A puristic analysis of population balance -1 - Semantic Scholar
Chemical
Engineering Science,1973,
Vol. 28, pp. 1423-1435.
Pergamon Press.
Printed
in GreatBritain
A puristic analysis of population balance -1 D. RAMKRISHNA Department
of Chemical
Engineering,
Indian Institute ofTechnology,
Kanpur 16, U.P., India
and J. D. BORWANKER Department
of Mathematics,
lndian Institute of Technology,
(Received
27 September
Kanpur 16, U.P., India
1972)
Abstract-The genera1 description of particulate systems is considered from the point of view of what are known as product densities associated with stochastic point processes developed essentially by physicists. It is shown that the so-called population balance equation is actually one of an infinite sequence of equations satisfied by the product densities, all of which are necessary for a puristic description of a particulate system. The population density in the population balance equation is shown to be the expected density, fluctuations about which can be calculated by solving a successive sequence of product density equations. Population balances are pointed out to be adequate for large populations for which fluctuations about mean quantities are negligible. Small populations, however, require higher order product density equations. An illustrative example is considered for the application of product densities. Analytical solutions have been presented for the two leading product densities. The coefficient of variation about the expected total population is derived which shows large fluctuations for small populations.
INTRODUCTION
I
and Katz [ l] have provided a framework for the analysis of particulate systems presenting an equation for the particle phase in a multivariate number density that has come to be known as the population balance equation. The individual particle state is characterized by a vector belonging to a state space and the multivariate density represents the number of particles per unit volume of this state space; the state space may include physical space in conjunction with more abstract spaces representing the “internal” coordinates of the particle. Scalar versions of the population balance equation [2-41 had appeared prior to the work of Hulburt and Katz. In fact, these equations date back even further to the efforts of Landau and Rumer[5] who were concerned with energy distribution of elementary particles in cascade processes. The population density in these analyses is a real, continuous function of its arguments, sufficiently smooth to justify the derivation of the population balance equation. Moreover, the HULBURT
total population, which refers to all particles regardless of their individual states is a deterministic function of time. This determinism arises from the averaging of statistical variations in individual particle behavior over a large number of particles. Landau and Rumer are careful to state that the population at any instant is the expected number of particles. Fredrickson et al. [6] also qualify their population density likewise. The total population is clearly an integralvalued quantity, whereas the population balance equation regards this as a continuous (and even differentiable) function of time. However, the corresponding expected quantity is not subject to the constraint of being integral valued so that the variables appearing in the population balance equation must be regarded in this light. Expected quantities are suggestive of fluctuations about them, which raises a question about the completeness of the population balance equation in describing the particulate system. The answer in part to this question lies in the averaging of statistical variations in individual particle
1423
D. R A M K R I S H N A and J. D. B O R W A N K E R
behavior over a large number of particles. Thus for sufficiently "large-sized" populations, the population balance equation may provide for an adequate description of the system. However, questions arise such as how large is "large"? H o w does one investigate smaller-sized populations? The answers to these questions lie in the theory of stochastic point processes, which it will be the aim of this paper to expound. Stochastic point processes are essentially concerned with distribution of discrete points in a uni- or multidimensional continuum. Particulate systems generally encountered in chemical engineering comprise large-sized populations• There are, however, several situations in which "smaller" populations are involved. Coalescing dispersions in the absence of particle breakage eventually lead to small populations. The objective of pre-sterilization of fermentors is to deplete the population of contaminating microorganisms to near extinction. The operation of continuous cultures under low holding times eventually culminates into a small population system. Besides, even in dealing with larger populations, one need hardly stress the desirability for awareness of a larger perspective. ANALYSIS
R a n d o m population density Since the spirit of extension to the vectorial case frequents the treatment of the scalar situations we shall confine discussion to the latter. We denote the actual population density by n(x, t) where n(x, t)dx represents the number of individuals between x and x + dx in the entire system. The particle state x is assumed to take on all positive values. The total number of particles is given by
N(t)= fo n ( x , t )
dx.
(1)
Since the total number is an integral-valued quantity the density n ( x , t ) must be a linear combination of Dirac delta functions 8(x--xj), with integral coefficients rfs, which implies that there are rj particles with state exactly x~ for
each j. The total number of particles would then be the sum of the rfs. At this point it becomes necessary to assume for the mere sake of initial simplicity that no more than a single particle can be identified with a definite state x. This is a physical condition, which is true in many situations of interest while being obviously untrue in some others such as the agewise description of population growth i n which multiplets are accommodated. Thus if there are v particles in the population with states xl, xz . . . . x~ then
n(x,t) = ~ 8(x-xj)
(2)
j=l
where j ~ k implies xj ~ xk. Clearly from (1) and (2) N ( t ) = v.
Probability densities Janossi[7] defines a probability density function Jr(x1, Xz. . . . . xv; t) such that Pr {n(x, t ) = ~ 8 ( x - x j ) } = J~dxldx2 . . . dx~ j=l
(3)
represents the probability that there are at time t, v particles in the population with one particle each between x~ and x~+ dx~ for i = 1, 2 . . . . . v. Since no two particles of the same state are distinguishable, the normalization of the Janossi density is expressed by
where we have arbitrarily ordered xl ~ x 2 . . . ~< x~. The same condition may also be rewritten as
1. /)
•
f° 0
f° f= dxd 0
=l.
(5)
0
The result (5) is obtained from (4) as follows. The positive region of the v-dimensional space is divided into v! regions given by {0
Engineering Science,1973,
Vol. 28, pp. 1423-1435.
Pergamon Press.
Printed
in GreatBritain
A puristic analysis of population balance -1 D. RAMKRISHNA Department
of Chemical
Engineering,
Indian Institute ofTechnology,
Kanpur 16, U.P., India
and J. D. BORWANKER Department
of Mathematics,
lndian Institute of Technology,
(Received
27 September
Kanpur 16, U.P., India
1972)
Abstract-The genera1 description of particulate systems is considered from the point of view of what are known as product densities associated with stochastic point processes developed essentially by physicists. It is shown that the so-called population balance equation is actually one of an infinite sequence of equations satisfied by the product densities, all of which are necessary for a puristic description of a particulate system. The population density in the population balance equation is shown to be the expected density, fluctuations about which can be calculated by solving a successive sequence of product density equations. Population balances are pointed out to be adequate for large populations for which fluctuations about mean quantities are negligible. Small populations, however, require higher order product density equations. An illustrative example is considered for the application of product densities. Analytical solutions have been presented for the two leading product densities. The coefficient of variation about the expected total population is derived which shows large fluctuations for small populations.
INTRODUCTION
I
and Katz [ l] have provided a framework for the analysis of particulate systems presenting an equation for the particle phase in a multivariate number density that has come to be known as the population balance equation. The individual particle state is characterized by a vector belonging to a state space and the multivariate density represents the number of particles per unit volume of this state space; the state space may include physical space in conjunction with more abstract spaces representing the “internal” coordinates of the particle. Scalar versions of the population balance equation [2-41 had appeared prior to the work of Hulburt and Katz. In fact, these equations date back even further to the efforts of Landau and Rumer[5] who were concerned with energy distribution of elementary particles in cascade processes. The population density in these analyses is a real, continuous function of its arguments, sufficiently smooth to justify the derivation of the population balance equation. Moreover, the HULBURT
total population, which refers to all particles regardless of their individual states is a deterministic function of time. This determinism arises from the averaging of statistical variations in individual particle behavior over a large number of particles. Landau and Rumer are careful to state that the population at any instant is the expected number of particles. Fredrickson et al. [6] also qualify their population density likewise. The total population is clearly an integralvalued quantity, whereas the population balance equation regards this as a continuous (and even differentiable) function of time. However, the corresponding expected quantity is not subject to the constraint of being integral valued so that the variables appearing in the population balance equation must be regarded in this light. Expected quantities are suggestive of fluctuations about them, which raises a question about the completeness of the population balance equation in describing the particulate system. The answer in part to this question lies in the averaging of statistical variations in individual particle
1423
D. R A M K R I S H N A and J. D. B O R W A N K E R
behavior over a large number of particles. Thus for sufficiently "large-sized" populations, the population balance equation may provide for an adequate description of the system. However, questions arise such as how large is "large"? H o w does one investigate smaller-sized populations? The answers to these questions lie in the theory of stochastic point processes, which it will be the aim of this paper to expound. Stochastic point processes are essentially concerned with distribution of discrete points in a uni- or multidimensional continuum. Particulate systems generally encountered in chemical engineering comprise large-sized populations• There are, however, several situations in which "smaller" populations are involved. Coalescing dispersions in the absence of particle breakage eventually lead to small populations. The objective of pre-sterilization of fermentors is to deplete the population of contaminating microorganisms to near extinction. The operation of continuous cultures under low holding times eventually culminates into a small population system. Besides, even in dealing with larger populations, one need hardly stress the desirability for awareness of a larger perspective. ANALYSIS
R a n d o m population density Since the spirit of extension to the vectorial case frequents the treatment of the scalar situations we shall confine discussion to the latter. We denote the actual population density by n(x, t) where n(x, t)dx represents the number of individuals between x and x + dx in the entire system. The particle state x is assumed to take on all positive values. The total number of particles is given by
N(t)= fo n ( x , t )
dx.
(1)
Since the total number is an integral-valued quantity the density n ( x , t ) must be a linear combination of Dirac delta functions 8(x--xj), with integral coefficients rfs, which implies that there are rj particles with state exactly x~ for
each j. The total number of particles would then be the sum of the rfs. At this point it becomes necessary to assume for the mere sake of initial simplicity that no more than a single particle can be identified with a definite state x. This is a physical condition, which is true in many situations of interest while being obviously untrue in some others such as the agewise description of population growth i n which multiplets are accommodated. Thus if there are v particles in the population with states xl, xz . . . . x~ then
n(x,t) = ~ 8(x-xj)
(2)
j=l
where j ~ k implies xj ~ xk. Clearly from (1) and (2) N ( t ) = v.
Probability densities Janossi[7] defines a probability density function Jr(x1, Xz. . . . . xv; t) such that Pr {n(x, t ) = ~ 8 ( x - x j ) } = J~dxldx2 . . . dx~ j=l
(3)
represents the probability that there are at time t, v particles in the population with one particle each between x~ and x~+ dx~ for i = 1, 2 . . . . . v. Since no two particles of the same state are distinguishable, the normalization of the Janossi density is expressed by
where we have arbitrarily ordered xl ~ x 2 . . . ~< x~. The same condition may also be rewritten as
1. /)
•
f° 0
f° f= dxd 0
=l.
(5)
0
The result (5) is obtained from (4) as follows. The positive region of the v-dimensional space is divided into v! regions given by {0
