Models based on population balance approach

by M. Miccio

Dept. of Industrial Engineering (Università di Salerno)

Prodal Scarl (Fisciano)

UNIVERSITÁ DEGLI STUDI DI SALERNOFACOLTÀ DI INGEGNERIA

Rev. 4.5 of November 9, 2012

Population Balance models

The population balance equation (PBE) is a statement of continuity for particulatesystems, originally derived in 1964.

Population balances are of key relevance to a very diverse group of scientists,including astrophysicists, high-energy physicists, geophysicists, colloid chemists,biophysicists, materials scientists, chemical engineers, and meteorologists.Chemical engineers have put population balances to most use, with applications inthe areas of crystallization; gas-liquid, liquid-liquid, and solid-liquid dispersions;liquid membrane systems; fluidized bed reactors; aerosol reactors; and microbialcultures.

Engineers encounter particles in a variety of systems. The particles are eithernaturally present or engineered into these systems. In either case these particlesoften significantly affect the behavior of such systems.

This modeling approach provides a framework for analyzing these dispersed phasesystems and describes how to synthesize the behavior of the population particlesand their environment from the behavior of single particles in their localenvironments

2

Population Balance models

Ramkrishna provides a clear and general treatment of population balanceswith emphasis on their wide range of applicability. New insight intopopulation balance models incorporating random particle growth, dynamicmorphological structure, and complex multivariate formulations with a clearexposition of their mathematical derivation is presented. PopulationBalances provides the only available treatment of the solution of inverseproblems essential for identification of population balance models forbreakage and aggregation processes, particle nucleation, growthprocesses, and more. This book is especially useful for process engineersinterested in the simulation and control of particulate systems. Additionally,comprehensive treatment of the stochastic formulation of small systemsprovides for the modeling of stochastic systems with promising new areasof applications such as the design of sterilization systems and radiationtreatment of cancerous tumors.

Doraiswami Ramkrishna, Purdue University, West Lafayette, Indiana, U.S.A.

“Population Balances. Theory and Applications to Par ticulate Systems in Engineering ”, Academic Press, ISBN: 0-12-576970-9,

Pages: 355, Publication Date: 8 August 2000, Price: £83.99

3

Population Balances

Formulation

The Population Balance Equation was originally derived in 1964, when two groups of researchers studying crystal nucleation and growth recognized that many problems involving change in particulate systems could not be handled within the framework of the conventional conservation equations only, see Hulburt & Katz (1964) and Randolph (1964) .

They proposed the use of an equation for the continuity of particulate numbers, termed population balance equation, as a basis for describing the behavior of such systems.

This balance is developed from the general conservation equation:

Input - Output + Net Generation = Accumulation

4 § 4.4 in Himmelblau D.M. and Bischoff K.B., “Process Analysis and Simulation”, John

Wiley & Sons Inc., 1967 (Collocazione: 660.281 HIM 1 )

Elements necessary for a population balance Model

5

Element General description Example

ENTITY • specific of each “particulate system”;• crystallization, biochemical processes, polymerization, leaching, comminution, and aerosolsare just some examples;• entities are “discrete” and not continuous;• entities are “countable”, but may be infinite in number

Yeast cell in a fermenter

PROPERTIESof ENTITY

• in general, entities have a specified set of m properties ζi with i = 1,2...m• the properties ζi will depend on the application;• typical examples are the entity's size diameter, entity's chemical composition, entity's age

Cell mass,cell age, etc.

ENTITYDISTRIBU-TION

• the function ψ(t, x, y, z, ζ1, ζ2, ..., ζm) represents the entity distribution• t is time• x, y, z are the spatial coordinates• it is similar to a probability density function

Number of cells per unit fermenter volume, unit cell mass, unitcell age, etc.

Entity distribution function

6

( ) 1d,,d,dz,dy,dxt,,,,z,y,x m1m1 =ζζζζψ∫Ω

KK

Congruence constraint

where Ω is the (3+m)-dimensional space of the independent variables:

3 spatial coordinates (x, y, z )

m properties (ζi with i = 1,2...m)

time t may be one additional independent variable

Physical meaning

is the fraction of entities at time t that are:• contained in the infinitesimal volume dV=dxdydz• characterized by values of properties in the ranges ζ1÷ζ1+dζ1, . . . , ζm÷ζm+dζm

( ) m1m1 d,,d,dz,dy,dxt,,,,z,y,x ζζζζψ KK

Net Generation

Accumulation = Net Generation

Generation (Positive and Negative) terms are defined as :

HYPOTHESES:

• large, arbitrary, time-varying sub-space

• closed sub-space, with no input or output of entities

• a balance can be written in the usual way:

(I)

7

R(t) ⊆ Ω

One dimension

a(t) b(t)

x

Multi-dimensional

where:R(t) is a time-variable region of the space Ωf(•) is a scalar functionl stands for any of the non-time variables x, y, z, ζ1, ζ2…., the sum Σ is over all such variables

ddt

f x, t( )a t( )

b t( )

∫ dx =∂f x, t( )

∂t+ d

dxdxdt

f x, t( )

dx =∂f x, t( )

∂ta t( )

b t( )

∫ dx+ f b t( ), t db t( )

dt− f a t( ), t

da t( )dta t( )

b t( )

∫

( )( )

( )dRf

t

fdRf

dt

dl

lt

ffdR

dt

d

tRtR

tR ll∫∫ ∫ ∑

•∇+∂∂=

∂∂+

∂∂=

•

Leibnitz formula

8dt

dll all oftor column vec theis=•

The total derivates with respect to time included in the Leibnitz formula can be explained in a physical way:

vi is the change rate with respect to the time or kinetics of the property ζi

The Equation (I) becomes:

As the region R(t) is arbitrary, the necessary condition for that equation to be true is that the integrating term is null

vvvv ii

zyx dt

d

dt

dz

dt

dy

dt

dx ====ζ

;;;

∂ψ∂t

+ ∂∂x

xv ψ( ) + ∂∂y

yv ψ( ) + ∂∂z

zv ψ( ) + ∂∂ iζ

( ivψ)+ M − Ni=1

m

∑

R t( )∫ dR = 0

∂ψ∂t

+ ∂∂x

xv ψ( ) + ∂∂y

yv ψ( ) + ∂∂z

zv ψ( ) + ∂∂ iζ

viψ( ) + M − N = 0i=1

m

∑

This is the generalized population balance model on «microscopic scale »

(in x, y, z coordinates)9

Generalized population balance model on «microscopic scale»

Often ψ spatial dependence is not known or not desired, while only averaged value on system geometric volume V is requested.The volume-averaged entity distribution is:

Integrating over the whole volume the generalized population balance model :

Let’s examine the various terms:

∫=V

dVV

ψψ 1dxdydzdV =

∂ψ∂t

+ ∇• vr

ψ( ) + ∂∂ iζ ivψ( ) + M − N( )

i=1

m

∑

V

∫ dV = 0

I II III IV

∂ψ∂t

dV = d

dtψ dV − ∇• sv

ur

ψ( ) dVV

∫V

∫V

∫

V

10

Switch to «macroscopic scale»

The term (I) can be obtained by an inversion of the multi-dimensional Leibnitz formula, restricted to the spatial coordinates only:

S

V

According to Gauss divergence theorem , the term V becomes:

The term II is transformed by the same Gauss theorem

The terms I and II (included V) can be combined together:

∇• svur

ψ( ) dVV

∫ = nr

• svur

ψ( )S

∫ dS

∇• v→

ψ

V

∫ dV = nr

• vr

ψ( )S

∫ dS

11

S

V

∂ψ∂t

+ ∇• vr

ψ( )

dV = d

dtψ dV

V

∫ + nr

• vr

− vS

uru( )ψ dSS

∫V

∫

VII II

In the term VI the surface S is considered as the sum of three contributions:

S = S1+ S2+ S*

Where:• S1 is inlet surface to the system• S2 is the outlet surface from the system• S* is the remaining part of the surface (impenetrable)

The term VI becomes :

nr

• vr

− vS

uru( )ψ dS

S∫ =

= nr

• vr

− vr

S( )1S +

2S∫ ψdS+ n

r

• vr

− vr

S( )ψd *S*

S∫ = −

1v 1ψ

1S∫ d 1S +

2v 2ψ d 2S + 0

2S∫

VII VIII IX

12

S1

S*S2

nr

• vr

− vr

S( ) = 0

The term IV yields:

The terms VIII and IX can be simplified as follows:

where Q1 and Q2 are volumetric flowrates through S1 and S2, respectively.

The term III is integrated over the volume V and yields:

1v 1ψ

1S∫ d 1S =

1ψ 1v

1S∫ d 1S =

1ψ 1Q ; 2v 2ψ

2S∫ d 2S =

2ψ 2v

2S∫ d 2S =

2ψ 2Q

∂∂ iζ ivψ( )

i=1

m

∑V

∫ dV = ∂∂ iζ ivψ( ) dV =

V

∫i=1

m

∑

= ∂∂ iζ ivψ dV =

V

∫∂

∂ iζ iv ψ dVV

∫

= V

∂∂ iζ iv ψ( )

i=1

m

∑i=1

m

∑i=1

m

∑

( ) ( )NMVdVNMV

−=−∫13

Further Hypothesis : ψ1 and ψ2 are averaged and, therefore, constant, respectively on S1 and S2

After replacing all the terms:

Dividing by V:

V∂∂t

ψ( ) + V∂

∂ iζ iv ψ( ) + V M − N( )i=1

m

∑ − 1Q1ψ + 2Q

2ψ = 0

∂∂t

ψ( ) + ∂∂ iζ iv ψ( )

i=1

m

∑ + M − N = 1Q1ψ − 2Q

2ψ

1

V

ACC PSEUDO/CONV GEN IN OUT

14

This is the population balance model on «macroscopic scale»

(without x, y, z coordinates)

Population balance model on «macroscopic scale»

Example No. 1Vapor Deposition in a gas-solid fluidized bed

15

ch.14 pag.136, Daizo Kunii and Octave Levenspiel, Fluidization Engineering , 2nd. ed. Butterworth-Heinemann, 1991

• Macroscopic approach

• Steady state

• Constant inventory of particles in the fluidized bed

• All particles having the same chemical composition

• Particles having spherical geometry

• Particles having constant density

• Negligible elutriation

• Negligible solid-solid and solid-wall attrition

16

Hypotheses

NOMENCLATURE

• W = bed mass (inventory of particles)

• F0 = mass feed rate of fresh particles

• F1 = mass discharge rate of processed particles

scheme of the fluidized bed

O. Levenspiel, D. Kunii, T. Fitzgerald The processing of solids of changing size in bubblin g fluidized bedsPowder Technology, Volume 2, Issue 2, December 1968, Pages 87-96

AIR+

VAPOR

AIR

W

p(d)

T, P

F0 p0(d)

F1 p1(d)

&m IN

&mOUT

Entity : solid particles

Property : particle diameter d

Objective : → p(d) [=] m -1

p(d)dd is the mass fraction of particleswith diameter between d and d+dd

17

Vapor Depositionin a gas-solid fluidized bed

Population Balance approach:the generation term

18

( ) ( ) ( )

( ) ( )change)property unit()time(

mass][

d

ddpW3

d

1

t

dd3

6d6

ddpW

d

1

dt

dV

d6

ddpWdN p

2

3p

pp

3p

⋅=ℜ⋅⋅⋅=

=⋅⋅⋅⋅⋅

⋅=⋅⋅⋅⋅⋅

⋅=dd

dd

d

d ρππρ

ρπρ

No. particles in

[d; d+dd]

rate of particle volume change

particle

density

particlesize

interval/

ℜ d( ) is the rate of particle size change with time (property kinetics)

The overall generation

19

N tot = N d( )0

∞∫ dd =

=3⋅ W ⋅ p d( ) ⋅ ℜ d( )

d0

∞∫ dd =[ ] generated mass

time

Overall Mass Balances

on the solid phase with ref. to the whole fluid bed:

IN – OUT + GEN = 0

0NFF tot10 =+−1.

20

This is NOT a Population Balance eq.

on the gas phase with ref. to the whole fluid bed:

&m IN − &mOUT − N tot = 01 bis.

Population Balance approach:the mass balance

F0 ⋅ p0 d( ) − F1 ⋅ p1 d( ) − W ⋅ ℜ d( ) dp d( )dd

− W ⋅ p d( ) dℜ d( )dd

+3 ⋅ W ⋅ p d( ) ⋅ ℜ d( )

d= 0

per unit time and with ref. to the size interval [d;d+ dd] :

( ) ( ) ( ) ( ) ( ) ( )0

d

ddddpW3

t

ddpW

t

ddpWdddpFdddpF dddd100 =ℜ⋅⋅⋅+

⋅⋅−

⋅⋅+⋅−⋅ +dd

dd

21

IN with feed

OUT with discharge

entering the control volume[d; d+dd] from a lower size

leaving the control volume[d; d+dd] to a larger size

generation in the interval[d; d+dd]

d d+dd

t t+dt

control volumein the property space

AIR+

VAPOR

AIR

W

p(d)

T, P

F0 p0(d)

Back mixing: p1(d) = p(d)

“monosize” feed: p0(d) = δ(d-D0)

22

F1 p1(d)

Further Hypotheses

&m IN

&mOUT

DIRAC DELTA FUNCTION

(UNIT IMPULSE)

δ(d)=0 for d≠D0

δ=∞ for d=D0

0 D0d

( ) 1d Dd0

0 =−δ∫∞

d

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) 0dpWd

ddpF

0ddpd

dW3

d

dddpWdpdWddpFddpF

0d

ddpW3

d

ddpW

d

dpdWdpFdpF

00

100

100

=−ℜ

⋅

=ℜ⋅⋅+ℜ⋅⋅−⋅ℜ⋅−⋅−⋅

=ℜ⋅⋅⋅+ℜ⋅−ℜ⋅−⋅−⋅

dd

dd

ddddd

d

d

d

dwith ref. to an infinitesimal size interval located immediately below the feed size D0: [D0; D0-dd]

( )( ) ( )

( )( )

( )( ) ( )

( )

( )

( ) ( )( )

( ) ( )0

D

0

0

00

Dp

00p

D

0

00

Dp

00p

D

0

00

D

1d

d

Dd

Dddp

dpWdd

dpF

0dpWdd

dpF

0

00

00

ℜ=

ℜ−δ

⇒

−δ=⇒

⋅=ℜ

⋅

=−ℜ⋅

∫

∫∫

∫∫

=

=

d

dd

dd

integration over [0, D 0] gives p(D 0):

( ) ( ) 0DW

FDp

0

00 ≠

ℜ⋅=4.

23

Population Balance approach:integration of mass balance eq.

0 D0d

→ p(d)dd = 0

manipulation and integration over [D 0, d] with d>D 0 give :

( )( )( )( )

( ) ( )( )( )

( )0

d

d3

d

dd

dp

ddpd

dW

F d

D

d

D

dp

Dp

d

D

1

0000

=⋅+ℜℜ−−

ℜ⋅− ∫∫∫∫

ℜ

ℜ

dd

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0ddpd

dW3

d

dddpWdpdWddpFddpF 100 =ℜ⋅⋅+ℜ⋅⋅−⋅ℜ⋅−⋅−⋅ d

d

ddddd

( )( )( )

( )( ) 0

D

dln3

d

Dln

dp

Dpln

d

d

W

F

0

00d

D

1

0

=⋅+ℜ

ℜ++ℜ

− ∫d

24

Population Balance approach:integration of mass balance eq.

with ref. to an infinitesimal size interval [d;d+ dd] located above the feed size D 0:

Taking the exponential, replacing eq. 4and then solving for p(d) gives:

Population Balance approach:entity distribution eq. p(d)

( )∫

⋅⋅ℜ⋅

=′ℜ′

−d

0D

1

)d(

d

W

F

30

30 e

D

d

dW

F)d(p

d

25

( ) ( )0

00 DW

FDp

ℜ⋅=4.

( )( )

( )( )

( )

ℜℜ=

ℜ− ∫

003

30

d

D

1

Dp

dp

D

d

d

Dln

d

d

W

F

0

d

3.

where d’ is an integration variable

Population Balance approach:congruence condition for p(d)

26

( ) 1d dp0D

=∫∞

d

( ) 1deD

d

dW

F

0

d

0D

1

D

)d(

d

W

F

30

30 =

∫⋅⋅

ℜ⋅∫∞ ′ℜ

′−

d

d

( )∫∞ ′ℜ

′− ∫

⋅ℜ

⋅=0

d

0D

1

D

)d(

d

W

F3

30

0 ded

d

D

FW d

d

2.

Population Balance approach:Final equations

( ) ( )d

d

ddpW3FF

0D

01 d∫∞ ℜ⋅=−1.

27

( )∫

⋅⋅ℜ⋅

=′ℜ′

−d

0D

1

)d(

d

W

F

30

30 e

D

d

dW

F)d(p

d

3.

( )∫∞ ′ℜ

′− ∫

⋅ℜ

⋅=0

d

0D

1

D

)d(

d

W

F3

30

0 ded

d

D

FW d

d

2.

Population Balance approach:Integro-differential equations

28

More generally, size distributions are time dependent.Ex.:rate of change of the distribution (shape) depends on the wholedistribution

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 0.2 0.4 0.6 0.8 1

s

x

x(s,t) under consideration

agglomeration of these may form a particle of

size s breakage of these may form a particle of size s

breakage of x(s,t) affects the distribution

d

Example No. 2Sublimation in a gas-solid fluidized bed

29

• Macroscopic approach

• Batch operation (dynamical system)

• “Active” particles (i.e., sublimating particles) different from the particles constituting the fluidized bed

• Perfect mixing

• All “Active” particles maintain the same composition

• Particles having spherical geometry

• Particles having constant density

• Negligible elutriation

• Negligible solid-solid and solid-wall attrition

30

Hypotheses

NOMENCLATURE• W0 = overall mass of sublimating solid particles at

time 0

• W(t) = overall mass of sublimating solid particlesat time t

schemeof the fluidized bed

Air

+

Vapor

Air

Population Balance approach:batch sublimation in fluidized bed

31

Entity : mass of sublimating solid particles

Property : particle diameter d

Objective : → m(d,t) [=] kg/m

m(d,t) ∆∆∆∆d is the mass of sublimating particles in thebed with diameter between d and d+ ∆d

Population Balance approach:batch sublimation in fluidized bed

32

control volume in the “property space ”

d d + ∆d

tt + ∆t

[ ] [ ] dt)t,d(mtt)t,d(mtd

)d(d)t,d(m3t

dt

d)t,d(mt

ddt

d)t,dd(m ∆−∆+=∆ℜ∆−∆−∆

∆+∆+

d

d

d

d

−1m ss

mm

m

kg

[ ]s

m

t

d)d( :where ==ℜ

d

d

mass Balanceon the size interval ∆d and in the time interval ∆∆∆∆t

Population Balance approach:batch sublimation in fluidized bed

dt

)t,d(m

d

)d(d)t,d(m3

dt

d)t,d(m

ddt

d)t,dd(m ∆

∂∂=ℜ∆−−

∆+∆+

d

d

d

d

t

)t,d(m

d

)d()t,d(m3

ddt

d)t,d(m

ddt

d)t,dd(m

∂∂=ℜ−

∆

−∆+

∆+d

d

d

d

( )[ ]t

)t,d(m

d

)d()t,d(m3

d

d)t,d(m

∂∂=ℜ−

∂ℜ∂

Let us divide by ∆t For ∆t 0

Let us divide by ∆d

For ∆d 0

PDE subject to:IC: for t=0 m(d,0) = mo(d)BC: for d=0, ∀t m(0,t) = 0

Population Balance approach:batch sublimation in fluidized bed

( ) ( )[ ]

d

)d()t,d(m3 M

d

d)t,d(m

vt

)t,d(m

tm

1i i

i

ℜ⇒

∂ℜ∂

⇒ζ∂ψ∂

∂∂

⇒∂ψ∂

∑=

( ) ( )∫= maxd

0dt,dmtW d

Comparison with the population balance model on «macroscopic scale»

Total mass at time t:

Population Balance approach:batch sublimation in fluidized bed

Particular cases for sublimation kinetics

[ ]

t

)t,d(m)t,d(hm3)t,d(hm

d

)t,d(mhd

hd)d(

dbt

d

6

d3

s

kg db

t

d

6t

m

32

p

33

pp

∂∂=+−

∂∂−

−=ℜ

π−=πρ

=π−=πρ=

d

d

d

d

d

d

1. Volume 2. Surface

[ ]

t

)t,d(m

d

)t,d(km3

d

)t,d(mk

k)d(

dct

d

6

d3

s

kg dc

t

d

6t

m

22

p

23

pp

∂∂=+

∂∂−

−=ℜ

π−=πρ

=π−=πρ=

d

d

d

d

d

d

Yeast cell distribution in a fermenter

36