Intermediate storage and the operation of periodic processes under equipment failure

Compurers them. Printed in Great

I i/12, pp. 1235-1243,

Engng, Britain.

Vol. 13, No. All rights reserved

INTERMEDIATE OF PERIODIC

1989 Copyright

cws-1354189 S3.00 + 0.00 6 1989 Pergamon Press plc

STORAGE AND THE OPERATION PROCESSES UNDER EQUIPMENT FAILURE E. S. LEE and

G. V.

REKLAITIS

School of Chemical Engineering, Purdue University; West Lafayette, IN 47906, U.S.A. (Received

for publication

19 June 1989)

model is developed to study holdup requirements in the intermediate storage which connects two batch/semicontinuous units that are susceptible to equipment failure and repair. Three cases are investigated: that is upstream failure, downstream failure and both failure modes. In each case, general results concerning the periodicity of the required storage volume, effects of first failure on initial holdup and the volume calculation are presented. Using Fourier series constructions, simple analytical results to predict the limiting minimum volume of intermediate storage as a function of the frequency of failure and duration of repair times are obtained. The general analytical expressions encompass all three of the above failure and repair cases and reduce in the limit to accommodate the batch failure case.

Abstract-A

1.

INTRODUCTION

Intermediate storage has an important role in improving operating efficiency by decoupling the periodic operation of adjacent batch or semicontinuous units. In addition, batch operations are susceptible to high levels of processing variability and operator vagaries and error. These kinds of process parameter variations can also be mitigated by intermediate storage if an adequate size of storage facility and an appropriate level of initial holdup are chosen. For the basic l-l system which consists of two batch/semicontinuous units with a single storage tank between them, analytical models relating the required storage volume to the batch sizes, transfer rates, cycle times and delay times of the up- and downstream stages have been developed by Takamatsu et al. (1984) and Karimi and Reklaitis (1983). Also, the various types of parameter variations and required conditions which any parameter variation must satisfy in order that the system can continue to operate in periodic fashion without process interruptions or storage overflows have been investigated by Oi et al. (1979) and Karimi and Reklaitis (1985). Over a long operating horizon, process uncertainties grow to include not only parameter variations but also larger processing disruptions, such as batch failures and equipment failures. It is of interest, therefore, to examine the role of intermediate storage in mitigating these uncertainties. Batch failure arises when batches of material from upstream and/or downstream stages are lost or must be bypassed because of failure to meet required product specifications. In this case, it can be shown that for the l-l system only upstream batch failure will have an effect on the required intermediate storage capacity. Equipment failure arises when operating units are forced down because of equipment breakdown or are inten-

tionally shutdown for maintenance. In this case, the effects of failure on the required capacity of intermediate storage are dependent not only on the frequency of failure (time between failures) but also on the duration of repair (repair time). In previous work, Lee and Reklaitis (1989) have studied process availability for the batch failure case in a singleproduct l-l system. Analytical expressions were obtained for the required limiting volume of intermediate storage as a function of the frequency of batch failure. The effects of both deterministic and stochastic variations in the failure parameters on the required size of intermediate storage were also studied and appropriate dynamic operating policies were developed. In this paper, we investigate the case of the l-l system involving equipment failure and repair, with the goal of predicting the storage capacity required to attain a specified system availability. The analysis is developed in three parts. First, we consider the l-l system in which the upstream equipment is susceptible to failure while the downstream equipment is not. This case may occur, for example, if the upstream batch unit is connected to the intermediate storage vessel by a transfer unit which is susceptible to failure while intermediate storage is directly connected to a downstream batch unit whose operation is so reliable as not to be susceptible to failure. Then we develop the converse case, namely, reliable upstream unit and unreliable downstream system. Finally, we show how these results can be extended to treat the more usual case involving both types of failure. It is demonstrated that the resulting general analytical predictive expressions reduce in limiting cases to include batch failure and, of course, equipment failure either up-or downstream of the storage facility. These average case results form the basis for an analysis of the 1235

E. S. LLE and G. V.

1236

effects of variations times which is the 2.

PROCESS

in the failure rates subject of a subsequent MODEL

FOR

l-l

and repair paper.

RE~KLAITIS

where:

SYSTEM

A schematic diagram of the process under consideration is shown in Fig. 1. This system, denoted consists of one upstream stage as a l--l system, and one downstream slage, both employing batch; semicontinuous units. We denote by C’, the upstream volume over the cycle time 0,. by VZ the downstream volume over the cycle time wZ The basic assumptions are as follows: 1. Batch units operate with fixed batch sizes and cycle times. 2. In the normal case (without failure), the average productivities of both stages are equal. into or out of the intermediate 3. When flowrates storage vessel occur, they are constant rates. size of the storage tank is equal to 4. The required the maximum hold up in Lhe tank. 5. The frequency of failure c/) and repair time of the unit (d) are fixed. failure (ky) will occur at the 6. The first equipment ;>th batch from time t = 0. exist least integers /?, : such that /31w 7 = 7. There the existence of BLWL? i.e. this in turn assures an extended least common multiple (LCM) of (11, that is 0 = LCM((v) f. ro:) = [I, c,,r = pZc+ 2.2. The cycle time of a batch unit is the sum of the filling time (tr), processing time (t,), discharge time (t,) and cleaning time (t,). For the l-1 system, the fractional time of inflow to the storage tank is given by .Y, = z~/w, and the fractional time of outflow from the storage tank is given by .Y> = I,-.‘o,. Let the starting time of inflow from the upstream unit to the downstream be denoted as t,,, and the outflow unit as tZo. Furthermore, let .r,. .rr denote the ratio of these times to their respective cycle times. Then )I, = t,,lw, and y2 = tzoj w2. In the general case involving N upstream and downstream batch units, the holdup volume of the intermediate storage tank can be expressed as follows (Karimi and Reklaitis, 1985): c;(t)

i\ = c ,=I

’c,F,(s - t;,) dT = V(O) + I(t). s ,I

cos 2n rr(z,)

- cos 2nrI(u,

- )‘,)

+ cos 2n rI.l’ij

(2)

=;[lM,->‘,/+-=,l + (2% -. l)(_r,

=,)I,

(3)

where u,=mod The intermediate required initial

1.1

z, = mod(x,

,

( W,

storage tank holdup volume I,‘* = max i V(O)

3.

UPSTREAM

+ J;,

1).

>

2

I(t) -

holdup volume and will be given by: +

V(0).

min I(f ).

EQUIPMENT

(4)

FAILL!RE

Let y, denote the frequency of upstream unit failure and d, the repair time of the failed unit. For this case, d, is in general different from w,. Consequently the cycle time of the actual stream flow in the upstream unit is not the same throughout the campaign. Therefore we cannot apply (if directly. To apply these general results, we assume that there exists a set of hypothetical upstream flows as shown in Fig. 2. The sum of these hypothetical upstream flows will represent the actual composite upstream flow. If we denote by o,T the cycle time of Ihe hypothetical upstream flows, by s: the fractional time of inflows to the storage tank by the hypothetic upstream flows. Then:

(6) We also denote by V, the upstream volume over the cycle w, , by P; the downstream volume over the cycle stream volume over w1 and by t‘: the hypothetical the cycle time (tit. then: I’, = (Z, (0, .,- 11

(7)

LTZ= C’,tl&r2, v:

= U,tu:r:

(8) = I’,.

(9)

As in the batch failure cast, WC need throttling of the downstream flow LO compensate for the deficit in the flow from the upstream unit due to failure. This can be accomplished by reducing UZ or reducing the filling fraction .x1. If reduction of* the downstream Rowrate is selected, then D> can be expressed as follows:

Intermediate storage and operation of periodic processes Normal

1

Upstream

Hypothetical

9

--------

flow

1237

operation

Equlpmcnt

failure

flows

Time

,

x;cy,-

-

I) -

y:

Fig. 2. Hypothetical flows. Let i = 1 refer to the upstream unit and i = 2 refer to the downstream unit and superscript j refer to thejth hypothetical upstream flow. Then, from (1) and (10). The holdup volume formula Z(t) can be expressed as follows:

expressed

as follows: t,,,=a,w:

+cc,w,

tIll,”=a,w:

+a,w,,

+x:w:,

(12) (13)

where a, and a2 = integer,

3.1.

Minimum

and maximum

holdup $or

U, > D2

For the case U, 3 uz, the increases or decreases in the holdup volume are determined by the time of beginning and stopping of inflow from the upstream unit. That is Z(r) is a piecewise linear function with multiple local optima, all of which are corner points with the following properties:

Then from the given value of the cycle time, the frequency of failure, repair time and t,, defined via (13), we can calculate the u,, y, and z, values corresponding to a local minimum point as shown in Table 1 (refer to Appendix B for details) where n =0 n =

if x,+y*<

1 if x2+y2a

1,

alp +a;

1,

alp + a2 2

c I,

1

and

1 p=pi. Note that these compact results require the imposition of the additional assumption: where i,_i = integer. To determine the search domain, we have to investigate the periodicity of V(t). As shown in Appendix A, the holdup in the intermediate storage V(t) is a periodic function with period vessel, R = LCM(o 7, 02)_ As a consequence of this periodicity, the times at which local maximum and minimum values of the holdup volume can occur can be

8. The greatest common measure of the cycle time of hypothetical upstream unit and the cycle Table

I. Variable

I 1

2

j 0

to y, -2 -

and function

j$

I & 0,

values

for minimum

2’;

Y:

ezW’ w:

X?-+Y’l

-%+Yf+*

holdup 4

a,p

+ c&D + n

1238

E. S. LEE and G. V. time

of the downstream 1. (WY, wr)=

unit

is

1. i.e.

GCM

This assumption can be relaxed at the expense of considerable additional algebraic complexity (see Lee, 1988 for details). For the upstream unit, using the reduced variables and relationship (3): ;‘,

-2

7,

,go h(”

-2

[I I- i J

1

:,Y;.z:)=

0:

=

2ci,x:

1 i(l

[

WI WI

x:

01 w:

&f-j--_y:

WI

OJ3

a,.--J--;

,=”

-Y,)S

As derived in Appendix C, h (z+ , yz, z2) value as follows: 1. If a,p

+y

2x,(x,p

z2 ) =

I

can

)I

obtain

+ xi + .Ul)‘~ Finally, choosing the minimum (18), it follows that:

s).

value of either

-x,(1

- kp) - s],

(18) (17) or

(19)

where Zy = min(.y’. px,), s = max(O. x, + ~1~-- 1). By the same strategy as used in the minimum holdup case, we can derive the following expression for maximum holdup which must occur at time I,,, given by expression (12):

the

+ cJ,cu,.w,(y, -

1)

1 -(r,-2+x,)3

iI

1 (20)

where Zp = max(y”

+ y ‘)

-72 )=2(x2

-

Il(a,p

+ 2s.

- p + .qp, O),

(15)

+Y’>

where

y’ is the remainder

when

by P. Let

Z(t-)

minimum

denote

a local

(16)

x2 + yz are divided of I(t)

3 1 V man= min I(r --) = min C z c, U,w$

=I.I*

T,X2,=I

a,p

and

(u, , y, ,

Then from (14) together with (15) and following simplifications can be derived:

z,).

(16)

the

+ y’ -- x, 3 0:

Z(f _) =

- I7~c+x,(r,p

+4”)

+ t7*wz.Kz(xL -tY>)

-

02w2s

I(t -) is an increasing function of q and a decreasing function a,. Therefore let %,=w?-1 and x2=0. Then: V 111111 = u zw,(-q+px~-

x*3” i-Y:

2. For

let

.q_v

L

-2x~(x,+y~)+2~~+2s,

For

+ Y’ -

of xi and CI:. Thus

+ y’ - x2 < 0, then:

W%,Y,,

I.

V n,in= I;izw,(--.x*

function

(14)

- 2.X2(X, fya) 2. If a,p

-

- x2 z 0, then:

h(%,Y2,

I(t -) is an increasing r, = az = 0. Then:

V min= o,w,[z;

2a,s

we

REWLAITIS

cc,p +Y’

-x2

< 0:

f(t -) = - U2w,(x, +

+ xz.yr - s).

- l)(a,p

u~o,x,~x,+y2)

+y’)

- UJWo,X,

(17)

3.21. Rerietv ofstrategY. The analysis for this case is similar in strategy to the previous case but differs in some of the fine details. According to (3), for fixed X, and z, values, only the u; and IQ variables. which represent the hypothetical upstream flow and downstream flow time fraction within the stream cycle time are time dependent variables. In the CT, b c: case, within the search domain [a = LCM(cuT, cuz)] there are 7, - 1 different u; values and each different u\ will appear IZ/LCM(w : ~w2) times. As far as zq is concerned, there exist as many as O/LCM((r, y, u+) different n2 values and each different u2 will appear 7, - I times. One important thing to observe is that each different u’, will match with all of the different u2 values. This indicates that the ?1 and xl variables are independent. i.e. we can sclcct the maximum or minimum r,. ,x2 independently to calculate the holdup volume. In the c’, cz 0, case, the local maximum/minimum point is determined by the downstream flow time and alI u2 values are the same but all of the 14: values are different. Therefore the time variable (m,) which connects the hypothetical upstream flow and downstream flow is a dependent variable. 3.2.2. Minimum and max-imum holdup. In this case, the local maximum and minimum holdup values occur at the time of beginning and end of outflow to the downstream unit. respectively. Thus:

- uzw2s + u,w,x,az 1 +(I

x [

-_yJS I

where

1,,,,, = t, + a3 co2,

(211

~,,rl = r0 + a,uz

(22)

0 < ct3G pz -

1.

+ .Y~W’,

Intermediate storage and operation of periodic processes We can follow a similar procedure as was employed for the U, 2 oz case, but now using equations (1 l), (21) and (22). It can be shown that the Vminand Y max expressions are the same as (17) and (20), respectively, but the variables 2; take on different values:

Z;=max K 3.3.

y’,p.r,],

= min[(z)

Z;

E@ect

g

z>

(23)

1.

(y”--p)+x,p,o

of first failure

to the upstream failure case. We need to define several variables to describe the downstream equipment failure unit, specifically, yz refers to the frequency of downstream equipment failure and d, denotes the repair time of the failed unit. We again represent the downstream unit as a sum of hypothetical flows. Let superscript * refer to the variables for the hypothetical downstream flows. Then the cycle time and fractional flow time of each of the hypothetical flows are:

(24)

wz’=(y*-

k;

x:=x

Under assumption 6, the first equipment failure (k ;) will occur at the y th batch. Suppose we relax this restriction in order to study the effect of k; on holdup volume. For the U, -E 0, case, the local minimum time expression which includes the ky variable becomes:

1239

(27)

1)w2+d1,

(>

2x9

*2

(28)

a2

v; = u*w:x:

= v,.

Also the throttled flow and holdup sions become as follows: 0,x,

= U,(Y, -

volume

expres-

1)x?,

(29)

r,i,=%~:+(%-<)~,, where c = y, The zJ, and y< values minimum point become:

Where j>k’;-

y,=O for 1. Then

kp.

corresponding

j
and

to

a

y,=d,/w:

local

for

y,--2

cos 2n I-I(u< - y’; ) = cos2nI-I

V”(t)= [

+(j+I)+ 1

.

cos2nII(u<--y’,)=cosZaII [

ci3T-j2 w:

w: 1

(26)

From (25) and (26), we can see that the < variable (in other words k;) does not affect the cos 2nII(u< - yji) term. Likewise, the kp variable does not affect the cos 2nII(uj, - z{) terms. So from (2) and (4), we can find that the first equipment failure variable k; has no effect on the size of intermediate storage. It will only have an effect on the initial holdup requirement.

4. I.

DOWNSTREAM

EQUIPMENT

c

,=o

(25)

As cos 2n II has a period of 2n II, we can add w, /w : to (25). Because of periodicity, it is sufficient to merely consider j in the range 0
4.

Of course, Z(r) has period R = LCM(w,, w:). To derive the holdup volume, the most desirable way is to use the results from the upstream equipment failure case. From (1), the general holdup volume expression for the upstream equipment failure case,

FAILURE

Strategy

The strategy for derivation of the holdup volume for the downstream equipment failure case is similar

I s d0

F; (T - t;,,) dr I

-

&(T - tzo) dz.

(31)

i 120 For

the downstream vd(t) =

‘F,(r)dr s0

unit: -“T’ j-0

F’,(r

-

&)dr.

(32)

S’i,

To convert (31) to (32), we have to shift the time argument r to r + 2t2, and then to move the time origin from 0 to r - 2t,,. Finally, interchanging subscripts 1 and 2 and setting Vd(t) = -V”(t), expression (3 1) becomes:

s ho

V”(t)

=

F,(r)dr

0

+

F, (r + 72 -

120)

d7

2

- jTo F',(r - &,) dz.

(33)

The h(ui, yi, z,) expression of the second term in (33) is different from the general expression which is given in (3) except when tz,, = 0. The time expression for local minimum and maximum are also different. Consequently we have to derive holdup volume expressions from the beginning even though the deviation procedure is similar to that of the upstream failure case.

E. S. LEE and G. V. REKLAITIS

1240 4.2.

where

Holdup

4.2.1. For iTr,> U, case. maximum time expressions

The local become:

minimum

and

Z;J = min[($)_vf,pr,].

u,

Z$ = max

t Ill,”=a,w,

li

+x,w,, S’=max

where

For minimum holdup, u1 = 0 which is the same as the normal case (no equipment or batch failure) as far as the first term of (30) is concerned. Let p” = l/w: and .x2 + yZ = k (1 /us) + y ‘. then the second term of (30) is given as follows: 1. For

0
h(u$,y’z,z:)=

-_I@jfl@

+x:1

where

2. For

a, > (yZ - 2)0,

q

!

(.Y”---p)

+ps,.

0

1,

!

For details, set Lee (1988). In 4.2.2. For 8_,< lY_, mw. minimum and maximum time

this case, expressions

the local become:

rmin= r, n1; + CC2 CO,+ .Y?(0, + tz” ) + J’(c+,/c~~) for miniFor this case, U, = a,( I /CD~ mum holdupand U, = ~,(l/to,)+ (.y’+ S)(cu,jw,) for maximim holdup, which is the same as in the normal case. On the other hand, u> = (CC:+ .x~ -i _I*~)((u~/u~) for the minimum holdup and ui = (ctz$- yl) (co: ‘w r ) for the maximunl holdup, tllat is, x1 and X~ are independent variables. ‘Thus wc can derive the second term of (30) using the same procedure as in the 0, > U, case. We find that the minimum and maximum holdup volumes arc the same as given in (34) and (35). rcspcctivcly. but again different Zy and Zi values must be defined: z:

= rnirl[J~‘, p.\-,I.

Z!

= max[l.”

- 1’ +p_~-, , 0]

+ d: 5.

+c+2x-G-1. If we use these equations in (30), we can find that with respect to the a, values there are seven monotonically increasing and decreasing regions within the search domain R. In each region, we must select the appropriate CL~values to minimize r(t). Finally, by comparing these local minimum expressions, we can determine the global minimum expressions. For the maximum holdup case, U, = x1 and again the first term of (30) is the same as the normal case. As far as the second term is concerned, we can apply a similar strategy to that which we employed for the minimum holdup case. In this fashion we obtain the following expressions:

UP

AND

DOWNSTREAM

EQUIPMENT

FAILURE

The up and downstream equipment failure case is the most general case of equipment failure. For this case, we also assume that there are (y, --- if hypothetical upstream flows and (yZ - 1) hypothetical downstream flows. This corresponds to the (7, - 1) - (YZ - 1) system with equal delay times. The cycle time and fractional flow time of each hypothetical stream is given by (5). (h), (27) and (28). Also the throttled flow rates of upstream and downstream units are:

Finally, follows:

the holdup

volume

expressions

is given

as

If we compare (27) with (31) and (32), we find that the first term of (37) is the same as the first term in the upstream failure case and the last term of (37) is the same as the second term in the downstream failure case. Thus we can apply the results obtained

Intermediate storage and operation of periodic processes from the analysis of the upstream equipment failure case and the downstream equipment failure case to the combined case. Within the search domain w:)], we obtain the following maximum N=(W:, and minimum volume expressions: v,i,=~*2W*[Z~d+X:tYZ-I)(kp

+Yr-2) - x2&

-

I) -

~‘1, (38)

V nIPI =~,w,(r,-I)[x,--x:(y,--++x,)l

+ UZWJZ,“”+ xf(yz For

For

l)(kp

+ S) -S’]*

(39)

U, 3 0>2: Zyd = min[y’,

pxf

(yz -

l)],

Zy” = max[y”

--p

+px:(y,

-

l), 01.

_ 1 Kg2>Y,Px:(Y,> 1) 1 1. K-2>

o,
Zud = min

Zzd = max

g

(Y”-~)+px:(y,-l),

0

These holdup volume expressions can readily be applied to multistage serial systems through the following simple modification. First, any general multistage serial system can be collapsed to a series of l-l subsystems by combining batch trains located between storage vessels. Then we can apply (38) and (39) successively to each of these l-1 subsystems, providing that the throttled flow rates of each upstream subsystem are used and updated in the immediately following subsystem.

1241

Reklaitis (1985) and Z’*(y)I,, = a denotes the size for the one batch failure case. Then by applying (19) and (20) with y = cc and d, = w, , the volume requirement for normal operation will be V* = 1458 kg, V(0) = 0. If we consider batch failure, we can apply the upstream equipment failure results with d, = oi. In this case, we have to throttle the downstream rate to i?& = 182 kg h-l. The volume requirement for the batch failure case will be V* = 2589 kg and Y(0) = 0 kg. For the upstream equipment failure case, from the given data, yi = 8 and dr = 15 h and the hypothetical cycle time and fractional flow time of the upstream unit become w: = 78 h and x: = l/78. The throttled flowrate of the downstream unit becomes uz = 168 kg h-l. From (19) and (20) the holdup volume and the required initial holdup become V* = 3200 kg and V(0) = 0, respectively. For the downstream equipment failure case, y2 = 7 and d2 = 7 h. Then from (27) and (28), w: = 37 h and x; = 4/37, from (29), (34) and (35) US = 257 kg h-‘, V* = 2576 kg and V(0) = 1038 kg. Finally, for the combined upstream and downstream equipment failure case, from (36) U, = 208 kg h-‘. The volume required for the combined case will be V* = 4100 kg, with V(0) = 840 kg. Comparing the normal case with the combined case, the downstream flow rate is the same but the operating time is less than for the normal case due to the failure of equipment. Consequently, the total production is less than for the normal case. However, the required volume is increased by three times and also an initial holdup is required. 7. CONCLUSIONS

6. EXAMPLE

Assume that for the upstream unit, the time for filling with raw material is 2 h, the reaction time is 6 h and the discharge time is 1 h. Discharge to the downstream unit starts 1.5 h after discharge from the upstream unit. Assume it takes 4 h for discharge to the downstream unit and after a 1 h interval, discharge will start again. Suppose that the upstream unit will fail on average every 8th batch and it will take 15 h to repair that unit. On the other hand, for the downstream unit, the equipment will fail on average every 7th batch and it will take 7 h to repair. Finally, suppose the flowrate of the upstream batch transfer pump will be 1500 kg h-i. Then, from the given data, k’, = 1500 kg, w, = 9 h, w2 = 5 h, y, = 0.3, X, = 4, x,=0.8. If there is no failure, the flowrate of the downstream unit will become U, = U,(x,x,)= 208 kg h- ‘. The size of the intermediate storage tank for normal operation can be obtained from the upstream equipment failure equations by using: k-*(Y)

Here normal

V*(Y)

I;.,= cc =

denotes inor

operations

which

v*(Y)

lnDr+ v,

the size of the tank for was derived by Karimi and

A comprehensive analysis of the deterministic periodically operated l-l system subject to equipment failure was presented. The analytical expressions for the limiting volume for upstream equipment failure, downstream equipment failure and the combined case are quite simple and can be useful for design and for simulation purposes. The expressions for the upstream equipment failure case include two special cases. namely, the results of the batch failure case which were reported earlier and the normal nonfailure case. The effects of the timing of the first equipment failure on the intermediate storage volume was investigated and was shown to only affect the initial holdup requirements. This result is important for any subsequent variability analysis for the 1-l system. The results for the 1-l system are readily applicable to a serial system providing that appropriately throttled flows are used throughout the process. NOMENCLATURE c, = Coefficient of unit i 4 = Repair time of unit i F = Flow rate function of a batch unit.

E. S. LEE and G. V. REKLAITIS

1242 h(u,,y,,

z,) = Function defined in equation (1) I(r) = Function defined in equation (1) i = Integer variable j = Integer variable k; = Index of first equipment failure flow in upstream how n = Integer value defined in Table 1 p = Defined as l/p, for equipment failure 1 pfl =w,* .s = s’ = ! = 1, = I~ = lp =

Variable defined in equation (19) Variable defined in equation (34) Time Time required to discharge a batch unit Time required to filling a batch Time required to process a batch tz. = Starting time of outflow to the downstream c, = Input or output flow rate of a unit i U, = Throttled flow rate of a unit i

U< = Variable as

if

hypothetical

jth

stream

defined

mod V, I/* x, x*

= = = =

y’ y” y; y’,

= = = =

z = z{ =

Greek

letters z, = Integer variable a’ = Variable defined in Table 1 u” = Dummy variable p, L Characteristic integer of a unit i of failure of unit i ;‘, = Frequency 6 = Variable defined in equation (20) [zy=ki R = Period of holdup variations w, = Cycle time of unit i streams o* = Cycle time of hypothetical

Subscripts i,] max min *

and = = = =

APPENDIX Proof of PeriodicitJ~

A qf‘ V[r)

From (1 I). I(Z) is a function uf N,, _r, and z,. From assumption 7, there exists fi, such that:

Then,

Batch size of a unit i Intermediate storage size Fractional flow time of unit i Fractional flow of hypothetical streams of unit i Value defined in Appendix C Value defined in equation (20) Fractional delay time of a unit i Fractional delay time of hypothetical jth stream for unit i Variable for equipment failure defined in equation (23) and (24) mnd(x, + y,, 1) Variable of hypothetical jth stream defined as mod(s; +y’,, 1)

Zy L

Karimi I. A. and G. V. Reklaitis, Intermediate storage in noncontinuous process involving stages of parallel units. AIChE JI 31, 44-52 (1985). Intermediate storage and Lee E. S. and G. V. Reklaitis, availability of periodic processes under batch failure. Comput. chm,. Errgng 13, 491~498 (19X9). Lee E. S., Intermediate storage and availability of periodic process. Ph.D. Thesis. Purdue IJniversity. West Lafayette (1988). Oi K., H. Itoh and 1. Muchi, improvement of operational flexibility of hatch units by a design margin. Conlpul. them. Engng 3, 1777184 (1979). Takamatsu T.. L. Hashimoto, S. Hascbe and M. Oshima, Design of a flexible batch process with intermediate storage tanks. Ind. Engng Cliem. Procrrs DP.F. Del-. 23, 40-a (1984).

from

R = LCM(v>;.

(0, ) = p,
the definition

of u,.

Since 2, and ?;, have a fixed value. I(r) = 1(t + a). So I(f) is a periodic function whose period is il. APPENDIX

B

There are 7, - 1 hypothetical upstream streams and each such stream has the equal delay time property. Consquently the fractional delay time of the .jth hypothetical upstream unit becomes

From the definition of z,. z; = mod(s: and 0 _(.j & 7, -- 2. 0 < x, +J O
+ .v’,, 1). AS < :‘, ~ 1 and

;‘I = x: + J; = (8, + j) ;+. of unit i

From

the definition

of i+:

superscripts

Unit ij Maximum Minimum Hypothetical

streams

in supported Acknowledgements --This research was part by the National Science Foundations under Grant No. CBT-85 18175 and by the Korea Steel Chemical Company. REFERENCES

Karimi I. A. and G. V. Reklaitis, Optimal selection of intermediate storage tank capacity in a periodic batch/ semicontinuous process. AIChE JI 29, 588-596 (1983).

As a, has a variation range of O~. Therefore, jl, = +. If we define r; = mod(oczw,. 8, ), then a,(to, :j3,) = j/j, + r;. wherej is an integer. If we define p r 1‘B, then: u2 = mod[r + z,p i j + cr:p. I] = r,p + CL;/)+ n.

Intermediate APPENDIX Proof

storage and operation

of periodic

processes

1243

As CC;’is a dummy variable, cry can be replaced Furthermore, as 0 < CQJI+ y’ -z 1, it follows that:

C

of (15) and (16)

[cos 2nrI(u, - z*) - cos 2nrI(z,) h&3 YzZ2) = F I “_, n2rI*

by OL,.

5 Icos2n~(u,-y,)=~-~(n,p+y’)+(a,p+y’)~. “_, nw By the same procedure:

- cos 2n fI(U, - y*) + cos 2n rIyJ. Let x,+y,=kp+y’, then y,-kp+y’-xx,, p = l//3,. 0 c y’ -Cp and k is integer. Then: cos 2nTI(u, ~ z>) = cos 2nII[(cr, -k

where

+ a;)p -y'].

As the cosine has a period of 2nlT, integer n’ ~ n can be eliminated and eI - k + a; can be merged with a;. As a; = -p,, for O<~~;
tu’)

-_la,p +Y’--21

+t%P

+Y'--x,Y, we can

Since zr takes on the values x2 + y, or x2 + y, - 1, easily determine that:

“Z, A

I

---iCOS2n~z,=6-~X2+~*)+(X*+y~)*-22S,

a’ 1 1 c --ccos2nrIy,=~-~*+,v:, .=,n’rv where s = max(O, x2 + y, - 1). From (2) together with the above equations, we obtain (15) and (16).