september 29, 1999 department of chemical engineering university of south florida tampa, usa

September 29, 1999

Department of Chemical EngineeringUniversity of South Florida

Tampa, USA

Population Balance Techniquesin Chemical Engineering

by

Richard Gilbert&

Nihat M. Gürmen

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

Part I -- Overview


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 coalescence or breakage.


(Dis)advantage of PBT

Advantage

• Analysis of complex dispersed phase system

Disadvantage

• Difficult integro-partial differential equations


Application Areas

• colloidal systems

• crystallization

• fluidization

• microbial growths

• demographic analysis


Origins of population balances: Demographic Analysis

• Time = t• Age =

Tampan(,t)

Ni(,t) No(,t)

EmigrationImmigration

Birth RateDeath Rate


A Mixed Suspension, Mixed Product Removal (MSMPR) Crystallizer

Qi, Ci, ni

Qo, Co, n

Particle SizeDistribution(PSD)


Information diagram showing feedback interaction

MassBalance

GrowthKinetics

NucleationKinetics

CrystalArea

PopulationBalance

Growth Rate

NucleationRate

PSD

Feed

Growth Rate

Supersaturation

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


Part II -- Mathematical Background


Two common density distributions by particle number

Size, LPo

pula

tion

Den

sity

, n(L

)

Normal Distribution

Popu

latio

n D

ensi

ty, n

(L)

Size, L

Exponential Distribution


Exponential density distribution by particle number

Size, L

Cum

ulat

ive

Popu

latio

n, N

(L)

Size, LPo

pula

tion

Den

sity

, n(L

)

dLL

0zddL

N1N1

N1 is the number of particles less than size L1

n1 is the number of particles per size L1

L1 L1

n1


Normal density distribution by particle number

Size, L

Cum

ulat

ive

Popu

latio

n, N

(L)

0

LzddL

Ntotal = Total number of particles

Lmax Size, LPo

pula

tion

Den

sity

, n(L

)

Ntotal

Lmax


Normalization of a distribution

f L dL f L dLL

L

( ) ( )min

max

zz 100

normalized

f L n L

n L dL( ) ( )

( )z

0

One way to normalize n(L)

Size, L

Nor

mal

ized

Pop

ulat

ion

Den

sity

, f(L

)

Lmax

1

0

Area under the curve


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.

L L f L dLz ( )0

2 2

0

zL L f L dLc h ( )

c vL

. . 2

where 2 (variance) is

L


Moments of a distribution

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

m L L f L dLj

j

z 10

b g ( )

Mean, = the first moment about zeroL

Variance, 2 = the second moment about the mean


Further average properties: Skewness and Kurtosis

j-th moment, j, of a distribution f(L) about mean

j

j

L L f L dL

zc h ( )

Skewness, 1 = measure of the symmetry

about the mean (zero for symmetric distributions)

1

33

2

4

22 3

FHGIKJKurtosis, 2 = measure of the shape of

tails of a distribution


Part III -- Formulation of Population Balance Technique


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

Check these Assumptions


Basic Definitions

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

3 external (spatial) coordinatesm internal coordinates (size, age, etc.)

Total number of particles is given by

N t n S t dSS

( ) ( , )Space S


The particle number continuity equation

Accumulation Input Output Generation

R1

S

ddt

ndR B D dRR R1 1

a subregion R1 from the Lagrangian viewpoint


x y z x jj

m

1

ddt

n R t dR B D dRR R

,b g b g1 1

z z

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

d xdt

v v ve i

Convenient variable and operator definitions


Applying the product rule of differentiation to the LHS

FHG

IKJznt dR n d x

dtR R11

FHG

IKJ

LNMM

OQPPz z

ntdR n d x

dtdR

R R1 1

FHG

IKJ

LNMM

OQPPz

nt

d xdt

n dRR1

ddt

n R t dRR

,b g1

z


Substituting all the terms derived earlier

nt

v n v n D B dRe iR

1

0

As the region R1 is arbitrary

nt

vn D B 0


nt x

v ny

v nz

v nx

v n D Bx y zjj

m

j

10

In terms of m+3 coordinates

Averaging the equation in the external coordinates

nt

v n nd V

dtB D Q n

Vik k

k

bg b gln

Micro-distributed Population Balance Equation

Macro-distributed Population Balance Equation


B - D terms represent the rate of coalescence

conventionally collision integrals are used for B and D

B v C u v u n u n v u duv

12 0

,

D v n v C v v n v dv

,0

the rate at which a bubble of volume u coalesces with a bubbleof volume v-u to make a new bubble of volume v is

a death function consistent with the above birth function would be


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

C x y K x y x ya, 1 3 1 3 1 3 1 3

2) Shear flow

C x y K x ya, 1 3 1 3 3

Coagulation kernel, C(x,y)


Simplifications for a Solvable System

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

nt x

v ny

v nz

v nL

v n D Bx y z L 0

nt

v nL

Gn D Be 0

Growth rate G is at most linearly dependent with particle size => G G aLo 1


Moment Transformation

Defining the jth moment of the number density function as

m x t n x t L dLj e ej, ,

0

Averaging PB in in the L dimension

L nt

v nL

Gn D B dLje

LNM OQP z bg0

0


mt

v m B jG m am B Dje j

jj j 0 0

0 1( )

j = 0,1,2,...

Microdistributed form of moment transformation

Macrodistributed form of moment transformation

mt

m d Vdt

B jG m amQ m

VB Dj

jj

j jk j k

k

(log ) ( ) ,0 00 1

j = 0,1,2,...


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


Part IV -- Examples


Example 1 : Demographic Analysis

Tampan(,t)

Ni(,t) No(,t)

• neglect spatial variations of population• one internal coordinate, age

EmigrationImmigration

Set up the general population balance equation?


Example 2: Steady-state MSMPR Crystallizer

Qi, Ci, ni

Qo, Co, n

The system is at steady-stateVolume of the tank : VOutflow equals the inflowFeed stream is free of particles

Growth rate of particles is independent of sizeThere are no particles formedby agglomeration or coalescnce

Derive the model equations for the system.


ReferencesBOOK

• 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


THE END

september 29, 1999 department of chemical engineering university of south florida tampa, usa

Documents