Ind. Eng. Chem. Res. 1987,26, 2413-2419
(N)= column matrix with elements N ; of dimension n x
fit'=total molar flux
1
ri = reaction rate of species i (r) = column matrix with elements ri of dimension n X 1 xi = mole fraction of species i (x) = column matrix with elements xi of dimension n X 1 (Y)= column matrix defined by eq 11 of dimension 2n X 1 z = distance across film [O] = square null matrix of dimension n X n Greek Symbols [A] = partitioned matrix defined by eq 14 of dimension 2n X 2n
C = summation over index Superscripts ( 2 ) = first derivative of (x)
(a) = second derivative of (x)
Subscripts
i, j , k = index indicating component number n = total number of components less one Matrix Notation
[I
= square matrix
( ) = column matrix [ I-' = inverse of square
matrix
2413
Literature Cited Beek, J. AZChE J. 1961,7(2), 337. Bird, R.B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; Wiley; New York, 1960;p 571. Bronson, R. Matrix Methods-An Introduction; Academic: New York, 1970. Burghardt, A.; Krupiczka, R. Znz. Chem. 1975,5,487,717. Cussler, E. L. Multicomponent Diffusion; Elsevier: Amsterdam, 1976;Chapter 9. Delancey, G. B. Chem. Eng. Sci. 1974,29,2315. Delancey, G.B.; Chiang, S. H. Ind. Eng. Chem. Fundam. 1970,9(3), 344. Krishna, R.; Standart, G. L. AZChE J. 1976,22,383. Krishna, R.;Standart, G. L. Chem. Eng. Commun. 1979, 3, 201. Lee, S. T.; Delancey, G. B. Chem. Eng. Sci. 1974,29,2325. gentarli, I. Master Thesis, Boiaziqi University a t Istanbul, 1985. Smith, L. W.; Taylor, R. Znd. Eng. Chem. Fundam. 1983,22(1),97. Stewart, W. E. NACA Technical Note 3208, 1954. Stewart, W. E.;Prober, R. Znd. Eng. Chem. Fundam. 1964,3(3),224. Taylor, R.Ind. Eng. Chem. Fundam. 1982,21(4),407. Toor, H.L. AZChE J. 1964,10(4),460. Toor, H.L. Chem. Eng. Sci. 1965,20,941. Wei, J. J. Catal. 1962,1, 526. Wilke, C. R. Chem. Eng. Prog. 1950,46(2),95.
Received for review September 2, 1986 Accepted August 3, 1987
On the Liquid Flow Distribution in Trickle-Bed Reactors R. 0. Fox Department of Chemical Engineering, Kansas State University, Munhattan, Kansas 66506
The previous analytical expression for the liquid flow distribution in trickle beds found by maximizing t h e so-called configurational entropy is reviewed, and a novel derivation is offered, more closely modeling data found in the literature. The novel distribution is a n analytical expression describing the liquid flow distribution recently found by employing a Monte Carlo simulation of the very large lattice model (VLLM), thus reducing this computationally time-consuming procedure t o a simple equation. The resultant distribution is a function of the total flow rate through the trickle bed, the bed permeability as expressed by the number of open channels through the bed, and the minimum liquid flow rate through a given channel. 1. Introduction The objective of this paper is to review the current theory describing the liquid flow distribution in trickle-bed reactors based on an entropy maximization procedure (Crine and L'Homme, 1984;Crine and Marchot, 1983) and to offer a novel theoretical derivation of the same distribution. The latter better describes the recent experimental findings of Ahtchi-Aliand Pedenen (1986). These authors developed the very large lattice model (VLLM) and conducted Monte Carlo simulations to study the resultant liquid flow distribution. The novel distribution presented in this work reduces these time-consuming Monte Carlo simulations to a simple analytical expression. The liquid flow distribution in a trickle-bed reactor determines the degree of irrigation of the packing and, in turn, can have a considerable effect on the overall performance. To date, most design procedures for these reactors rely on empirical models which are difficult to apply with any certainty to novel operating conditions. It is therefore of some interest for the advancement of the design of trickle-bed reactors to develop a priori theoretical models which relate the behavior of the system a t the level of a single packing element to the overall performance. The theory proposed by Crine and Marchot (1981a,b) based on percolation theory is an attempt to model the 0888-5885/87/2626-2413$01.50/0
trickle bed starting on the level of a single particle; these results are then averaged over the entire trickle bed by using the liquid flow distribution (also see Crine and L'Homme (1984)). They proceed by calculating local transport properties which are functions of the local liquid flow rate. These local properties are averaged over the entire trickle bed by using the liquid flow distribution derived by maximizing the "configurational entropy" of the system. The liquid flow distribution is thus obviously an important quantity in this theory since it determines the "average" behavior of the trickle bed. In the second section of this paper, the derivation presented by Crine and Marchot (1983) of the liquid flow distribution is critically reviewed. The resultant distribution is in the form of a classical exponential distribution, a decreasing function of the flow rate. Recently, Ahtchi-Ali and Pedersen (1986) conducted an experimental investigation of the liquid flow distribution and found that the exponential form did not correspond to their findings. The experimental distributions exhibited a relative maximum which cannot be modeled by using a liquid flow distribution that is a strictly decreasing function of the flow rate. It is possible, however, to derive an a priori liquid flow distribution exhibiting a relative maximum; this is the topic of the third section. In the final section, the two 0 1987 American Chemical Society
2414 Ind. Eng. Chem. Res., Vol. 26, No. 12, 1987
resultant distributions are juxtaposed in order to study their forms as a function of the parameters controlling their shapes. There it is also shown that the results from the computer simulation performed by Ahtchi-Ali and Pedersen (1986) can be described by using the distribution derived in section 3. This fact represents a considerable savings of time since the computer simulations are quite lengthy and must be repeated for each new set of the controlling parameters. 2. Review of Previous Derivation of Liquid Flow Distribution Crine and Marchot (1983) present a derivation for the liquid flow distribution based on an entropy maximization procedure. The object of the present section is to quickly review the foundations of this approach. A more comprehensive discussion of derivations based on the entropy maximization procedure can be found elsewhere (e.g., Fox and Fan (1986)). The variables, li, are defined as the number of active bonds-a term taken as synonymous to open channels in the sequeal-with flow rate, iAq, where Aq is the size of a hypothetical flow packet. If the total flow rate to the bed is fixed and equal to rAq, then the following constraints must apply to the variables, I,: r
r
Cli = n
1=0
Cili = r
,=O
are incorporated by using Lagrangian multipliers, yielding (Crine and Marchot, 1983) li = exp(a - 1 + pi) i = 0,1,2,...,r (3) where a and p are the two Lagrangian multipliers. The only difference between this expression and the one which appears in the original paper (Crine and Marchot, 1983) is the upper limit on the values of i discussed earlier. In their paper, Crine and Marchot (1983) use an upper limit of infinity, disregarding the fact that the total flow rate is equal to r. Evaluating the partial sums resulting from the application of the constraints is a straightforward task. However, since it is considerably easier to work with integrals in place of summations, first let Aq tend to zero and define a continuous variable, q, equal to the value of iAq in this limit. Doing so, entirely equivalent expressions for the liquid flow distribution and the two constraints are found:
If the two constraints are applied to determine the values of the two constants, a and b, in this distribution, we discover that both constraints cannot be satisfied unless n is greater than or equal to two. For this case, the resultant expression is
(1)
These constraints reflect, respectively, the fact that the total number of open channels is equal to n, and the total number of flow packets is equal to r. The upper bound of the summation in these constraints has been set equal to r to reflect the fact that there can be no channels with a flow rate greater than r. In the work of Crine and Marchot (1983), the summations appear with an upper limit of infinity which, in principle, is allowable since all values of li with an index greater than r must be equal to zero. This notation shall be avoided to illustrate a slight deviation in the form of the liquid flow distribution resulting when the Constraints are applied to the distribution found by using the entropy maximization procedure. Note that the value of r can be arbitrarily large but that the product, r b q , must be equal to total flow rate through the trickle bed. Although the origins of the distribution are a bit obscure, Crine and Marchot (1983) next present an expression, derived through combinatorial analysis, for the number of possible spatial configurations, i.e., n! lo!1I!. ..I,!
(6) When n is large and the variable representing the flow rate is written in continuous form, this unnormalized distribution is the same as that given by Crine and Marchot (1983). Note that neither eq 5 nor eq 6 depends on the size of an individual flow packet. The distribution found by using the entropy maximization procedure is a strictly decreasing function of the flow rate. In a recent experimental investigation of the liquid flow rate distribution in a trickle bed, Ahtchi-Ali and Pedersen (1986) found that this distribution does not represent their experimental data. Their data indicate that the liquid flow rate distribution has a relative maximum that becomes more and more pronounced as the fraction of open channels increases. The object of the next section is to derive a liquid flow distribution based on the random division of the flow packets among the n open channels. The resultant distribution has many of the properties which Ahtchi-Ali and Pedersen (1986) found lacking in the distribution derived in this section.
Note that the form of this expression is exactly the same as those used throughout the entropy maximization literature (Shannon, 1948). Combinatorially, this expression represents the number of ways that the n open channels can be divided into r different categories where the ith catagory represents the flow rate, iAq. Obviously, the two constrains given by eq 1 must be applied to ensure that the limits on the total number of flow packets and on the total number of open bonds are observed. However, it is not quite clear why Crine and Marchot (1983) decided to divide the channels among the flow rates in this manner. An alternative derivation which proceeds by dividing the r flow packets among the n open channels is offered in the next section. The next step in the derivation as presented by Crine and Marchat is to maximize eq 2 with respect to the flow rates, l j , subject to the two constraints. The constraints
3. Novel Derivation of Liquid Flow Distribution The present derivation begins by dividing the crosssectional area of the trickle bed into n open flow channels and N - n closed channels where N is the total number of open plus closed channels. Note that the value of n can be determined from the liquid flow structure morphology proposed by Crine and L’Homme (1984) based on percolation theory. The value of n is an increasing function of the percolation parameter, p , and related to the fraction of open channels, G1,by n = NGI (7) Note also that n is independent of the liquid flow rate through the bed and determined solely by the structure of the packing before any liquid is added. The next step in the derivation is to discretize the flow through the cross section of the bed during a given time
c=-
where, for large n, (9) = Q / n
= Q2/n2
Ind. Eng. Chem. Res., Vol. 26, No. 12, 1987 2415 interval. The flow rate, Q, shall be held constant and specified independently by the operator of the trickle bed. The flow rate can then be divided into r small packets of size Aq. During the given time interval, r such packets pass through the n open channels in the cross section of the trickle bed. Each of the n channels is numbered, and the flow rate through the ith channel is denoted by q ior, equivalently, by ri, where ri has an integer value between 0 and r. Obviously, n
Cri = r i=l
The question now of interest is, what is the joint probability distribution, f(rl,r2,...,rn),of the ris? To answer this question, an assumption must be made concerning the probability that a single given flow packet chooses a given flow channel through which to pass. Since the flow channels have been arbitrarily numbered, unless there is a particular spatial inhomogeneity in the liquid flow, this probability should be independent of which channel is chosen. Furthermore, if this probability is assumed to be independent of the number of packets already flowing through the given channel, it then follows that the probability of choosing channel i is given by
i = 1,2,...,n (9) Thus, a given packet will choose with equal probability from among each of the open channels. Note that this assumption may be rather tenuous since it is hardly likely, for hydrodynamic reasons, for a packet to choose a channel where many other packets are already flowing. However, handling of the case where this assumption is relaxed requires a more difficult analysis than that presented here. The assumptions used in the present derivation are, however, consistent with those employed by Crine and Marchot (1983) and are equivalent to the Monte Carlo simulations used by Ahtchi-Ali and Pedersen (1986). With these assumptions, the reader will now recognize the classical case from statistics involving the placement of r objects in n boxes where each box has the same probability of receiving a given object. The joint probability for ri can immediately be written as pi = l / n
While the form of this multinomial distribution is very similar to that employed by Crine and Marchot (1983),the random variables and thus the meaning in the present case are different. Here, the random variable, p i , denotes the flow rate through channel i, while in the distribution appearing in the work of Crine and Marchot (1983), the variable, li, denotes the number of bonds with flow rate i. In the present derivation, the constraints concerning the total flow being equal to r and the total number of channels being equal to n are contained in the distribution itself. A random variable equal to the number of channels with flow rate k can be defined by utilizing a function whose value is one when a given channel has flow rate k and zero otherwise. The number of channels with flow rate k , l k , is then the sum of these functions over all values of i, n lk
= Clpi(ri) i=l
k = 0,1,2,...,r
(11)
where Ilkl(ri)is the function described above. Note that lk is a random variable and a function of the random variables, ri,whose distribution is given by eq 10. It would be a rather formidable task to attempt to find the distribution of the Lk’S, defined above, starting from, the distribution in eq 10. A more modest, yet fruitful, undertaking is to find the expected values of the random variables, lk. The resultant expected distribution of liquid flow rates can then be compared to the distribution derived by Crine and Marchot (1986) for the most probable distribution of liquid flow rates. To find the expected value of l k , the definition of lk in terms of ri and the multinomial distribution can be employed. This calculation results in an unnormalized binomial distribution (Appendix A),
with the following mean and variance:
r
(k)= n
c> = L ( l -
i)
(13) n Another interesting perspective concerning this distribution can be gained by deriving an expression for the probability distribution of the flow rate through an individual channel independent of the flow rates through the other channels. Since the channels are arbitrarily numbered, the joint probability distribution reduces to the desired distribution by summing over the first n - 1 variables under the constraint n-1
C r i = r - rn i=l
(14)
Using the multinomial theorem, the resultant expression is
Note that, although this expression has exactly the same form as the one found for the expected distribution of the number of channels with a given liquid flow rate, the meanings of the two distributions are not the same. The latter distribution is the probability of observing a given flow rate in any particular channel, while the former is not a probability distribution at all. Since the two forms are identical, however, this system is “ergodic” in the sense that observing an individual flow channel a t several different times will give the same result as that found by observing the mean distribution of flow rates through all the open channels is a t a given instant. Up to this point, since the flow rate is usually thought of as being continuous, it may seem somewhat arbitrary to divide the flow rate into small packets. There is some evidence as to the existence of a minimum liquid superficial velocity, L,, found in the work of Crine and L’Homme (1984). This minimum liquid flow rate may be used in the present model in place of Aq. However, further experimental investigation of the liquid flow distribution is necessary to establish this relationship. In the sequel, the form of the flow distribution derived in this section is shown to be quite dependent upon the value of L,. In the limit where n is much larger than one, the binomial distribution reduces to a Poisson distribution with parameter equal to the total flow rate through the bed divided by the number of open channels, i.e., _l k -- e - r l n ( r / n ) k k = O,l,...,+m n k!
2416 Ind. Eng. Chem. Res., Vol. 26, No. 12, 1987
Note that, unlike the distribution found by Crine and Marchot (1983), the distributions found in this section exhibit a maximum for certain values of k . A continuous distribution not including the artificial discretization of the flow rate can be found from the binomial distribution by writing the factorials in terms of continuous functions, in this case gamma functions, and defining a variable q equal to kL,,
For this distribution, the mean and variance are, respectively,
C is a constant of normalization approximately equal to one for values of Q/L, greater than 20; ita exact value can be found by normalizing eq 17. Note that the variance depends on L,; this was not the case with the exponential distribution given in eq 5. The liquid flow distributions derived thus far are not entirely equivalent to actual distribution in the trickle bed since the closed channels whose flow rate is equal to zero have not been included. If the closed channels are included the distribution becomes l k = ( N - n)ak,o + f ( k ) (18) where bk,O is a Kronecker delta, then the normalized flow distribution function, f f k , defined by
or
is thus
The first term in this expression represents the closed channels, while the second is the unnormalized distribution for the open channels found earlier. In the sequel, the following forms for “ k and “ ( q ) shall be of interest:
Equation 22 was found by entropy maximization. Equations 21 and 23 result from the derivation given in this section for the discrete and continuous cases, respectively. The comparison of these expressions will be postponed until the next section. The distributions given in eq 21 and 23 compare quite favorably with those found experimentally by Ahtchi-Ali and Pedersen (1986)) and eq 21 is equivalent to the results of a Monte Carlo simulation of the VLLM. These authors did not find a dependence on L, in their results since L, was implicitly assumed equal to one and the value of n was varied by manipulating the percolation parameter. In concluding the discussion of the derivation of the liquid flow distribution in a trickle bed, note that these distributions are quite independent of the percolation parameter, p , for a given value of n. Thus, while the value of the percolation parameter will determine the value of n, or equivalently of Gl, found in these distributions, this parameter is completely independent of the liquid flow rate through the bed. There seems to be some confusion in the literature concerning this point. In particular, Ahtchi-Ali and Pedersen (1986) state that high values of p correspond to high flow rates. The reason for this confusion may lie in the method which Ahtchi-Ali and Pedersen (1986) used to fix the flow rate through the bed in their computer simulation. In their work, they randomly added flow packets into the system until all of the open channels a t the top of the bed had received a t least one packet. By use of this method, the total number of flow packets and thus the total flow rate are random variables whose values are a function of the number of open channels a t the top of the bed. To better understand the dependence of the total flow rate on the value of n (and thus of p ) , the expected total number of packets which must be added to the bed before all n open channels have a t least one packet can be calculated. To this end, let mk be the number of flow packets that must be added to the system between the time when the (k - 1)th or kth channels receive their first packet. Note that mk does not include the first packet added to either the (k - 1)th or kth channel. During this time, there are k - 1 channels out of n having a t least one packet. Therefore, the probability of choosing an open channel with no packets is equal to (n- k + l)/n. The probability of choosing mk channels having a t least one packet before choosing one channel with no packets is then (Appendix B)
mk = 0,1,2,...,+-
Adding together the total number of flow packets injected into the system up to the point where all open channels have a t least one packet results in a value for r, n
r=n
ff(q) = (1
-
+
+ kE= lm k
(25)
The expected value of r can be found from the distribution of mk (Appendix B), n
(r)= n
+ C (mk) k=l
n
=n=
Cl/k k=l
The mean irrigation rate, ( j ) ,defiled as the mean number
Ind. Eng. Chem. Res., Vol. 26, No. 12,1987 2417 I .o
=.01.0=128.
-GI
0.9 G,; . I , 0 = 2880
N = 6400 L,= I .o
GI = . I , 0 = 2 8 8 0 .
N = 6400 L,= I .o
G , = 2. 0 = 7040
0.8
G , = .2.0 = 7040.
0.7
G, = .4.0 = I3 120.
0.6
GI = .4.0 = I3 120.
0.5 GI = .6.0 = 18944.
-G,
0.4
= .6.0 = 18944.
+GI
0.3
= .8.0 = 2 1504.
0.2
0.I 0
I
2
3
5
4
q, l o c a l
flow
6
7
8
9
IO
0.0 0
rate
Figure 1. Liquid flow distribution, cyk, as a function of the local flow rate, q, for various values of the fraction of open channels, Gl, and of the total flow rate, Q, as found from eq 21.
of packets flowing through any given channel, is thus a function of p through G1,i.e.,
GzIn (NG,)
(27) Note that this expression is also dependent on the size of the bed used in the simulation through the value of N . Thus, if the cross-sectional area of the bed were doubled, the mean irrigation rate would increase by the addition of an amount almost equal to In 2 when using this method. These results in no way imply that, in general, high values of p correspond to high flow rates since the former is a property of the packing and the latter a property controlled independently by the operator of the experiment. It is only due to the fact that Ahtchi-Ali and Petersen (1986) chose to use a method dependent upon p that they found any correspondence between the flow rate and the percolation parameter. 4. Comparison of Liquid Flow Distributions In this section, the relative shapes of the liquid' flow distributions given by eq 21 and 22 are compared as functions of the fraction of open channels, GI. Since the present results are to be compared with those found in the work of Ahtchi-Ali and Pedersen (1986), the total flow rate, Q, is allowed to increase with increasing values of G,; however, to yield a wider spread in the resulting distributions, the values used here will be approximately 4 times larger. The chosen value of N is 6400, corresponding to a square cross section with 80 X 80 channels. The value of the percolation parameter, p , corresponding to a given value of Gz is not given here. The interested reader can find further information concerning this relationship in the work of Crine and Marchot (1981a) or Ahtchi-Ali and Pedersen (1986). In Figure 1,the liquid flow distribution found from eq 21 is presented. Note that, as expected, for almost all values of Gz,this distribution exhibits a maximum with respect to q. Almost all of the weight of the distribution of q equal to zero results from the closed channels. This is in sharp contrast to the results in Figure 2 for eq 22 where a large portion of the open channels have zero flow rate. The results in Figure 1 can be compared with the bond distribution obtained from the VLLM in Figure 6 of Ahtchi-Ali and Pedersen (1986). Except for the fact that the mean flow rate in Figure 1 is approximately 4 times larger, the two distributions have the same form. The distribution in Figure 2 found from eq 22 is seen
0.4
0.3
0.2
0.I
0.0 00
20
40
60
80
100
q. l o c a l f l o w r a t e
Figure 3. Effect of the parameter L, on the liquid flow distribution found from eq 23.
to be a strictly decreasing function of q. As Gzincreases, this distribution spreads out due to the much larger variance of the exponential distribution as compared to the binomial. In the present case, since N is large, for G,close to one, the variance of the exponential distribution is Figure 2 is approximately equal to the square of the variance of the distribution in Figure 1 [eq 6 and 131. Ahtchi-Ali and Pedersen (1986) have noted that the strictly decreasing distribution given in Figure 2 (their Figure 7) does not represent their experimental findings as well as the distribution found by Monte Carlo simulation of the VLLM. Therefore, since the distribution in Figure 1 (eq 21) is equivalent to the Monte Carlo simulation of the VLLM, we can assert that this distribution also compares favorably with experimental findings. An important parameter implicitly overlooked in the Monte Carlo simulations of Ahtchi-Aliand Pedersen (1986) is L,. This parameter controls the relative width of the liquid flow distribution as seen from Figure 3. The units in the figures for L, and Q must be the same in order for eq 23 to be valid. The value of Q is set by the operator of the trickle bed, and L, is postulated to be a physical parameter of the system that can be determined by fitting eq 23 to experimentaldata. Since, as can be seen in Figure 1,the value of Gl approximately correspondsto [ l - a(O)], its value can be determined independently of the value of L,. Note that L, could be a function of the physical properties of both the liquid and the packing particles as well as of the total flow rate through the trickle bed. 5. Conclusions In this work, a novel derivation of the liquid flow dis-
2418 Ind. Eng. Chem. Res., Vol. 26, No. 12, 1987
tribution in a trickle bed has been presented and shown to be equivalent to the distribution found through a Monte Carlo simulation of the VLLM by Ahtchi-Aliand Pedersen (1986). The resultant distribution has the form of a binomial distribution plus an additional part representing the closed flow channels in the trickle bed. In contrast to the previous analytical expression, derived by Crine and Marchot (1983) and based on an entropy maximization procedure, the distribution derived in this work exhibits a relative maximum with respect to the flow rate and is dependent upon the value chosen for the minimum liquid flow rate through a single channel. The latter quantity controls the spread or variance of the distribution, larger values correspondingto larger variances. This dependency was not found in the simulations of Ahtchi-Ali and Pedewen (1986)because they implicitly assumed that the value of the minimum liquid flow rate was constant and equal to one while increasing the number of flow packets passing through the bed. However, if their results were renormalized fixing the total flow rate at a constant value while decreasing the size of a given flow packet, their distributions should become more and more peaked around the mean flow rate through a single channel. It is important to remember that the percolation parameter as used by Crine and Marchot (1981a) only determines the static properties of the packing and does not affect the flow rate distribution except through the value of n, the number of open channels. Thus, it is necessary to derive separately an expression for the liquid flow distribution. One need only know the number of open channels and nothing about percolation theory to understand and use this derivation. This fact seems to often not be well understood in the literature where references to the effect of the percolation parameter on the total flow rate can be found (Ahtchi-Ali and Pedersen, 1986). Percolation theory, as used by Crine and Marchot (1981a,b), only determines the number of open paths through which liquid may flow. Once these channels are open, it is then a separate matter to determine how much liquid flows through each channel. The liquid flow distribution is, however, the important quantity in determining how the local behavior can be averaged over the entire trickle bed. In this work, a rational derivation of this distribution has been presented. Much work remains to be done to determine if the local particle behavior can be understood and modeled as strictly a function of the local flow rate over the particle, and thus whether a liquid flow distribution in any form can be used to average the particle behavior over the entire trickle bed. Only if these properties exist can there be any hope of a priori design of trickle beds based on the models presented in this work and its predecessors. Acknowledgment This material is based upon work supported under a National Science Foundation Graduate Fellowship. Nomenclature a = Lagrangian multiplier in continuous distribution b = Lagrangian multiplier in continuous distribution G, = fraction of open channels IlkJ(r,) = function equal to one when ri is equal to k and zero otherwise 1, = number of active bonds or open channels with local flow rate, iAq I(q) = number of active bonds or open channels with local flow rate, q L , = minimum liquid flow rate (=Aq) m k = random variable defined in section 3
n = number of open channels or active bonds N = total number of channels or bonds p L= probability that a given flow packet chooses channel i q = local liquid flow rate Q = total liquid flow rate to the trickle bed r = total number of flow packets added to the bed per unit time r, = number of flow packets passing through channel i per unit time (.) = expected value = variance Greek Symbols a = Lagrangian multiplier in discrete distribution
= fraction of channels with local flow rate, kAq a ( q ) = fraction of channels with local flow rate, q @ = Lagrangian multiplier in discrete distribution r(.)= gamma function 6,, = Kronecker delta Aq = size of a flow packet "k
Appendix A: Evaluation of the Expected Value of lk The expected value of the random variable, l k , defined as n
where 1 if ri = k Ifkl(ri) = 0 if ri # k
(
can be found by using the distribution f ( r l7 - 2 , . ..,rn) =
'!
fi(L>"
rl!r2!,..rn!j=in
?rj = r (A3)
j=l
By the definition of the expected value, this yields
?
rj = r-k
j=1, j#i
.
.
.
.
.
c...cc...c rl
(r - k ) !
fi (A) n
r8.1ri+l rn rl!...ri-l!ri+l...rn!j=l,j # i
Appendix B: Derivation of the Probability Distribution for mk The random variable, mk, has been defined to be equal to the number of flow packets added to the system between the times when the ( k - 1)th and kth channels receive their first packet. During this time, there are k - 1 out of n channels which have received a t least one packet and n - k + 1out of n which have yet to receive a packet. Thus, choosing randomly from among the n channels, the probability of picking an open channel is ( n - k + l ) / n .
Ind. Eng. Chem. Res. 1987,26, 2419-2423
Dividing eq B2 by N yields an expression for the mean irrigation rate,
The probability distribution of mk is the probability of choosing mk nonempty channels followed by an open channel, n-k+l k - 1 mk dmk)
=
(
mk
2419
n
( j }= G I C l / k
)(T)
k=l
c(:"of
= 0,1,2,...,+m
> G1
The expected value of mk is thus (m,) = 0 n-k+l k = 2,3,...,n (mk) = k - 1 Adding the n extra flow packets which go to add the first packet to each of the n open channels, the expected value of r is " n - k + l (r)= n + C k-2 k-1
ds
= Gl ln
