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An alternative to the concept of the interval of quiescence (IQ) in the Monte Carlo simulation of population balances
Chemical Engineering Science 54 (1999) 5711}5715
An alternative to the concept of the interval of quiescence (IQ) in the Monte Carlo simulation of population balances M. Song *, X.-J. Qiu Department of Chemical Engineering, University of Dortmund, 44221 Dortmund, Germany CDI 1111 West Holly Street, Bellingham, WA 98225-2922, USA Received 1 July 1998; accepted 10 February 1998
Abstract This work presents an alternative concept to the interval of quiescence in the Monte Carlo simulation of population balances. An example is given to show that the present method is consistent with the concept of the interval of quiescence. Both concepts can be equally used in the Monte Carlo simulation of population balances if the population is not very large. For a large population, the present concept shows some advantages. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Population balance equation; Interval of quiescence; Monte Carlo simulation; Breakage process
1. Introduction The concept of the interval of quiescence (IQ) was originally proposed by Kendall (1949, 1950) almost a half-century ago for simulating the behavior of properties that obey the birth-and-death model in biological populations. He called the simulation based on this concept the arti"cial realization of the birth-and-death process. Such a process involves the random appearance of new individuals and the disappearance of existing individuals with respective transition probabilities or &frequency functions' which continuously change distribution of the total population. Consequently, a &birth' event will add one to the population while a &death' event will delete one from it. It is naturally assumed that during an in"nitesimal time interval dt, multiple events will occur with probability of O(dt). Kendall (1949, 1950) de"ned the &interval of quiescence' (IQ) as the period between successive events during which the number and properties of individuals would remain constant. Clearly, it is a random quantity due to the stochastic nature of the events themselves. Kendall showed that the interval of quiescence has an exponential distribution with a coe$-
*Corresponding author. Present address: R.A.S. Industries Ltd., 8020-128th Street, Surrey, BC, Canada, V3W 4E9.
cient parameter in the exponent which depends on the number of individuals at the beginning of the interval and the frequency function (or say, transition probability). Following Kendall's pioneering work, Shah et al. (1976) derived the expression for the cumulative probability distribution function of the interval of quiescence as
+ , O F(q"A )"1!exp ! a (x(t#q , x , t)) dq , R H G H G (1) where M is the number of kinds of monoparticle phenomena such as breakage and nucleation, N is the number of particles, and a is the frequency function of the ith G monoparticle phenomenon. Eq. (1) describes the probability that any one of monoparticle phenomena to any one of particles happens in time q under the state of population at any time t. Based on this concept, Shah et al. (1977a) presented a general Monte Carlo simulation procedure for particulate systems. This method was utilized by Ramkrishna et al. in numerous simulations covering various scienti"c and engineering areas (Shah et al., 1976; Shah et al., 1977b; Bajpai et al., 1977; Habbard and Ramkrishna, 1979; Sampson and Ramkrishna, 1979; Sweet et al., 1987). Gupta et al. (1979) employed this procedure for simulating the behavior of solid}liquid
0009-2509/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 9 9 ) 0 0 0 9 9 - 8
5712
M. Song, X.-J. Qiu/Chemical Engineering Science 54 (1999) 5711}5715
suspensions under the e!ect of shear. This procedure was adopted by Bapat et al. (1983) for simulating the mass transfer process in liquid}liquid dispersions. Das (1996) proposed a Monte Carlo simulation algorithm for a drop breakage process based on this concept as well. The interval of quiescence is a profound concept. Ramkrishna (1981) established the precise mathematical connection between the population balances and the Monte Carlo simulations based on the interval of quiescence. However, as analyzed below, there are some disadvantages if it is used in the Monte Carlo simulation. This contribution proposes an alternative concept to it for the Monte Carlo simulation of population balances. It will be shown that there is a consistency between the present concept and the concept of the interval of quiescence.
2. Monte Carlo simulations based on the concept of IQ For simplicity, we assume a system of only particle breakage with transition function a(x). Then the cumulative distribution function of the interval of quiescence is given by
, F(q"A )"1!exp ! a(x )q . R G G
(2)
This describes the probability that a breakage happens to any one of particles after a quiescence interval following time t. The simulation algorithm developed by Shah et al. (1976, 1977a) can be described as follows: 1. Determine the interval of quiescence, denoted by ¹, by which two subsequent events are separated. A random number, say r , is generated by computer, which obeys the uniform distribution on the interval
[0, 1). According to Eq. (2), ¹ is determined by ln (1!r ) . ¹" ! , a(x ) G G
(3)
2. Determine to what particle an event (breakage) happens. We can readily get the probability that, given that a particulate event has occurred after a quiescence interval ¹ following time t, breakage happens to the ith particle as a(x ) G . p "P+the ith particle breaks up "A , ¹," G R , a(x ) G G
Then another random number, say r , is generated. If the following inequality follows: G\ G p (r ) p , (5) H H H H we identify that the event happens to the ith particle. This algorithm is straightforward and physically based. However, there are some drawbacks in the Monte Carlo simulation algorithm based on the concept of the interval of quiescence: (1) ¹ could be very small. (2) All particles have to be identi"ed even though some particles are of the same characteristic properties. The larger the population, the more serious these two drawbacks become. (3) The time step in the numerical simulation is irregular.
3. An alternative to the concept of IQ Like popular population simulations such as with the method of classes (Marchal et al., 1988), we divide the population into ¸ classes, C , C , 2 , C , according to *
Table 1 Comparison of simulation results between the present concept and the concept of the interval of quiescence 1s
(4)
2s
3s
x
IQ
Our concept
IQ
Our concept
IQ
Our concept
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.99 1.0
0.1446 0.2676 0.3732 0.4642 0.5472 0.6207 0.6804 0.7302 0.7770 0.8125 1
0.1463 0.2653 0.3706 0.4667 0.5477 0.6161 0.6787 0.7310 0.7771 0.8140 1
0.2406 0.4226 0.5628 0.6693 0.7536 0.8187 0.8687 0.9043 0.9332 0.9525 1
0.2455 0.4236 0.5644 0.6724 0.7545 0.8182 0.8673 0.9050 0.9337 0.9538 1
0.3233 0.5402 0.6867 0.7909 0.8620 0.9093 0.9418 0.9628 0.9779 0.9870 1
0.3224 0.5381 0.6886 0.7908 0.8602 0.9084 0.9406 0.9627 0.9782 0.9871 1
M. Song, X.-J. Qiu/Chemical Engineering Science 54 (1999) 5711}5715
the characteristic property (e.g. volume here). All particles in one class are identi"ed the same. Our alternative to the concept of the interval of quiescence is very simple and direct, which can be described as follows. 1. Determine the number of particles that are going to break up during the period from t to t#Dt. Assume that the total number of particles in class C is G N . Then the number of particles that are going to break G up can be calculated by N "N a(x ) Dt. G G G
(6)
2. Specify particles to break up
Or, the initial cumulative distribution function of particles based on number is given by
0, x)1, F (x, 0)" L 1, x'1.
r )N ![N ] (7) G G specify one particle more to break up where [N ] is G the integer part of N . Otherwise, no more is to be G speci"ed to break up. In this consideration N ![N ] is the probG G ability that one particle more is going to break up. Therefore, the statistical average number of particles undergoing breakage within a time step should be [N ]#+N ![N ],1 G G G #+1!(N ![N ]),0"N . G G G
(8)
4. Consistency between two concepts Eq. (9) is the number-based population balance equation
*n(x, t) "!a(x)n(x, t)# l(y)a(y)b(x, y) n(y, t) dy, *t V
(11)
The breakage frequency is assumed to be directly proportional to its volume with the coe$cient of unity, that is, a(x)"x.
(12)
All particles undergo the uniform breakage if they are selected to break up, that is, x B(x, y)" y
First, specify [N ] particles to break up. Then G generate a random number, say r , which is uniformly distributed on the interval [0, 1). If
5713
(13)
or 1 b(x, y)" , y
(14)
where x and y are the volume of the daughter and mother particle, respectively, and B(x, y) is the number-based cumulative function of daughter particles. Clearly, in this case l(x)"2.
(15)
We have solved this problem with the two algorithms. Fig. 1 gives the simulation results with our concept, which show the variation of the cumulative distribution function with time. Because we can not see any di!erence in simulation results between these two methods if we draw curves on a "gure, we give Table 1 in which we show the comparison of results at three instants (1s, 2s and 3s) between the two discussed methods. Table 2 gives the results of the total number in the system at three instants. From these two tables we can
(9)
where x is the volume of particles, n(x, t) is the number density, a(x) is the breakage frequency function, l(y) is the number of daughter particles from a mother particle of volume y, and b(x, y) is the number-based distribution function of daughter particles of volume x from a mother particle of volume y. We take a simple case to show the consistency between these two simulation methods. Assume that there are 10,000 particles of volume x"1 initially in a system. Then the initial number-density distribution function is given by
#R, x"1, f (x, 0)" and L 0, xO1
>
f (x, 0) dx"1. L (10)
Fig. 1. Simulation results of the cumulative distribution function of particles with the present concept.
5714
M. Song, X.-J. Qiu/Chemical Engineering Science 54 (1999) 5711}5715
Table 2 Comparison of simulation results from the two concepts compared with theoretical values
Total number of particles Number of initial particles remaining unbroken
1 2 3 1 2 3
s s s s s s
IQ
Our concept Theoretical value
20041 29972 40245 3658 1338 490
20012 30055 40112 3661 1340 490
¹M "
3679 1353 498
observe very close agreement of simulation results between the two concepts. According to the concept of the interval of quiescence, the probability that an initial particle with breakage frequency of unity (due to the volume of 1) remains unbroken at instant t is e\R. From the viewpoint of statistical average the number of initial particles remaining unbroken should be N "N e\R,
(16)
where N is the total number of initial particles. We calculate the values of this parameter at three instants, which are listed in Table 2 as theoretical values. The results of the number of initial particles remaining unbroken from both simulations are listed in Table 2 as well. We can see that the results from both concepts are in good agreement with theoretical values. Actually, according to the algorithm presented in this contribution, the number of initial particles remaining unbroken at any instant t can be calculated by N "N (1!Dt)R DR.
near 0 and 1 even though the probability is small. If r is near 1, ¹ can be big (e.g., ¹"4.6;10\ at r "0.99, and the probability that it happens is 1%). ¹ can be very small if r is very close to 0 (e.g., ¹"1.0;10\ at r "0.01, and the probability that it happens is also 1%). The statistical average time step in the IQ concept can be obtained from Eq. (3) as (20)
Note that the uniform distribution of r on the interval [0, 1) has been used. In our concept the time step is Dt"0.01.
6. Concluding remarks This paper presents an alternative to the concept of the interval of quiescence in the Monte Carlo simulation of population balances. The present concept is simple and direct. It is shown that the concept presented in this contribution is consistent with the interval of quiescence. Both concepts can be equally used in the Monte Carlo simulation of population balances if the population is not very large. The simulation results are in very close agreement between these two methods. If the population is large such as in grinding and comminution engineering, our concept shows some advantages: (1) The calculated time step Dt can be much bigger. (2) We do not need to identify the particles of the same value of particle status. (3) The time step in the numerical simulation can be regular.
(17) Notation
If we let Dt approach zero, we have N " lim N (1!Dt)R DR"N e\R. (18) D R This is consistent with the theoretical value, given by Eq. (16).
A R a B b E F
5. Time step
F L
In the algorithm presented in this paper we can utilize the regular time step. Taking t"1.0 and Dt"0.01, the relative di!erence between two methods is approximately
f L
"(1/e)!(1!0.01)" E" +0.5%. (1/e)
ln (1!r ) dr "1.0;10\. ! , a(x ) G G
(19)
In the IQ concept ¹ can be very irregular. Assume 10,000 particles in a system with a(x)"1. It is possible that r is
¸ M N N n P p
state of population at instant t transition probability of mono-particle event cumulative breakage distribution function breakage distribution function relative error cumulative distribution function for interval of quiescence cumulative distribution function of particles based on number distribution function of particles based on number number of particle classes number of kinds of monoparticle phenomena number of particles initial number of particles number density of particles probability probability value
M. Song, X.-J. Qiu/Chemical Engineering Science 54 (1999) 5711}5715
r ¹ ¹M t x y
random number value of interval of quiescence average time step in the IQ concept time value of particle state value of particle state
Greek letters Dt l q
calculated time step number of daughter particles per breakage quiescence time following time t
References Bajpai, R.K., Ramkrishna, D., & Prokop, A. (1977). E!ect of drop interactions on the performance of hydrocarbon fermentors. Biotechnology and Bioengineering, 19, 1761}1772. Bapat, P.M., Tavlarides, L.L., & Smith, G.W. (1983). Monte Carlo simulation of mass transfer in liquid}liquid dispersions. Chemical Engineering Science, 38, 2003}2013. Das, P.K. (1996). Monte Carlo simulation of drop breakage on the basis of drop volume. Computers and Chemical Engineering, 20, 307}313. Gupta, S.K., Saraf, D.N., & Pandley, B.P. (1979). Colloid Polymer Science, 257, 663.
5715
Habbard, J.L., & Ramkrishna, D. (1979). Proceedings of the 73rd Annual American Institute of Chemical Engineers Meeting, Paper No. 70e, San Francisco. Kendall, D.G. (1949). Stochastic processes and population growth. Journal of Royal Statistical Society, Series B, 11, 230}240. Kendall, D.G. (1950). An arti"cial realization of a simple &&birth-anddeath'' process. Journal of Royal Statistical Society, Series B, 12, 116}126. Marchal, P., David, R., Klein, J.P., & Villermaux, J. (1988). Crystallization and precipitation engineering, I. An e!ective method for solving population balance in crystallization with agglomeration. Chemical Engineering Science, 43, 59}67. Ramkrishna, D. (1981). Analysis of population balance } IV. The precise connection between Monte Carlo simulation and population balances. Chemical Engineering Science, 36, 1203}1209. Sampson, K.J., & Ramkrishna, D. (1979). Proceedings of the 73rd Annual American Institute of Chemical Engineers Meeting, Paper No. P-101, San Francisco. Shah, B.H., Borwanker, J.D., & Ramkrishna, D. (1976). Monte Carlo simulation of microbial population growth. Mathematical Bioscience, 31, 1}23. Shah, B.H., Ramkrishna, D., & Borwanker, J.D. (1977a). Simulation of particulate systems using the concept of the interval of quiescence. American Institute of Chemical Engineers Journal, 23, 897}904. Shah, B.H., Ramkrishna, D., & Borwanker, J.D. (1977b). Simulation of bubble populations in a gas #uidized bed. Chemical Engineering Science, 32, 1419}1425. Sweet, I.R., Gustafson, S.S., & Ramkrishna, D. (1987). Population balance modelling of bubbling #uidized bed reactors: Well-stirred dense phase. Chemical Engineering Science, 42, 241}351.
An alternative to the concept of the interval of quiescence (IQ) in the Monte Carlo simulation of population balances M. Song *, X.-J. Qiu Department of Chemical Engineering, University of Dortmund, 44221 Dortmund, Germany CDI 1111 West Holly Street, Bellingham, WA 98225-2922, USA Received 1 July 1998; accepted 10 February 1998
Abstract This work presents an alternative concept to the interval of quiescence in the Monte Carlo simulation of population balances. An example is given to show that the present method is consistent with the concept of the interval of quiescence. Both concepts can be equally used in the Monte Carlo simulation of population balances if the population is not very large. For a large population, the present concept shows some advantages. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Population balance equation; Interval of quiescence; Monte Carlo simulation; Breakage process
1. Introduction The concept of the interval of quiescence (IQ) was originally proposed by Kendall (1949, 1950) almost a half-century ago for simulating the behavior of properties that obey the birth-and-death model in biological populations. He called the simulation based on this concept the arti"cial realization of the birth-and-death process. Such a process involves the random appearance of new individuals and the disappearance of existing individuals with respective transition probabilities or &frequency functions' which continuously change distribution of the total population. Consequently, a &birth' event will add one to the population while a &death' event will delete one from it. It is naturally assumed that during an in"nitesimal time interval dt, multiple events will occur with probability of O(dt). Kendall (1949, 1950) de"ned the &interval of quiescence' (IQ) as the period between successive events during which the number and properties of individuals would remain constant. Clearly, it is a random quantity due to the stochastic nature of the events themselves. Kendall showed that the interval of quiescence has an exponential distribution with a coe$-
*Corresponding author. Present address: R.A.S. Industries Ltd., 8020-128th Street, Surrey, BC, Canada, V3W 4E9.
cient parameter in the exponent which depends on the number of individuals at the beginning of the interval and the frequency function (or say, transition probability). Following Kendall's pioneering work, Shah et al. (1976) derived the expression for the cumulative probability distribution function of the interval of quiescence as
+ , O F(q"A )"1!exp ! a (x(t#q , x , t)) dq , R H G H G (1) where M is the number of kinds of monoparticle phenomena such as breakage and nucleation, N is the number of particles, and a is the frequency function of the ith G monoparticle phenomenon. Eq. (1) describes the probability that any one of monoparticle phenomena to any one of particles happens in time q under the state of population at any time t. Based on this concept, Shah et al. (1977a) presented a general Monte Carlo simulation procedure for particulate systems. This method was utilized by Ramkrishna et al. in numerous simulations covering various scienti"c and engineering areas (Shah et al., 1976; Shah et al., 1977b; Bajpai et al., 1977; Habbard and Ramkrishna, 1979; Sampson and Ramkrishna, 1979; Sweet et al., 1987). Gupta et al. (1979) employed this procedure for simulating the behavior of solid}liquid
0009-2509/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 9 9 ) 0 0 0 9 9 - 8
5712
M. Song, X.-J. Qiu/Chemical Engineering Science 54 (1999) 5711}5715
suspensions under the e!ect of shear. This procedure was adopted by Bapat et al. (1983) for simulating the mass transfer process in liquid}liquid dispersions. Das (1996) proposed a Monte Carlo simulation algorithm for a drop breakage process based on this concept as well. The interval of quiescence is a profound concept. Ramkrishna (1981) established the precise mathematical connection between the population balances and the Monte Carlo simulations based on the interval of quiescence. However, as analyzed below, there are some disadvantages if it is used in the Monte Carlo simulation. This contribution proposes an alternative concept to it for the Monte Carlo simulation of population balances. It will be shown that there is a consistency between the present concept and the concept of the interval of quiescence.
2. Monte Carlo simulations based on the concept of IQ For simplicity, we assume a system of only particle breakage with transition function a(x). Then the cumulative distribution function of the interval of quiescence is given by
, F(q"A )"1!exp ! a(x )q . R G G
(2)
This describes the probability that a breakage happens to any one of particles after a quiescence interval following time t. The simulation algorithm developed by Shah et al. (1976, 1977a) can be described as follows: 1. Determine the interval of quiescence, denoted by ¹, by which two subsequent events are separated. A random number, say r , is generated by computer, which obeys the uniform distribution on the interval
[0, 1). According to Eq. (2), ¹ is determined by ln (1!r ) . ¹" ! , a(x ) G G
(3)
2. Determine to what particle an event (breakage) happens. We can readily get the probability that, given that a particulate event has occurred after a quiescence interval ¹ following time t, breakage happens to the ith particle as a(x ) G . p "P+the ith particle breaks up "A , ¹," G R , a(x ) G G
Then another random number, say r , is generated. If the following inequality follows: G\ G p (r ) p , (5) H H H H we identify that the event happens to the ith particle. This algorithm is straightforward and physically based. However, there are some drawbacks in the Monte Carlo simulation algorithm based on the concept of the interval of quiescence: (1) ¹ could be very small. (2) All particles have to be identi"ed even though some particles are of the same characteristic properties. The larger the population, the more serious these two drawbacks become. (3) The time step in the numerical simulation is irregular.
3. An alternative to the concept of IQ Like popular population simulations such as with the method of classes (Marchal et al., 1988), we divide the population into ¸ classes, C , C , 2 , C , according to *
Table 1 Comparison of simulation results between the present concept and the concept of the interval of quiescence 1s
(4)
2s
3s
x
IQ
Our concept
IQ
Our concept
IQ
Our concept
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.99 1.0
0.1446 0.2676 0.3732 0.4642 0.5472 0.6207 0.6804 0.7302 0.7770 0.8125 1
0.1463 0.2653 0.3706 0.4667 0.5477 0.6161 0.6787 0.7310 0.7771 0.8140 1
0.2406 0.4226 0.5628 0.6693 0.7536 0.8187 0.8687 0.9043 0.9332 0.9525 1
0.2455 0.4236 0.5644 0.6724 0.7545 0.8182 0.8673 0.9050 0.9337 0.9538 1
0.3233 0.5402 0.6867 0.7909 0.8620 0.9093 0.9418 0.9628 0.9779 0.9870 1
0.3224 0.5381 0.6886 0.7908 0.8602 0.9084 0.9406 0.9627 0.9782 0.9871 1
M. Song, X.-J. Qiu/Chemical Engineering Science 54 (1999) 5711}5715
the characteristic property (e.g. volume here). All particles in one class are identi"ed the same. Our alternative to the concept of the interval of quiescence is very simple and direct, which can be described as follows. 1. Determine the number of particles that are going to break up during the period from t to t#Dt. Assume that the total number of particles in class C is G N . Then the number of particles that are going to break G up can be calculated by N "N a(x ) Dt. G G G
(6)
2. Specify particles to break up
Or, the initial cumulative distribution function of particles based on number is given by
0, x)1, F (x, 0)" L 1, x'1.
r )N ![N ] (7) G G specify one particle more to break up where [N ] is G the integer part of N . Otherwise, no more is to be G speci"ed to break up. In this consideration N ![N ] is the probG G ability that one particle more is going to break up. Therefore, the statistical average number of particles undergoing breakage within a time step should be [N ]#+N ![N ],1 G G G #+1!(N ![N ]),0"N . G G G
(8)
4. Consistency between two concepts Eq. (9) is the number-based population balance equation
*n(x, t) "!a(x)n(x, t)# l(y)a(y)b(x, y) n(y, t) dy, *t V
(11)
The breakage frequency is assumed to be directly proportional to its volume with the coe$cient of unity, that is, a(x)"x.
(12)
All particles undergo the uniform breakage if they are selected to break up, that is, x B(x, y)" y
First, specify [N ] particles to break up. Then G generate a random number, say r , which is uniformly distributed on the interval [0, 1). If
5713
(13)
or 1 b(x, y)" , y
(14)
where x and y are the volume of the daughter and mother particle, respectively, and B(x, y) is the number-based cumulative function of daughter particles. Clearly, in this case l(x)"2.
(15)
We have solved this problem with the two algorithms. Fig. 1 gives the simulation results with our concept, which show the variation of the cumulative distribution function with time. Because we can not see any di!erence in simulation results between these two methods if we draw curves on a "gure, we give Table 1 in which we show the comparison of results at three instants (1s, 2s and 3s) between the two discussed methods. Table 2 gives the results of the total number in the system at three instants. From these two tables we can
(9)
where x is the volume of particles, n(x, t) is the number density, a(x) is the breakage frequency function, l(y) is the number of daughter particles from a mother particle of volume y, and b(x, y) is the number-based distribution function of daughter particles of volume x from a mother particle of volume y. We take a simple case to show the consistency between these two simulation methods. Assume that there are 10,000 particles of volume x"1 initially in a system. Then the initial number-density distribution function is given by
#R, x"1, f (x, 0)" and L 0, xO1
>
f (x, 0) dx"1. L (10)
Fig. 1. Simulation results of the cumulative distribution function of particles with the present concept.
5714
M. Song, X.-J. Qiu/Chemical Engineering Science 54 (1999) 5711}5715
Table 2 Comparison of simulation results from the two concepts compared with theoretical values
Total number of particles Number of initial particles remaining unbroken
1 2 3 1 2 3
s s s s s s
IQ
Our concept Theoretical value
20041 29972 40245 3658 1338 490
20012 30055 40112 3661 1340 490
¹M "
3679 1353 498
observe very close agreement of simulation results between the two concepts. According to the concept of the interval of quiescence, the probability that an initial particle with breakage frequency of unity (due to the volume of 1) remains unbroken at instant t is e\R. From the viewpoint of statistical average the number of initial particles remaining unbroken should be N "N e\R,
(16)
where N is the total number of initial particles. We calculate the values of this parameter at three instants, which are listed in Table 2 as theoretical values. The results of the number of initial particles remaining unbroken from both simulations are listed in Table 2 as well. We can see that the results from both concepts are in good agreement with theoretical values. Actually, according to the algorithm presented in this contribution, the number of initial particles remaining unbroken at any instant t can be calculated by N "N (1!Dt)R DR.
near 0 and 1 even though the probability is small. If r is near 1, ¹ can be big (e.g., ¹"4.6;10\ at r "0.99, and the probability that it happens is 1%). ¹ can be very small if r is very close to 0 (e.g., ¹"1.0;10\ at r "0.01, and the probability that it happens is also 1%). The statistical average time step in the IQ concept can be obtained from Eq. (3) as (20)
Note that the uniform distribution of r on the interval [0, 1) has been used. In our concept the time step is Dt"0.01.
6. Concluding remarks This paper presents an alternative to the concept of the interval of quiescence in the Monte Carlo simulation of population balances. The present concept is simple and direct. It is shown that the concept presented in this contribution is consistent with the interval of quiescence. Both concepts can be equally used in the Monte Carlo simulation of population balances if the population is not very large. The simulation results are in very close agreement between these two methods. If the population is large such as in grinding and comminution engineering, our concept shows some advantages: (1) The calculated time step Dt can be much bigger. (2) We do not need to identify the particles of the same value of particle status. (3) The time step in the numerical simulation can be regular.
(17) Notation
If we let Dt approach zero, we have N " lim N (1!Dt)R DR"N e\R. (18) D R This is consistent with the theoretical value, given by Eq. (16).
A R a B b E F
5. Time step
F L
In the algorithm presented in this paper we can utilize the regular time step. Taking t"1.0 and Dt"0.01, the relative di!erence between two methods is approximately
f L
"(1/e)!(1!0.01)" E" +0.5%. (1/e)
ln (1!r ) dr "1.0;10\. ! , a(x ) G G
(19)
In the IQ concept ¹ can be very irregular. Assume 10,000 particles in a system with a(x)"1. It is possible that r is
¸ M N N n P p
state of population at instant t transition probability of mono-particle event cumulative breakage distribution function breakage distribution function relative error cumulative distribution function for interval of quiescence cumulative distribution function of particles based on number distribution function of particles based on number number of particle classes number of kinds of monoparticle phenomena number of particles initial number of particles number density of particles probability probability value
M. Song, X.-J. Qiu/Chemical Engineering Science 54 (1999) 5711}5715
r ¹ ¹M t x y
random number value of interval of quiescence average time step in the IQ concept time value of particle state value of particle state
Greek letters Dt l q
calculated time step number of daughter particles per breakage quiescence time following time t
References Bajpai, R.K., Ramkrishna, D., & Prokop, A. (1977). E!ect of drop interactions on the performance of hydrocarbon fermentors. Biotechnology and Bioengineering, 19, 1761}1772. Bapat, P.M., Tavlarides, L.L., & Smith, G.W. (1983). Monte Carlo simulation of mass transfer in liquid}liquid dispersions. Chemical Engineering Science, 38, 2003}2013. Das, P.K. (1996). Monte Carlo simulation of drop breakage on the basis of drop volume. Computers and Chemical Engineering, 20, 307}313. Gupta, S.K., Saraf, D.N., & Pandley, B.P. (1979). Colloid Polymer Science, 257, 663.
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Habbard, J.L., & Ramkrishna, D. (1979). Proceedings of the 73rd Annual American Institute of Chemical Engineers Meeting, Paper No. 70e, San Francisco. Kendall, D.G. (1949). Stochastic processes and population growth. Journal of Royal Statistical Society, Series B, 11, 230}240. Kendall, D.G. (1950). An arti"cial realization of a simple &&birth-anddeath'' process. Journal of Royal Statistical Society, Series B, 12, 116}126. Marchal, P., David, R., Klein, J.P., & Villermaux, J. (1988). Crystallization and precipitation engineering, I. An e!ective method for solving population balance in crystallization with agglomeration. Chemical Engineering Science, 43, 59}67. Ramkrishna, D. (1981). Analysis of population balance } IV. The precise connection between Monte Carlo simulation and population balances. Chemical Engineering Science, 36, 1203}1209. Sampson, K.J., & Ramkrishna, D. (1979). Proceedings of the 73rd Annual American Institute of Chemical Engineers Meeting, Paper No. P-101, San Francisco. Shah, B.H., Borwanker, J.D., & Ramkrishna, D. (1976). Monte Carlo simulation of microbial population growth. Mathematical Bioscience, 31, 1}23. Shah, B.H., Ramkrishna, D., & Borwanker, J.D. (1977a). Simulation of particulate systems using the concept of the interval of quiescence. American Institute of Chemical Engineers Journal, 23, 897}904. Shah, B.H., Ramkrishna, D., & Borwanker, J.D. (1977b). Simulation of bubble populations in a gas #uidized bed. Chemical Engineering Science, 32, 1419}1425. Sweet, I.R., Gustafson, S.S., & Ramkrishna, D. (1987). Population balance modelling of bubbling #uidized bed reactors: Well-stirred dense phase. Chemical Engineering Science, 42, 241}351.