Ind. Eng. Chem. Res. 1994,33,137-145
137
Stochastic Modeling of Devolatilization-Induced Coal Fragmentation during Fluidized-Bed Combustion Wei-Yin Chen,' Ganesh Nagarajan, and Zhao-Ping Zhang Department of Chemical Engineering, Anderson Hall, University of Mississippi, University, Mississippi 38677
Bao-Chun Shen and L. T. Fan Department of Chemical Engineering, Durland Hall, Kansas State University, Manhattan, Kansas 66506
The breakage of coal particles into smaller fragments during the devolatilization stage mostly occurs randomly due to the heterogenous structure of coal, thereby necessitating a stochastic approach for modeling. In the present work, the master equation approach has been proposed for predicting the statistics of the size distribution of the coal particles during their stepwise degradation. The particlesize distribution has been lumped into a limited number of states, each representing a particular volume range. The master equation and the equations for the means, variances, and covariances of the random variables, each representing the number of particles in the individual states in the system, have also been derived from the stochastic population balance. Simulation has been performed with a stiff differential equation solver to predict the dynamic particle number statistics a t any time. By fitting the model to experimental data, the transition intensity function is found to be inversely proportional to the square of particle radius.
Introduction The dynamics of coal fragmentation profoundly affects the performance of a combustor in a variety of ways. Under the condition of external diffusion control, fragmentation enhances the rate of combustion (Chirone et al., 1991), reduces bed carbon load, and, in turn, decreases the loss of efficiency due to the overflow of carbon with bed solids. On the other hand, fragmentation increases the exposed surface carbon, and, therefore, the rate of abrasion of fines of elutriable size, which increases the loss of efficiency owing to carryover. Recently, Chirone et al. (1991) reviewed their 15 years of research on coal fragmentation during fluidized combustion. Most of their experiments were conducted with a 4-in. i.d. electrically-heated sand bed. It was discovered that the initial stage of fragmentation is caused either by buildup of pyrolysis-generatedvolatiles (hydrocarbons and water; see, e.g., Howard, 1981;Ohet al., 1989),accompanied by an increase in pressure, or by thermal stress. This fragmentation process takes place almost simultaneously with the pyrolysis and is called primary fragmentation. A relatively slow fragmentation process may also occur because of disintegration of the solid matrix through oxidation of carbon to carbon oxides. In addition, attrition causes fragmentation during combustion. The behavior of primary fragmentation depends largely on the coal constituents, particle size, particle shape, and temperature. This is evident from the experimental observations by Chirone and Massimilla (1988) and also from an available high-speed film (Bergbau-Forschung Co., 1976). Individual particles of a German bituminous coal with 1-2-mm diameter were heated in air on a metal screen with a heating rate of 1500 "C per second. The film was taken at a speed of 2500 exposures per second. The first four segments of this film demonstrated the devolatilization and fragmentation phenomena of samples from different seams, and the other three segments recorded similar phenomena of various petrographic constituents of coal known as macerals. It has been demonstrated that
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samples from various seams may behave quite differently, and fragmentation starts before tar evolution. It is also shown that fusinite, a particular maceral, undergoes relatively little fragmentation during pyrolysis, while vitrinite, a major maceral in most types of United States coal, undergoes extensive primary fragmentation. Since coal is a heterogeneous mixture with varying compositions of macerals, the fragmentation phenomenon is bound to lead to fluctuations in particle-size distributions and emission of pollutants. The particle-size distribution in a circulating fluidizedbed coal combustor could also profoundly influence the emission of nitrogen compounds. It has been shown that for a Montana lignite char the rate of evolution of nitrogen during oxidation can be proportional to that of carbon during combustion (Song et al., 1982). This suggests that combustion of large coal particles could result in delayed HCN evolution, which eventually contributes to high NzO production (Kramlich et al., 1989). The coal fragmentation during fluidized-bed combustion is an example of a complex dynamic system which is mesoscopic in nature. For such a mesoscopic system, the number of variables characterizing the detailed microscopic dynamics of the system, such as the chemical composition, temperature distribution, size, shape, momentum, or position of each coal particle during fragmentation, is numerous. For practical purposes, however, only a limited number of variables describing the macroscopic nature of the system can be investigated for most cases due to lack of detailed microscopic information of all the system variables. Consequently, an exceedingly large number of the degrees of freed.omexist in describing the macroscopic properties of this mesoscopicsystem, and fluctuations persist as natural phenomena rather than exception (see, e.g., Keizer, 1987;and van Kampen, 1992a). It is extremely difficult, if not impossible, to provide a microscopic description, e.g., the quantum or molecular level description, to the system; moreover, the deterministic or macroscopic description cannot generate the inherent probabilistic characteristics, such as fluctuations, of the system. On the other hand, the stochastic modeling, analyzing the system according to probabilistic laws, is
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138 Ind. Eng. Chem. Res., Vol. 33, No. 1, 1994
often appropriate to study the complex dynamics of the mesoscopic system. This is particularly true for systems where the number of entities is small (Fox and Fan, 1990~). One well-known example in physics is the Brownian motion description of colloidal particles. This branch of mathematics has been adopted by and rigorously developed for various biological (Hill, 1977; Renshaw, 19911, chemical (Nicolis and Prigogine, 1977; Oppenheim et al., 1977; van Kampen, 1992b), and physical (Lavenda, 1991) systems. Recent advances in the fluidized-bed combustion technology have extended the operating sizes of coal particles up to about 60 mm in diameter (La Nauze, 1986;Thomas et al., 1986). For a combustion system with such coarse particles, the number of particles is limited and the fluctuation can be significant, and the stochastic approach is a viable alternative for modeling it. Some attempts were made to model stochastically fragmentation of solid particles in process systems. Nassar et al. (1987) resorted to the Kolmogorov equation in treating particle degradation in a semicontinuous flow system. Wei et al. (1977) adopted the Markov-chain approach in the analysis of catalyst attrition and deactivation. This paper resorts to the master equation approach to stochastically treat the devolatilization-induced coal fragmentation during fluidized-bed combustion. This approach is based on a probability balance of the particle numbers in different size ranges. It does not require detailed information concerning volatiles pressure and temperature distribution; nevertheless, the dynamics of particle-sizedistribution is lumped into a set of parameters, namely, transition intensity functions, which define the probability of particle transition from one size range to another. Unlike the Kolmogorov equation expressed in terms of the conditional probability density function, the master equation is expressed in terms of the probability density function related directly to the initial particle size distribution. The analytical methodology below follows those of Oppenheim et al. (19771, van Kampen (1992b), and Fox and Fan (1990a,b). Model Formulation
For simplicity, the particle-size distribution is lumped into a limited number of equally-spaced states, each representing a particular weight (or volume) range. The number of particles in each state is a time-dependent random variable. Assumptions. A stochastic model for the type of random fragmentation under consideration has been derived by imposing the following assumptions. 1. Particles’ states are assigned according to their volumes. Each particle breaks into two particles, one at state j and the other at k with j + k = i and j < k where i,j , and k can be any number from 1to C with C representing the highest state. The addition property, j + k = i, holds because the states are equally spaced on the volume scale. 2. The transition of a particle from one size range to another is Markovian, Le., the change in the particle distribution depends solely on the present condition of population and not on its past conditions. 3. The transition intensities (probabilities of specific transitions per unit time) are considered to be temporally homogeneous, i.e., they are time independent. 4. The process is a batch process. No particle is fed into or exits from the system during the operation. Derivation. Let the probability that a coal particle at state i will split into two fragments, one at state j and the other at state k , with j + k = i and j < k, during the time interval, ( t ,t + At), be
aijAt + o(At) (1) where aij represents the transition intensity, i.e., the probability that a particle undergoes fragmentation from state i to state j per unit time, and o(At) satisfies lim -- - 0 (2) A t 4 At It should be mentioned that the transition intensity depends on the properties of coal and operating conditions (Chirone et al., 1991). Also let (3)
where [i/23represents the largest integer less than or equal to i/2. Then, the probability that a coal particle will remain at state i during the time interval, (t, t + At), is 1 = aiAt + o(At)
(4) The random variable, N i ( t ) ,i = 1, 2, ..., 1, represents the number of coal particles in state i at time t , and the random vector, NO)= [ N l ( t ) Nz(t), , ...,Nl(t)I,represents the size distribution of coal particles at time t with its realization vector, n(t) = [ n l ( t ) ,nz(t),..., nl(t)l. Furthermore, we define
P,,(t) E P ( N ( t )= n) 5 [probability that the coal particles have the particle-size distribution, n, at time t3 ( 5 ) and
-6, = [61,,
62,,
**a,
(6)
6,,1
where the ith element, &a, is a Kronecker delta, i.e., 6,, = 1and asp = 0 for a # 6. The probability balance gives rise to
P,(t
+ At) =
[probability that all particles remain intact, i.e., the particle-size distribution remains at n during the time interval between t and ( t A t ) ] +[probability that the particle-size distribution is transformed into n due to a single breakage of particles at all possible states in the time interval between t and ( t + A t ) ]
+
=Q1+
QP
(7)
where
e [iPI
The lower limit of summation over i in the above expressions is 2 instead of 1because state 1represents the smallest particle size. Note that eq 9 includes the possibility of breakage of a particle into two particles in the same size range ( i e . ,j = k, and i = 2j). Formulation of eqs 8 and 9 also assumes that the breakage of each particle is independent of each other. Substituting eqs 8 and 9 into eq 7 and taking the limit At 0 give rise to
-
Ind. Eng. Chem. Res., Vol. 33, No. 1, 1994 139
e d -[P,(t)l = -CniaiPn(t)+ dt i=Z
This is the master equation for the system of interest; mathematically, it is a set of differential equations. Each differential equation in this set corresponds to a realization value of n. Consequently, the number of total equations is large when the number of states or particles is large. Thus, it is not always easy to solve this set of equations for the complete joint probability distribution. Nevertheless, in practice, it often suffices to determine only the expressions for the first and second momenta of the resultant particle-size distribution. These moments, in turn, yield the expressions for the means, variances, and covariances that can be correlated or compared with the experimental data. Mean. Given that N ( t ) = n = (nl,nz,...,ni, ...,ne),ni(t) may change its value during the time interval, ( t ,t + At), owing to breakage of coal particles in state i or higher. Because of the assumptions in connection with eqs 1-3, ni(t)may decrease by one with a probability of niaiAt + o(At)
= -aiE[Ni(t)]+
c
qiE[Nj(t)] j=i+l ]#Zi
Since
Ni(t + At) - Ni(t) E[Ni(t+ A t ) ] - E[Ni(t)l (19) At At taking the limit of eq 18 as At 0 yields
I=
E[
-
(11)
increase by one with a probability of I
njaj,iAt+ o(At)
(12)
i=i+l .j&i
increase by two with a probability of nziazijAt
+ o(At)
(13)
or remain unchanged with a probability of c
1- ([nisi
+ Cnj'~j,i+ nzia,,i]Atj + o(At)
(14)
j=i+l
or
]#21
Thus, the conditional expected change of Ni(t) during the time interval, ( t , t + At), can be expressed as
E[Ni(t+ A t ) - N i ( t ) ( N ( t ) = n l = -niaiAt
+[
2
This is a set of C equations governing the mean numbers of particles in different states. These equations correspond to the deterministic macroscopic equations. Variance. The variance of the number of coal particles in state i is evaluated through the expression (see, e.g., Casella and Berger, 1990a)
+
njaj,iAt]+ 2nziazi,iAt o(At)
(15)
jfi+L
I#%
Dividing this expression throughout by At, we obtain
~ a r [ ~ , ( t=) E l [ N , ~ ( ~-)(E[N,(~)I)' I (21) The second term on the right-hand side of this equation can be calculated by solving eq 20, whereas the differential equations governing the first term can be derived from a procedure similar to that in obtaining eq 20. Given that N ( t ) = n = ( n l ,nz,...,mi, ...,ne),in a sufficiently small time interval of ( t ,t + At), ni2(t)may decrease by (2ni - 1)with a probability of
+ o(At)
(22) This . . is caused by a decomposition of one particle at state niaiAt
L, z.e.,
= -nisi +
c njaji j=i+l
+ 2nzia2i,i+ At
j#Zi
The condition, N(t) = n,can be removed by the following definition of expected conditional value:
Ni(t + At) - Ni(t)
E[
At
.
I=
n:(t
(16)
+ A t ) - n:(t) = (ni- 1)' - n:
or
n:(t + A t ) - n:(t) = -(2ni - 1) Similarly, ni2(t)may increase by (2ni + 1)with a probability of I
n j q i A t + o(At) j=i+i ]#21
Consequently, eq 16 becomes
which is caused by decomposition of a particle at a higher state; and increase by (4ni + 4) with a probability of
140 Ind. Eng. Chem. Res., Vol. 33, No. 1, 1994
n2ia2i,iAt+ o(At) (24) which is caused by a decomposition of a particle at state 2i; or remain unchanged with a probability of 1 - {[nisi
+
e
njajc+ n2ia2i,ilAt) + o(At)
j=i+l
j#2i
Thus, the conditional expected change of Ni2(t)during the time interval, (t, t + At), is
E[{N?(t+ At) - N?(t)j(N(t)= nl
+
= -(2ni - l)niaiAt (2ni+ 1 )
c e
njajiAt
j=i+l
jf2i
+ (4ni+ 4)n2ia2i,iAt+ o(At) (26) Dividing both sides of this equation by At leads to
{N?(t + At) - N?(t))lN(t)=n At
1
+
= -(2ni - l)niai (2ni+ 1)
-
+ 4a,,i{E[Nj(t)N,i(t)l+ E[N,j(t)l) (31)
(25)
by taking the limit of the resultant expression as At 0. "his is a set of C equations governingthe variances (through eq 21) of particle numbers at different states. The terms on the right-hand side of eq 31 indicate that this equation is coupled with the covariances, which are derived below. Covariance. The governing differential equation for the cross-moment between the numbers of coal particles in states i and j for which i
