T

#I w,, = cross-over frequency

= major time c:onstant of process = defined in Equation 53

References

(1) Buckley, P. S., “Techniques of Process Control,” p. 95, Wiley, New York, 1964.

(2) Cohen, G. H., Coon, G. A., Tram. ASME 75,827 (1953). (3) Coon, G. A., ZSA J . 11, 77 (September 1964); 81 (October

1964); 81 (November 1964). (4) Coughanowr, D. IR., Koppel, L. B., “Process Systems Anal-

ysis and Control,” MkGraw-Hill, New York, 1965. (5) Hougen, J. O., Chem. Eng. Progr. Monogrujh Ser., No. 4, 60

(1964). (6) Kuo, B. C., “Analysis and Synthesis of Sampled-Data Control

Systems,” p. 328, Prentice-Hall, Englewood Cliffs, N. J., 1963. (7) Laspe,C. G.,ZSAJ. 3,134(1956).

(8) Ragazzini, J. R., Franklin, G. F., “Sampled-Data Control

(9) Tou, J. T., “Digital and Sampled-Data Control Systems,”

(10) Truxal, J. G., “Control System Synthesis,” p. 546, McGraw-

Systems,” McGraw-Hill, New York, 1958.

p. 318, McGraw-Hill, New York, 1959.

Hill. New York. 1955. (11) T’sypkin, Y.’Z., “Sampling Systems Theory,” Vol. 2, p. 555,

(12) Williams, T. J., Chem. Eng. Progr. Symp. Ser., No. 36, 57, 100 Macmillan, New York, 1964.

(1961). (13) Zbih.. No. 46. 59.1 (1963). (14) Ziegier, J. G., Nichols, N. B., ZSA J. 11, 73 (June 1964);

(15) Ziegler, J. G., Nichols, N. B., Trans. ASME 64, 759 (1942). (16) Zbid., 65, 433 (1943).

75 (July 1964); 63 (August 1964).

RECEIVED for review October 6, 1965 ACCEPTED March 7, 1966

OPTIMA,L DESIGN OF MULTIPLE-EFFECT EVAPORATORS BY DYNAMIC PROGRAMMING

S E l J l I T A H A R A A N D L E O N A R D I . S T l E L

Syracuse University, Syracuse, iV. Y.

Optimal design procedures have been established for multiple-effect evaporators by dynamic program- ming, The exact dynamic programming solution for the minimum evaporator area requires two state varia- bles and one decision variable. A simplified iterative procedure has one state variable and one decision variable. ‘The per cent saving in area over the equal area design increases with increasing number of stages. The criterion that A,/AT, is constant does not result in minimum area when the over-all heat trans- fer coefficient is a function of temperature. Design procedures are presented for determining the minimum annual operating costs and optimum number of effects. The simplified dynamic programming method can be useld to determine the optimum arrangement of effects of varying evaporator type for minimum initial cost.

N RECENT years optimization techniques, including linear and I nonlinear programming, dynamic programming, and gradient search methods, have found wide utility in the design of chemical processes. Dynamic programming is particularly well suited for the optimization of stagewise processes. I t has been applied for the optimal design of chemical reactors (7) , cross-current extractors (2), and mass transfer separation processes ( 5 ) . There is currently increased interest in the design of evaporators. particularly for saline water conversion. Therefore, in this study optimal design procedures have been established for multiple-effect evaporators by dynamic pro- gramming.

Bonilla (3, 4) has developed a simplified method for the calculation of the minimum total area for multiple-effect evaporators. In the design of evaporators having effects of equal area the following equation is used :

where m is any effect and M is the total number of effects.

Equation 1 is applied recursively until the area in each of the effects is the same. Bonilla (3) obtained an expression similar to Equation 1 for the determination of the temperature drops in the effects which result in minimum total area. He differen- tiated the total area with respect to the temperature drop in each effect, assuming that the boiling point rise is negligible and that the heat transferred and over-all coefficient in each effect do not vary with temperature. Equation 2 resulted :

M c A T , 4ic ( 2)

j = 1 A T , =

j = 1 5l/gi/Ls. The design procedure is similar to that for the equal area case, except that the successive values of the temperature drop in the mth effect are determined from Equation 2, and the calculations are completed when A m / A T m is the same in each effect.

Optimal Design by Dynamic Programming

Equation 2 obtained by Bonilla is not a general solution for the determination of the minimum area in multiple-effect evaporators because of the simplifications made in its develop-

VOL. 5 NO. 3 J U L Y 1 9 6 6 309

coolant

Vl

Figure 1. Multiple-effect evaporation system

ment. The assumption that Urn is independent of tempera- ture is not met by most commercial evaporators in which the over-all coefficient is a strong function of temperature. The dynamic programming solution of the minimum area problem does not require that these assumptions be made.

Dynamic programming requires a stagewise structure for the problem, with each stage characterized by an input vector, x,, an output vector, x,-,, a decision vector, y,, and a return function, R,, which is a function of the input and decision vectors. The input and output vectors, which completely specify the state of the system entering and leaving the stage, respectively, are related by the transformation equation,

xn-1 = Fn(xn,Yn) (3)

In dynamic programming, each stage is considered sepa- rately, and, therefore, the calculations for the over-all optimiza- tion are enormously reduced. The dynamic programming technique is based on the principle of optimality, which specifies that for a particular input, x,, the best nth stage de- cision is that y n which maximizes (or minimizes) the total return (or cost) for the remaining n stages. The nth stage decision vector transforms the input vector x, to x,-, through Equation 3. To apply this technique, the optimum decision for each possible input to the last stage (n = 1) is determined. This procedure is followed for the preceding stages until the optimum decision with respect to each possible input to the first stage is determined.

For the problem of the minimum area in a multiple-effect evaporator (Figure 1) the decision vector is the temperature in the nth effect, T,, and the input vector (Vn+l,Tnt.l) repre- sents the quantity and temperature of the vapor entering the nth effect. The return function is the area of the nth effect,

(4) Vn+, An+,

un(Tn+, - Tn) A , =

Since the temperature of the last effect, T,, is fixed by the con- denser temperature, there is no decision and the return is

The transformation equations are derived from heat and mass balances over the last effect:

Vzh2 + LzCp(Tz - Ti) = ViXi

Lz = L1 + Vl

( 6 )

(7 )

Since L1 is fixed by an over-all balance, Equations 6 and 7 define LZ and Vl for a particular input ( VZ, T2).

For the nth effect, the minimum total area for effects 1 through n is

The transformation equations which are used to obtain V, and L,+l for an input vector ( V,+,, T,+,) are :

Vn+lXn+l + Ln+lcp(Tn+l - T n ) = VnXn

Ln+1 = Ln + Vn

(9)

(1 0)

and

The heat and mass balances for the n - 1st effect serve to define L, as follows :

Ln = gn(Vn,Tn) (11)

For a particular Vn+,, and T,+l, the decision T, fixes L,+1 and Vn through Equations 9, 10, and 11. The optimal decision T,* for each input vector is determined in this manner. If the return function is established to be unimodal, a Fibonacci search procedure (8) can be used. By the use of the set of Tn*, the functional relationship Ln+1 = gn+l( Vn+l, Tn+d can be established for use in the transformation equations for the n + 1st effect. This functional dependence is most easily established through a least squares procedure. To reduce the storage requirements which can be a problem for a large number of effects, f, and T,* can also be fitted to V,+1 and T,+, by a numerical procedure.

For the Nth effect, LNS1, TNCl, and Ti (the temperature of the inlet liquid) are known. Therefore, for a particular V N + ~ , the heat and mass balances

VN+lhN+l + LN+ICp(Ti - TN) = V N ~ N (12)

LN+l - - VN + LN = VN + gN(VN,TN) (13)

provide two equations with two unknowns, VN and TN. minimum over-all area is

The

A* = min fN(VN+,) = (vN+1)

A flow chart of the computational procedure is presented in Figure 2.

Simplified Dynamic Programming Procedure

The exact dynamic programming solution for the minimum area problem is complex because of the two state variables re- quired and the unusually difficult transformation equations which arise from the fact that both a liquid and vapor stream are entering each stage. This procedure requires considerable

310 l & E C P R O C E S S D E S I G N A N D D E V E L O P M E N T

t

Vn+lhn+l + L n + l C p ( T n + l -T n ) =VnXn

Ln+l = Ln + vn vn - - hn (Vn+l,Tn+l,Tn)

1 (V ,T 'N+P N+I UN(TN+l-%) + 'N-1 N N T o t a l area = min fiu(VN+l.) = min

{'N+13 {'N+l)

Figure 2. Flow chart for rigorous dynamic programming solution

VOL. 5 NO. 3 J U L Y 1 9 6 6 311

machine storage locations and computing time. In addition, the resulting transformation equation is dependent upon the accuracy of the numerical approximation technique.

The heat and mass balances serve to define the heat trans- ferred in each effect, q,, as functions of the temperature drops in the effects. The value of q, does not vary appreciably with changes in the temperature drops. A greatly simplified dynamic programming procedure can be used if it is assumed that q, is independent of the temperature drops. An initial temperature distribution is assumed, the heat and mass bal- ances are applied to determine the initial set of q,, and the dynamic programming solution is performed keeping the q, constant. After the optimal temperature distribution is established for these q,, a new set of q, corresponding to the new temperature distribution is determined and the procedure is repeated. The calculations are terminated when the qn’s are constant. This procedure is similar to that used by Bonilla (3) with Equation 2, but does not require the assumption that the over-all coefficient is independent of temperature.

In the simplified dynamic programming procedure, the decision vector is AT, and the input vector En (the tempera- ture of the vapor leaving the n + 1st effect minus the tempera- ture of the vapor leaving the last effect). For saline water conversion, the over-all coefficient in the nth effect can be assumed to vary linearly with temperature, as follows (7) :

Un = a + bT,

Other functional dependences for the over-all coefficient are possible, including U, as a function of AT, or of both AT, and T,. If Equation 15 is used, the return function for the nth effect is

For the last (n = 1) effect,

and for the nth effect

The solution of Equation 18 involves a search procedure for the optimum AT, for each En. For a one-dimensional process the Fibonacci method is very efficient (8) if the return function can be established to be unimodal. Both the exact and simplified dynamic programming solutions can be applied equally well to the case of backward flow of the liquid stream. A flow chart of the computations required for the simplified dynamic programming solution is presented in Figure 3.

The effect of the boiling point rise can be easily considered in the simplified dynamic programming procedure. For each iteration that a temperature distribution is obtained, the boiling point rise is calculated from a solids balance over an effect. The boiling point rise reduces the effective temperature drop across an effect, and, therefore, the area required is increased. The boiling point rise in each effect is maintained constant through a dynamic programming solution, and the method of solution is the same as described previously.

Example

A problem has been solved for a varying number of effects for a feed of 100,000 pounds per hour of saline water to be

concentrated from 3.5 to 7.0%. The steam temperature is 249’ F. and the last effect operates a t 115’ F. The entering liquid feed is a t 110’ F. The heat losses and boiling point rise are considered negligible, and the latent heat of vaporiza- tion and specific heat are taken as equal to those of water. The over-all coefficient was assumed to vary according to the following equations applicable for propeller calandria evapora- tors (6):

U, = 2.14 Tn - 64.3 (100’ F. < Tn < 175’ F.) (1 9) U , = 2.88 Tn - 183.1 (175’ F. < Tn < 220’ F.)

All calculations were performed on a digital computer. For two effects a negligible decrease (0.07%) in the total

area was obtained by the simplified dynamic programming procedure over that for the equal area case obtained from Equation 1. The answer obtained by the dynamic program- ming method was confirmed by a direct Fibonacci search for the intermediate temperature after the unimodality of the return was established. The dynamic programming solution converged in four iterations, starting with either the equal area temperature distribution or equal temperature drops in each effect, with the final 4,’s differing from the previous values by less than 0.01%. An attempt was made to use Bonilla’s method, Equation 2, to solve this problem by holding U, constant for each iteration and readjusting both q, and U, after each step. However, a slightly larger area (l.75yo) was obtained than the equal area solution, and the optimum solution obtained by dy- namic programming did not meet the requirement that A,/A T , be constant.

For three stages the minimum area solution obtained by both the rigorous dynamic programming method and the simplified method was virtually identical and represented a l.5Oy0 im- provement over the equal area case. The calculations for the simplified method are summarized in Table I. The return function given in Equation 16 was found to be unimodal, and in the Fibonacci search 1 5 experiments over a tempera- ture interval of 50’ F. produced an uncertainty of 0.05’ F. in the optimal AT,. Bonilla’s method again produced a slightly higher total area than the equal area case (1.09%), and the criterion that An/ATn be constant again was not met by the optimal dynamic programming solution. These results indicate that Bonilla’s method cannot be used when the over-all coefficient varies widely with temperature, as in Table I.

The minimum area was also calculated for four to 10 effects by the simplified dynamic programming procedure. The initial temperature distribution in each case was estab- lished by dividing the total temperature drop by the number of effects. The Fibonacci search procedure was similar to that for three effects. The per cent decrease in area over the equal area solution varied from 2.71% for four effects to 11.1% for 10 effects. In Figure 4 the total areas for two to 10 effects are presented for both the minimum area and equal area solutions; the savings in area can be significant as the number of effects increases.

The calculations were repeated for an inlet flow rate of 500,000 pounds per hour, all other conditions the same as in the previous example. For two to 10 effects the minimum areas were approximately five times larger than that for a flow of 100,000 pounds per hour and the per cent decrease in area over the equal area design was virtually the same as obtained previously.

Optimum Number of Effects Bonilla (3) pointed out that the minimum area solution does

not necessarily correspond to minimum steam consumption.

312 I & E C P R O C E S S D E S I G N A N D D E V E L O P M E N T

APPROXIMATE ATi's b I I CALCULATE HEAT LOADS, Q~

AT; AT^ * ... AT^,^ * AT: A~ ... - - - 'n - - - - i

= C N - ATN

* * IzN AT; A~ A~ ... AT: AT^ ... ATN-l I

3. [RECALCULATE HEAT LOADS, Q' 1

+ o r a - END

Figure 3. Flow chart for simplified dynamic programming solution

VOL. 5 NO. 3 J U L Y 1 9 6 6 313

Third effect ( n = 1) Temp., O F. A T , O F.

Table I. Results of Iterative Procedure Iteration

7 2 3 4 5 115 115 115 115 115 44.67 64.11 62.88 62.92 62.92

q, B.t.u./hr.

U, B.t.u./hr. sq. ft. O F. Temp., O F. A T , ' F. q, B.t.u./hr. Area, sq. ft. Area/AT U, B.t.u./hr. sq. ft. F.

y4,;T ft.

Second effect ( n = 2)

First effect ( n = 3) Temp., O F. A T , F. q, B.t.u./hr. Area, sq. ft. Area/AT U , B.t.u./hr. sq. ft. O F.

159.67 179.11 177.88 177.92 177.92 44.67 34 * 87 35.13 35.15 35.15 13,139,494 13,157,458 13,178,316 13,176,282 13,176,282

1 .141.67 32.48 375

204.33 213.97 213.01 213.07 213.07 44.67 35.03 35.99 35.94 35.94 22,572,827 23,554,926 23,479,533 23,482,564 23,482,564

1.521.63 42.34 430

Total area, sq. ft. 4,078.05 Steam, lb./hr. 24,792.90

12000

I1000

loo00

- BOO0 t:

0 - eooo

U

u)

2 ~ 7000

2

a c

0 c e000

5000

4000

3000

I I I I I I I I I 2 3 4 5 6 7 8 9 1 0

EFFECTS

Figure 4. minimum area methods

Total area vs. number of effects for equal and

In the example considered, the steam required was approxi- mately 1% greater than the equal area requirement, in- dependent of the number of effects. In order to establish the temperature drops which result in minimum annual operating costs including both the costs of steam and capital depreciation, the rigorous dynamic programming method has to be used. For the nth effect ( n # N ) the optimal n-stage return is

cn(Vn+1, Tn+,) =

where E represents the annual fixed charges per square foot. For the Nth stage the total cost is

where S represents the annual cost per pound of steam. Equations 20 and 21 can be applied for varying number of

effects to determine the optimum number of effects. A stage for the condenser can be added after the last stage, and the functional equation for this stage will represent the minimum cost for the condenser for each input temperature drop. For this case the total temperature drop is the steam tempera- ture minus the temperature of the condenser cooling water, and the complete function to be minimized is the sum of the annual fixed charges on the condenser, the annual fixed charges on the evaporators, and the annual steam costs.

The optimum number of effects can also be determined for the minimum area design obtained by the simplified dynamic programming procedure. The number of stages is determined for which the following objective function is a minimum:

Since the saving in area increases with increasing number of stages while the percentage increase in steam consumption remains fairly constant, the optimum cost for minimum area design will be substantially smaller than that for equal area design if the optimum number of effects is large.

Optimal Design for Effects of Varying Type

An interesting problem for which dynamic programming is ideally suited is the determination of the minimum initial capital cost when evaporators of varying types are used in the effects and the arrangement is not predetermined. In the dynamic programming solution of this problem, a new decision variable representing a choice of evaporator types must be included. Since the cost per square foot and temperature variation of the over-all coefficient are different for each evaporator type, the solution of the following functional equation for the minimum cost provides knowledge of the best evaporator type for each input temperature:

314 l & E C PROCESS D E S I G N A N D D E V E L O P M E N T

where k l represents the cost coefficient for each evaporator type, p t is an exponent, and U,, is the over-all coefficient for each evaporator type.

If the choice is between two types of evaporators, the solution of Equation 23 will involve the consideration of two functional equations. For each xn and t , the optimal A T , is obtained by a Fibonacci search. By the use of the optimal AT,, the evapo- rator type t is chosen that produces the smallest value of K n ( x , ) . For the last ( n = 1) effect, the choice is between the two values

An example has been solved for the determination of the minimum cost from Equation 23 for four effects, for the inlet and outlet conditions used in the minimum area example. One type of evaporator to be considered is the propeller calandria type with over-all coefficients given by Equation 19. The alternate type is the forced circulation evaporator with over-all coefficients which vary with temperature as follows ( 6 ) :

U,, = 1.54 ( T o F.) + 311.54 (25)

For the propeller calandria evaporator kl = 200 and 61 = 0.75, and for the forced circulation type kz = 281.2 and pa = 0.75 ( 6 ) .

The optimum arrangement was found to be one propeller calandria evaporator followed by three forced circulation evaporators. The optimal temperature distribution is AT4 = 34.97’ F., AT2 = 27.03’ F., A T 2 = 32.49‘ F., and A T 1 = 39.51’ F. The comparative costs for the optimum arrange- ment and for all of one type with minimum area design are:

All propeller calandria evaporators $171,979 All forced circulation evaporators 150,021 Optimum arrangement 147,694

Conclusions

The simplified dynamic programming procedure, Equations 17 and 18, can be used to determine the optimum temperature distribution for minimum area in multiple-effect evaporators with temperature-dependent over-all coefficients. Bonilla’s criterion that A , / A T , is constant in all of the effects does not yield the exact solution to this problem. The saving in area over the equal area design increases with the number of effects and depends on the inlet and outlet conditions and the evapora- tor type. The simplified dynamic programming procedure can also be used to determine the optimum arrangement of evaporators of varying types.

Ac knowledgment

Seiji Itahara was the recipient of a NASA Fellowship. Numerical computations were performed at the Syracuse University Computing Center and were supported in part by National Science Foundation Grant GP-1137.

Nomenclature

a = constant in Equation 15 A = heat transfer area in an effect, sq. ft. b = constant in Equation 15 c = optimal cost function for multiple-effect evaporators,

C = annual fixed charges for multiple-effect evaporators,

C, = heat capacity of liquid stream, B.t.u./hr. ’ F. D = condensate from an effect, lb./hr. E = annual fixed charges on an effect per unit heat transfer

f = optimal return function for minimum area design, sq. ft. F = transformation function g = transformation function for minimum area design k = cost coefficient for an evaporator K = optimal return function for varying types of evaporators,

L = liquid stream inlet to an effect, lb./hr. p = cost exponent for an evaporator q = rate of heat transfer in an effect, B.t.u./hr. R = return function for a stage S = annual steam cost, $ hr./lb. yr. 7’ = temperature, ’ F. AT = temperature drop across heating surface of an effect,

U = over-all coefficient for an effect, B.t.u./hr. sq. ft. ’ F. V = vapor from an effect, lb./hr. x = state vector y = decision vector for a stage

gS/yr.

$/yr.

area, $/sq. ft. yr.

$

’ F.

GREEK LETTERS X Z

= latent heat of vaporization, B.t.u./lb. = temperature drop between vapor stream inlet to effect

and that leaving last effect

SUBSCRIPTS AND SUPERSCRIPTS i = property of liquid stream inlet t o first effect j = parameter in Equations 1 and 2 m = effect in a n evaporator system M = total number of stages n = stage or effect N = first stage or effect t = evaporator type * = optimal value

literature Cited

(1) Aris, Rutherford, “Optimal Design of Chemical Reactors,”

(2) Aris, Rutherford, Rudd, D. F., Amundson, N. R., Chem. Eng.

( 3 ) Bonilla, C. F., Trans. Am. Znst. Chem. Engrs. 41, 529 (1945). (4) Zbid., 42, 407 (1946). (5) Mitten, L. G., Nemhauser, G. L., Can. J. Chem. Eng. 41,

187 (1963). (6) Perry, R. H., Chilton, C . H., Kirkpatrick, S. D., eds., “Chemi-

cal Engineers Handbook,” 4th ed., McGraw-Hill, New York, 1963

Academic Press, New York, 1961.

Sci. 12, 88 (1 960).

-,

(7) Standiford, F. C., Bjork, H. F., Advan. Chem. Ser., No. 27,115

(8) Wilde, D. J., “Optimum Seeking Methods,” Prentice-Hall, (1 960).

Englewood Cliffs, N. J., 1964.

RECEIVED for review October 1, 1965 ACCEPTED April 18, 1966

VOL. 5 NO. 3 J U L Y 1 9 6 6 315