40
Ind. Eng. Chem. Process Des. Dev. 1904, 23, 40-48
Design of a Flexible Batch Process with Intermediate Storage Tanks Takelchlro Takamatsu, Iorl Hashlmoto, Shlnjl Hasebe, and Masahko O’Shlma +
Department of Chemical Engineering, Kyoto University, Kyoto 606 Japan
The problem of the design of a flexible batch process which consists of several batch stages and intermediate storage tanks is dealt with. The variations related to the operation schedule and to the batch size of each batch stage are considered as uncertain variations in the batch process. The quantitative relationship between the size of the variations and the necessary volume of the storage tank is derived. By use of this relationship, the design problem which makes not only every variation within a given range allowable but also minimizes the construction cost is mathematically formulated. When the cycle time or the batch size of each subprocess takes only discrete values, this design problem is effectively solved by applying the Dynamic Programming Approach.
Introduction Many batch processes are utilized for producing products with high added values. Such processes usually necessitate so-called sophisticated operations, such as reactions with complex reaction paths and/or long reaction times, and the processing of solid materials which are very difficult to handle. In such a sophisticated batch operation, there are many uncertainties. Often there are cases in which the processing time and/or cleaning time of each batch unit becomes shorter or longer than the scheduled ones. It also happens that the amount of material processed in a batch, that is, the batch size, often changes from the preassigned nominal value. When each batch unit is connected in a series, the outlet flow from the precedent batch unit directly becomes the inlet flow to the next unit. Therefore, the variations in processing time and/or batch size in a certain unit affect the operation schedule and the batch size of the whole process. That is, these variations cause problems such that the operation schedules and/or the batch sizes of other batch units have to be readjusted. From the practical viewpoint, however, such readjustments are unfavorable because usually operators are forced to do some excessive work which sometimes leads to dangerous mis-operations. In some batch units, it might be impossible to amend the preassigned schedule, i.e., it is absolutely necessary to strictly operate those batch units in accordance with the fixed schedule. In designing a batch process in which the variations mentioned above might easily occur, it is indispensable to develop some countermeasures so as to decrease the unfavorable effect caused by the Variations. As a promising countermeasure to avoid the propagation of such unfavorable influences on the other batch stages, it is most effective to install an intermediate storage tank between batch stages. By installing a storage tank between two batch stages, the outlet flow from the precedent stage is stored in the tank and does not immediately become the inlet flow to the batch stage following the tank. Therefore, the storage tank installed between two batch stages can absorb any influence due to the variation in the operation which occurred in the previous batch stage and prevent the propagation of the influence on the batch stage following the tank. In order to determine the rational values of the storage tanks and to analyze the role of the tanks in assuring the smooth operation of the whole process, two different ap-
proaches have been proposed so far. One is the “stochastic approach” and the other is the ”deterministic approach”. By taking a stochastic approach, Smith and Rudd (1964) showed that the variation in processing times affects the performance of a batch process. They solved simple problems by using queuing theory. Simulation was then presented to solve the more complicated problems. Ross (1973) also studied the problem of how to determine the optimal tank volume so as to assure the smooth operation of the whole process by using the Monte Carlo simulation technique. Oi et al. (1979) took a deterministic approach and analyzed the flexibility of a process consisting of parallel batch units, a storage tank, and a continuous section. In this study, the allowable range of the fluctuations of processing times of batch units was derived so that the operation of the continuous section following the tank was not affected by the variation in the operation of any batch unit before the tank, insofar as these variations remain in the allowable range obtained. In our previous work (Takamatsu et al., 1982), the optimal design procedure was developed for a batch process consisting of many batch stages and intermediate storage tanks without taking into account the uncertainties in each operation. In this paper, we deal with the problem of how to design a flexible batch process in which the storage tanks never overflow nor run out of stored material, and moreover the preassigned regular operations are kept in other batch stages, even though the batch size and the processing and cleaning times of a batch stage vary due to various uncertain causes and/or mis-operations. Description of the Problem A general single product batch process consists of many batch stages and intermediate storage tanks as shown in Figure 1. In this paper, the problem of how to design a flexible batch process which can absorb unfavorable effects due to some uncertainties is considered. In order to clarify the problem, the following assumptions are first introduced: (i) The capacities of the feed and discharge pumps of the batch stages are known. (ii) Each batch stage consists of one or more identical items of batch equipment in parallel. Every batch item in a batch stage is periodically operated with the same cycle time by delaying its starting moment at equal intervals. Figure 2 shows an illustrative example of the operation schedule of a batch stage consisting of two batch items in parallel. As shown in Figure 2, we define the following terminologies: For each batch item in batch stage i, the time it takes to pass through all of the steps such as the
0196-4305/84/1123-0040$01.50/0 0 1983 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 1, 1984 41 feed ...
stage i batch stage i.1
Figure 1. Schematic diagram of a general batch process.
.1
4
i
product cycle time of the -r subprocess
Figure 3. A subprocess consisting of two batch stages.
the starting moment of the inflow from the last batch unit in a subprocess to a tank (i.e., the starting moment of the inflow to a tank) only are considered in this paper. The pbatch-item 1p . L ranges of delay and advance in the starting moments of batch item 2 -..the inlet flow to and the outlet flow from each storage tank 7 p from the original scheduling time are moreover assumed +stage cycle time(N/~,)-* H filling to be given a priori. H processing c3 cleaning ,minimal q r l e lime of When a variation in the starting moment of the inflow the batch item (w,) --? W dtxhorging i waiting cycle time of the to the tank occurs in a certain batch, the process is prec, batch item (W,)sumed to be operated according to the operation schedule Figure 2. Operation schedule of batch stage i ( N , = 2). as shown in the following. In every periodical operation after the variation has occurred, the starting moment of filling, processing, etc., is called “the cycle time of the batch the inflow to the tank is shifted from the originally item, WT. The time required for a batch of material to scheduled starting moment by the size of the variation. On pass completely through the steps excluding the waiting the other hand, the starting moment of the outlet flow step is called “the minimal cycle time of the batch item, from the tank is not changed but is kept exactly the same win. For batch stage i with Ni batch items in parallel, the as that of the original schedule. The variation in the time interval Wi/Niis called “the stage cycle time”. The starting moment of the inflow to the tank is defined as amount of product produced in this time interval is called “allowable” if and only if the tank does not overflow nor “the batch size of the batch stage, Si”. The “processing run out of stored material even though the whole process capacity of the batch item, ci,” in batch stage i is defined is periodically operated according to the readjusted by schedule mentioned above. As for the variation in the starting moment of the outlet ci = S i / W i (1) flow from the tank, the “allowable variation” is similarly defiied as the variation in the starting moment of the inlet (iii) The minimal cycle time of the batch item, wi, is a flow to the tank. (vii) In regard to the variations in batch function of the batch size of the batch stage, Si. The processing capacity of the batch item, ci, is a monotonically sizes, the upper limit of these variations is known a priori. Moreover, the upper and low bounds of the total sum of increasing function of the batch size of the batch stage, the variations in the batch size of a batch that is flowed Si.(iv) The process does not cause any volume changes into and discharged from a subprocess over the whole in the flow, and the equipment size of the batch item and production period are all known. the batch size of the batch stage are measured by the same unit. (v) Possible places where an intermediate storage When the variation in the batch size of a batch of material that is flowed into the tank occurs, the process is tank can be installed are known. A series of batch stages presumed to be operated according to the preassigned between two consecutive tanks is called “a subprocess”. schedule as shown in the following: In every periodical If two batch stages are directly connected in a series, the operation after the variation in the batch size has occurred, outlet flow from the former batch stage directly becomes the batch sizes of a batch that is flowed into and from the the inlet flow to the latter batch stage. Therefore, the tank are identical with the original batch sizes before the batch sizes and cycle times of both batch stages have to variation occurred. The starting moments in the inflow be identical. The batch size and cycle time are called “the to and outflow from the tank are not changed and follow batch size of its subprocess” and “the cycle time of its the preassigned schedule, irrespective of whether a variasubprocess”, respectively. Figure 3 shows an operation tion in the batch size has occurred or not. The variation schedule of a subprocess which consists of two batch stages. in the batch size of a batch that is flowed into the tank In the process shown in Figure 1, many different kinds is defined as “allowable” if and only if the tank does not of variations may occur in its real operation due to various overflow nor run out of stored material even though the uncertainties. However, here we consider only the two whole process is operated according to the schedule exmost essential, different kinds of variations, that is, the plained above. variation in the operation schedule and that in the batch size. As for the variation of the batch size in a batch of material discharged from the tank, the “allowable variation” In order to make the discussion clearer, the following is defined in a way similar to the case of the variation in assumptions are further introduced: (vi) Even though the batch size of a batch flowing into the tank. variations in the duration times of the operation steps such The flexibility of a batch process may be evaluated by as the filling, processing, discharging and cleaning of a many different kinds of measures. In this paper, it is certain batch unit in a subprocess occur due to some unassumed that the flexibility of a batch process could be certain causes, the operation of the subprocess can be measured by the size of the regions of “allowable continued by readjusting the operation schedules of the variations” in the operation schedule and the batch size. other batch units contained in the same subprocess. The probelm of the design of a flexible batch process Therefore, the variations in the starting moment of the is essentially multi-objective. It is necessary to simultaoutflow from a tank to the first batch unit in a subprocess neously solve two optimization problems which are mu(i.e., starting moment of the outflow from a tank) and in stage i
b
42
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No 1, 1984
4Subprocess 1
Figure 4. A process consising of two subprocesses.
tually contradictory: one is how to minimize the investment cost of the process, and the other is how to maximize the flexibility of the process. These two objectives related to completely different attributes, flexibility and cost, cannot be evaluated by the same parameters. Consequently, this design problem has to be handled either as an optimization problem of how to increase the flexibility of the process within a given available budget for construction, or as that of how to decrease the construction cost of the process which has a given flexibility. In this paper, the problem of how to design a batch process which has a certain desired flexibility so as to minimize the required construction cost is dealt with. In other words, the problem is how to minimize the construction cost of a batch process when the ranges of the allowable variations with respect to the operation schedule and/or the batch sizes are given a priori. A Process Consisting of Two Subprocesses and a Tank In this section, a process consisting of two subprocesses and a tank as shown in Figure 4 is taken up, and it is assumed that both subprocesses are operated with fixed cycle time for a long period. Under the above assumption, the relationships between variables, such as the number of parallel batch items and the batch size in each batch stage, the volume of the storage tank, etc., are shown. The batch size of every batch stage in a subprocessmust be equal to the batch size of the subprocess: i.e. (i E GI;j = 1, 2, ...,K ) (2) S, = SI where SI is the batch size of subprocess j , GI is a set of suffixes representing the order of batch stages in subprocess j , and K is the number of subprocesses. In order to satisfy the given production requirement per unit time, P, the processing capacity of each batch item, c,, and the number of batch item, N,, in each batch stage i must satisfy the following relationship (i = 1, 2 , ..., B ) P I N,.c,(S,) = N,.S,/w,(S,) where B is the number of batch stages and w,is the minimal cycle time of the batch item in batch stage i. By using the above relationship, the minimum number of parallel batch items in each batch stage is given as a function of the batch size of the batch stage: i.e. Nz = llP~*(SL)/SLIl (3) where llxll is the minimum 2 x . Batch sizes and cycle times of both subprocesses must be determined so as to satisfy the material balance of the whole process: i.e. (4) P = &/W1= S 2 / W 2 where W,is the cycle time of subprocess j . The input function to and the output function from the tank, u1 and u2,are defined as u,(t) = U,: iW,It IiW,+ i?,/U, (5) = 0: otherwise
0'
u(t) = VO +
Subprocess 2
= 1, 2; i = 0, 1, 2, ...)
where U, is the capacities of the feed 0' = 1) and the discharge 0' = 2) pumps of the tank per unit time. Then, the holdup of the tank, u ( t ) , is given by
1'[.1(7 0
- tl)-
U2(7
- t 2 ) ]d7
(6)
where VO is the initial inventory in the storage tank, tl is the starting moment of the inflow from the subprocess 1 to the tank in the first cycle, and t2is the starting moment of the discharge from the tank to subprocess 2 in the first cycle. The tank does not overflow nor run out of stored material if and only if the holdup of the tank satisfies the following inequalities for any time t 0 5 u(t) I v (7) where V is the volume of the storage tank. In our previous paper (Takamatsu et al., 1982), the relationship which v, tl, and t2must satisfy under the condition that U1 = U2 and V' = 0 hold was drived. The relationship between V, tl, and t 2without presuming the condition mentioned above is derived as follows. Theorem 1. It is presumed that the batch sizes and cycle times of both batch stages are determined so as to satisfy the given production requirement, and that the initial inventory in a storage tank is given a priori. Then both subprocesses can be operated in a steady cyclic condition without causing either the overflow or the exhaustion of stored material in the tank if and only if the capacity of the storage tank, V, and the starting moment of the inflow to and the starting moment of the discharge from the tank in the first cycle, tl and t 2 satisfy the following inequalities (1 - P/U2)S2 - VO - (1- b)(l - h).G.C.M.(Sl,S,) I ( t 2- tl)P (sa) 5 V - VO - (1 - P/Ul)S,
+ (1 - b ) ( l - r).G.C.M.(S,,S,) (8b)
where b = P/min(U1, U2)
(84
h = mod[V'/G.C.M.(S1,S2), 11
(8d)
r = mod[(V- VO)/G.C.M.(&,S,), 11
(8e)
G.C.M.(X,Y) is the extended greatest common measure of X and Y (= the maximum rational number which is a common aliquot part of the rational numbers X and Y. For example, G.C.M.(48,36) = 1 2 and G.C.M.(4.8,3.6) = 1.2; see Takamatsu et al. (1982)); mod(X,Y) = X trunc(X/Y).Y and trunc(X) = the largest integer IX. The proof of Theorem 1 is shown in Appendix I. Minimum Capacity of the Storage Tank In this section, a process consisting of two subprocesses and a tank as shown in Figure 4 is considered again, and the minimum capacity of the storage tank is derived such that the tank does not overflow nor run out of stored material even though the variations of the operation schedule and/or the batch size of a subprocess may occur. We number the variations which occurred in the process according to their occurrence times. Here, occurrence times are defined as follows. For the delay in the starting moments of the inflow to and the outflow from the tank, it is defined as the time when the inflow to or the outflow from the tank is scheduled. For the advance in the starting moments of the inflow to and the outflow from the tank, it is defined as the time when the inflow or the outflow actually starts, respectively. The occurrence time of the variation in the batch size is defined as the time when the inflow or the outflow of a batch of material with varied batch size to or from the tank begins.
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 1, 1984
By using four kinds of variables, the ith variation which occurred in the process can be expressed in terms of an array such as (At', AT', Ad, AS'), where Ati, AT' are the amounts of variations in the starting moments of the inflow to and the outflow from the tank for the ith variation, respectively, and As', AS' are the amounts of variations in the batch sizes of a batch which was flowed into and was discharged from the tank for the ith variation, respectively. The size of the ith variation in the starting moment is measured by the magnitude of the delay or advance from the time schedule operated after the (i - 1)th variation occurred; i.e., the starting moments of the inflow to and the outflow from the tank after the variation has occurred are shifted by
J t u 2 ( -~ t2)d7
( t , It < t, + At') (lob)
where t , is the time when the first variation takes place. For any t , ul(t) and u z ( t ) are functions which take nonnegative values. Therefore, u ( t ) defined by eq 10 satisfies the following relationship for any t
2 Atj and 2 ATj
j=1
43
At') - U ~ ( T- tz)] d7
j=l
from the original schedule in which no variations are assumed to occur. Here, the positive value is taken for the delay, and the negative one is taken for the advance in starting moment. The variations in the batch sizes are measured as deviations from the preassigned batch sizes of both subprocesses. The positive value is taken for the increase and the negative one is taken for the decrease in the batch size. Here it is assumed that each variation consists of only one kind of variation. For every i, an array for ith variation (At', AT', Asi, ASi) has only one nonzero element. For example, if the ith variation which occurred in the process is the delay of the starting moment of the inflow to the tank, At' takes a positive value, and all of the other variables, such as AT', As', and AS' are equal to zero, i.e., ith variation can be expressed by the array (At', 0, 0, 0). If two kinds of variations occur at one time, for example, variations in batch size and in the starting moment of the inlet flow into the tank occur simultaneously, these variations are handled as if two variations expressed by the arrays (At', 0, 0, 0) and ( 0, 0, As'", 0), have occurred. From the assumption which was introduced. in the previous section, the sum of amounts of each kind of variation satisfies the following constraints
The following result can be derived from the above inequalities. When u ( t ) defined by eq 10a satisfies eq 7 for any t , the variation At' is allowable if and only if u ( t ) defined by eq 1Oc satisfies eq 7 for any t. This result was derived by Oi et a!. (1979) for the process consisting of parallel batch units, a storage tank, and a continuous section. By substituting tl for tl + At' in eq lOc, eq 1Oc is equal to eq 6. Therefore, in order to continue the operation of the process without overflow or exhaustion of stored material irrespective of the variation At', variables tl, tZ,and V must be chosen in such a way that eq 8 is satisfied even though tl At' is substituted for tl in eq 8. Even for the other variations related to an operation schedule, such as the advance of the starting moment of the flow into the tank, and the delay and/or advance in the starting moment of the outlet flow from the tank, we can derive the same kind of result as shown here. Next, we consider the variation of the batch size of a subprocess. If the batch size of some batch that is flowed into the tank increases by As' at t = t,, the holdup of the tank changes with time as follows
+
u(t) =
k
AtL ICati IAtU
(k = 1, 2, ...)
(94
(k = 1, 2, ...)
(9b)
i=l
k
ATL ICAP IATU i=l
k
AsL ICAS' IAsu
(k = 1, 2, ...)
i=l
(9c)
k
ASL IC A S ' IASu i=l
(k = 1, 2, ...)
(9d)
These lower and upper bounds have to be determined by taking into account many kinds of factors, such as the frequency and the magnitude of the variations and/or the characteristics of the operation of the process, such as a property in which its operation schedule is easy or not to rearrange. We first consider the condition that the variables tl, t z , and V must satisfy so that the tank does not overflow nor run out of stored material even though a delay of the starting moment of the inflow to the tank occurs. In this case, the holdup of the tank changes with time as follows u(t) =
u(t) =
V" + AS1 +
Jt[U1(7
- t l ) - u2(7 - t 2 ) ] d7
(tb
5 t)
(1lc) where tb = t , + (3, + As')/U,. For any t, u ( t ) determined by eq 11satisfies the following relationship
V" +
Jt[u1(7
- tl) - u2(7 -
As'
t2)]d7 Iu ( t ) IV"
+
+ J t [ u l ( 7 - tl) - u2(7 - tz)] d r
Therefore, if u ( t ) given by eq l l a satisfies eq 7 for any t , the variation in the batch sue of a batch that is flowed into the tank, As', is allowable if and only if u ( t ) determined by eq l l c satisfies eq 7 for any t. By substituting VO for V" + As1 in eq llc, eq l l c is equal to eq 6. Therefore, As' is allowable if and only if eq 8 is satisfied even though P + As' is substituted for V" in eq 8.
44
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 1, 1984
Even for the other variations, such as the decrease in the batch size of a batch of material that is flowed into the tank, and the increase and/or decrease of the batch size of a batch discharged from the tank, we can derive the same kind of result by repeating a similar discussion. The holdup of the tank after the nth variation is over, un(t), is given by n
un(t) = VO
+ r =AS' l
-
AS') -
L t [ u l ( -~ tl - ?Atr) - uZ(7- tz - ? A T ) ] dT (12) r=l
1=1
n
n
n
i=1
1=1
r=l
+ C(As' - AS'); tl + Eat'; and t2 + EAT'
in eq 12, respectively, eq 12 is equal to eq 6. Consequently, we can derive the following theorem from the above discussion and Theorem 1. Theorem 2. It is assumed that the ( n - 1)th variation is allowable. Then the nth variation (At", AT', As", ASn) is allowable if and only if the following relationship is satisfied (1- P/u,)S, - (1- b)(i - ~?.G.c.M.(S,,S,) - vo CSn I(t2- tl + C F ) P (13a) 5
v - vo - E s ~- (1 - P / U , ) S , + (1 - b ) ( l -
r2. G.C.M.(S~,SJ (i3b)
where h' = mod[(VO
+ (R"+ r").G.C.M.(S1,Sz) + (As"
where
h" = mod[(VO
r=l
ET"
subject to Si =
E(AT - At') r=l
The left-hand side of eq 13a is a monotonically decreasing function with respect to ESn,and the right-hand side is a monotonically increasing function with respect to ET".Therefore, eq 13a is satisfied for any variations restricted by eq 9 if and only if eq 13a is satisfied for the variations such that ES" = AsL - AS" and ET' = ATL - AtU hold. Similarly, eq 13b is satisfied for any variations restricted by eq 9 if and only if eq 13b is satisfied for the variations such that E S n = Asu - ASL and ET" = ATu - AtL hold. By substituting the above values for CS" and ET' in eq 13 and by eliminating tz - tl in eq 13, the constraint that the capacity of the tank must satisfy is derived as follows (1- P/Ul)Sl + (1- P/U2)S2 - (1 - b)(2 - h'?. G.c.M.(~,,~,)- vo + (ASU - ASL) + tu + ATU A t L - ATL)P5 (R"+ br'?.G.C.M.(S1,Sz) (14) where h " = mod[(VO + AsL - AsU)/G.C.M.(&,&, 11
R" = trunc[(V - VO - Asu r" = mod[(V - V" - Asu
+ ASL)/G.C.M.(S1,Sz)I
+ ASL)/G.C.M.(S1,&,), I]
If the ranges of variations of operation schedules and batch sizes and the batch sizes themselves are given for both subprocesses, the value of the left-hand side of eq 14 can be calculated. Therefore, the minimum capacity of the tank is given as a minimum value of
B
K-1
i=l
j=1
P.I. = CN,.Pi(Ei) +
n
=
+ AsL - ASu)/G.C.M.(Sl,Sz), 11
The proof of Theorem 3 is shown in Appendix 11. Mathematical Formulation In previous sections, relationships among the variables in the process consisting of two subprocesses and a tank are clarified. By using the results obtained in previous sections, the optimal design problem for a general batch process as shown in Figure 1 can be mathematically formulated as follows. Find the optimal number of parallel batch items in each batch stage, Ni, the optimal equipment size of each batch and the optimal volumes of intermediate storage item, Ei, tanks, Vi, so as to minimize
r' = mod[(V - VO - ESn)/G.C.M.(Sl,SJ, 11 n
+
Q = [(l- P / U l ) S l (1 - P/U2)Sz - VO (AS" AsL) + (At" + A T u - AtL - ATL)P]/G.C.M.(S1,Sz) (1 - b)(2 - h'?
+ ESn)/G.C.M.(~,,Sz),11
CS" = AS' - AS')
- ASL)
which satisfies eq 14. Theorem 3. When the batch sizes and the cycle times of both subprocesses are determined so as to satisfy the given production requirement, and the ranges of variations are also given by eq 9, the minimum tank volume is given by the following equation V = (trunc(Q) + min[mod(Q,l)/b,l]~-G.C.M.(&,~~) + VO + (AS' - ASL) (15)
+
By substituting VO, tl, and t z for
V'
VO
c qj( Vj)
Sj = P-Wi (i E G j ;j = 1, 2, ..., K ) Ei= Si+ ASiM Ni = IIP*wi(Si)/SiII
(16)
(17) (18)
(3)
Vj = (trunc(Qj) + min[mod(Qj, l)/bj, 11)-
G.C.M.(S~,~;.+,) + vjo + ( A S ~ U - A S ~ L )
Qj
(19)
= [(l- P/Uj?Sj + (1- P/U,d)Sj+1 - Vi0 + (AS," AsjL) (Atj" + ATj" - At? - ATjL)P]/G.C.M.(Sj, S,,,) - (1 - bj)(2- hj") (20)
+
bj = P/min(Uj',U,")
(21)
hj" = mod[(Vjo + AsjL - AS,u)/G.C.M.(S,,SI+l), 11 (i = 1, 2, ...,B; j = 1, 2, ..., K - 1) (22) where pi, q j are monotonically increasing functions and ASiMis the maximum value of the variation in the batch size of batch stage i. Subscript j of the symbols Vj, V ,: U t , Ujd,Atju, AtjL, ATju, ATjL, Asj", As?, ASju, and ASP, means that these symbols are related to storage tank j (see Nomenclature). Solution Method From eq 3, 17, and 18, the number of parallel batch items and the equipment size of the batch item in each batch stage can be obtained as functions of the batch size of the subprocess. From eq 19, the volume of the tank can be given as a function of the batch sizes of subprocesses before and after the tank. The batch size of a subprocess is given as a function of the cycle time of the subprocess. Therefore, by assuming the cycle time of all of the subprocesses,all of the variables
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 1, 1984 45
the initial holdup in each storage tank VIo = 0 (m3)
ASIM = 0.05Si; AS,' = 0.2S1; AslL = -0.23,; Ask' = 0.2Sk+l; AskL = -0.23k+l; At,' = AT,' = 4 (h); AtlL = ATkL = -2 (h) (i = 1, 2, 3, 4; j = 1, 2, 3; k = 1, 2)
Figure 5. Relationships among the variables.
in the process and the performance index can be calculated. For the process shown in Figure 1, Figure 5 shows how all the variables are uniquely determined as functions of the cycle times of the subprocesses. In Figure 5 , each vertex corresponds to a variable or a set of variables. An arc leading from vertex A to vertex B means that the variable corresponding to vertex B is a function of the variables corresponding to vertex A. Therefore, the optimal solution of the problem formulated above can be obtained by letting the cycle times of all of the subprocesses be free variables and by performing the search in the feasible domain of these free variables. Since the number of parallel batch items and the volume of the storage tank determined from eq 3 and 19 are not continuous functions of batch sizes, the searching procedure has to be performed by applying a certain direct search method. So far, it has been assumed that the cycle time of each batch stage can be chosen arbitrarily. But in a real batch process there are many cases where the cycle time of each batch stage can take only discrete values. For example, assume that the process is on a 24-h job and the minimum cycle time of a subprocess is 23 h. In this case, we may take 24 h as the cycle time of the subprocess even though the process can by cyclically operated in the interval of 23 h. We next consider the case where the cycle time of each subprocess can take only discrete values. From Figure 5, it can easily be understood that the part of the performance index related to subprocess j is a function of the cycle time of subprocess j and that the other part related to storage tank j is a function of the cycle times of both subprocess j and j + 1. Therefore "Dynamic Programing" can be easily applied to obtain the optimal solution. Numerical Example In this example, the whole process is assumed to have the same structure as shown in Figure 1. It is here assumed that the item of equipment in stage 5 is operated continuously with a fixed processing rate so as to satisfy the production requirement. It is also assumed that there is a sufficient supply of raw material for batch stage 1 and that the cycle time of each subprocess can only take discrete values which are multiples of 2 h. The performance index is given by 3
r=l
]=1
P.I. = CN1*~,*E,O7 + C0.6V,0.7
= 1, 2,3)
the ranges of variations
P.I
4
0'
(23)
where a, = u4 = 1.5; a2 = 1.0; a3 = 2.0. Other data are given as follows: the minimal cycle time of a batch item in each stage W,(S,) = yr + 2.08, (i = 1, 2, 3, 4) where y1 = 50; yz = 20; y3 = 30; y4 = 8 (h); the capacity of each pump and production requirement ulf= Uld = 30; uZf= u2d= u3f= 20; u3d= p = 0.1 (m3/h)
A continuous unit can be regarded as a batch unit the cycle time of which is equal to the time required for filling the batch unit. Therefore, the capacity of the storage tank between batch stage 4 and the continuous section can be calculated by using eq 19. By calculating the performance index given by eq 23 for all of the combinations of the cycle time values of subprocess 1, 2, and 3, and by selecting the minimum value among them, the optimal solution can be obtained. However, this approach is inefficient and time-consuming, especially in high dimensional problems. In such a case, dynamic programming can play an essential role in reducing searching numbers of the cycle time values for finding the optimal solution. In order to formulate the problem, the following functions are first introduced
For each of the values of the cycle time of subprocess 2, the cycle timepf subprocess 1 is determined so as to minimize fl(Wl,W2). Then
becomes a function of the cycle time of subprocess 2. Next, for each value of the cycle time of subprocess 3, the optimal cycle time of subprocess 2 is searched so as to minimize
By calculating the value of
for all of the candidates of the cycle time values in subprocess 3, and by selecting the minimum value among them, the optimal solution can be obtained. That is, the minimum value of the performance index given by eq 23 can be obtained by calculating the formula
In this problem, the perforpance index takes the minimum value for Wl =28 (h), Wz = 18 (h) and W3= 10 (h). Then, the batch sizes of subprocesses and the-value of the performance index are calculated as follows: S1= 2.8 (m3); Sz = 1.8 (m3); S3= 1.0 (m3);P.I. = 21.49. The optimal values of other design variables are shown in Figure 6. In order to show how the solution changes depending upon whether variations are taken into account or not, the optimal solution for the case where no variations are assumed to occur in the process is also shown in Figure 7. The optimal solution for this case can be easily obtained by applying the exact same optimization procedure as that mentioned above, assuming that all of the upper and lower bounds on the variations are zero.
46
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 1, 1984
stase’
st3ge2
Ian*:
slcge 3
tank2
stage4
lank3
Figure 6. Equipment sizes of batch units and volumes of tanka.
slagel
stage2
tank1
stage 3
stage4
tank3
Figure 7. Result without any consideration on variations.
Figure 8. Range of allowable variations.
Needless to say, the value of the construction cost for the process shown in Figure 7 (P.I. = 18.78) is smaller than that of the optimal process shown in Figure 6 (P.I. = 21.49). However, the process shown in Figure 7 has no flexibility. On the other hand, the optimal process obtained by taking into account variations has a favorable flexibility; in other words, it can absorb the variations shown in the beginning of this section, though it requires slightly higher construction cost. Now consider the case where the design variables of the process are already given. It is easily understood from eq 13 that whether or not the overflow or exhaustion of stored material in tank 1will occur after the nth variation is over depends upon the values of t z - tl + ET” and CS”. Therefore, by graphically displaying the range of a pair of values of t , - tl +ET” and CS” which satisfy eq 13, whether the variations that occurred in the process are allowable or not, can easily be judged. When the process is designed as shown in Figure 6, the allowable range of variations is given in Figure 8. In that figure, the abscissa shows the value of t z - tl + ET”,and the ordinate shows the value of CSn. The rectangle described in broken lines shows the preassigned range of variations. The black circle O Lin Figure 8 corresponds to the ith variation given by the following (Atll, ATl’, Asll, ASI1) = (0, 0, 0.2 m3, 0) (At12, AT12, As,,, AS12) = (0, 3 h, 0, 0)
(At13, ATI3,As13, AS13) = (0, 0, 0, -0.2m3)
For the above variations, Figure 8 shows that tank 1 does not overflow nor run out of stored material even if these variations occur. Conclusion The problem of the design of a flexible batch process which consists of several subprocesses composed of a certain number of batch units and intermediate storage tanks has been dealt with. In order to consider the design problem of a flexible batch process, it is first necessary to clarify the following two points. One is the way of defining
“flexibility” itself and the kind of measure to be used to evaluate the magnitude of the flexibility. The other is determining what performance index has to be used in order to express the two kinds of objectives which are mutually cont,radictory; minimization of the construction cost and maximization of the flexibility of the process. As uncertain variations in the batch process, we considered the variations related to the operation schedule of a subprocess and the variations of its batch size. The “allowable variation” in a subprocess is defined as the one which does not cause the overflow nor the running out of the stored material in the tank, even though the operation schedules and the batch sizes of the other subprocesses are not readjusted. The larger the volume of the storage tank installed between subprocesses becomes, the larger the range of the allowable variation becomes. In this study, we derived the quantitative relationship between the size of variations in the operation schedule and in the batch sizes and the necessary volume of the intermediate storage tank so as to make those variations allowable ones (Theorem 3). By using this theorem, the volume of the tank which used to be estimated only through empirical knowledge could be determined more rationally. Presuming that the range of the allowable variations is given a priori, we mathematically formulated the problem of the design of a batch process which makes not only every variation within the given range “allowable” but also minimizes the construction cost. We then proposed an effective solution method. In order to search for the optimal solution of this design problem, we cannot help but depend upon direct searching with respect to the free variables the dimension of which is equal to the number of subprocesses. When the cycle time or the batch size of each subprocess takes only discrete values, the optimal solution of the problem can be easily obtained by applying a Dynamic Programming Approach. Here, we took up a batch process consisting of a certain number of batch stages and storage tanks and derived several interesting results. The results obtained can be applicable to a more general process which consists of batch stages, continuous sections, and storage tanks, as demonstrated by using an example. An approach to the problem of the design of a flexible batch process was proposed. The definition of flexibility introduced in this paper only represents an aspect of the manifold attributes with which the so-called “flexible process” has to be essentially endowed. In this paper, it is assumed that only “hard countermeasures are available in order to increase the flexibility of the process. In other words, the problem of how to rationally estimate the design margin of the design variables is considered so as to absorb every unfavorable effect due to various uncertain variations. Needless to say, “soft” countermeasures such as readjusting the operation schedule or installing sophisticated control systems are also very effective in enhancing the flexibility of the process. From now on, much effort has to be devoted to developing effective design methods which enable the design of flexible processes which are fully supported by both countermeasures.
Appendix I (Proof of Theorem 1) Case I U1 L U,. In this case, the holdup of !he tank, u ( t ) , takes on local minimum values at t = tl + iW1, where i = any nonnegative integer. Therefore u ( t ) 2 0 holds for any t if and only if the following inequality is satisfied for any nonnegative integer i.
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 1, 1984 47
u(tl
+ iWl) =
V" +
isl- 1
tl+iWl
0
u2(r- t2)d r I 0
(i = 0, 1, 2,
...)
(-41) The time when the depletion of the outlet flow from the tank, i.e., S&uZ(r- t2) d r exceeds V" + is1 is given by t 2 + trunc[(V" + iSl)/S2]-W2+mod(Vo + iSl,S2)/U2 Therefore eq A1 is satisfied if and only if the following inequality holds tl
+ iWl It2 + trunc[(V" + iS1)/S2]-W2 + mod(Vo + iS1,S2)/U2 (i = 1, 2, ...)
(A2)
Let L, M, H, and R be integers which satisfy the following equations L = &/G.C.M.(Sl,S2) = Wl/G.C.M.(fil,W2) (A3)
M = S2/G.C.M.(Sl,S2) = W2/G.C.M.(@l,W2)
H = trunc[ VO/G.C.M.(Sl,S2)] R = trunc[(V- V")/G.C.M.(&,S,)]
(A4) (A5) (A6)
By using eq A3, A4, A5, and 8d, eq A2 is rewritten as t2- tl 1 {iL - trunc[(H h iL)/Ml.M).Wl/L mod(H + h + iL,M).G.C.M.(Sl,S2)/U2 (A7)
+ +
Since both H + iL and M are integers, we obtain (AS) trunc[(H h iL)/Ml = trunc[(H iL)/Ml
+ + + mod(H + h + iL,M) = mod(H + iL,M) + h
(A9)
From eq A7 to A9, the following equation is derived (t2- t 1 ) / ~ . c . ~ . ( W l , WI2 ) mod(H + iL,M)(l- P / U 2 )- H - hP/U2 (A10) Equation A10 is satisfied for any integer i if and only if eq A10 is satisfied for some i which maximizes the value of mod(H iL,M). In the previous paper (Takamatsu et al., 1982), it was shown that max [mod(H + iL,M)] = M - 1 (All)
+
i
From eq A10 and All, we can derive eq 8a. We next derive the condition in which the storage tank does not overflow. u j t ) takes on local maximum values at t = tl + (SI/Ul) + i Wl, where i = any nonnegative integer. Therefore V I v ( t ) holds for any t , if and only if the following inequality is satisfied for any nonnegative integer i v I u(tl + S , / U , + iWl) = V" (i 1)S1-
s,
+ +
t;+(S,/ Cl)+ig1
u2(r- t2)d r
(i = 0, 1, 2, ..) (A12)
Equation A12 is satisfied if and only if the time when the depletion of the outlet flow from the tank becomes V" + (i + l)S1- V, is earlier than or equal to tl + (Sl/Ul) + iWl, That is, the tank does not overflow if and only if the following inequality is satisfied for any nonnegative integer i tl + S1/ul+ iWlIt2 + trunc{[V" + (i 1)S1- v]/S,).W~+ mod[V" + (i + 113, - V,S2]/U2- 6i (i = 0, 1, 2, ...I 6413) where 6i = W2- S 2 / U 2 ;mod[V" + (i + l)Sl - V,S2] = 0
+
= 0; otherwise
From eq A3 to A6, eq A13 is rewritten as t2 - tl I iWl + Sl/Ul - trunc{[(i l)L - R - r]/M). W 2- mod[(i + l)L - R- r,Ml.G.C.M.(S1,S2)/U2 ai (A141 If r which is determined by eq 8e is not equal to zero, eq A14 is rewritten as
+
+
(tz - ~I)/G.C.M.(W~,W~) I mod[(i + l)L - R - 1,MI x (1- P/U2) - L + R + 1 - (1 - r)P/Uz+ LP/U1 (A15) From the fact that G.C.M.(L,M) = 1 min{mod[(i + l)L - R - 1,W) = 0 1
Then, it can be concluded that eq A15 is satisfied for any integer i if and only if the following equation is satisfied (tz - tl)/G.C.M.(Wl,W2) IR - L + LP/U1 rP/U2 + 1 - P / U 2 (A16)
+
If r = 0, eq A14 can be rewritten as ( t 2 - tl)/G.C.M.(W1,W2) I R - L LP/Ul 6i/G.C.M.(@l,18'2) mod[(i l)L - R,Ml(l - P / U 2 ) (-417) For any integer i
+
+
+
+
~JG.C.M.(W~,WJ+
mod[(i + l ) L - R,Ml(l- P / U 2 ) I 1 - P / U 2
This inequality shows that eq A17 holds for any integer i if eq A16 holds. From eq A16, eq 8b can be derived. Case 11. U1< U p By using steps similar to those which were applied for Case I, we can derive eq 8a and 8b. (Q.E.D.) Appendix 11. (Proof of Theorem 3) The minimum value of the necessary tank volume can be obtained by minimizing R I' + r '' under the constraint of eq 14. Since br" < 1, R" must be equal to or greater than trunc(Q). If mod(Q,l) < b, trunc(Q) + mod(Q,l)/b is the minimum value of R" + r"which satisfies eq 14. If mod(Q,l) I b, R" must be greater than trunc(Q). Therefore, trunc(Q) + 1 is the minimum value of R"+ r" for this case. Consequently we obtain eq 15. (Q.E.D.) Nomenclature B = number of batch stages b = P/min(U,,Uz) c, = processing capacity of the batch item in batch stage i Ei = equipment size of a batch item in batch stage i Gj = a set of suffixes representing the order of batch stage in subprocess j K = number of subprocesses Ni= number of parallel batch items in batch stage i P = production requirement per unit time pi = cost function of a batch equipment in batch stage i q ' = cost function of intermediate storage tank j = batch size of batch stage i Si= batch size of subprocess i ASL, AS",(AS'?,ASju) = lower and upper bounds of the s u m s of amounts of variations in the batch size of a batch dischar ed from the tank (tank j ) AsL,A 8 , (AsjL, AsjU) = lower and upper bounds of the sums of amounts of variations in the batch size of a batch which is flowed into the tank (tank j ) ASiM = maximum value of the variation of the batch size of batch stage i As,,A S L= amounts of variations in the batch sizes of a batch which is flowed into and discharged from the tank for the ith variation t , , t z = starting times of the inflow to and the discharge from the tank in the first cycle
di
48
Ind. Eng. Chem. Process Des. Dev. 1984, 23, 48-52
A P , A P , (AT:, AT,") = lower and upper bounds of the sums
of amounts of variations in the starting moment of the discharge from the tank (tank j ) AtL, AtU, (At,L, At,u) = lower and upper bounds of the sums of amounts of variations in the starting moment of the inflow to the tank (tank j ) At', A T = amounts of variations in the starting moment of the inflow to and the discharge from the tank for the ith variation Ul, (v,q = capacity of the feed pump of the tank (tank j ) per unit time U2,(U,d) = capacity of the discharge pump of the tank (tank j ) per unit time u l , u2 = input function to and output function from the tank V, ( V ) = volume of the intermediate storage tank (tank j ) p, (+,O) = initial holdup of the tank (tank j ) W , = cycle time of the batch item in batch stage i
wj = cycle time of subprocess j wi = minimum cycle time of the batch item in batch stage i G.C.M.(X,Y) = extended greatest common measure of X and
Y trunc(X) = largest integer 5 X mod(X,Y) = X - trunc(X/Y).Y llXll = minimum integer t X
Literature Cited 01, K.; Itoh, H.; Muchi, I. Comput. Chem. Eng. 1979, 3 , 177. Ross, R. C . Hydrocarbon Process. 1973, 52, 7 5 . Smith, N. H.; Rudd, D. F. Chem. Eng. Sci. 1964, 79, 403. Takamatsu, T.; Hashimoto, I.; Hasebe, S. Ind. Eng. Chem. Process D e s . Dew. 1982, 21, 431.
Received for review July 22, 1982 Revised manuscript received February 17, 1983 Accepted March 18, 1983
Physical and Thermodynamic Properties of Dichlorosilane Jeng-Shla Cheng, Carl L. Yaws,* Larry L. Dlckens, and Jack R. Hopper Department of Chemical Engineering, Lamar University, Beaumont, Texas 777 10
George Hsu and Ralph Lutwack Jet Propulsion Laboratory, Pasadena, Californie 9 1 109
Dichlorosilane has major use in the semiconductor industry for the epitaxial growth of high-grade silicon. I t also is involved in a new process in the production of solar-cell grade silicon. Pertinent physical and thermodynamic properties, i.e., critical properties, vapor pressure, heat of vaporization, heat capacity, density, Viscosity, surface tension, and thermal conductivity, are presented in concise graphical form with texts for rapid use.
Introduction Dichlorosilane is a chemical material encountered in the semiconductor industry. It is involved in a new process in the manufacture of solar-cell grade silicon (Yaws et al., 1977). Physical and thermodynamic property data for dichlorosilaneare most beneficial tQ the design and operation of process equipment as is used in the electronics industry. The physical properties such as vapor pressure, critical properties, latent heat of vaporization, heat capacity, density, viscosity, etc., have been compiled or calculated in this article for rapid use in research, development, and manufacture. Critical Properties Values of critical temperature, T,, critical pressure P,, and critical volume, V,, for dichlorosilane (Table I) were calculated by Lydersen's structural contribution method with derived critical property increments for silicon (Reid et al., 1977). This method produced only 2.3% error for T,and 3.4% error for V, when compared with the experimental values of trichlorosilane and it produced 0% error for T,, V,, and P, when compared with the known values of silicon tetrachloride. The calculated values for the critical properties are also within reasonable agreement (4% for T,, 0.2% for P,, and 14% for V,) of calculated Russian values (Lapidus et al., 1970). From these values, z, was derived from its definition, its value being the same as that derived by the Garcia-Barcena boiling point method (Reid et al., 1977). 0196-4305/84/1123-0048$01.50/0
Table I. Critical Constants and Physical Properties of Dichlorosilane identification formula state (std. cond.) molecular weight boiling point, Tb,"C melting point, T,, "C critical temperature, T,, "C critical pressure, P,, atm critical volume, V,, cm3/g-mol critical compressibilityfactor, 2, critical density, p , , g/cm3 acentric factor (a)
dichlorosilane
SiH,Cl, gas 101.008 8.3 -122.0 178.9 44.9' 228.3' 0.276' 0.4424= 0.1107
' Estimated. Vapor Pressure The vapor pressure of dichlorosilane (Figure 1)has been determined from -80 to 30 "C (Stock and Somieski, 1919; Bailar et al., 1973). The experimental data were extended over the entire liquid range by use of the YSSP vapor pressure correlation (Yaws, 1977) B log P, = A + - + C log T + DT + ET2 (1)
T
where P, is the vapor pressure of saturated liquid in mm Hg, T is temperature in K, and A, B , C, D, and E are correlation constants derived with a generalized leastsquares computer program. Average absolute deviation was about 1%for the experimental data points. 0 1983 American Chemical Society
