OPTIMAL DESIGN OF MULTIPLEIEFFECT EVAPORATORS WITH VAPOR BLEED STREAMS 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, aV.Y. Dynamic programming has been used to establish procedures for the determination of the optimal annual operating expenses and optimal number of effects for multiple-effect evaporators in which a vapor bleed stream from each effect is used to preheat the feed stream. An iterative procedure which can b e easily implemented on the computer is described in detail. The function to b e minimized includes the costs of steam, capital depreciation, maintenance, power, and labor. An example is presented for a saline water system, and the results are compared with the equal area solution. The effects of the final condenser and of variable approach temperatures in the heat exchangers are also discussed. TAHARA
and Stiel (1966) have shown that dynamic pro-
I gramming can be an effective technique for the optimal design of simple multiple-effect evaporator systems. They presented procedures for the determination of the minimum over-all area of the evaporators, the optimal number of effects, and the optimal arrangement of effects of varying type. For minimum area design the savings in area increased with increasing number of effects. In the design of large scale multiple-effect evaporators such as are used for saline water conversion, factors are usually present which were not considered in the previous study. In these systems a fraction of the vapor overhead from each effect is bypassed to a condenser for preheating of the feed stream, while the remainder of the vapor is sent to the succeeding effect. All condensate streams from the effects and preheat condensers are also used in the preheat exchangers. The usual design procedure (Sherwood, 1963) for systems of this type is to assume that the areas in the evaporators and the approach temperatures in the heat exchangers are equal ; thus the exchanger areas, vapor bleed rates, and the steam requirement are uniquely fixed. This procedure can be followed for varying number of effects and the optimum number established, but it does not necessarily result in the minimum over-all cost possible for the specified conditions. I n the present study, dynamic programming has been applied to establish optimal design procedures for multiple-effect systems employing condensate and vapor bleeds for preheat. A diagram of a system of this type is shown in Figure 1. Optimal Design by Dynamic Programming
Itahara and Stiel (1966) presented both exact and simplified dynamic programming formulations for the optimal design of simple multiple-effect evaporator systems. The exact dynamic programming formulation required complex transformation equations and a large number of computer storage locations. T h e simplified procedure was found to reduce computer storage requirements significantly and to produce essentially the same results as the exact formulation. For the design of multiple-effect evaporators with vapor bleed an exact dynamic programming formulation is even more difficult because of possible feedback effects on the evaporating brine caused by the preheating of the cold feed stream. Present address, American Cyanamid Co., Bound Brook, N. J . 6
l&EC PROCESS DESIGN AND DEVELOPMENT
However, a simplified dynamic programming formulation for this case can be readily established. In the design of multiple-effect evaporators with vapor bleed, the amount of vapor overhead from each effect is essentially the same and does not vary appreciably with changes in the temperature drops. Therefore, in the simplified procedure the vapor overhead rates from each effect are made exactly equal as a first approximation :
v, = LS+lIV- L1
(11
Using this set of vapor rates, which is now assumed to be independent of temperature, the optimal temperature distribution is determined by dynamic programming. Following this step, a new set of V,’S is calculated using the optimal temperature distribution in the heat and material balance equations. This procedure is terminated when the set of V,’s remains constant over two successive iterations. This computational scheme is similar in construction to the authors’ previous method as applied to a simple multiple-effect evaporator system (Itahara and Stiel, 1966). The single-state variable is 2 , [the temperature of the vapor entering the nth effect minus the temperature of the vapor leaving the last ( n = 1) effect] and the decision variable is AT,, the temperature drop across the evaporator. The cost function to be minimized is represented by the annual operating expense, which includes the cost of steam, capital depreciation, maintenance, power, and labor. For X effects, the variable costs are the steam cost and a percentage of the evaporator and exchanger costs, while the other components of the operating expenses are fixed. T h e cost function for the nth stage (n N ) involves the evaporator and exchanger costs. For the last effect (n = iV), the cost function includes both evaporator costs and the cost of the steam. T h e input and output variables, which completely specify the states of the system entering and leaving the stage, respectively, are related through the stage transformation equation. I n dynamic programming, each stage is considered separately, and, therefore, the calculations for the over-all optimization problem are enormously reduced. T h e calculations are based on the principle of optimality, which specifies that for a particular input variable, Z, the best nth-stage decision is that AT,* which minimizes the total cost for the remaining n stages. This nthstage decision transforms the input variable to the output variable.
*
N '
' TN
---1
r
I
I
,
,
&1-j
_-_ brine
I
Figure 1. Multiple-effect evaporation system with vapor bleed
TL
T h e equations pertinent for the formulation of the problem are derived for effects 1, n, and N (see Figure 1). T h e boiling point rise caused by the dissolved solids is considered to be linear with respect to concentration. T h e boiling point rise for each effect is uniquely determined for each iteration by the set of vapor rates, since the boiling point rises are assumed to be independent of temperature. T h e over-all heat transfer coefficient in the evaporators is assumed to vary linearly with temperature (Standiford and Bjork, 1960).
Un = a
+ bTn
+ Ln+lcp(Tn+1 - T n ) = Vnhn
+ BPRn - BPRn+11
- VBn+1Xn
+I
(13)
L.V+ICP
The log mean temperature difference, which is identical in each of the two exchangers, is
- 1
APT The state input variable for the nth effect is Z, and the corresponding transformation equation is
(3)
T h e heat required in both exchangers is
LN+lCp[Tn+l- T n
T X , = [Tn+l - BPRn+l - A P T ]
(2)
with a and 6 as constants. Initially the approach temperatures in the heat exchangers are assumed to be equal. T h e heat balance around the nth effect is
(Vn+l - vBn+dXn+l
Solving for TX,,,
(4)
For n = 1, the cost function for each state input variable 21(21 = T2 - T I ) does not require a decision, A T I , since the temperature in the last stage ( T I )is specified:
while the total heat provided by the condensates is
(LN+I- Ln+J
c p [Tn+l
-
Tn
+ BPRn - BPRn+lI
(5)
T h e vapor bleed rate is given by the difference between Equations 4 and 5,
VBn+1 =
Ln+Icp[Tn+I- T n
+ BPRn - BPRn+lI
An +I
(6)
+
De, [ E X ~ I ] ~ ' "De, [EX21]"')
(16)
For the nth effect the cost function is
Substitution of the value of VBn+linto Equation 3 yields
which shows that the initial assumption of equal Vn's is valid for constant h and small boiling point rise differences. T h e areas of the two exchangers are:
(17) where
Un = a
+ bTn = a + b ( Z n - ATn + T I - BPRn+l)
(18)
If the cost function is established to be unimodal, the best search procedure for the determination offn(Zn)is a Fibonacci search (Wilde, 1964). T h e heat balance around the first (Nth) effect is
VsXs
- L.v+lCp ( T N- T L ) = VNXN
(19)
where T Lis the temperature of the inlet liquid to the Nth effect. There are no exchanger calculations to be made and the final cost function can be written directly for each Z,. (which is a known quantity), VOL. 7
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and Z'v-1 = T.v
- TI
(23)
A flow chart for the computations required for the simplified dynamic programming solution is presented in Figure 2. T h e principle of optimality applies only if no feedback effects occur (Aris, 1964). Whenever a n input state vector is affected by a subsequent decision or set of decisions, the dynamic programming method cannot be simply applied. Although the evaporating liquid stream and preheating feed stream run countercurrently, in the one-state variable formulation, ZS, the input state vector to the Nth stage, is independent of the decisions made from stages N to 1. E N is equal to the difference between the fixed, intensive parameters T8 (the steam temperature) and TI (the final effect temperature). Consequently, the one-state variable formulation with constant approach temperature not only reduces the dimensionality but eliminates the feedback conditions which may be present in a n exact dynamic programming formulation.
Example. A problem has been solved for the determination of the optimal annual operating expenses and the optimum number of effects for a large scale multiple-effect system .by the simplified dynamic programming procedure. This problem was chosen because the solution for equal area design is available for comparison (Sherwood, 1963). I t is desired to produce 17,000,000 gallons per day of fresh water from a feed containing 3.5% dissolved solids. T h e exit brine contains 7% dissolved solids. T h e steam temperature is 212' F. and the final effect operates at 115' F. A 5' F. minimum approach temperature [which was found by Sherwood (1963) to be close to the optimal value for the equal area design] is used in the heat exchangers. T h e over-all heat transfer coefficient can be expressed as (Sherwood, 1963) : Un = 172
+ 2.33 T ,
(OF.)
(24)
As in the assumption in the equal area design, the latent heat of vaporization is taken as constant a t 1000 B.t.u. per lb. T h e boiling point elevation of sea water is taken as 0.70' F. for the inlet stream and varies linearly with concentration, independent of temperature. T h e objective functions to be minimized are similar to Equations 17 and 20 :
where MAY are the fixed costs (Sherwood, 1963) and D, is the steam cost equivalent to 12.18 cents per million B.t.u. or $0.1218 X per pound of steam used in this example. T h e results for the total capital investment and annual operating expenses for 7 to 11 effects are presented in Table I along with the equal area values. T h e optimal temperature drops and evaporator areas for each case are presented in Table 11. The cost function was found to be unimodal for 10 effects, and unimodality was assumed for the other cases. Therefore, a Fibonacci search procedure (Wilde, 1964) was used to establish the optimal temperature drops. In the Fibonacci search 15 experiments over a temperature interval of 30' F. produced an uncertainty of 0.05' F. in the optimal AT,'s. T h e dynamic programming solution converged in four iterations, with the final V,'S differing from the previous values by less than 0.01%.
f
Figure 2. Flow chart for dynamic programming method with constant approach temperatures 8
l&EC PROCESS DESIGN A N D DEVELOPMENT
Table 1. Total Capital Investment and Annual Operating Expense for Varying Number of Effects with Constant Approach Temperatures of 5' F. Annual Operating Expense Total Capital Inuestment Equal area Equal area Dynamic (Sherwood, (Shemood, Dynamic IVO.of programming 7963) 7963) Stages programming 11 $7,472,963 $7,490,000 $1,585,338 $1,595,000 1,572,075 1,588,000 10 6,930,299 6,999,000 1,571,466 1,585,000 9 6,392,612 6,445,000 1,587,922 1,602,000 8 5,858,980 5,924,000
7
5,329,335
5,393,000
1,628,540
1,643,000
Table II.
Evaporator Areas and Temperature Drops by Dynamic Programming, Constant Approach Temperatures of 5' F.
EffectNumber
No. of 1
Stages
2
113,797 111,903 8.95 B 9.33 10 A 112,552 110,537 B 10.37 9.90 9 A 111.429 109.205 B 11.63 11.06 110,314 108,100 8 A 12.45 B 13.20 7 A 109,342 106,723 B 15.21 14 25 a A. Evaporator areas, sq. ft. B. Temperature drop, F.
11
A=
3 109,173 8.69 108,812 9.50 107.254 10.57 105,090 11.93 105,082 13.40
4
110,000 8.23 108,204 9.07 106.421 10.06 105,439 11.19 102,288 12.82
5 107,490 8.04 105,437 8.85 104.224 9.74 102,130 10.90 101,032 12.20
6 105,810 7.83 105,171 8.49 103,830 9.33 101,738 10.39 98,971 11.76
7 105,373 7.56 104,045 8.23 101.084 9.15 98,739 10.18 122,362 ii.60
8
104,760 7.33 101,676 8.09 100.120 8.85 124,439 10.02
9 102,562 7.22 100,882 7.85 125.144 8.91
70 11 101,213 129,624 7.07 7.12 126,566 7.97
O
I t can be seen from Table I that for the optimal design the minimum number of effects is nine, similar to the equal area design. However, eight effects may be the most desirable (Sherwood, 1963), because an increase of $16,500 in the annual operating expense is offset by a saving of over $500,000 in the total capital investment. For this example, the equal area design is close to the dynamic programming solution, and the optimal solution provides a saving of $65,000 in the total capital investment and $14,000 in the annual operating expense for eight effects. I n the above formulation the approach temperatures in the heat exchangers have been specified as constant and equal. Therefore the vapor bleed rates depend only on the temperature drops in the evaporators and are not considered in the optimization process. T h e simplified dynamic programming procedure can be applied for varying constant values of the approach temperatures to determine the optimum value. For the conditions of the above example the optimal design procedure was applied for integer approach temperature values from 1' to 5' F. T h e results are compared in Figure 3. T h e optimum approach temperature is approximately 2' F. For all values of the approach temperatures, the difference between the optimal solution and the equal area solution was of the same order of magnitude as that shown in Table I for an approach temperature of 5' F. For this example the assumption of equal approach temperatures was tested by a dynamic programming procedure using two decision variables. As in the previous case, a onestate variable formulation can be used in which the vapor rates, V ~ Sare , assumed to be independent of the temperature drop during any one iteration. This assumption is most nearly exact when the difference AP, - APn+l is close to zero. For the nth effect, the heat balance equation is identical to Equation 3, while the total heat provided by the condensate flow is given by Equation 5 . T h e heat required in both of the exchangers is
LI+I c p
[(Tn+l
- BPRn+1 - APn.1) ( T , - BPR, - AP,)]
W v)
$
1,620,000
X
w 1,610,000 0
f
2
1,600,000
W
a
0 ~
1,590,000
a
3
z
5
1,580,000
1,570,000
1,560,000
I 1 [-----r) --
1,550,000
7
a NUMBER
Figure 3. stages
IO
9 OF
II
STAGES
Annual operating expense vs. number of
(27)
T h e vapor bleed rate is given by the difference between Equations 5 and 27,
T h e exchanger areas are
T h e log mean temperature differences are
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(T,+I - BPR,+i - TXn) Tn+1 - BPRn+1 In AP,
AT1.m.s =
- AP,
"1
[
(35)
The input state variable for the nth effect is Z, and the corresponding transformation equation is the same as Equation 15. The cost function is
perature of the first effect. Consequently, the excess is taken up by the auxiliary condenser with its own separate cooling water stream. T h e cooling water to the primary condenser is the feed stream to the evaporator system. The heat balance for the primary condenser is
VpX
=
L N + ~[Ti c ~ - BPRi
- APi
-
Tcwl
(40)
and for the auxiliary condenser
+ De, IEx2nIPez} f
De, [EX1nIPez
V,,X fn-l(Zn-1))
(36)
= (Vi
- Vp)X = LauxCp[Ti- AP,,, - Tcw1
(41)
where AP,,, is the approach temperature in the auxiliary condenser and
where U , is again given by Equation 24. T h e cost function for n = iV - 1 is
Ve.,
+
v p =
VI
T h e area of the primary condenser is
steam costs
+ fN=2(Z.v-2)}
(43) (37) and for the auxiliary condenser
Equation 37 includes the steam costs, since the steam rate V , is directly influenced by AP,-, the decision at stage N - 1,
VSX = V.vX
+ LN+iAPN
(38)
(44) where
T h e cost function for the Nth stage is given by
(45)
and where M.v is again the fixed costs for N total stages. Equations 27 to 39 were used to determine the optimal temperature drops and approach temperatures for the conditions of the above example. The objective functions for stages 1 to N - 1 were found to be unimodal and a variation of Wilde's "two-dimensional dichotomous search" (Wilde, 1964) was used for the two-decision variable minimization. The approach temperature in the final condenser was set equal to 2' F., which was found to be near the optimum for the constant approach temperature case. For effects 1 to N - 1 the approach temperatures were found to be approximately equal, as shown in Table I11 for nine effects. However, for stage IC'- l the optimum value of AP,. was considerably lower (1.12' F.) because of its direct influence on the steam rate. The optimum annual operating expense for the variable approach case was virtually identical to the optimum value for the constant approach case with the approach temperatures equal to 2' F., indicating that the variable approach solution is not required. Final Condenser Section. I n the above analysis the approach temperature in the final condenser, AP1, has been considered fixed. I n an actual optimization of a vapor bleed system a stage can be added in the dynamic programming solution to determine the optimum value of AP1. A typical condenser system is shown in Figure 1. In general, the amount of latent heat available in the final vapor stream, V I , is in excess of the heat required by the feed stream a t the tem-
Table 111.
The resulting cost function is fO(Z'0)
= ,
r
(47) where R is the pumping cost which depends on the cooling water flow rate, L,,,. T h e dynamic programming solution including Equation 47 proceeds as before, except that 2,' = T,+l - Tcw and Ti is no longer fixed. The evaporator system is now designed to operate optimally between T , and
Tcw. The dynamic programming procedure outlined in Figure 2 can be used for the optimal design of large scale multiple-effect evaporators with vapor bleed. For a particular system, the optimal design procedure is dependent on the applicable cost structure, while the resulting equal area design is independent of all costs considered. Therefore, the savings to be gained for the optimal design procedure depend on the particular conditions of the problem. I n the example considered, the cost function is relatively insensitive to changes in AT,, so that only a small saving is obtained over the equal area design. In addition, cost penalties may be associated with the optimal
Evaporator Areas, Temperature Drops, and Approach Temperature for 9 Effects Eject Number I 2 3 4 5 6 7 8
_____ Evaporatorareas,sq.ft. A T , F. Approach temp., F. O
~
105,612 12.36 2.00
105,211 11.22 3.60
105,545 10.61 3.14
~~~
10
l&EC PROCESS DESIGN A N D DEVELOPMENT
105,414 10.04 3.20
104,408 9.64 3.16
104,432 9.20 3.15
102,494 8.97 3.11
101,575 8.69 3.12
9
124,951 8.59 1.12
design solution because of the use of unequal areas. However, because of the large flow rates involved in desalination systems the optimal solution may result in significant savings. For a particular problem the optimal design procedure presented in this paper should be employed and compared with the equal area solution to determine the savings involved. Acknowledgment
Seiji Itahara was the recipient of a NASA Fellowship. Numerical computations were performed a t the Syracuse University Computing Center and Jvere supported in part by National Science Foundation Grant GP-1137. Nomenclature
constant in Equation 3
a
=
AP
= variable approach temperature in heat exchangers,
APT
= approach temperature in all heat exchangers,
b B PR
= = = = = = = = = = = =
C
CP
D
EX1 EX2
f
K L M
P Q
’ F.
=
Q’
=
R
= = =
7’ TX
’ F. constant in Equation 3 boiling point rise, O F. cost function heat capacity of liquid stream, B.t.u./hr. cost coefficient, dollars area of vapor-bleed condenser, sq. ft. area of condensate lieat exchanger, sq. ft. optimal cost function for minimum cost, dollars percentage factor liquid stream inlet to an effect, lb./hr. fixed costs for a particular .l’-effect system cost exponent rate of heat transfer in vapor-bleed condenser, B.t.u./hr. rate of heat transfer in condensate heat exchanger, B.t.u./hr. pumping cost temperature, ’ F. intermediate temperature between vapor bleed condenser and condensate heat exchanger, ’ F.
AT
= temperature drop across heating surface of an effect,
’ F. ATl,nL,= log mean temperature difference U = over-all coefficient for an effect, B.t.u./hr. sq. ft. ’ F . U1 = over-all coefficient for vapor-bleed condenser, B.t.u./ hr. sq. ft. ’ F. U2 = over-all coefficient for condensate heat exchanger, B.t.u./hr. sq. ft. ’ F. V = vapor from an effect, lb./hr. VB = vapor-bleed rate from an effect, lb./hr. GREEKLETTERS x = latent heat of vaporization, B.t.u.,/lb. 2 = temperature drop between vapor stream inlet to an effect and that leaving last effect 2‘ = temperature drop between vapor stream inlet to an effect and cooling water temperature SUBSCRIPTS ASD SUPERSCRIPTS = auxiliary condenser aux cw = cooling Lsater ev = evaporator ex = heat exchanger L = liquid stream entering system n = number of effect N = first effect and also total number of effects P = primary condenser S = steam literature Cited
h i s , Rutherford, “Discrete Dynamic Programming,” Blaisdell Publishing Co., New York, 1964. Itahara, Seiji, Stiel, L. I., IND.ENG. CHEM.PROCESS DESIGN DEVELOP. 5. 309 11966). Sherwood, T.’ K., ~‘‘ A Course in Process Design,” MIT Press, Cambridge, Mass., 1963. Standiford, F. C., Bjork, H. F., Adzsan. Chem. Ser., N o . 27, 115 (1960). kiilde, D. J., “Optimum Seeking Methods,” Prentice-Hall, Englewood, Cliffs, N.J., 1964. RECEIVED for review January 16, 1967 ACCEPTED September 5, 1967
A N I N T E R M E D I A T E T E M P E R A T U R E FUEL C E L L Operation on Hydrogen and Oxygen W I L L I A M B. M A T H E R , J R . , A N D A L L E N N. W E B B Texaco Research Center, Beacon, .V.Y. 12508
Fuel cells containing a phosphoric acid paste electrolyte have been operated continuously for up to 3 months on hydrogen and oxygen at 200’ C. Cell power densities as high as 110 mw. per sq. cm. have been attained, although a more reproducible power density i s 65 to 70 mw. per sq. cm. The intermediate temperature fuel cell paste electrolyte consists of boron phosphate dispersed in phosphoric acid. It i s thermally stable, rejects carbon dioxide, has high ionic conductivity and good mechanical properties, and does not need regeneration with water.
o FULFILL the ultimate goal of fuel cell operation on hydroTcarbon fuels the temperature range of 100’ < T < 300’ C. is desirable. This range is chosen to minimize hydrocarbon cracking and deterioration of materials encountered above 350’ C., and to increase electrode activity above levels possible below 100’ C . Operation above 100’ C. also permits better water control in the fuel cell which is operated a t atmospheric pressure.
T h e intermediate temperature fuel cell, here described, uses phosphoric acid as the electrolyte. Phosphoric acid has a high conductivity (0.6 ohm-lcm.? for ortho-HsPOd a t 200’ C.) a t elevated temperatures, conducting via a proton transfer mechanism (Greenwood and Thompson, 1958). Carbon dioxide, formed in the oxidation of all carbonaceous fuels, is virtually insoluble in phosphoric acid. Phosphoric acid is chosen over other acids because of its good temperature stability VOL. 7
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