Ind. Eng. Chem. Fundam. 1983, 22, 206-208
206
w = denotes the values of the cooling water Registry No. Propane, 74-98-6.
Literature Cited
"Kagaku Kogaku Benran"; Kagaku Kogaku Kyokai, Maruzen: Tokyo, 1958; p 54. Leutner, H. W.; Stokes, G. S. Ind. Eng. Chem. 1961, 53, 341-342. Matsuda, 8.; Katagiri, Y.; Terasawa, S. Sekiyu Gakkai Shi 1966, 11, 186-189. .. ..
Anderson, J. E.: Case, L. K., Ind. Eng. Chem. Process Des. Dev. 1962, 1, 161-165. Durand, J. L.; Brandmaier, H. E. I n "Kinetics, Equilibria, and Performance of High Temperature System, Proceedings, 2nd Conference", Bahn, G. S., Ed.; Gordon and Breach Science: New York and London, 1963; p 115. Freeman, M. P.; Skrivan, J. F. AIChE J . 1962, 8 , 450-454. Hougen, 0. A.; Watson, K. M.; Ragatz, R. A. "Chemical Process Principles"; Wiley: New York. 1954; p 306.
Shirotsuka, T.; Hirata, A.; Kawasaki, K.; Chao, MwKen Kagaku Kogaku 1969, 33, 662-668. Takahashi, T.; Nakashio, F. Kagaku Kogaku 1972, 3 6 , 774-781.
Received for review January 28, 1982 Revised manuscript received January 12, 1983 Accepted January 19, 1983
Intermediate Tank as a Way of Increasing Plant Availability Norbert0 Plcclnlnl* and Ugo Anatra Dipatiimento di Scienza dei Materiali e Ingegnerh Chimica, Politecnico di Torino, 10129 Torino, Ifah
If the availability A and repair time 7 of an upstream unit U and a downstream unit D are equal ( A = A D and the two units will probably break down the same number of times in a given period. It can readily be deduced that a tank installed between them will increase the overall availability and must be kept half-full. If, however, A = 1 and A D C 1, the tank should be empty: if A C 1 and A D = 1, it should be full. In all other cases,its level cannot be immediately defined, and in any event its optimum size and the desirability of its installation in economic terms must be established. These three questions are closely interrelated. They are examined in analytical terms in the present paper. In particular, equations are presented for the determination of tank size and filling level as a function of the two parameters, increased availability and increased cost. 7u = 7D),
Introduction Over the past few decades the desire to achieve economies of scale has led to the building of increasingly large chemical and petrochemical plants. These, in turn, have been increasingly integrated, with the result that the same premises may house not only different manufacturing processes but also the raw materials for its various sections. It is clear that a mandatory shutdown of one or more of such processes will lead to considerable economic losses owing to the quantities involved. Studies have therefore been directed to the question of plant availability and maximum utilization (Cherry et al., 1978; Henley and Hoshino, 1977). In the case of complex installations, of course, to increase availability beyond a certain point may require modifications that would involve prohibitive costs. This paper shows that the insertion of a tank between two plants connected in series can enhance the availability of the complex as a whole. In particular, the calculations required to dimension such a tank so as to optimize both availability and costs are described. Increased availability through the installation of a storage tank has not received very much attention in the literature (Henley and Hoshino, 1977; Ross, 1973). The only paper whose methodology bears on the present discussion is that of Henley and Hoshino (1977). Storage T a n k s i n Chemical Plants Attention will be solely directed to upstream and downstream units, U and D, connected in series with output flows Fu and F D , respectively. Installation of an intermediate tank between the two will be desirable when it is intended to increase the availability of the entire system if, as often happens, the availability of U and/or D is less than unity. A tank can thus decrease plant unavailability, Q. For a repairable system, this is defined so 0196-431 318311022-0206$01.50/0
that Q/(1- Q) is the ratio of expected downtime to uptime. Breakdown of Plant U. Si-opose that plant U goes out of action and no longer sends a flow to D. If this flow is indispensable for production, the whole system will become unavailable. To prevent this eventuality, it is merely necessary to interpose a permanently full tank whose capacity is such that it can feed D for as long as U is out of operation. The tank will, of course, tend to empty. It will only be restored to its original level if D breaks down, or in some independent manner, for example by increasing the capacity of U. Breakdown of Plant D. Now assume that D breaks down. In this case, U must also be shut down, and with it all the parta upstream, since D is no longer able to andle output flow Fu. Production must thus be brought to a halt. To prevent this, an empty tank is interposed to take the flow from U. This time the tank will tend to fill. It will only be emptied when U breaks down, or in some independent manner, e.g., by increasing the capacity of D. Extreme Cases. If availabilities A U , A D and repair times 7u, 7D are equal, the two plants will break down the same number of times on average over a given period. In this case, it is obvious that the storage tank, irrespective of its size, must be kept half full. In addition, if AU = 1, and A D < 1, the tank must be kept completely empty, whereas it must be always full when AU < 1and AD = 1. In all intermediate cases, it cannot immediately be said where the level of the tank should be set. Nevertheless, the question of its optimum dimension, and whether or not its installation is justified, economically speaking, can be examined. These three questions are closely interlinked. Influence of a T a n k on System Availability We can first examine the decrease in unavailability Qs, brought about by the insertion of a tank (Figure 1). Three 0 1983 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 22, No. 2, 1983 207
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s2 Figure 2. Two units with intermediate tank A" = 1 and A D < 1. W2 subsystem comprising downstream unit D and tank T.
situations must be considered (I) AU < 1 and AD = 1; (11) AU = 1 and AD < 1; and (111) AU < 1 and AD < 1. (I) A < 1 and A = 1. Here the tank augments the availability of U. Plant U and tank T can be regarded as a single complex, W1, with F T as the T output flow rate. Kwl is the probability that Fu will not be reinstated in a time interval Bu. Assuming, as usual, an exponential distribution for the Fu reinstatement time (Henley and Hoshino, 1977; Rasmussen, 1975), Kwl is given by
Kw1= exp(
-2)
= exp(
-FTTU E)
Y
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-
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..- - - - - - WI
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- 1
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s1 Figure 1. Two units with intermediate tank: A" < 1 and AD = 1; W1 subsystem comprising upstream unit U and tank T.
/
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----
,
--
W2
,
kJ 5'
/
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S3 Figure 3. Two units with intermediate t a n k A" < 1 and A D < 1.
(111) A u < and AD < 1. Since plant availability is not usually equal to unity, expressions 1 and 9 are insufficient, and in general as opposed to extreme cases they must be applied simultaneously. In Figure 3, their individual importance will depend on whether the availability of U in greater or less than that of D. To facilitate the calculation, the Qsl and Qs2expressions can be rewritten in terms of availability. So as to determine the total availability As3
we can readily calculate the increase in availability provided by tank insertion AA = (1 - Kwi)X + (1 - KwJY = X + Y -KwiX - Kw2Y (13) where
(1)
where V = tank volume and f = degree of ffing. It follows that
Qwi = KwiQu (2) If X is used to indicate the failure rates, the following relationships are also valid rw1 =
Tu
Awl = Xu exp(
-2)
(3) (4)
For the complete system S1 (= U + D + T), we have (5) Qsi = KwiQu + QD - Kw~QuQD = + AD (6) (7) si = O w i T w i + ADTD)/(AWI +AD) Since Kwl is less than unity by virtue of (l),it follows that Qsi < Qs (8) = 1 and A D < 1. Here T increases the avail(11) A ability of D (Figure 2), W2 is the single complex D + T, and Kw2the probability that F D will not be reinstated in
In effect, eq 1,9,14, and 15 show that the AA is more than zero; i.e., insertion of the tank leads to an actual increase in availability. Expressions 1,9, and 13 are applicable when T is filled to the extent f V. This will be the case when the mean time between failures (MTBF) of either U or D is much longer than the time required to refill or empty T when function is restored. If the tank is not immediately brought back to the preset level, f itself will vary according to the failure frequency of U and D. Kwl and Kw2will then be calculated as follows. The ratio between U and D failures is calculated on the assumption that (MTBF)Dis greater than (MTBF),, from
In statistical terms, U should fail z1 times before D fails. QW1,therefore, passes from
0,. Assuming an exponential distribution for TD as before, we have
prior to the first U failure, to Qw2
= KWPQD
(10)
As in the first case, the same conclusion is reached Qsz Qs (11)
prior to the first D failure. The mean unavailability can be expressed in the form
208
Ind. Eng. Chem. Fundam., Vol. 22, No. 2, 1983
Mutatis mutandis, when (MTBF)U is greater than (MTBF)D, we can write
(20) In other words, if T is brought back to its initial level only when the second unit fails, expressions 18 and 20 take the place of expressions 2 and 10. Cost-Effectiveness of the Insertion of a Storage Tank An increase in A means higher production, and a larger profit. On the other hand, the cost of a tank influences the cost of the system as a whole. If t is taken as the planned production of the system per year and P the profit per m3 of product and C is the annual depreciation of the tank per m3, the greatest benefit will obviously be obtained by maximizing the expression G = (X + Y - KwiX - Kw2Y)tP- CV (21) Differentiation with respect to V and f will give the Vo and fo values for which (21) is maximal
This system of transcendental equations, however, does not permit immediate evaluation of the pros and cons of installing the tank. It may thus be more practical to employ eq 22 and 23 suitably re-elaborated to allow them to be displayed graphically. An example is given in Figure 4 of two cases in which input data for AD,Au, TD, TU,y d F T were used to correlate f and V with profit, tP, and gam obtainable, G. More precisely the diagram can be drawn by considering, for example, the following values of the parameters in eq 22 and 23: AU = 0.7; AD = 0.9; Fu = F T = 10 m3/h; TU = TD = 10 h; C = $50 mT3. A generic value of V (say 115 m3) can be put forward for insertion in eq 22 in an explicit form. The degree of Tiling (f, is found to be 0.57. Insertion of these two values in eq 23 permits determination of tP value ($2.2 X 106) that satisfies the equation itself. Calculation of G is obviously carried out with eq 21. The result is G = $3 X lo4. The interpretation of these figures is as follows: in the case of a plant whose profit is $2.2 X lo6,it is advisable to insert a storage tank with an optimum volume of 115 m3 and a degree of filling of 0.57. The resulting gain will be $6.5 X 104. Conclusions It is clear that insertion of a storage tank between two units connected in series can augment the availability of
Pitfit,
$
Dclrtc
tf
liliiil
Figure 4. Correlation between degree of filling and tank volume as a function of profit (tP),and gain obtainable ( G ) . In both cases TU = rD = 10 h; Fu = FT = 1 m3/h;C = $50 m-3. The broken lines represent losses.
the entire system. This intuitive idea has been developed analytically. Eguations that can be employed to dimension such a tank and its degree of filling, as a function of both availability and cost, have also been developed. Acknpw ledgment The authors wish to thank Dr. D. Barone for having suggested the subject of this study. Nomenclature A = plant availability C = tank cost D = downstream unit F = flow rate f = degree of filling G = gain K = probability of no flow repair in time interval 0 K = mean probability of no flow repair in time interval 0 MTBF = mean time between failures P = profit Q = plant unavailability Q = mean plant unavailability S = whole plant comprising U and D units S1 = layout in Figure 1 S2 = layout in Figure 2 S3 = layout in Figure 3 T = tank t = outputfyear U = upstream unit V = tank volume W1, W2 = subsystem U + T (Figure 1); subsystem D + T (Figure 2) X , Y = coefficient defined by relationships 14 and 15 zl,zz,=coefficient defined by relationships 16 and 19 0 = time interval h = failure rate 7 = time to repair Literature Cited Cherry, D. H.; Grogan, J. C.; Holmes, W. A,; Perris, F. A. Chem. Eng. Progr. w a , 7 4 , 55. Heniey, E. J.; Hoshino, H. Ind. Eng. Chem. Fundam. 1977, 18, 439. Rasmussen, N. C. Nuclear Regulatory Commission, NRC Report No. WASH 1400, NUREG 751014, Washington, DC, 1975. Ross, R. C. Hydrocarbon Process. 1973, 55(8),75.
Received for review February 8, Accepted January 3,
1982 1983
