ir/engineering/all.ansys/ansys 12...

Documents