Minimization of Capital Investment for Batch Processes Calculation of Optimum Equipment Sizes Yashwant R. Loonkar and Jerome D. Robinson American Cyanamid Co., W a y n e , N . J . 07470
A procedure is described for calculating the sizes of interdependent pieces of equipment which minimize the capital investment of a batch processing plant. Power-law-type cost expressions are presented for both true batch a n d semicontinuous units a n d combined to form an over-all objective function. Ordinary calculus i s used to establish necessary conditions for optimum equipment sizes. These conditions lead t o a set of one-dimensional algebraic equations, solvable by simple iterative techniques. A sample calculation illustrates the procedure and demonstrates the savings realizeable in equipment costs.
A
major objective in the design of processing plants is to meet specified monthly or yearly production rates at a minimum initial capital investment. For maximum profitability, raw material and operating costs must also be minimized. This paper deals with the first aspect for batch processing, the minimization of capital investment. In the design of batch plants, the selection of equipment sizes which minimize the initial capital investment is made difficult by interrelations between many pieces of equipment. Consider a simple example of a batch reactor with feed and discharge pumps. The production capacity depends not only on the reactor size and residence time but also on the pumping time required for feeding and discharge. For a given production capacity, a larger reactor is required when very small pumps are specified; the reactor size is reduced if larger pumps are used. Obviously, a balancing of the sizes is required to obtain the desired throughput a t a minimum equipment cost. The logistics of a batch plant can be complex and may include overlapping cycles, multiple use of individual equipment in each cycle, and parallel units for certain sections of the plant. There are also “job shop” plants, in which several products are produced using the same equipment. I n the analysis presented here, we consider only batch plants used to make just one product. The special cases of overlapping batch cycles, multiple use of equipment, and parallel units are also analyzed. We define the following terms frequently referred to in this paper. True Batch Equipment. Equipment specified by volume and not throughput or processing rate-e.g., reactors, tanks, tray dryers. Semicontinuous Equipment. Equipment specified by throughput or processing rate-e.g., pumps, heat exchangers. Semicontinuous Subtrain. A combination of more than one semicontinuous unit used in series. All equipment included in semicontinuous train must operate for the same length of time.
Consider a general batch processing train which is used to make one product. Such a train has M true batch pieces of equipment and N semicontinuous pieces. The over-all batch cycle time, T , is made up of the sum of residence times in true batch equipment and the time required for various semicontinuous subtrains. If there are R semicontinuous trains and M true batch units, the over-all batch or cycle time is 34 n
T=
c t,+E
1 = 1
8,
- 1
The production capacity of the plant is also related to the product volume per batch and the over-all batch time.
W = ViT I t follows, therefore, that a knowledge of plant production capacity, W , and individual batch times (all t, and 0,) completely determines the equipment sizes. V is not necessarily equal to the volume handled by various pieces of equipment. All batch times (ti and 0!), however, are based on actual volumes being handled a t various stages of the process. The actual volumes handled are proportional to the product batch volume, V; the proportionality constants are determined by particular “recipes” and physical properties of process fluids. Figure 1 shows a typical batch train. There are three true batch units, five semicontinuous units, and three semicontinuous subtrains ( M = 3, N = 5 , R = 3). Generalized Cost Expressions
Expressions for the installed cost for each piece of equipment are required for minimization of plant equipment cost. Such expressions are readily obtained using the exponent rule for cost estimation (Bauman, 1964). The resulting cost expressions may be classified into one of two forms. (Specific examples of these expressions are given in the Appendix.) True Batch Units
c, = a , V a ‘
(2)
Ind. Eng. Chern. Process Des. Develop., Vol. 9, No. 4, 1970 625
Semicontinuous Units
Ck = b k (V/&)”
the class of problems solvable by geometric programming (Duffin et al., 1967). However, the “degree of difficulty” for this problem is 2M + N - 2, a very large number in general. Geometric programming loses its advantages as the degree of difficulty becomes large. The analysis presented here is based on ordinary calculus. Solving Equation 6 for V and substituting into Equation 5 yields
(3)
Here, is the time associated with the kth semicontinuous unit. The N Bk’s are not all independent, since there N ) with R are only R semicontinuous subtrains ( R independent 8,’s-that is, -
0 k = & (0,) For example, the batch train shown in Figure 1 has
01
02 -
= 01 =
(4)
v
’
Let
02
83 = 82
04 85
= 03 = 03
Then
Statement of Problem
Given a “recipe” specifying a sequence of operations (including residence times, etc.) for a batch plant, and a preliminary process design of various pieces of equipment, find the equipment sizes which minimize the total installed equipment cost of the plant. The plant must produce a specified quantity of product per time periode.g., cubic feet per year. The total equipment cost for a batch plant is given as
I =
N
M
M
.V
t = 1
k = l
ca,VuL+E b k ( V / & ) P k
T o establish necessary conditions for a minimum in I ,
set
aI _ -0,
L=l,2,...,R
yielding
(5)
Since the production rate is fixed, the following equality constraint must be satisfied. M i = 1
Note that
R
W = V / T = V/(c t,
+E8,)
-
(6)
a0k 801,
, = 1
The problem then is to minimize I by choice of R 8,’s with Equation 6 satisfied. (The “true batch” residence times, tr’s, are fixed by the recipe and cannot, in general, be adjusted by the design engineer.)
- (1, 0,
e, #
Let
Analysis
The multivariable minimization problem defined above is most easily solved through calculus. I t also falls into
t FROM STORAGE
_-
PUMP e,
PUMP
e,.
8,.
01
HEAT EXCHANGER
-
REACTOR
8,.
11
81
FEED TANK t2
TRAY DRYER
t3
Figure 1. Typical batch train
626
Ind. Eng. Chem. Process Des. Develop., Vol. 9,No. 4, 1970
0L
Equation 9 is now rewritten as M
R
Since the right-hand side of Equation 12 (or 9) is independent of L ,
A , = A I = A S= . . . . . = A (a constant) Equation 12 can now be written as M
Step 2. Calculate the R BL's from Equation 16. Step 3. Add up the 8 ~ ' sand t,'s to get a new T ( T ' ) and compare with the original guess ( T O ) . If 1 T' - T O 1 2 6 , where t is a prescribed convergence criterion, T o is corrected by some suitable procedure and steps 1 and through 3 are repeated. If IT' - Tal < 6 , the problem is solved. Since the left-hand side of Equation 16 will usually have more than one term containing O L , a numerical procedure will be needed for the second step above. For example, the expression for o2 of Figure 1 is
b2TP202-P2-+ b3TB3&-P3-= A R
An efficient procedure both for correcting T o and for the second step above is outlined by Muller (1956). The procedure uses a second-degree Lagrangian polynomial to fit the last three function iterations. The root of the polynomial is then used as a next guess. Convergence is generally very fast.
or .M
R
Illustration
Since R
T L = l
Consider the batch train shown in Figure 1. Assume that the required production rate is 50 cu feet per hour and that the cost expressions (in dollars) in the range of interest are
41
OL=C t, , = l
then 41
M
, = I
t = l
Combining Equations 11 and 15
L = 1, 2, . . . , R Equation 16 represents a set of equations that are satisfied when the objective function, I , is a t a minimum. Solution
Special Case. For the special case where a , = a constant (usually 0.6), Equation 16 reduces to
@k
= y,
(17) L = 1, 2 , . . . , R The optimal semicontinuous subtrain times, oL's, are determined explicitly from Equation 17. With these times, the optimal sizes of all pieces of equipment for the entire plant are fixed. The particular situation in which the individual equipment cost expressions may be considered linear in the region of interest (y = 1) has been analyzed (Ketner, 1960). General Case. Because of the nonlinearities in the equation represented by 16, an explicit solution for the optimal oL's is not attainable. However, Equation 16 can readily be solved numerically (all equations are one dimensional). An algorithm for doing so is as follows: Step 1. Guess T ( T o ) and calculate A according to Equation 15.
The fixed times are given in Table I along with initial guesses of the 0,'s and the optimal values calculated using the set of equations represented by 16. Also included are the total batch times, volumes, and total costs. As can be seen, the net saving in equipment cost is $46 x lo', or approximately one third of the initial estimate. Since one cannot generally expect to buy pumps, autoclaves, etc., in odd sizes, the "optimal" choices are not always feasible. However, the optimal sizes serve as a guide and the standard sizes nearest these are chosen. The total cost of the standard units will be close to the calculated minimum and the savings will remain substantial. Discussion
A general equipment cost model for batch processing plants has been defined and used to obtain expressions ~~~~
Table I. Results of Illustrative Example Initial Guess, Hr
3
ti
3
1 6
t? t3 81
02 Bi
Total batch time, T Volume) batch, V Total Equipment cost, I
0.26 0.50 0.50 11.25 568 ft' $162 x
Calculated Optimal Values, Hr
1
io3
6 0.106 1.40 3.29 14.80 740 f t ' $116 x 10'
Ind. Eng. Chern. Process Des. Develop., Vol. 9, No. 4,
1970 627
for optimal equipment sizes. The optimal sizes are determined by solving these expressions using one-dimensional iterative procedures. Although the “optimal” sizes will not generally be available as standard items, they serve as a guide for choosing such items. The “cost savings” are appreciable and in general the optimal choice of sizes does not adversely affect the cost of operation. In the development of the cost model and subsequent analysis, no mention was made of overlapping batch cycles, multiple use of equipment in a cycle, or parallel units. The particular case of overlapping batch cycles is readily solved, as demonstrated below. Divide the over-all batch plant into several small sections called subbatch trains, such that the cycle time for a subbatch train is independent of the cycle times of the train immediately preceding and immediately following it. Then,
T = rnax
(T
Appendix
Expressions for the installed cost of three specific pieces of equipment are developed t o demonstrate the applicability of Equations 2 and 3. Reactor and Tank Costs. Using the exponent rule, the cost of a reactor (or tank) is given by
CR = C R h (VR V40R
(-4-1)
with
VR = FRV Hence,
}
where 2’ = over-all batch cycle time and T , = cycle time for j t h subbatch train. The optimum sizes of semicontinuous equipment are determined by the relation: -
The cost expressions for semicontinuous units are handled in exactly the same fashion as for the true batch units.
Pump Cost. Pump costs are given by
(-4-4)
i
t,
0 = T , = I
where 19, = optimum time for each semicontinuous equipment in the j t h subbatch train, t, = time for the ith true batch equipment in the j t h subbatch train, and S = number of true batch equipment in the j t h subbatch train. The results of this analysis are also valid for batch trains in which a particular unit is used for more than one operation. The cycle time must include the time of operation of each unit. However, if the units are numbered sequentially according to use in the cost model, the second and subsequent times the unit appears, it is given a zero cost coefficient. Also, whenever more than one semicontinuous subtrain uses the same semibatch unit, the times associated with these subtrains must be equal. For example, consider the system shown in Figure 1. Assume that the same pump will serve to feed the reactor and to empty the feed tank. Then, bi = 0 82 = B ,
Here, X = head x (volume to be pumpeditime required for pumping) x (conversion factor/efficiency). Assuming constant efficiency and for required delivery of constant head,
x
a
(VPIBPI
(A-5)
Since
Vp = FpV we have
X
a
(V’Bi.)
Then
Heat Exchanger Cost. Heat exchanger costs are given by CH = cH.9 ( A H / A H S ) @ ~ (-4-7) The area of an exchanger can be estimated as
R=2
(-4-8)
and the nonzero elements of the array a & / d B i are
Parallel units in batch plants can also be handled by the analysis presented. All that is required is modification of the cost coefficient for each unit in parallel (a, or b b ) . For example, suppose there are to be P identical reactors in parallel rather than the one shown in Figure 1. In our analysis, the P reactors in parallel are counted as one true batch unit with one true batch time, t,. However, the cost expression for the unit is the cost for all P reactors and is given by
P a , ( V PIffb or a: V f f ’ where
a; = pl 628
ffi
Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 4,1970
where Q = heat transferred per batch, U = over-all heat transfer coefficient, and AT = mean temperature driving force in the exchanger. Since
V H= FHV
Exponents CYR, B p , B H , etc., are evaluated in the neighborhood of V R ~ X s, , and AH\, respectively. Acknowledgment
The authors are grateful to their colleague, M. R. Newberger, for his help in writing this paper.
Nomenclature
a, a, A AH -
I M N R
cost coefficient for ith true batch unit
a,W”‘ constant defined by Equation 15 heat transfer area of exchanger cost coefficient for hth semicontinuous unit bkWPk cost of exchanger of A Hft2 cost of exchanger of A H Sft2 cost of ith unit cost of pump and drive of X hp cost of pump and drive of XShp cost of reactor of VR ft’ cost of reactor of VS ft3 proportionality constants relating product volume and volume handled by heat exchanger, pump, reactor, respectively total equipment cost for batch plant number of true batch units number of semicontinuous units number of semicontinuous subtrains
t , = residence time for ith true batch unit T = over-all batch cycle time
v = product volume per batch w = plant production capacity ffi
OL 7,
= cost exponent for ith true batch unit
cost exponent for hth semicontinuous unit time associated with hth semicontinuous unit = time required for Lth semicontinuous subtrain = cycle time for j t h subbatch train
Literature Cited
Bauman, H. C., “Fundamentals of Cost Engineering in the Chemical Industry,” pp. 42-76, Reinhold, New York, 1964. Duffin, J. R., Peterson, E. L., Zener, C. M., “Geometric Programming,” pp. 11, 83, Wiley, New York, 1967. Ketner, S., American Cyanamid Co., unpublished work, 1960. Muller, D. E., M a t h . Table A i d s C o m p . 10, 208 (1956). RECEIVED for review November 24, 1969 ACCEPTED May 8, 1970
Flow Pulsation Generator for Pilot-Scale Studies Charles R. Milburn’ and Malcolm H. I. Baird Chemical Engineering Department, M c M a s t e r Uniuersity, Hamilton, Ontario, Canada
An air-pulsing technique involving no mechanical moving parts was applied t o water 4 - and 2-inch pipes a t time-average Reynolds numbers from 3000 t o 66,000. The frequency (0.35 to 1.1 cycles per second) and displacement (up to 2.5 feet) are sufficient to cause flow reversal. A numerical technique for predicting the pulsator
flows in
performance has been developed, but its accuracy is limited by uncertainties about In i t s present form, the technique gives a conservative estimate of
two-phase flow.
pulsation intensity.
FLUID
PULSATION as an aid to processing has been of interest to researchers ever since the first patent on pulsed extraction was taken out by Van Dijck (1935). Pulsed heat transfer has been studied widely since the first paper by Martinelli (Martinelli et al., 1943), and has been reviewed by Lemlich (1961). In the last ten years, the effects of pulsation on other processes such as absorption (Ziolkowski and Filip, 1963), distillation (McGurl and Maddox, 1967), and fluidization (Massimilla et al., 1966) have been investigated. Such work has been reviewed by Baird (1966). In the majority of cases, pulsation increased the rate of the process compared with unpulsed operation. However, fluid pulsation has rarely been applied industrially in chemical engineering, with the exception of pulsed solvent extraction. The reason for this anomaly is that extraction is unique in requiring small amplitudes,
Present address, Ashland Oil Co.. Ashland, Ky.
41101
less than 0.5 inch as a rule, for a good improvement in efficiency. Other processes such as heat transfer require higher intensity pulsations before any appreciable improvement is seen. The generation of intense pulsations on a large scale introduces problems of cavitation and mechanical moving parts, and the question of power consumption becomes important. Air pulsing, first proposed by Thornton (1954) for pulsed solvent extraction, provides an answer to the first two problems. The air acts as a barrier between the process liquid and any moving parts (piston or bellows) and prevents rapid changes in pressure. Baird (1967) developed an air-pulsing technique which requires no moving parts and depends instead on the unstable alternating discharge of air and liquid from an orifice. This technique has been used by Baird et al. (1968) to pulse a solvent extraction column at low amplitudes. As the operating frequency is close to the natural frequency of the system, the power consumption for given intensity is near minimum. Ind. Eng. Chern. Process Des. Develop., Vol. 9, No. 4, 1970
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