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September 29, 1999 Department of Chemical Engineering University of South Florida Tampa, USA

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in Chemical Engineering

by

Richard Gilbert

&

Nihat M. Gürmen

September 29, 1999

Department of Chemical Engineering

University of South Florida

Tampa, USA

Part I -- Overview

R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

What is the Population Balance Technique (PBT)?

PBT is a mathematical framework for an accounting procedure for particles of certain types you are interested in.

The technique is very useful where identity of individual particles is modified or destroyed by coalescenceor breakage.

R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

(Dis)advantage of PBT

Advantage

- Analysis of complex dispersed phase system

Disadvantage

- Difficult integro-partial differential equations

R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

Application Areas

- colloidal systems
- crystallization
- fluidization
- microbial growths
- demographic analysis

R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

No(q,t)

Tampa

Immigration

Emigration

n(q,t)

Birth Rate

Death Rate

Origins of population balances: Demographic Analysis- Time = t
- Age = q

R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

A Mixed Suspension, Mixed Product Removal (MSMPR) Crystallizer

Qi, Ci, ni

Particle

Size

Distribution

(PSD)

Qo, Co, n

R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

Growth

Kinetics

Growth Rate

Supersaturation

Mass

Balance

Feed

Nucleation

Kinetics

Population

Balance

PSD

Nucleation

Rate

Crystal

Area

(from Theory of Particulate Processes, Randolp and Larson, p. 3, 1988)

Information diagram showing feedback interactionR. Gilbert & N. Gürmen, v.1.0, Tampa 1999

Part II -- Mathematical Background

R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

Population Density, n(L)

Size, L

Size, L

Exponential Distribution

Normal Distribution

Two common density distributions by particle number

R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

Exponential density distribution by particle number

N1

Cumulative Population, N(L)

Population Density, n(L)

N1

n1

Size, L

Size, L

L1

L1

N1 is the number of particles less than size L1

n1 is the number of particles per size L1

R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

Ntotal

Lmax

Size, L

Normal density distribution by particle numberNtotal = Total number

of particles

Cumulative Population, N(L)

Lmax

Size, L

R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

Normalized Population Density, f(L)

0

Lmax

Size, L

Normalization of a distributionOne way to normalize n(L)

normalized

Area under

the curve

R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

Average properties of a distribution

The two important parameters of a particle size distribution are

* How large are the particles?

mean size,

* How much variation do they have with respect to the mean size?

coefficient of variation, c.v.

where 2(variance)is

R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

Mean, = the first moment about zero

Moments of a distributionj-th moment, mj, of a distribution f(L) about L1

Variance, 2= the second moment about the mean

R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

Skewness, 1 = measure of the symmetry

about the mean (zero for

symmetric distributions)

Kurtosis, 2= measure of the shape of

tails of a distribution

Further average properties: Skewness and Kurtosisj-th moment, j, of a distribution f(L) about mean

R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

Part III -- Formulation of Population Balance Technique

R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

Assumptions

Basic Assumptions of PBT(Population Balance Technique)- Particles are numerous enough to approximate a continuum
- Each particle has identical trajectory in particle phase space S spanned by the chosen independent variables
- Systems can be micro- or macrodistributed

R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

Basic Definitions

Number density function n(S,t) is defined in an (m+3)-dimensional spaceS consisted of

3 external (spatial) coordinates

m internal coordinates (size, age, etc.)

Total number of particles is given by

SpaceS

R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

The particle number continuity equation

a subregion R1 from the Lagrangian viewpoint

R1

S

R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

Convenient variable and operator definitions

where

is the set of internal and external coordinates spanning the phase space R1

R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

Applying the product rule of differentiation to the LHS

R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

Substituting all the terms derived earlier

As the region R1 is arbitrary

R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

Micro-distributed Population Balance Equation

Averaging the equation in the external coordinates

Macro-distributed Population Balance Equation

R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

B - D terms represent the rate of coalescence

conventionally collision integrals are used for B and D

the rate at which a bubble of volume u coalesces with a bubble

of volume v-u to make a new bubble of volume v is

a death function consistent with the above birth function

would be

R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

C(x,y) : the rate at which bubbles of volumes x and y collide and coalesce.

in the modeling of aerosols two of the functions used for C(x,y) are where Ka is the coalescence rate constant

1) Brownian motion

2) Shear flow

R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

Simplifications for a Solvable System

- dynamic system => t
- spatially distributed => x, y, z
- single internal variable, size => L

Growth rate G is at most linearly dependent

with particle size =>

R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

Moment Transformation

Defining the jth moment of the number density function as

Averaging PB in in the L dimension

R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

Microdistributed form of moment transformation

j = 0,1,2,... ³ k

Macrodistributed form of moment transformation

j = 0,1,2,... ³ k

R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

If Assumptions Do Not Allow Moment Transformations

- You have to use other methods of solving PDEs like
- method of lines
- finite element methods

difficult if both of your

variables go to infinity

R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

Part IV -- Examples

R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

Example 1 : Demographic Analysis

- neglect spatial variations of population
- one internal coordinate, age q

Ni(q,t)

No(q,t)

Immigration

Emigration

Tampa

n(q,t)

Set up the general population balance equation?

R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

Qo, Co, n

Example 2: Steady-state MSMPR CrystallizerThe system is at steady-state

Volume of the tank : V

Outflow equals the inflow

Feed stream is free of particles

Growth rate of particles is

independent of size

There are no particles formed

by agglomeration or coalescnce

Derive the model equations for the system.

R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

References

- BOOK
- Randolph A. D. and M. A. Larson, Theory of Particulate Processes, 2nd edition, 1988, Academic Press
- PAPERS
- Hounslow M. J., R. L. Ryall, and V. R. Marhsall, A discretized population balance for nucleation, growth, and aggregation, AIChE Journal, 34:11, p. 1821-1832, 1988
- Hulburt H. M. and T. Akiyama, Liouville equations for agglomeration and dispersion processes, I&EC Fundamentals, 8:2, p. 319-324, 1969
- Ramkrishna D., The prospects of population balances, Chemical Engineering Education, p. 14-17,43, 1978

R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

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