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Sizing of intermediate storage for variabilities in noncontinuous processes with parallel units
Comput. them. Engng, Vol. 12, No. 6, pp. 561-572, 1988 Printed in Great Britain. All rights nserved
0098~1354/88 s3.00 + 0.00 Copyright 0 1988 Pergamon Press plc
SIZING OF INTERMEDIATE STORAGE FOR VARIABILITIES IN NONCONTINUOUS PROCESSES WITH PARALLEL UNITS T. 0. ODI and I. A. Department of Chemical
Engineering,
Northwestern
KARIMI~
University,
Evanston,
IL 60208, U.S.A.
(Received 9 February 1987; final revision received 22 October 1987; received for publication 3 November 1987) Abstract-Noncontinuous processes involving batch/semicontinuous operations are frequently subject to variations in processing parameters. As a counter-measure, intermediate storage is usually installed in such processes to eliminate the effects of these variations. Patterns of parameter variations commonly arising in processes involving stages of nonidentical, parallel units are analyzed. Conditions which guarantee that continuity of operation is maintained in the presence of the variations are derived. These conditions are used to develop storage sizing expressions for accommodating the worst case variations with a given degree of confidence. The sizing expressions are analytical and are in terms of parameters which can be easily estimated. An example of a noncontinuous process is used to illustrate the sizing procedure.
INTRODUCTION
Noncontinuous processes are usually labor intensive and involve unsteady-state operations. They are particularly suitable for complex operations such as handling of solid materials, etc. The complexity of these operations, coupled with their unsteady-state nature, render the noncontinuous processes highly susceptible to fluctuations in key processing parameters such as transfer flowrates, batch sizes or batch start times. The effect of these variations is to introduce delays of operation, as stages either wait or are blocked due to the tardiness or busyness of other stages. Often variations originating at one point in a chain of stages will propagate along the chain perhaps growing in magnitude or causing similar or different additional variations. An effective way of mitigating the unfavorable effects of these variations is to install intermediate storage between adjacent stages in the system. Due to the increasing importance of noncontinuous processes in recent years and the role of intermediate storage in enhancing their flexibility, several works (Henley and Hoshino, 1977; Karimi and Reklaitis, 1985a,b,c; Ross, 1973; Takamatsu et al., 1984) have studied the design and operation of intermediate storage in such processes. A survey of different approaches has been reported by Karimi (1984). The most common approach, called the deterministic approach, has been used to analyze different patterns of variations to arrive at conditions which ensure the continuity of operation and then estimate the required size of intermediate storage based on these conditions. Karimi and Reklaitis (1985~) used this approach to obtain impressive sizing results for serial processes with a single unit in each stage. t Author
to whom all correspondence
should be addressed. 561
Takamatsu et al. (1984) analyzed a special case of processes with identical units in a stage, in which the batch start time of a unit in a stage is delayed by a fixed time interval relative to the earlier unit. The aim of this paper is to extend the deterministic approach to size intermediate storage in general noncontinuous processes employing nonidentical, parallel units in each stage. In contrast to the approach of Takamatsu et al. (1984), no special operating policy will be assumed. In this paper, we first present conditions which simultaneous variations in a system with nonidentical parallel units must satisfy to ensure continuous operation. These conditions are then combined with theoretical results for holdup to derive analytical sizing expressions to accommodate sets of arbitrary variations. The expressions are simple and of two types. One is for predicting the amount of initial inventory required to accommodate the variations and the other for estimating the appropriate size of storage. Each expression consists of two parts: one part depends on the nature of the variations while the other depends on the nominal values of the parameters of the process. The former part is evaluated from the stochastic distributions characterizing individual variations. PROCESS MODEL
A general noncontinuous process consists of mulparallel employing nonidentical, tiple stages batch/semicontinuous (B/SC) units. Intermediate storage tanks may be available between successive stages of operation. In this paper, we focus our attention on a building-block configuration of two stages of nonidentical, parallel units separated by a storage vessel; as any general process can he studied by considering pairs of successive stages of operation.
562
T. 0.
001
and I. A.
KARIMI
Fig. 1. Schematic diagram of a parallel process. This building block configuration will be referred to as the L-M system, as it employs L upstream and M downstream B/SC units as shown in Fig. 1. In this section, we first describe a fixed parameter model in terms of the nominal or average values of different process parameters. Then we indicate the type of variations which can arise in tbese parameters and present a model with variabfe parameters. The motivation behind this is that later we will use the results for processes with fixed parameters in our analysis of the variable parameter processes, Fixed parameter model In this model, all parameters of the L-M process are fixed at their nominal or average values. We make the following assumptions regarding its operation. (1) a typical cycle of a batch unit consists of steps such as filling, processing, emptying and preparation and/or waiting and each subsequent cycle of the unit is a repetition of the above steps; (2) a semicontinuous (SC) unit processes material in batches of fixed size at a constant rate. After processing a batch of material, it remains idle for a fixed interval of time and then resumes operation in a periodic manner; (3) all units operate in a nonstop manner and both stages have equal nominal productivities, i.e.
where N is the total number of units, L + M. (4) the discharge rates of all upstream units and the feed rates of all the downstr~m units are constant and known. Let us first define the following key process parameters for unit i. Nominal cycle time wi. For a batch unit i, oi = T, + TB + T, + TP, where Tf, TB, T,, Tp denote its filling time, processing time, emptying time and
preparation and/or waiting time, respectively. For a SC unit, it is given by at, = T, + Td, where T, and T, are the processing and idle times of the unit, mspeetively. Nomin~i tr~s~e~~ractio~ n,. The average fraction of the cycle time of unit i, during which transfer of materia1 occurs between the unit and the storage. For a batch unit: 1 B i c L, L+l<‘i
(la&)
For a SC unit, xi denotes the processing fraction and is given by: xi = T&o, UC) Nominal frowrate Ui. The average rate at which material is transferred between unit i and the storage. Nomi~ai batch size Vi. The average amount of material processed by unit i in one cycle. Clearly Vi is either U, T, or Vi T, or UiTs depending on the type of unit and its stage in the system. Hence: v, = l_J,x,cO,.
(2)
Nominal trarqfer Jlowrate fiction unit i, it is given by:
,F;ft). For each
ciui lo,=5 t <(I + X,)Wi,, (3) i 0 (t$:x&&
!I$ = g, u(t
- t,)k;:(t - Cd.
(4)
Sir&g of intemediate where u(t) is a unit step function. Note that the process described by equation (4) is completely periodic only for I 3 mfrx(t,,) and not for t 6 mp(tio). Integrating equation (4), we get,
u(z - t,)i;,(.r - ta) dt
c(z
f G my(tio),
(5a)
t 3 max0,,),
(56)
- rio) dr
where min (&,) = 0 and V(0) = V,,. For the fixed parameter system, Karimi and Reklaitis (1985a) analyzed equation (5b) and obtained analytical expressions for estimating storage size V* and initial inventory VO. Types of variations A detailed description of the types and sources of variations arising in non continuous processes was presented by Karimi and Reklaitis (1985b). Two types of variations were identified: elementary and composite variations. Elementary variations are simple variations in starting moments, transfer flowrates and transfer fractions. Composite variations are those that can be represented as combinations of elementary variations. A starting moment variatior or revision occurs when a unit does not start at its scheduled starting time; an advance if earlier and a delay if later. Let Ati denote the amount of delay (A?# 2 0) or advance (At! < 0) in the j th revision of unit i, its revised schedule of starting moment after n revisions is given by: b,=l,P+Iw,
l>n
(6)
where
th = tm+
5 At:.
(7)
/=I
A flowrate variation is said to occur in a cycle if the rate of transfer of material between a unit and the storage deviates from its nominal value. A transfer fraction variation occurs when the time required for transferring a batch of material between the storage and a unit deviates from its nominal value. An interval of a variation is defined as the interval during which it is actually occurring. We will denote the starting and ending moments of&h variations by aj and b,, respectively. A set of variations is termed as overlapping if it contains at least two variations whose intervals overlap. The variations in a set are conventionally sequenced chronologically according to the ending moments of intervals of variations (Karimi and Reklaitis, 1985b).
storage
263
ALLOWABILITY
OF VABIATKONS
In this section, we derive sufficient conditions that will make common sets of overlapping variations allowable in the L-M system. These conditions wilI be useful in the development of expressions for sizing the intermediate storage to accommodate variations later in this paper. Before analyzing the variations, we state two important definitions introduced by Oi et al. (1979). Continuubiliry. A holdup function, V(t), t 3 0, V(0) = VO, is said to be continuable if it satisfies 0 Q V(t) d V, where V* is the storage capacity. Allowability. A set of variations in process parameters is allowable if the holdup function associated with it is continuable. Let us begin with the fbllowing definitions:
H,(r) = holdup in the storage during the intervaf Q, d t g 6, associated with nth variation, Y” (t) = holdup in the storage after the completion of the nth variation (r > b.) assuming that nominal periodic operation continues with no further variations, V(n,t) = holdup in the storage for a set of n completed variations, k;,,k&Q*ka = numbers of starting moment revisions,. flowrate variations and transfer fraction variations for the unit i, respectively, k = total number of elementary variations, k = k,, + k,* f k13, sum of /ci, starting moment re6k,, =cumulative visions [See equation (711. Our analysis will be subject to the following assumptions: 1. The variations in process parameters are not permanent in nature and the process resumes nominal periodic operation -after their te~i~tion if no further variations occur. 2. The holdup Y(0, t) before the occurrence of any variation in the system is ,continuable [note that V(O,r) is given by equation (5a)]. 3. Each starting moment revision occurring in unit i, denoted by At;, n = 1,2,. . . , is limited to the range, IAt:] < (I - x,)w,. Assumption 3 is reasonable because the magnitude of the variations is expected to be small. Especially in the case of batch units with small transfer times, this range of values is sufficiently large. It is also valid for SC units having moderate values of processing fractions. We now state two important lemmas which will be used in our analysis. Lemma 1. The holdup function in the intermediate storage after k starting moment revisions is given by:
r I
vk(t) =
v, f i
~if7
i= I Jru,,
--t&,)d7
564
T. 0. ODI and I. A. KAMMI bl
a1
b
aLI
2
Fig. 2. Schematic diagram of two variations with overlap.
where, k,2 = k, = 0, Vi and kth variation occurs in unit j. For the proof of Lemma 1, see Odi (1986). Lemma 2. If the nth delay of starting moment for unit i, At;, occurring in the interval, a, < t d b, is such that 0 < Arl< (1 - x&q, then 4
Fi(7--zt,)d7=0 a,
0 < 1 < a2
V(2, t) = H*(t),
a2 < t < b2
I’(2, t) = v2(t),
t > b2
where Y2 (t) is given by Lemma 1 with k = 2. Clearly, V(2, t) is continuable for 0 < t < a,. Using Lemma 1, Hz(t), a2 < t
Lemma 2, Hz(t) = v*(t), a2 < t d b2. we conclude that V(2,r) is continuable if
C”(r) and v2(t) are both continuable. Therefore the two variations are allowable if the holdup functions after their terminations are continuable. One can show that this result is true for any overlap pattern of these variations or for that matter any two elementary variations. The simplest patterns of overlaps are those involving two elementary variations. We discuss a pattern of two overlapping delays only as patterns involving these variations are the most difficult to analyze. Transfer fraction and transfer flowrate variations are easily handled in a general manner and will be considered in the analysis of multiple variations. Multiple elementary variations Here we extend our analysis to sets of multiple elementary variations with overlaps. The analysis will be done by induction. We first assume that a set bf n arbitrary elementary variations ivith overlaps, Y,,, with a,,,, b,,,, m = 1,2,. , as the respective starting and ending moments of m th variation has already occurred in the L-44 system and the holdup V(~,I) associated with it is continuable. We also assume that ‘p, has been sequenced such that b, < b2 d . . . < b,, Next we form another set of variations, Y,,,, by superimposing on Y,, the (n + 1)st variation such We also assume that the latter is that b,
OdtGa,+l
v(n + 1, t) = H,, , (f) = W, +c.AU’ I Qz+‘ I (t
t) -a,+l).
a,+,
xjwj.
t ah+,
expressions, we above From the get, ‘ if c,ACJ~J~+‘>O and I’@, t) < H”+, (t) G t’ “+(t) v(n, t) B H,, , (1) > V”+ ’ (t) if cjAU;j*+ ’ d 0. Since I’(n,t) is already continuable, we deduce that H,, , (t) is continuable if V”+ ’ (t) is assumed to be continuable. Therefore, given that Yy, is allowable and the holdup function after the (n + 1)st variation is continuable, then Y,+, is allowable. The same
Sing of intermediate storage
sequenced as 1,2 , . . , n is ahowable if the holdup function after each of the subsets: 111. n], occurring alone in the sysfl,2),. ..,{1,2,..., tem is continuable.
result can be proved for the transfer fraction variation (Odi, 1986). Delay of starting moment. The analysis of this variation differs from the other two elementary variations in that it is difficult to study this variation in a general manner because the expression for H, + , (t ) depends on the overlap patterns of the variations in YJn+i* This is especially true when the overlaps involved are those of delays themselves. Therefore, we consider specific but commonly occurring patterns of overlaps and derive the condition for ~lowability. The earlier result for two delays is very significant because simultaneous overlap of many delays is not less likely. With these considerations in mind, let us assume that Y” is entirely made up of delays, as their overlaps present the major difficulty in the analysis. For the present discussion, let us also assume that a,du,+,db,fori== 1,2, . . n, which is equivalent to saying that the (n + I)st delay starts later than all the previous delays but overlaps with some say last k delays in Ym_A schematic diagram of this pattern of overlap is shown in Fig. 3. (Note that a,, i = 1,2,. . . n are not shown). First note that V(n, t) = V” (t) for t 3 u,,as shown in Appendix B. Now,
Dhxssion The importance of the results in Propositions 1 and 2 is that they relate the allowability of a set of variations to the holdup function after the termination of each variation in the set. EIence, we do not have to worry about the actual holdup function associated with each interval of variation. The holdup in an interval of variation is nonperiodic and thus difficult to analyze. In contrast, the holdup function after the termination of a variation is periodic and thus easier to analyze. Note that Proposi-. tions 1 and 2 are applicable to ‘arbitrary sets of variations. In view of these facts, the storage size to accommodate any set of variations can be obtained by analyzing the hotdup function after each variation in the set. Thus the results obtained from the preceding analysis gives a very convenient basis for sizing storage in the L-M system. It is noteworthy that the results ‘pertaining to multiple elementary variations with or without overlap are valid for multiple composite variations. This is because, as shown by Karhni and Reklaitis (1985b), the latter can be decomposed into the former. Thus, the holdup function after the te~ination of a set of arbitrary variations in an L-M system is given in the following proposition. Z+opodtion 3. For a set of arbitrary multiple variations which can be decomposed into k elementary variations, the holdup function after the kth elementary variation is given by:
, Fj(t - tin,,) dr. i %+I where ei, is the number of starting moment revisions that unit i has undergone. Since V(n, t) = V”(t), with the help of Lemma 1, we get, %+,(t)
= V@, t) -
I f&+,(t)=
VR(t)-
&(r -t,n,,)dT $
AV{
an+1*Ft ~fb,+,
i-1
Then from Lemma 2, H,+,(t) = I’“+* (8). Thus, given the allowability of Pm, !Y”+, is allowable if I’“+’ (t) is continuable. For a cast in which all the variations in Yy,, , have the same starting moment, the above analysis is still valid. Although general results could not be obtained analytically for arbitrary overlap patterns of delays, our simulations of various configurations and magnitudes of these variations indicate that the results for tbe special cases considered here should hold true for most other cases. The following proposition summarizes the results of our analysis. Proposition 2. For an L-M system, a set of n elementary variations with overlaps which can be
-
b,
.I...
b,
565
bn-k
.
.
.
.
bn-k+1
.
.
.
.
.
.
.
+
c,m,(k, - k,)min(xi,
“t
FAT --t,,,)dr s ‘Lib
1
,
(8)
where Al’!=
(U,+AU{)(x,+Ax’;)wj-
v;..
(9)
The convention under which the expression was derived is that both a flowrate variation and a transfer fraction variation are thought to occur in a cycle of a unit. If only one of them occurs, the other one is assumed to have a zero value. This convention in effect ensures that the resultant of the two variations in a cycle is always a batch size variation with
.
.
bn--l
4-z
Fig. 3. Schematic diagram of Yr”+ , . ,Z,bE
xii- A.+)AI@
,
“n +1
C.A.C.E.
1 j-i
c
4
bIt+1
566
T. 0. ODI and I. A. KANMI
constant cycle time as in the first term in the parenthesis of equation (8). An explanation of the rest of the terms in the parenthesis in equation (8) is available in Karimi and Reklaitis (1985b). The proof of equation (8) follows directly from the one given for the l-l system by Karimi and Reklaitis (1985b).
By choosing V * and V, to satisfy equation (13a,b) we can make V’(t) continuable.. Since the allowability condition requires continuability of all V’ (t), i Q k, V * and V, which can accommodate all k variations are given by the. following proposition. Proposition 4. V * and V, required to accommodate a set of k elementary variations are:
STORAGE SIZING FOR L-M SYSTEMS
V*= In this section, we develop analytical expressions for determining the size of storage for an L-M system undergoing a specified range of process parameter variations. We begin with the elementary variations and later proceed to the general case. Sizing under elementary variations The allowability conditions (Propositions 1 and 2) constitute the basis behind our subsequent development of expressions for sizing the storage to accommodate a set of arbitrary variations. We first consider cases in which only one type of elementary variations is occurring in the L-M system. Later, we generalize the results to include sets of arbitrary, random variations. Starting moment revisions. Suppose that the system is subject to only starting moment revisions and k revisions have occurred with Iq, for unit i. As k,I = !c,~= 0, the expression for V’ (t) in Proposition 3 reduces to:
P(t) =
V,+AV,,+
VO= -AV,+
2 Vj(l-xx,). i-1 5 Vi(l -xi), i-L+,
(14a,b)
where AV, and AV, satisfy AV, < AV < AV,. Later we will show how AV, and AV, can be estimated analytically from the statistical distributions describing the variations. For sets of only flowrate variations or only transfer fraction variations, we only need to increase V* and V, for the fixed parameter system (Karimi and Reklaitis, 1985a) by AV* and A VO respectively as follows: AV,=max
(0, -AV,)
and AV*=AV,+AV,,
(15a,b)
where AV,
v, + 1(t),
N
42
c
c
c,w,x,AV;dAV,
i-In-1
is for flowrate variation and
where
1(t)= 5 From equation
’ Fi(7 - ta,,) dr.
i-l I l&, (10) it follows that:
k,s
N
(11)
AV,
c c ciw,xiAV~
fraction
variations.
V* = max V’ (t) = VO+ V_ I and
SIZING UNDER GENERAL VARIATIONS
min V”(t) = V, + Vmin, I where V_ = max I(t) and Vtin = min I(t).
I
I
(12a,b)
Since negative holdup is not permitted, V, 2 - V,,. Using the upper and lower bounds on V,,,,, and V,, as derived in Appendix C, we obtain:
Sizing under arbitrary variations is the same as sizing under multiple but mixed elementary variations. The approach that we adopt is a direct extension of our treatment of multiple elementary variations of the same type. The general holdup function of Proposition 3 can can be rewritten as: V’(t)=
V,+AV,+
5 W i-I s ‘UiI
- te,,) dr,
where V*=
V,,+AV+
f’oa -AI’+
f V,(l -xi), ir I f irI_+
v,(l-xx,),
AV,=
Wa,b)
2 E c,AV; ,..I 1 n-1 + cioi(kn - kn) min (x,, xi
1
where (13c)
First, let us simplify the third term in V”(t) as we did for the starting moment revisions. Thus to make
Sizing of intermediate storage v’(t)
continuable, I’*= V,>
we require:
I’,+Av+ -AI’+
f I’,(1 -xi), i-l
(17a,b)
5 I’,(1 -xi), i-L+1
where.
Notice that this simplifies our analysis considerably, as we have lumped the effects of all variations into a single term AV. All other terms in equation (17a,b) are independent of variations. This makes it easy to calculate V* and V,, required to make all V’(t), i C k, continuable because now we can simply postulate bounds AV, and AV, on AV defined by equations (17a, b) and (18) to obtain the following result. Proposition 5. V + and V, required for the allowability of a set of arbitrary variations satisfying AV,
561
to the starting moment revisions, the number of cycles that a unit completes in T is itself a RV. To simplify matters, we assume that unit i completes NI = trunc (T, wi) cycles of operation in time T. This is a very reasonable assumption, especially when T is sufficiently large. Thus the maximum numbers of elementary variations that unit i can undergo are ki, = Ni - 1 starting moment revisions, kD = kfl = N, flowrate and transfer fraction variations. Clearly, should one be able to estimate these a priori, the actual numbers of variations should be used. Our development of the bounds AV, and AVu for various types of variations will employ the following three results from the theory of probability. The first result concerns the variance of a sum of independent RVs (IRVs), the second is useful in determining the expected value of a product of two IRVs while the third deals with the distribution of the extreme values of partial sums of IRVs. For these lemmas, let X,, X,,. .., X,, by any IRVs. Lemma 3. Let
aixi
Y = i
V*=
V,+AV,+
*=I
f v,(l-xi) i-l
with arbitrary
constants
a,, then
and V,a
-AVL+
5
Vi(l-xi).
(190)
i-L+1
E[Y]=
k aiE[Xi] i=l
and
a*(Y)=
Lemma 4. For any functions STATISTICAL
ESTIMATION
OF BOUNDS
Throughout our discussion of sizing expressions, we postulated upper and lower bounds on sums of different types of variations. The primary reason behind this is that the allowability condition requires continuability of holdup after the completion of each variation. This, in principle, presupposes our knowledge about the time of occurrence and the magnitude of each variation. Thus, if we are dealing with a prespecified set of variations, the determination of A V,_and A Vu poses no difficulty. On the other hand, in a design problem, this cannot be done, as such information is not available. A more desirable approach is to estimate the bounds A VL and AV,,, which were postulated or used in a deterministic manner, using a stochastic approach. To achieve this, a few simplifications are in order. First, we assume that elementary variations, Atj, AU{ and Ax! are independent random variables (RVs) with zero means and standard deviations o(t,), o(U) and a(~,) respectively. Second, the elementary variations of a type are identically distributed for a unit, thus the variance of a particular variation depends only on the unit in which it occurs. Third, we base our analysis on a given number of variations of each type. Suppose that it is desired to maintain the continuity of operation for a stretch of time of length 2’. Now, the number of variations that a unit undergoes during time T is dependent on the number of cycles and the probability of occurrence of the variation. But, due
%(X,)h(Xd
=
i a: 02 (X,). i-I
g and h:
~kW,)l~MX,Il
Lemma 5. Let S,=
;: xi, k = I,n, i=l
then for Xis symmetric
about 0 (Woodroofe,
1975):
Let us first develop the bounds for the case when only starting moment revisions are occurring. The treatment for the other elementary variations is similar so we indicate only the final results. Starting moment revisions. Recall that in this case, k,, = Nr - 1, k,2 = k,, = 0 and AV was defined by equation (13c). Let
x= -
2 civit,,,_
,,/o+
i-1
Clearly, tqN,_I;~ are IRVs because Atj’s are IRVs thus X is a RV. Let us assume that both advance delay are equally likely, hence At{ is a symmetric with E[At{] = 0. By a straightforward application Lemma 3 to X [equation (13c)] and ru,, [equation we obtain: E[X]=
5 qV,t.&+=E[AV], is,
a*(X)
=
2 (N, - 1) Vfa*(t,)/o$,
r=l
and and RV of (7)],
T. 0.
568
ODI
and
where a2(r,) = &(At{) for all j. Since AV is a RV, we cannot estimate A VL and A Vu with absolute certainty and must specify a confidence level p such that Pr[Av, < AV d Av,] = p. Notice that Avis a partial sum of IRVs, hence Lemma 5 is applicable. Here we mean all possible values of A V [equation (13c, 18)] for l
I.
A. K.uum
suggested earlier for a@,). Note that even though we are using the notation y, for all variations, the values of yi may not be the same for all variations. General uuriutions. Here the basic procedure for the development of bounds remains the same as that employed earlier for each of the elementary variations. Now, X will be defined as: x=,:,~,(~$,avi +AU~oimin(xi,xj+Ax~)-
If we design the storage to accommodate the positive and the negative deviations of AV with equal weight, we must have Al’,= E[AV] --E and Av,, = E[AV] + E. E can be easily evaluated by noting that X is a sum of IRVs. Since a linear combination of any IRVs tends towards a normal RV; if the number of summands is sufficiently large, X can be approximated fairly well by a normal RV. Then the bounds on AV are:
. I Since E[AVj] =0, and E[AU{] =0, defining B, = AUyW,min (x,, x, + Ax?) and applying Lemmas 3 and 4, we obtain:
Jwl = i C,V,bd% in I
AV,=E[X]-Z&Y), where Z denotes the standard normal variable. If we define Z, such that Pr(Z 2 Z.} = a, then in the above bounds, Pr{lZI > Z.} = (1 + p)/2 and a = (1 - p)/4. In practice, each o(ti) can be estimated from the operating data of the plant. If such data are not available, the following method suggested by Karimi and Reklaitis (198%) can be used. First assume a suitable distribution for Atj’s. Then postulate a pmbability, p. such that Pr{ - yiwi < ArJ 6 y,wi} = pi, i.e. the probability that any starting moment revision will be within + yi fraction of the nominal cycle time is pi. If we know the distribution function, then we can calculate a(r,) from p, and yi. For instance, if Ari’s are normally distributed then:
(20)
u2 (X) = 5 {Niu2(Vi) + u2(Bi) i-I
+ (Ni-
A?‘,, = E[X] + &a(X) and
Vitai,/o,
1) V~u2(fi)/o:}.
(21)
Since both A{ and Bi are RVs which are dependent on RVs AUi and AXi, as shown in Appendix D, we can obtain the following expressions for their variances:
Again, X is approximately a normal RV. Following the same procedure as for starting moment revisions, the bounds AV, and AV, for a given confidence, p, are obtained and the size of the storage required is then given by the following proposition. Proposition 6. V * and V, required to accommodate the worst case general variations in an L-M system with confidence, p, are: v* = ZZ,u(X)+
Yi”i
f V,(l -Xi), i-l
a&) = ___
z(l - PI)/2
The above procedure can be applied to the other two elementary variations for which the bounds can be expressed as: AV, = Z,o(X)
and
AVL = - Z,cr(X),
where
(1 -PI
a=---2--’ for flowrate
u(X) = 5 N,[V,U(U,)/U~]~ I- I
and
V,=E[X]+Z,u(X)+
t Vi(l -x,). i-L+,
(23a,b)
where u = (1 - p)/4, u(X) and E(X) are evaluated from equations (20-22). As shown earlier, u(ti), u(U,) and u(xi) can be estimated empirically when data not available. Having obtained u(U,) and a(~,), u(V,) can be easily estimated from equation (22a). In practice, it may be difficult to estimate u(U,) from plant data. This difficulty can be overcome by first estimating u(q) and then calculating u(U,) from equation (22a).
EXAMPLE
for transfer fraction variations. If c(Ui) and a(~,) they can be estimatqd as are not available, u(Ui) = Yi uilz(l -p&2 and U(xi) = YGJZ~I -p,),~r as
A noncontinuous process has two upstream batch units and a downstream SC unit separated by an
Sizing of intermediate storage Tabk
1. Nominal
Batch size
Transfer
v, W
@?0
XI
1 2 3
10,000 8ooo 8000
10.0
0.20
8.0 4.0
0.20 0.89
Cycle time
intermediate storage facility. The nominal operating parameters of this 2-l process are summarized in Table 1. It is desired to maintain the continuity of operation for a length of T = 160 h. Calculate the size of the storage and the initial inventory required to accommodate all variations arising in the system with probability of at least 90%. The statistical characteristics of the variations are as follows: Pr(lAtjl Q 0.05~~) = 0.99, Pr(lAhUjl 6 O.OSV,) = 0.99, Pr(lAx{l < 0.05~~) = 0.99. From the given data, yi = 0.05, pi = 0.99 and thus 2.575 for all variations. We use these to get a(t,), a(U,), a(~,) and estimate 02(Vi)=7.5 x 10-4Vf and a*(B,)=3.8 x lo-“Vf. Table 2 shows cr(ti), a( U,) and u (V,). Table 3 shows some intermediate results required in the calculation of u(X), which using equation (21), is 2480 kg. From equation (20), E [X] = 0.0. Since the degree of conlidence for design is p = 0.90, Z,, _-pj,4= 1.96. Therefore, from Proposition 6, V* = 25,000 kg and V, = 5740 kg will be required to maintain continuity of operation in the presence of randomly occurring variations. The results from the theory were compared with those obtained from the Monte-Carlo simulation for this example by using a simulation approach used by Karimi and Reklaits (1985~). The CPU time required to perform 2000 simulation runs of 160 h duration each was 831 s on the CDC Cyber 180/845. Y* = 22,400 kg and V,== 3600 kg were estimated from the simulation runs. This amounts to an overZ,, _p,j,2 = Z,,,,,,, =
Table 2. Standard deviations for Unit
parameters
Unit i
a)
I
example ,J(Ui)
i
W
(kg)
h? h-‘)
1 2 3
0.19 0.16 0.08
275 220 220
;: 44
Table 3. Calculation
unit
Avcragc cycks
O’(B)
i
N.
Ckd
1 2 3 Total
16 20 40
3.8 2.4 2.4 8.6
x x x x
10’ 10’ IO’ IO’
569
for example
fraction
Starting moment
Flowrate WLg
b-‘1
t&0
5000 5000 2247
0.0 1.2 0.6
design of 11.5% in our theoretical result as compared to the simulation. Using the V, and V* values from theory and simulating the system again, we find that the system operated uninterruptedly with a probability of 95% as compared to the design value of 90%. To get an idea about the sensitivities of V, and V* to changes in the magnitudes of parameter variations, we reworked this sample with yi = 0.1 and obtained u(X) = 4963 kg, Vs = 10,600 kg and V * = 34,700 kg. Thus, to accommodate these variations, V * and V,, increased by 39 and 85% respectively as compared to their values for the + 5% case. Similarly for y,= kO.25, u(X)= 124Okg, V* =20,lOOkg and V, = 3310 kg; which represent 19.4 and 42.3% reductions respectively in the values of V* and V, as compared to the + 5% case. COMPARISON
WITH
SIMULATIONS
To evaluate the performance of the analytical results of this paper, the sizing expressions were used to solve 20 problems. For each problem, V, and V* were obtained from the analytical results using the same values of variations as those employed in the simulation runs. To compare the two approaches, two criteria were employed. The first criterion is the CPU time required by the simulation approach. As shown in Table 4, enormous time (average 1018 s problem-‘) is required by the simulation approach as compared to, a negligible time for the analytical approach. The second criterion is the extra size of storage predicted by the theory over that by the simulation. On an average, the analytical approach, yields sizes about 11% higher than those of simulation. This is understandable because the analytical result is based on two bounds whose tightness contributes to the conservativeness of the model predictions. However, considering the times required for problem-specific simulations, reasonably good accuracy and the generality of the theoretical approach, the latter is clearly superior.
of cl(X)
for cxam~le
W(
v,,
Ckn)’ I.2 1.0 1.9 4.1
x x x x
106 10’ 106 106
(kal’ 5.7 x 4.6 x 9.4 x 19.7 x
’
105 IO’ IO’ IO’
570
T. 0. 001 and I. A. KAMMI Table 4. Comparison System L-M
2-l l-2 3-l 2-2 l-3 4-l 3-2 2-3 14 s-1 4-2 3-3 24 l-5 61 5-2 4-3 ?+I 2-5 l-6
Length of operation T(h)
between theoretical
Theoretical results’ v ‘(W v0 (W
160 125 150
5740 1770 2140
160
4460
150 150
2760 2480 1970 1230 2520 3140 3520 1390 2550 3210 2430 3110 1880 2020 3890 2710
loo
100 125 120 180
180 160 160 125 125 125 125 125 125
25,000 9850 11.600 22;OO0 11,800 11,500 7870 6160 10,800 13.500 12:900 6630 9910 12,100 12,100 14,200 10,600 11,500 17.200 10:900
method
and Monte-Carlo
simulation
Monte-Carlo %
S.illlUlZltiOU
Expected
over
v0 (W
V* (W
design
3600 1450 1060 2800 2150 1360 985 820 1940 1700 1810 640 1640 2250 1130 1450 880 1010 2440 1830
22,400 9140 9580 18,900 10,700 9123 6850 5710 10,200 12.000 12:100 6020 9060 10,500 10,900 13,400 9700 10,400 15,400 9480
11.5 7.7 21.1 16.6 10.7 18.0 15.0 8.0 7.0 11.0 7.0 10.2 9.5 15.0 11.4 5.3 9.6 10.0 11.7 15.3
continuity
(%)
;: 99
96 95 95 94 ;: 93 94 99 95 ;: 95 96 98
RcSUltS CPU time (s) 830 972 824 1020 963 930 916 1000 1012 1011 1040 1170 1200 1209 945 1020 1030 1040 1120 1120
‘Design based on 90% continuity.
The above discussion gives a relative evaluation of the effectiveness of our results. An absolute evaluation can be made by simulating the system designed by our approach and measuring its performance against the chosen design parameters like p. The percentage of the 2000 runs for each problem, for which there is no interruption in the process operation due to the tank being full or empty is the expected continuity. Table 4 shows that the expected continuity is always a few percent higher than the design value of continuity. Again, this is due to the worst case nature of the design. However, the important point is that the design is on the conservative side and thus safer. CONCLUSIONS
The role of intermediate storage in increasing the productivity of noncontinuous processes subject to random parameter variations has been analyzed. Sufficient conditions for ensuring uninterrupted process operation in the presence of common patterns of these variations were derived. These conditions were then used to develop analytical expressions for sizing the storage to accommodate given sets of variations. These expressions are analytical and expressed in terms of parameters which can bc easily specified. The superiority of the analytical procedure is evident
from the fact that the average simulation time required per problem was extensive (1018 s on CDC Cyber 180/845), while the analytical results yielded storage sizes reasonably close (within 11% on an average) to those predicted by simulation. The results of this paper obviate the need for extensive simulations for sizing storage and can be very useful in minimizing production losses due to process variations. They can also be used to assess the effectiveness of existing storage capacity in a plant. Thus this work constitutes a significant contribution
towards the development of accurate and systematic sizing procedures for designing intermediate storage in general noncontinuous processes with parallel, nonidentical units subject to variations. NOMENCLATURE q = Starting moment of ith variation b, = Ending moment of ith variation ci = A coefficient assigned to ith unit in equation
(3) Fi(f) = General flowrate function for ith unit H,(t) = Holdup function during the interval of nth variation i, j = Integers i(m) = Unit undergoing
mth variation 1(r) = Integral sum of flowrates as in equation (11) k = Number of variations k,, = Number of ith type of variation for ith unit, j = 1 starting moment, j = 2 flow rate, j = 3 transfer fraction I= Integer L = Number of units in the upstream stage M = Number of units in the downstream stage N = Total number of units; ‘L + &f N, = Number of cycles of ith unit I = Time tie= Initial starting moment of ith unit zi, = tiO+ sum of j starting moment revisions of ith unit T = Time units of uninterrupted operation Td = Idle time for semicontinuous ,unit T. = Time required to empty a batch unit Tr = Time required to fill a batch unit rs = Processing time for a batch unit T, = Processing time for a semicontinuous unit r, = Preparation time and waiting time, for a batch unit u(t) = Unit step function Vi = Nominal transfer flowrate of ith unit V(i, t) = Holdup profile (1 $0) for i completed variations V(t) = Holdup in the storage vessel V, = Initial inventory in the storage tank Vi = Nominal batch size of ith unit Y* = Capacity of storage vessel Vi(t) = Holdup function after completion of ith variation without any further variation x, = Nominal transfer fraction for ith unit as defined by equation (1) Z = Standard normal random variable
571
Sizing of intermediate storage APPENDIX
Greek letters a v, At/ AV
= = = =
Confidence level A set of k elementary variations Amount of jth starting moment revision for unit i Amount of change in storage holdup due to parameter variations as in equations (13~) and (18) AU/ = Amount of jth flow rate variation of unit i AI’: = Amount of jth batch size variation of unit i Axj = Amount of jth transfer fraction variation of unit i u(X) = Standard deviation of random variable X yz =.The maximum fraction by which a parameter can vary from its nominal value for unit i w, = Nominal cycle time of unit i p = likelihood of an event Subscripts L = Lower bound U = Upper bound Mathematical
Proof
A 2
of Lemma
Let A=
P”
F(r - t,)dr, a,< t
Symbols
E[X] = Expected value of random variable X max( ) = Maximum of the quantities within the theses min( ) = Minimum of the quantities within the theses mod(x, y) = x such that x = ky + I for some integer o
parenAPPENDIX
parenDerivation
k and
x
of H,,(t),
B a,,
, < t
For clarity in the following analysis, see Fig. 3. Let i(m) be the unit undergoing the mth variation, and Him) (t) denote the holdup in the interval, b, d t d b,,,, , , when only FMis occuring. Considering each subinterval contained in the interval a”+, d I d b,, , ,,the holdup H,(t) is given by:
Abbreviations B IRV RV SC
= = = =
Batch Independent random variable Random variable Semicontinuous
REFERENCES Henley E. J. and H. Hoshino, Effect of storage tanks on plant availability. Imf. Chem. Fundam. 16, 439 (1977). Karimi I. A., Analysis of intermediate storage in noncontinuous processing. Ph.D. Dissertation, School of Chemical Engineering, Purdue University, West Lafayette (1984). Karimi I. A. and G. V. Reklaitis, Intermediate storage in noncontinuous processes involving stages of parallel units. AZChE JI 31, 44(1985a). Karimi I. A. and G. V. Reklaitis, Deterministic variability analysis for intermediate storage in noncontinuous processes. Part I. AIChE JI, 31, 1516(1985b); Part II. AIChE JI 31, 1528 (1985c) Odi T. O., Sizing of intermediate storage for variabilities in noncontinuous processes. MS. Thesis, Northwestern University, Evanston (1986). Oi K., H. Itoh and I. Muchi, Improvement of Operational Flexibility of Batch Units by a Design Margin. Comput. &em. Engng 3, 177 (1979). Ross R., How much tankage is enough? Hydrocarbon Process 52, 75(1973). Smith N. and D. Rudd, On the effects of batch time variations on process performance. Chem. Engng Sci. 19, 403(1964). Takamatsu T., I. Hashimoto, S. Has&e and M. O’Shima, Design of a flexible batch process with intermediate storage tanks. Ind. Engng Chem., Process Des. Dee. 23, 40(1984). Woodroofe M., Probability with Applications, p. 312. MacGraw-Hill, New York (1975).
b
=
V(t) + b,_,Ctdb.
I
Fe, (T - hve,, ) dr.
Using Lemma 2:
Hi-“(t)=Hr)(t)=Y”(t)
b”-,
Also: @-2’(t)=
I@(t)-
i ~,v,,j (r - t<+,) dr m-l s *
n--2 r +Cm_, 6nFW(r - bhdd~ s b-l =A$-‘l(t)+ F.+-I, I x(r -tKm-,h;,_,,)dr. b,_,St =Sb._, Again using Lemma 2. ny)(t)=Hr-‘)(t)=
V”(t).
b,_19t
Similarly:
H(m’(t)=H;m+l)(t)= P(t). n m=n-k,...,n
b,
bb,,,,
T. 0. ODI and I. A. ILurrur
572 Clearly;
where
Ht-k’(t)=@+“+“(r)
= V”(r).
h-kd‘%+,dt~b,-k,, Hence when ‘Z’”is acting alone: Z&(t)=
Vn(r)
Q.E.D.
a”+, 41
APPENDIX
C
APPENDIX
Derivutions of E [At’;],
Derivations of m,‘” I(t) and max I(t)
,
I. E[AVl]
In order to derive expressions for the extreme values of Z(I), let us first represent the flowrate function F,(t) in terns of Fourier series @Carimi and Reklaitis, 1985a):
m 2ciui +~~,nnsmnnxicOs~x
.
‘-;
cai Substituting for Fi (r - r,,) in equation Integrating term by term, we get:
zw = it, gt
>
(Cl)
(11) for Z(r) and
and uz (Vi)
AV:=w,(c/i+AU;)(xi+~,)--iUix, =wiVi~~+w,x,AU;+w,aX~AU:.
(AV)’ = a’(Ax)* + b2(AV)’
~~2(Ax)2(ACJ,)2
+ 2abAxAlJ + 2aoAL’(Ax)*
+;:..w,iT
+ 2a~Axx(AU)~.
As E[Ax] = E[AU] = 0, c2 (x) = E[(Ax)‘] and .0’(b) = E[(AU)2]. Then taking expectations on both sides of the above equation and using Lemma 4, we get:
r-l
~[sin2nn(*-~)+sinn7rxi].
(CZ) U2(Vi) = Vf
Using the equality
@l)
For convenience, we will drop both the subscript, i, and the’ superscript, n, in the following analysis. Taking expectations on both sides of equation (Dl) and poticing ‘that E[AxALI].= E[Ax]E[ACZj = O E[Ax] = E[AU] = 0 and (Lemma 4). we obtain E[AV] = 0. Now, cz (I’) = E[(AV)2] - (E[AVD’= E[(AV)‘]. Defining a = oU and b = ox, from equation (Dl) we get:
- C,)
i-1
D
d (V,), E [B,], o* (Bi)
of productivities
assumption
we get:
d(x,)
(
-fT7 4
2. E[B,] and u* (B,) Dropping subscript, sion for Bi, we have
Defining ,=mod(+‘,
B=
l),
u2(xr) U2(Ui) XI
i, and superscript,
WXAV
ifAx 20
A(lo(x+Ax)
ifAxgO’
d(U,)
+vf
>
. 032)
k,* in the expres-
(D3ab)
Now, ~[Bl=~Pl~xOl P@x~O)+E[B(Ax~0] WAX s 0). From equation @3a, b), it is clear that, E [B] = 0 and u2 (B) = E (B*). Again, as was done for E[B], E[B*] = E[B*[Ax 201 fi(hr > 0) + B [B2 ]Ax < 01 Pr(Ax d 0). From equation
(D3),
W21dx.o]=T Xarimi and Reklaitis (1985a) derived simple bounds on the value of the second term for any value of yi. Using these, we obtain:
maxW)sAv+
i
t
i
i-L+,
V,(l-x,),
and
d(U)
o*(u) d(x)
uz+c12xz
Assuming a balanced design, i.e. I’r(Ax d 0) = l/2 and reinstating subscript,
V,(l-x,),
i-I
minZ(t)>AV-
VW(U)
Wa,b)
u2(Bi) = V;
pr(Ax >O)= i, we get, (D4)
0098~1354/88 s3.00 + 0.00 Copyright 0 1988 Pergamon Press plc
SIZING OF INTERMEDIATE STORAGE FOR VARIABILITIES IN NONCONTINUOUS PROCESSES WITH PARALLEL UNITS T. 0. ODI and I. A. Department of Chemical
Engineering,
Northwestern
KARIMI~
University,
Evanston,
IL 60208, U.S.A.
(Received 9 February 1987; final revision received 22 October 1987; received for publication 3 November 1987) Abstract-Noncontinuous processes involving batch/semicontinuous operations are frequently subject to variations in processing parameters. As a counter-measure, intermediate storage is usually installed in such processes to eliminate the effects of these variations. Patterns of parameter variations commonly arising in processes involving stages of nonidentical, parallel units are analyzed. Conditions which guarantee that continuity of operation is maintained in the presence of the variations are derived. These conditions are used to develop storage sizing expressions for accommodating the worst case variations with a given degree of confidence. The sizing expressions are analytical and are in terms of parameters which can be easily estimated. An example of a noncontinuous process is used to illustrate the sizing procedure.
INTRODUCTION
Noncontinuous processes are usually labor intensive and involve unsteady-state operations. They are particularly suitable for complex operations such as handling of solid materials, etc. The complexity of these operations, coupled with their unsteady-state nature, render the noncontinuous processes highly susceptible to fluctuations in key processing parameters such as transfer flowrates, batch sizes or batch start times. The effect of these variations is to introduce delays of operation, as stages either wait or are blocked due to the tardiness or busyness of other stages. Often variations originating at one point in a chain of stages will propagate along the chain perhaps growing in magnitude or causing similar or different additional variations. An effective way of mitigating the unfavorable effects of these variations is to install intermediate storage between adjacent stages in the system. Due to the increasing importance of noncontinuous processes in recent years and the role of intermediate storage in enhancing their flexibility, several works (Henley and Hoshino, 1977; Karimi and Reklaitis, 1985a,b,c; Ross, 1973; Takamatsu et al., 1984) have studied the design and operation of intermediate storage in such processes. A survey of different approaches has been reported by Karimi (1984). The most common approach, called the deterministic approach, has been used to analyze different patterns of variations to arrive at conditions which ensure the continuity of operation and then estimate the required size of intermediate storage based on these conditions. Karimi and Reklaitis (1985~) used this approach to obtain impressive sizing results for serial processes with a single unit in each stage. t Author
to whom all correspondence
should be addressed. 561
Takamatsu et al. (1984) analyzed a special case of processes with identical units in a stage, in which the batch start time of a unit in a stage is delayed by a fixed time interval relative to the earlier unit. The aim of this paper is to extend the deterministic approach to size intermediate storage in general noncontinuous processes employing nonidentical, parallel units in each stage. In contrast to the approach of Takamatsu et al. (1984), no special operating policy will be assumed. In this paper, we first present conditions which simultaneous variations in a system with nonidentical parallel units must satisfy to ensure continuous operation. These conditions are then combined with theoretical results for holdup to derive analytical sizing expressions to accommodate sets of arbitrary variations. The expressions are simple and of two types. One is for predicting the amount of initial inventory required to accommodate the variations and the other for estimating the appropriate size of storage. Each expression consists of two parts: one part depends on the nature of the variations while the other depends on the nominal values of the parameters of the process. The former part is evaluated from the stochastic distributions characterizing individual variations. PROCESS MODEL
A general noncontinuous process consists of mulparallel employing nonidentical, tiple stages batch/semicontinuous (B/SC) units. Intermediate storage tanks may be available between successive stages of operation. In this paper, we focus our attention on a building-block configuration of two stages of nonidentical, parallel units separated by a storage vessel; as any general process can he studied by considering pairs of successive stages of operation.
562
T. 0.
001
and I. A.
KARIMI
Fig. 1. Schematic diagram of a parallel process. This building block configuration will be referred to as the L-M system, as it employs L upstream and M downstream B/SC units as shown in Fig. 1. In this section, we first describe a fixed parameter model in terms of the nominal or average values of different process parameters. Then we indicate the type of variations which can arise in tbese parameters and present a model with variabfe parameters. The motivation behind this is that later we will use the results for processes with fixed parameters in our analysis of the variable parameter processes, Fixed parameter model In this model, all parameters of the L-M process are fixed at their nominal or average values. We make the following assumptions regarding its operation. (1) a typical cycle of a batch unit consists of steps such as filling, processing, emptying and preparation and/or waiting and each subsequent cycle of the unit is a repetition of the above steps; (2) a semicontinuous (SC) unit processes material in batches of fixed size at a constant rate. After processing a batch of material, it remains idle for a fixed interval of time and then resumes operation in a periodic manner; (3) all units operate in a nonstop manner and both stages have equal nominal productivities, i.e.
where N is the total number of units, L + M. (4) the discharge rates of all upstream units and the feed rates of all the downstr~m units are constant and known. Let us first define the following key process parameters for unit i. Nominal cycle time wi. For a batch unit i, oi = T, + TB + T, + TP, where Tf, TB, T,, Tp denote its filling time, processing time, emptying time and
preparation and/or waiting time, respectively. For a SC unit, it is given by at, = T, + Td, where T, and T, are the processing and idle times of the unit, mspeetively. Nomin~i tr~s~e~~ractio~ n,. The average fraction of the cycle time of unit i, during which transfer of materia1 occurs between the unit and the storage. For a batch unit: 1 B i c L, L+l<‘i
(la&)
For a SC unit, xi denotes the processing fraction and is given by: xi = T&o, UC) Nominal frowrate Ui. The average rate at which material is transferred between unit i and the storage. Nomi~ai batch size Vi. The average amount of material processed by unit i in one cycle. Clearly Vi is either U, T, or Vi T, or UiTs depending on the type of unit and its stage in the system. Hence: v, = l_J,x,cO,.
(2)
Nominal trarqfer Jlowrate fiction unit i, it is given by:
,F;ft). For each
ciui lo,=5 t <(I + X,)Wi,, (3) i 0 (t$:x&&
!I$ = g, u(t
- t,)k;:(t - Cd.
(4)
Sir&g of intemediate where u(t) is a unit step function. Note that the process described by equation (4) is completely periodic only for I 3 mfrx(t,,) and not for t 6 mp(tio). Integrating equation (4), we get,
u(z - t,)i;,(.r - ta) dt
c(z
f G my(tio),
(5a)
t 3 max0,,),
(56)
- rio) dr
where min (&,) = 0 and V(0) = V,,. For the fixed parameter system, Karimi and Reklaitis (1985a) analyzed equation (5b) and obtained analytical expressions for estimating storage size V* and initial inventory VO. Types of variations A detailed description of the types and sources of variations arising in non continuous processes was presented by Karimi and Reklaitis (1985b). Two types of variations were identified: elementary and composite variations. Elementary variations are simple variations in starting moments, transfer flowrates and transfer fractions. Composite variations are those that can be represented as combinations of elementary variations. A starting moment variatior or revision occurs when a unit does not start at its scheduled starting time; an advance if earlier and a delay if later. Let Ati denote the amount of delay (A?# 2 0) or advance (At! < 0) in the j th revision of unit i, its revised schedule of starting moment after n revisions is given by: b,=l,P+Iw,
l>n
(6)
where
th = tm+
5 At:.
(7)
/=I
A flowrate variation is said to occur in a cycle if the rate of transfer of material between a unit and the storage deviates from its nominal value. A transfer fraction variation occurs when the time required for transferring a batch of material between the storage and a unit deviates from its nominal value. An interval of a variation is defined as the interval during which it is actually occurring. We will denote the starting and ending moments of&h variations by aj and b,, respectively. A set of variations is termed as overlapping if it contains at least two variations whose intervals overlap. The variations in a set are conventionally sequenced chronologically according to the ending moments of intervals of variations (Karimi and Reklaitis, 1985b).
storage
263
ALLOWABILITY
OF VABIATKONS
In this section, we derive sufficient conditions that will make common sets of overlapping variations allowable in the L-M system. These conditions wilI be useful in the development of expressions for sizing the intermediate storage to accommodate variations later in this paper. Before analyzing the variations, we state two important definitions introduced by Oi et al. (1979). Continuubiliry. A holdup function, V(t), t 3 0, V(0) = VO, is said to be continuable if it satisfies 0 Q V(t) d V, where V* is the storage capacity. Allowability. A set of variations in process parameters is allowable if the holdup function associated with it is continuable. Let us begin with the fbllowing definitions:
H,(r) = holdup in the storage during the intervaf Q, d t g 6, associated with nth variation, Y” (t) = holdup in the storage after the completion of the nth variation (r > b.) assuming that nominal periodic operation continues with no further variations, V(n,t) = holdup in the storage for a set of n completed variations, k;,,k&Q*ka = numbers of starting moment revisions,. flowrate variations and transfer fraction variations for the unit i, respectively, k = total number of elementary variations, k = k,, + k,* f k13, sum of /ci, starting moment re6k,, =cumulative visions [See equation (711. Our analysis will be subject to the following assumptions: 1. The variations in process parameters are not permanent in nature and the process resumes nominal periodic operation -after their te~i~tion if no further variations occur. 2. The holdup Y(0, t) before the occurrence of any variation in the system is ,continuable [note that V(O,r) is given by equation (5a)]. 3. Each starting moment revision occurring in unit i, denoted by At;, n = 1,2,. . . , is limited to the range, IAt:] < (I - x,)w,. Assumption 3 is reasonable because the magnitude of the variations is expected to be small. Especially in the case of batch units with small transfer times, this range of values is sufficiently large. It is also valid for SC units having moderate values of processing fractions. We now state two important lemmas which will be used in our analysis. Lemma 1. The holdup function in the intermediate storage after k starting moment revisions is given by:
r I
vk(t) =
v, f i
~if7
i= I Jru,,
--t&,)d7
564
T. 0. ODI and I. A. KAMMI bl
a1
b
aLI
2
Fig. 2. Schematic diagram of two variations with overlap.
where, k,2 = k, = 0, Vi and kth variation occurs in unit j. For the proof of Lemma 1, see Odi (1986). Lemma 2. If the nth delay of starting moment for unit i, At;, occurring in the interval, a, < t d b, is such that 0 < Arl< (1 - x&q, then 4
Fi(7--zt,)d7=0 a,
0 < 1 < a2
V(2, t) = H*(t),
a2 < t < b2
I’(2, t) = v2(t),
t > b2
where Y2 (t) is given by Lemma 1 with k = 2. Clearly, V(2, t) is continuable for 0 < t < a,. Using Lemma 1, Hz(t), a2 < t
Lemma 2, Hz(t) = v*(t), a2 < t d b2. we conclude that V(2,r) is continuable if
C”(r) and v2(t) are both continuable. Therefore the two variations are allowable if the holdup functions after their terminations are continuable. One can show that this result is true for any overlap pattern of these variations or for that matter any two elementary variations. The simplest patterns of overlaps are those involving two elementary variations. We discuss a pattern of two overlapping delays only as patterns involving these variations are the most difficult to analyze. Transfer fraction and transfer flowrate variations are easily handled in a general manner and will be considered in the analysis of multiple variations. Multiple elementary variations Here we extend our analysis to sets of multiple elementary variations with overlaps. The analysis will be done by induction. We first assume that a set bf n arbitrary elementary variations ivith overlaps, Y,,, with a,,,, b,,,, m = 1,2,. , as the respective starting and ending moments of m th variation has already occurred in the L-44 system and the holdup V(~,I) associated with it is continuable. We also assume that ‘p, has been sequenced such that b, < b2 d . . . < b,, Next we form another set of variations, Y,,,, by superimposing on Y,, the (n + 1)st variation such We also assume that the latter is that b,
OdtGa,+l
v(n + 1, t) = H,, , (f) = W, +c.AU’ I Qz+‘ I (t
t) -a,+l).
a,+,
xjwj.
t ah+,
expressions, we above From the get, ‘ if c,ACJ~J~+‘>O and I’@, t) < H”+, (t) G t’ “+(t) v(n, t) B H,, , (1) > V”+ ’ (t) if cjAU;j*+ ’ d 0. Since I’(n,t) is already continuable, we deduce that H,, , (t) is continuable if V”+ ’ (t) is assumed to be continuable. Therefore, given that Yy, is allowable and the holdup function after the (n + 1)st variation is continuable, then Y,+, is allowable. The same
Sing of intermediate storage
sequenced as 1,2 , . . , n is ahowable if the holdup function after each of the subsets: 111. n], occurring alone in the sysfl,2),. ..,{1,2,..., tem is continuable.
result can be proved for the transfer fraction variation (Odi, 1986). Delay of starting moment. The analysis of this variation differs from the other two elementary variations in that it is difficult to study this variation in a general manner because the expression for H, + , (t ) depends on the overlap patterns of the variations in YJn+i* This is especially true when the overlaps involved are those of delays themselves. Therefore, we consider specific but commonly occurring patterns of overlaps and derive the condition for ~lowability. The earlier result for two delays is very significant because simultaneous overlap of many delays is not less likely. With these considerations in mind, let us assume that Y” is entirely made up of delays, as their overlaps present the major difficulty in the analysis. For the present discussion, let us also assume that a,du,+,db,fori== 1,2, . . n, which is equivalent to saying that the (n + I)st delay starts later than all the previous delays but overlaps with some say last k delays in Ym_A schematic diagram of this pattern of overlap is shown in Fig. 3. (Note that a,, i = 1,2,. . . n are not shown). First note that V(n, t) = V” (t) for t 3 u,,as shown in Appendix B. Now,
Dhxssion The importance of the results in Propositions 1 and 2 is that they relate the allowability of a set of variations to the holdup function after the termination of each variation in the set. EIence, we do not have to worry about the actual holdup function associated with each interval of variation. The holdup in an interval of variation is nonperiodic and thus difficult to analyze. In contrast, the holdup function after the termination of a variation is periodic and thus easier to analyze. Note that Proposi-. tions 1 and 2 are applicable to ‘arbitrary sets of variations. In view of these facts, the storage size to accommodate any set of variations can be obtained by analyzing the hotdup function after each variation in the set. Thus the results obtained from the preceding analysis gives a very convenient basis for sizing storage in the L-M system. It is noteworthy that the results ‘pertaining to multiple elementary variations with or without overlap are valid for multiple composite variations. This is because, as shown by Karhni and Reklaitis (1985b), the latter can be decomposed into the former. Thus, the holdup function after the te~ination of a set of arbitrary variations in an L-M system is given in the following proposition. Z+opodtion 3. For a set of arbitrary multiple variations which can be decomposed into k elementary variations, the holdup function after the kth elementary variation is given by:
, Fj(t - tin,,) dr. i %+I where ei, is the number of starting moment revisions that unit i has undergone. Since V(n, t) = V”(t), with the help of Lemma 1, we get, %+,(t)
= V@, t) -
I f&+,(t)=
VR(t)-
&(r -t,n,,)dT $
AV{
an+1*Ft ~fb,+,
i-1
Then from Lemma 2, H,+,(t) = I’“+* (8). Thus, given the allowability of Pm, !Y”+, is allowable if I’“+’ (t) is continuable. For a cast in which all the variations in Yy,, , have the same starting moment, the above analysis is still valid. Although general results could not be obtained analytically for arbitrary overlap patterns of delays, our simulations of various configurations and magnitudes of these variations indicate that the results for tbe special cases considered here should hold true for most other cases. The following proposition summarizes the results of our analysis. Proposition 2. For an L-M system, a set of n elementary variations with overlaps which can be
-
b,
.I...
b,
565
bn-k
.
.
.
.
bn-k+1
.
.
.
.
.
.
.
+
c,m,(k, - k,)min(xi,
“t
FAT --t,,,)dr s ‘Lib
1
,
(8)
where Al’!=
(U,+AU{)(x,+Ax’;)wj-
v;..
(9)
The convention under which the expression was derived is that both a flowrate variation and a transfer fraction variation are thought to occur in a cycle of a unit. If only one of them occurs, the other one is assumed to have a zero value. This convention in effect ensures that the resultant of the two variations in a cycle is always a batch size variation with
.
.
bn--l
4-z
Fig. 3. Schematic diagram of Yr”+ , . ,Z,bE
xii- A.+)AI@
,
“n +1
C.A.C.E.
1 j-i
c
4
bIt+1
566
T. 0. ODI and I. A. KANMI
constant cycle time as in the first term in the parenthesis of equation (8). An explanation of the rest of the terms in the parenthesis in equation (8) is available in Karimi and Reklaitis (1985b). The proof of equation (8) follows directly from the one given for the l-l system by Karimi and Reklaitis (1985b).
By choosing V * and V, to satisfy equation (13a,b) we can make V’(t) continuable.. Since the allowability condition requires continuability of all V’ (t), i Q k, V * and V, which can accommodate all k variations are given by the. following proposition. Proposition 4. V * and V, required to accommodate a set of k elementary variations are:
STORAGE SIZING FOR L-M SYSTEMS
V*= In this section, we develop analytical expressions for determining the size of storage for an L-M system undergoing a specified range of process parameter variations. We begin with the elementary variations and later proceed to the general case. Sizing under elementary variations The allowability conditions (Propositions 1 and 2) constitute the basis behind our subsequent development of expressions for sizing the storage to accommodate a set of arbitrary variations. We first consider cases in which only one type of elementary variations is occurring in the L-M system. Later, we generalize the results to include sets of arbitrary, random variations. Starting moment revisions. Suppose that the system is subject to only starting moment revisions and k revisions have occurred with Iq, for unit i. As k,I = !c,~= 0, the expression for V’ (t) in Proposition 3 reduces to:
P(t) =
V,+AV,,+
VO= -AV,+
2 Vj(l-xx,). i-1 5 Vi(l -xi), i-L+,
(14a,b)
where AV, and AV, satisfy AV, < AV < AV,. Later we will show how AV, and AV, can be estimated analytically from the statistical distributions describing the variations. For sets of only flowrate variations or only transfer fraction variations, we only need to increase V* and V, for the fixed parameter system (Karimi and Reklaitis, 1985a) by AV* and A VO respectively as follows: AV,=max
(0, -AV,)
and AV*=AV,+AV,,
(15a,b)
where AV,
v, + 1(t),
N
42
c
c
c,w,x,AV;dAV,
i-In-1
is for flowrate variation and
where
1(t)= 5 From equation
’ Fi(7 - ta,,) dr.
i-l I l&, (10) it follows that:
k,s
N
(11)
AV,
c c ciw,xiAV~
fraction
variations.
V* = max V’ (t) = VO+ V_ I and
SIZING UNDER GENERAL VARIATIONS
min V”(t) = V, + Vmin, I where V_ = max I(t) and Vtin = min I(t).
I
I
(12a,b)
Since negative holdup is not permitted, V, 2 - V,,. Using the upper and lower bounds on V,,,,, and V,, as derived in Appendix C, we obtain:
Sizing under arbitrary variations is the same as sizing under multiple but mixed elementary variations. The approach that we adopt is a direct extension of our treatment of multiple elementary variations of the same type. The general holdup function of Proposition 3 can can be rewritten as: V’(t)=
V,+AV,+
5 W i-I s ‘UiI
- te,,) dr,
where V*=
V,,+AV+
f’oa -AI’+
f V,(l -xi), ir I f irI_+
v,(l-xx,),
AV,=
Wa,b)
2 E c,AV; ,..I 1 n-1 + cioi(kn - kn) min (x,, xi
1
where (13c)
First, let us simplify the third term in V”(t) as we did for the starting moment revisions. Thus to make
Sizing of intermediate storage v’(t)
continuable, I’*= V,>
we require:
I’,+Av+ -AI’+
f I’,(1 -xi), i-l
(17a,b)
5 I’,(1 -xi), i-L+1
where.
Notice that this simplifies our analysis considerably, as we have lumped the effects of all variations into a single term AV. All other terms in equation (17a,b) are independent of variations. This makes it easy to calculate V* and V,, required to make all V’(t), i C k, continuable because now we can simply postulate bounds AV, and AV, on AV defined by equations (17a, b) and (18) to obtain the following result. Proposition 5. V + and V, required for the allowability of a set of arbitrary variations satisfying AV,
561
to the starting moment revisions, the number of cycles that a unit completes in T is itself a RV. To simplify matters, we assume that unit i completes NI = trunc (T, wi) cycles of operation in time T. This is a very reasonable assumption, especially when T is sufficiently large. Thus the maximum numbers of elementary variations that unit i can undergo are ki, = Ni - 1 starting moment revisions, kD = kfl = N, flowrate and transfer fraction variations. Clearly, should one be able to estimate these a priori, the actual numbers of variations should be used. Our development of the bounds AV, and AVu for various types of variations will employ the following three results from the theory of probability. The first result concerns the variance of a sum of independent RVs (IRVs), the second is useful in determining the expected value of a product of two IRVs while the third deals with the distribution of the extreme values of partial sums of IRVs. For these lemmas, let X,, X,,. .., X,, by any IRVs. Lemma 3. Let
aixi
Y = i
V*=
V,+AV,+
*=I
f v,(l-xi) i-l
with arbitrary
constants
a,, then
and V,a
-AVL+
5
Vi(l-xi).
(190)
i-L+1
E[Y]=
k aiE[Xi] i=l
and
a*(Y)=
Lemma 4. For any functions STATISTICAL
ESTIMATION
OF BOUNDS
Throughout our discussion of sizing expressions, we postulated upper and lower bounds on sums of different types of variations. The primary reason behind this is that the allowability condition requires continuability of holdup after the completion of each variation. This, in principle, presupposes our knowledge about the time of occurrence and the magnitude of each variation. Thus, if we are dealing with a prespecified set of variations, the determination of A V,_and A Vu poses no difficulty. On the other hand, in a design problem, this cannot be done, as such information is not available. A more desirable approach is to estimate the bounds A VL and AV,,, which were postulated or used in a deterministic manner, using a stochastic approach. To achieve this, a few simplifications are in order. First, we assume that elementary variations, Atj, AU{ and Ax! are independent random variables (RVs) with zero means and standard deviations o(t,), o(U) and a(~,) respectively. Second, the elementary variations of a type are identically distributed for a unit, thus the variance of a particular variation depends only on the unit in which it occurs. Third, we base our analysis on a given number of variations of each type. Suppose that it is desired to maintain the continuity of operation for a stretch of time of length 2’. Now, the number of variations that a unit undergoes during time T is dependent on the number of cycles and the probability of occurrence of the variation. But, due
%(X,)h(Xd
=
i a: 02 (X,). i-I
g and h:
~kW,)l~MX,Il
Lemma 5. Let S,=
;: xi, k = I,n, i=l
then for Xis symmetric
about 0 (Woodroofe,
1975):
Let us first develop the bounds for the case when only starting moment revisions are occurring. The treatment for the other elementary variations is similar so we indicate only the final results. Starting moment revisions. Recall that in this case, k,, = Nr - 1, k,2 = k,, = 0 and AV was defined by equation (13c). Let
x= -
2 civit,,,_
,,/o+
i-1
Clearly, tqN,_I;~ are IRVs because Atj’s are IRVs thus X is a RV. Let us assume that both advance delay are equally likely, hence At{ is a symmetric with E[At{] = 0. By a straightforward application Lemma 3 to X [equation (13c)] and ru,, [equation we obtain: E[X]=
5 qV,t.&+=E[AV], is,
a*(X)
=
2 (N, - 1) Vfa*(t,)/o$,
r=l
and and RV of (7)],
T. 0.
568
ODI
and
where a2(r,) = &(At{) for all j. Since AV is a RV, we cannot estimate A VL and A Vu with absolute certainty and must specify a confidence level p such that Pr[Av, < AV d Av,] = p. Notice that Avis a partial sum of IRVs, hence Lemma 5 is applicable. Here we mean all possible values of A V [equation (13c, 18)] for l
I.
A. K.uum
suggested earlier for a@,). Note that even though we are using the notation y, for all variations, the values of yi may not be the same for all variations. General uuriutions. Here the basic procedure for the development of bounds remains the same as that employed earlier for each of the elementary variations. Now, X will be defined as: x=,:,~,(~$,avi +AU~oimin(xi,xj+Ax~)-
If we design the storage to accommodate the positive and the negative deviations of AV with equal weight, we must have Al’,= E[AV] --E and Av,, = E[AV] + E. E can be easily evaluated by noting that X is a sum of IRVs. Since a linear combination of any IRVs tends towards a normal RV; if the number of summands is sufficiently large, X can be approximated fairly well by a normal RV. Then the bounds on AV are:
. I Since E[AVj] =0, and E[AU{] =0, defining B, = AUyW,min (x,, x, + Ax?) and applying Lemmas 3 and 4, we obtain:
Jwl = i C,V,bd% in I
AV,=E[X]-Z&Y), where Z denotes the standard normal variable. If we define Z, such that Pr(Z 2 Z.} = a, then in the above bounds, Pr{lZI > Z.} = (1 + p)/2 and a = (1 - p)/4. In practice, each o(ti) can be estimated from the operating data of the plant. If such data are not available, the following method suggested by Karimi and Reklaitis (198%) can be used. First assume a suitable distribution for Atj’s. Then postulate a pmbability, p. such that Pr{ - yiwi < ArJ 6 y,wi} = pi, i.e. the probability that any starting moment revision will be within + yi fraction of the nominal cycle time is pi. If we know the distribution function, then we can calculate a(r,) from p, and yi. For instance, if Ari’s are normally distributed then:
(20)
u2 (X) = 5 {Niu2(Vi) + u2(Bi) i-I
+ (Ni-
A?‘,, = E[X] + &a(X) and
Vitai,/o,
1) V~u2(fi)/o:}.
(21)
Since both A{ and Bi are RVs which are dependent on RVs AUi and AXi, as shown in Appendix D, we can obtain the following expressions for their variances:
Again, X is approximately a normal RV. Following the same procedure as for starting moment revisions, the bounds AV, and AV, for a given confidence, p, are obtained and the size of the storage required is then given by the following proposition. Proposition 6. V * and V, required to accommodate the worst case general variations in an L-M system with confidence, p, are: v* = ZZ,u(X)+
Yi”i
f V,(l -Xi), i-l
a&) = ___
z(l - PI)/2
The above procedure can be applied to the other two elementary variations for which the bounds can be expressed as: AV, = Z,o(X)
and
AVL = - Z,cr(X),
where
(1 -PI
a=---2--’ for flowrate
u(X) = 5 N,[V,U(U,)/U~]~ I- I
and
V,=E[X]+Z,u(X)+
t Vi(l -x,). i-L+,
(23a,b)
where u = (1 - p)/4, u(X) and E(X) are evaluated from equations (20-22). As shown earlier, u(ti), u(U,) and u(xi) can be estimated empirically when data not available. Having obtained u(U,) and a(~,), u(V,) can be easily estimated from equation (22a). In practice, it may be difficult to estimate u(U,) from plant data. This difficulty can be overcome by first estimating u(q) and then calculating u(U,) from equation (22a).
EXAMPLE
for transfer fraction variations. If c(Ui) and a(~,) they can be estimatqd as are not available, u(Ui) = Yi uilz(l -p&2 and U(xi) = YGJZ~I -p,),~r as
A noncontinuous process has two upstream batch units and a downstream SC unit separated by an
Sizing of intermediate storage Tabk
1. Nominal
Batch size
Transfer
v, W
@?0
XI
1 2 3
10,000 8ooo 8000
10.0
0.20
8.0 4.0
0.20 0.89
Cycle time
intermediate storage facility. The nominal operating parameters of this 2-l process are summarized in Table 1. It is desired to maintain the continuity of operation for a length of T = 160 h. Calculate the size of the storage and the initial inventory required to accommodate all variations arising in the system with probability of at least 90%. The statistical characteristics of the variations are as follows: Pr(lAtjl Q 0.05~~) = 0.99, Pr(lAhUjl 6 O.OSV,) = 0.99, Pr(lAx{l < 0.05~~) = 0.99. From the given data, yi = 0.05, pi = 0.99 and thus 2.575 for all variations. We use these to get a(t,), a(U,), a(~,) and estimate 02(Vi)=7.5 x 10-4Vf and a*(B,)=3.8 x lo-“Vf. Table 2 shows cr(ti), a( U,) and u (V,). Table 3 shows some intermediate results required in the calculation of u(X), which using equation (21), is 2480 kg. From equation (20), E [X] = 0.0. Since the degree of conlidence for design is p = 0.90, Z,, _-pj,4= 1.96. Therefore, from Proposition 6, V* = 25,000 kg and V, = 5740 kg will be required to maintain continuity of operation in the presence of randomly occurring variations. The results from the theory were compared with those obtained from the Monte-Carlo simulation for this example by using a simulation approach used by Karimi and Reklaits (1985~). The CPU time required to perform 2000 simulation runs of 160 h duration each was 831 s on the CDC Cyber 180/845. Y* = 22,400 kg and V,== 3600 kg were estimated from the simulation runs. This amounts to an overZ,, _p,j,2 = Z,,,,,,, =
Table 2. Standard deviations for Unit
parameters
Unit i
a)
I
example ,J(Ui)
i
W
(kg)
h? h-‘)
1 2 3
0.19 0.16 0.08
275 220 220
;: 44
Table 3. Calculation
unit
Avcragc cycks
O’(B)
i
N.
Ckd
1 2 3 Total
16 20 40
3.8 2.4 2.4 8.6
x x x x
10’ 10’ IO’ IO’
569
for example
fraction
Starting moment
Flowrate WLg
b-‘1
t&0
5000 5000 2247
0.0 1.2 0.6
design of 11.5% in our theoretical result as compared to the simulation. Using the V, and V* values from theory and simulating the system again, we find that the system operated uninterruptedly with a probability of 95% as compared to the design value of 90%. To get an idea about the sensitivities of V, and V* to changes in the magnitudes of parameter variations, we reworked this sample with yi = 0.1 and obtained u(X) = 4963 kg, Vs = 10,600 kg and V * = 34,700 kg. Thus, to accommodate these variations, V * and V,, increased by 39 and 85% respectively as compared to their values for the + 5% case. Similarly for y,= kO.25, u(X)= 124Okg, V* =20,lOOkg and V, = 3310 kg; which represent 19.4 and 42.3% reductions respectively in the values of V* and V, as compared to the + 5% case. COMPARISON
WITH
SIMULATIONS
To evaluate the performance of the analytical results of this paper, the sizing expressions were used to solve 20 problems. For each problem, V, and V* were obtained from the analytical results using the same values of variations as those employed in the simulation runs. To compare the two approaches, two criteria were employed. The first criterion is the CPU time required by the simulation approach. As shown in Table 4, enormous time (average 1018 s problem-‘) is required by the simulation approach as compared to, a negligible time for the analytical approach. The second criterion is the extra size of storage predicted by the theory over that by the simulation. On an average, the analytical approach, yields sizes about 11% higher than those of simulation. This is understandable because the analytical result is based on two bounds whose tightness contributes to the conservativeness of the model predictions. However, considering the times required for problem-specific simulations, reasonably good accuracy and the generality of the theoretical approach, the latter is clearly superior.
of cl(X)
for cxam~le
W(
v,,
Ckn)’ I.2 1.0 1.9 4.1
x x x x
106 10’ 106 106
(kal’ 5.7 x 4.6 x 9.4 x 19.7 x
’
105 IO’ IO’ IO’
570
T. 0. 001 and I. A. KAMMI Table 4. Comparison System L-M
2-l l-2 3-l 2-2 l-3 4-l 3-2 2-3 14 s-1 4-2 3-3 24 l-5 61 5-2 4-3 ?+I 2-5 l-6
Length of operation T(h)
between theoretical
Theoretical results’ v ‘(W v0 (W
160 125 150
5740 1770 2140
160
4460
150 150
2760 2480 1970 1230 2520 3140 3520 1390 2550 3210 2430 3110 1880 2020 3890 2710
loo
100 125 120 180
180 160 160 125 125 125 125 125 125
25,000 9850 11.600 22;OO0 11,800 11,500 7870 6160 10,800 13.500 12:900 6630 9910 12,100 12,100 14,200 10,600 11,500 17.200 10:900
method
and Monte-Carlo
simulation
Monte-Carlo %
S.illlUlZltiOU
Expected
over
v0 (W
V* (W
design
3600 1450 1060 2800 2150 1360 985 820 1940 1700 1810 640 1640 2250 1130 1450 880 1010 2440 1830
22,400 9140 9580 18,900 10,700 9123 6850 5710 10,200 12.000 12:100 6020 9060 10,500 10,900 13,400 9700 10,400 15,400 9480
11.5 7.7 21.1 16.6 10.7 18.0 15.0 8.0 7.0 11.0 7.0 10.2 9.5 15.0 11.4 5.3 9.6 10.0 11.7 15.3
continuity
(%)
;: 99
96 95 95 94 ;: 93 94 99 95 ;: 95 96 98
RcSUltS CPU time (s) 830 972 824 1020 963 930 916 1000 1012 1011 1040 1170 1200 1209 945 1020 1030 1040 1120 1120
‘Design based on 90% continuity.
The above discussion gives a relative evaluation of the effectiveness of our results. An absolute evaluation can be made by simulating the system designed by our approach and measuring its performance against the chosen design parameters like p. The percentage of the 2000 runs for each problem, for which there is no interruption in the process operation due to the tank being full or empty is the expected continuity. Table 4 shows that the expected continuity is always a few percent higher than the design value of continuity. Again, this is due to the worst case nature of the design. However, the important point is that the design is on the conservative side and thus safer. CONCLUSIONS
The role of intermediate storage in increasing the productivity of noncontinuous processes subject to random parameter variations has been analyzed. Sufficient conditions for ensuring uninterrupted process operation in the presence of common patterns of these variations were derived. These conditions were then used to develop analytical expressions for sizing the storage to accommodate given sets of variations. These expressions are analytical and expressed in terms of parameters which can bc easily specified. The superiority of the analytical procedure is evident
from the fact that the average simulation time required per problem was extensive (1018 s on CDC Cyber 180/845), while the analytical results yielded storage sizes reasonably close (within 11% on an average) to those predicted by simulation. The results of this paper obviate the need for extensive simulations for sizing storage and can be very useful in minimizing production losses due to process variations. They can also be used to assess the effectiveness of existing storage capacity in a plant. Thus this work constitutes a significant contribution
towards the development of accurate and systematic sizing procedures for designing intermediate storage in general noncontinuous processes with parallel, nonidentical units subject to variations. NOMENCLATURE q = Starting moment of ith variation b, = Ending moment of ith variation ci = A coefficient assigned to ith unit in equation
(3) Fi(f) = General flowrate function for ith unit H,(t) = Holdup function during the interval of nth variation i, j = Integers i(m) = Unit undergoing
mth variation 1(r) = Integral sum of flowrates as in equation (11) k = Number of variations k,, = Number of ith type of variation for ith unit, j = 1 starting moment, j = 2 flow rate, j = 3 transfer fraction I= Integer L = Number of units in the upstream stage M = Number of units in the downstream stage N = Total number of units; ‘L + &f N, = Number of cycles of ith unit I = Time tie= Initial starting moment of ith unit zi, = tiO+ sum of j starting moment revisions of ith unit T = Time units of uninterrupted operation Td = Idle time for semicontinuous ,unit T. = Time required to empty a batch unit Tr = Time required to fill a batch unit rs = Processing time for a batch unit T, = Processing time for a semicontinuous unit r, = Preparation time and waiting time, for a batch unit u(t) = Unit step function Vi = Nominal transfer flowrate of ith unit V(i, t) = Holdup profile (1 $0) for i completed variations V(t) = Holdup in the storage vessel V, = Initial inventory in the storage tank Vi = Nominal batch size of ith unit Y* = Capacity of storage vessel Vi(t) = Holdup function after completion of ith variation without any further variation x, = Nominal transfer fraction for ith unit as defined by equation (1) Z = Standard normal random variable
571
Sizing of intermediate storage APPENDIX
Greek letters a v, At/ AV
= = = =
Confidence level A set of k elementary variations Amount of jth starting moment revision for unit i Amount of change in storage holdup due to parameter variations as in equations (13~) and (18) AU/ = Amount of jth flow rate variation of unit i AI’: = Amount of jth batch size variation of unit i Axj = Amount of jth transfer fraction variation of unit i u(X) = Standard deviation of random variable X yz =.The maximum fraction by which a parameter can vary from its nominal value for unit i w, = Nominal cycle time of unit i p = likelihood of an event Subscripts L = Lower bound U = Upper bound Mathematical
Proof
A 2
of Lemma
Let A=
P”
F(r - t,)dr, a,< t
Symbols
E[X] = Expected value of random variable X max( ) = Maximum of the quantities within the theses min( ) = Minimum of the quantities within the theses mod(x, y) = x such that x = ky + I for some integer o
parenAPPENDIX
parenDerivation
k and
x
of H,,(t),
B a,,
, < t
For clarity in the following analysis, see Fig. 3. Let i(m) be the unit undergoing the mth variation, and Him) (t) denote the holdup in the interval, b, d t d b,,,, , , when only FMis occuring. Considering each subinterval contained in the interval a”+, d I d b,, , ,,the holdup H,(t) is given by:
Abbreviations B IRV RV SC
= = = =
Batch Independent random variable Random variable Semicontinuous
REFERENCES Henley E. J. and H. Hoshino, Effect of storage tanks on plant availability. Imf. Chem. Fundam. 16, 439 (1977). Karimi I. A., Analysis of intermediate storage in noncontinuous processing. Ph.D. Dissertation, School of Chemical Engineering, Purdue University, West Lafayette (1984). Karimi I. A. and G. V. Reklaitis, Intermediate storage in noncontinuous processes involving stages of parallel units. AZChE JI 31, 44(1985a). Karimi I. A. and G. V. Reklaitis, Deterministic variability analysis for intermediate storage in noncontinuous processes. Part I. AIChE JI, 31, 1516(1985b); Part II. AIChE JI 31, 1528 (1985c) Odi T. O., Sizing of intermediate storage for variabilities in noncontinuous processes. MS. Thesis, Northwestern University, Evanston (1986). Oi K., H. Itoh and I. Muchi, Improvement of Operational Flexibility of Batch Units by a Design Margin. Comput. &em. Engng 3, 177 (1979). Ross R., How much tankage is enough? Hydrocarbon Process 52, 75(1973). Smith N. and D. Rudd, On the effects of batch time variations on process performance. Chem. Engng Sci. 19, 403(1964). Takamatsu T., I. Hashimoto, S. Has&e and M. O’Shima, Design of a flexible batch process with intermediate storage tanks. Ind. Engng Chem., Process Des. Dee. 23, 40(1984). Woodroofe M., Probability with Applications, p. 312. MacGraw-Hill, New York (1975).
b
=
V(t) + b,_,Ctdb.
I
Fe, (T - hve,, ) dr.
Using Lemma 2:
Hi-“(t)=Hr)(t)=Y”(t)
b”-,
Also: @-2’(t)=
I@(t)-
i ~,v,,j (r - t<+,) dr m-l s *
n--2 r +Cm_, 6nFW(r - bhdd~ s b-l =A$-‘l(t)+ F.+-I, I x(r -tKm-,h;,_,,)dr. b,_,St =Sb._, Again using Lemma 2. ny)(t)=Hr-‘)(t)=
V”(t).
b,_19t
Similarly:
H(m’(t)=H;m+l)(t)= P(t). n m=n-k,...,n
b,
bb,,,,
T. 0. ODI and I. A. ILurrur
572 Clearly;
where
Ht-k’(t)=@+“+“(r)
= V”(r).
h-kd‘%+,dt~b,-k,, Hence when ‘Z’”is acting alone: Z&(t)=
Vn(r)
Q.E.D.
a”+, 41
APPENDIX
C
APPENDIX
Derivutions of E [At’;],
Derivations of m,‘” I(t) and max I(t)
,
I. E[AVl]
In order to derive expressions for the extreme values of Z(I), let us first represent the flowrate function F,(t) in terns of Fourier series @Carimi and Reklaitis, 1985a):
m 2ciui +~~,nnsmnnxicOs~x
.
‘-;
cai Substituting for Fi (r - r,,) in equation Integrating term by term, we get:
zw = it, gt
>
(Cl)
(11) for Z(r) and
and uz (Vi)
AV:=w,(c/i+AU;)(xi+~,)--iUix, =wiVi~~+w,x,AU;+w,aX~AU:.
(AV)’ = a’(Ax)* + b2(AV)’
~~2(Ax)2(ACJ,)2
+ 2abAxAlJ + 2aoAL’(Ax)*
+;:..w,iT
+ 2a~Axx(AU)~.
As E[Ax] = E[AU] = 0, c2 (x) = E[(Ax)‘] and .0’(b) = E[(AU)2]. Then taking expectations on both sides of the above equation and using Lemma 4, we get:
r-l
~[sin2nn(*-~)+sinn7rxi].
(CZ) U2(Vi) = Vf
Using the equality
@l)
For convenience, we will drop both the subscript, i, and the’ superscript, n, in the following analysis. Taking expectations on both sides of equation (Dl) and poticing ‘that E[AxALI].= E[Ax]E[ACZj = O E[Ax] = E[AU] = 0 and (Lemma 4). we obtain E[AV] = 0. Now, cz (I’) = E[(AV)2] - (E[AVD’= E[(AV)‘]. Defining a = oU and b = ox, from equation (Dl) we get:
- C,)
i-1
D
d (V,), E [B,], o* (Bi)
of productivities
assumption
we get:
d(x,)
(
-fT7 4
2. E[B,] and u* (B,) Dropping subscript, sion for Bi, we have
Defining ,=mod(+‘,
B=
l),
u2(xr) U2(Ui) XI
i, and superscript,
WXAV
ifAx 20
A(lo(x+Ax)
ifAxgO’
d(U,)
+vf
>
. 032)
k,* in the expres-
(D3ab)
Now, ~[Bl=~Pl~xOl P@x~O)+E[B(Ax~0] WAX s 0). From equation @3a, b), it is clear that, E [B] = 0 and u2 (B) = E (B*). Again, as was done for E[B], E[B*] = E[B*[Ax 201 fi(hr > 0) + B [B2 ]Ax < 01 Pr(Ax d 0). From equation
(D3),
W21dx.o]=T Xarimi and Reklaitis (1985a) derived simple bounds on the value of the second term for any value of yi. Using these, we obtain:
maxW)sAv+
i
t
i
i-L+,
V,(l-x,),
and
d(U)
o*(u) d(x)
uz+c12xz
Assuming a balanced design, i.e. I’r(Ax d 0) = l/2 and reinstating subscript,
V,(l-x,),
i-I
minZ(t)>AV-
VW(U)
Wa,b)
u2(Bi) = V;
pr(Ax >O)= i, we get, (D4)