OPTIMAL CONDENSER DESIGN B Y GEOMETRIC PROGRAM M I N G M O R D E C A I A V R I E L ' Bechtel Corp., San Francisco, Calif. DOUGLASS J. W l L D E Department of Chemical Engineering, Stanford University, Stanford, Calif.
Optimal design of tubular condensers can often b e accomplished efficiently by geometric programming, an optimization technique for handling nonlinear power functions. The method transforms the original problem, minimizing nonlinear capital and operating charges, subject to nonlinear heat transfer and design geometry constraints, into an equivalent concave maximization problem with linear constraints, suitable for solution by standard methods. Geometric programming provides insight into the optimal distribution of the annual cost among the various cost items. Ranges of these distributions can b e established by linear programming. Computational experience is described.
design of process equipment often involves finding numerical values for the design parameters minimizing a cost function, usually nonlinear, subject to design constraints. Recently a new optimization method called geometric programming has been developed which can handle very efficiently the design of optimal horizontal tube condensers by minimizing annual cost subject to certain design constraints. The emphasis in this article is on illustrating the optimization method and deriving certain general relations of optimality rather than designing an actual condenser, although the solution of a practical condenser design problem is given. Geometric programming is applicable when the function to be minimized is a "posynomial" (defined below), subject to inequality constraints, also expressed as posynomials. Engineering optimization problems based on design relations developed by dimensional analysis or by fitting power functions to experimental results, the case for most heat transfer equipment, are especially suitable for geometric programming. An important feature of geometric programming is that frequently a n optimal solution also reveals certain invariances. I n the case of a condenser design, it is shown that under certain conditions the distribution of the variable minimum annual cost among the various expenses is invariant-i.e., independent of fluctuating prices. Thus with certain simplifying assumptions, the thermal energy cost is always 43]/3%,, the fixed charges on the condenser invariably 53l/3%, and pumping cost 3I/3% of the optimal annual cost-no matter how the prices of steam, tubes, and electricity vary. Under less stringent assumptions these proportions can be confined within invariant ranges-42.9% to 50.0% for thermal costs, for example. Another important feature of geometric programming is its computational convenience. T o minimize a n unconstrained posynomial function of rn variables, conventional methods of calculus involve solving rn nonlinear equations. O n the other hand, if the function to be minimized contains 1 terms, geometric programming involves solving exactly m rn 1 linear equations in rn 1 variables, a far easier task. This becomes advantageous when inequality constraints make conventional methods cumbersome.
0
PTIMAL
+
+
+
1 Present address, Central Research Division Laboratory, Research Department, Mobil Oil Corp., Princeton, N. J.
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I & E C PROCESS D E S I G N A N D DEVELOPMENT
Geometric Programming
Geometric programming was developed by Zener, Duffin, and Peterson (5-7, 76-78) and introduced into the chemical engineering literature by Sherwood (73) and Wilde (75). I t is based on the well-known mean theorem (70), which states that the arithmetic mean of positive numbers is always equal to or greater than their geometric mean. Thus, ('i2)3.'1
+
1 (Y1y2)1'2
('/2)3.2
with equality if and only if y~ = yz. positive weights, 6i: such that 8,+62+
...
(1)
By defining a set of no
+s,,=
1
(2)
one can generalize Inequality 1 to 81Yl
+ 6zyz + . . . + 6n,yn,
2
cr1)81(3.2)8~
. . . ( y n o ) h (3)
with equality if and only if all y r are equal. The use of the above geometric-arithmetic mean inequality in process optimization was also discussed by Ferron (8). Taking now no arbitrary positive numbers PI, Pz, . . ., P,,, Inequality 1 becomes:
P = P1+Pz+
. . . +P,,> (P1/61)*1(P2/62)~2
.
,
. (Pn0/6n,)6*o
(4)
I n the general case the P, can be functions of several variables . . . , x,, and in this case the left-hand side of Inequality 4 is a function of the xt, i = 1, . . . ,m,while the right-hand side is a function of both the x i and 6 j , j = 1, . . ., no. T h e key to the use of geometric programming is the fact that Inequality 4 becomes a n equation only if its right-hand side is independent of X I ) x2, . . x,. Consider now the case where XI, X Z ,
. )
Pj
=
Pj(x1)alj(xz)e2j
.. .
pj
(x,,)Qmj;
j
=
>0
(5)
1, . . . , n o
The x i are nonnegative and the exponents aij are given real numbers. Lt'e can now write Inequality 4 as
The left member of Inequality 6 is called by Duffin and his
coworkers a posynomial, to distinguish it from a generalized polynomial in which the B j are not necessarily positive. Consider now the minimization of this posynomial. Zener and Duffin (78) shoived that the posycomial attains its minimum if Inequality 6 becomes a n equation. T h e first step in the optimization problem is to make the right member of Inequality 6 independent of the x I by selecting values for the 6 j such that
(7) T h e weights 6, should therefore satisfy Equations 2 and 7 , which represent a system of m 1 linear equations i n no unknowns. I n the fortuitous case where m 1 = no, there is a unique set of \\eights 6, which when substituted into Equation 6 yields the minimum value of the posynomial. I n the general case, where m 1 < no
+
+
+
and the unconstrained minimization of a posynomial is equivalent to the maximization of a product function V (the right member of Equation 8 ) ) subject to linear constraints, Equations 2 and 7. Duffin, Peterson, and Zener (7) extended geometric programming to the minimization of posynomials subject to inequality constraints of the form
2+
12
j = nk-
1
Pi,k
=
5
yk=
(9)
I n this case the following
where the Pj are also posynomials. relation is obtained :
where
1, . . . , p
1
aj
k = l , ...,p
(11)
j = nx-1 + I
and the \\eights 6 j satisfy the following set of relations: na
C6,=1 ~
j=1 n.
2
0lljSj
= 0,
i
=
1, . . ., m
(1 3)
. .,np
(14)
j=1
6 j 2 0, j = 1, .
Thus the mathematical problem of minimizing a highly nonlinear posynomial, subject to nonlinear posynomial constraints, can be accomplished by the generally simpler problem of maximizing a product function, subject to linear constraints. T h e relation between the terms of the posynomial objective function and the optimal weights 6 j * is given by
T h e optimal values of the variables xi, . . . , xm are found from Equation 15. Duffin, Peterson, and Zener (7) define the “degrees of difficulty’‘ of a geometric program by the relation Degrees of difficulty = n p - m
-
1
(16)
The reason for this concept becomes clear by considering the set of equations given by Equations 12 and 13. This set
+
consists of m 1 equations in n p variables. Thus if one has a geometric program in m variables and the total number of terms in the posynomials of the objective function and con1-Le., straints is m
+
n,=m+l
+
(1 7 )
then Equations 12 and 13 become a system of m 1 linear 1 unknowns, and by Equation 16 the program equations in m has zero degrees of difficulty. Problems with this property were the first to be presented by Zener (77) as geometric programs, and their solution was obtained simply by solving a square system of linear equations. The linear constraints, Equations and Inequalities 12 to 14, have also the important property of being independent of primal coefficients P j . Hence solution to a problem with zero degrees of difficulty is invariant in the sense that no matter what the numerical values of the coefficients P I are, the optimal variables 6 j are uniquely determined by solving the system given by Equations 12 and 13. Frcm Equation 15 we obtain for the optimal solution that the optimal variables 61*, associated with the terms of the posynomial objective function, represent the fraction, or weight? of each term in the optimal solution. If the objective function represents total cost to be minimized, then the variables 61*, . , ., ana* measure the relative contribution of the various cost items to the minimum cost. I n the case of zero degrees of difficulty, each term in the optimal objective function has a n invariant Lveight represented by the unique solution of the linear constraints. T h e mathematical importance of this property is that in geometric programs with zero degrees of difficulty the \\eight of each term in the objective function is independent of the coefficients P I . I n practical optimization problems, this property provides insight into the economic or engineering structure of the problem by revealing the invariant \\eights. Thus, an economic or engineering analysis to find the relative importance of the terms in the objective function can be made in the zero degrees of difficulty case without prior kno\\ledge of the numerical values of the coefficients. From a computational point of view, the invariant weights enable one to evaluate the optimal variables X I * , . . ., xm* from Equation 15 for any positive values of P j \vithout solving the programming problem again. I n practical situations this means that the optimal solution is easily adjusted for any change in the coefficients reflecting market fluctuations or altered design parameters. T h e concept of invariant iveights can be extended to the general case of geometric programming with a n arbitrary positive number of degrees of difficulty. I n the following we show that for any geometric program one can find a n invariant range of weights for the terms of the optimal objective function. In the case of zero degrees of difficulty, this range of weights reduces to a single point. Since the value of the product function V(6) depends on the given coefficients p j , the maximizing vector 6* is also generally a function of these coefficients. However, the optimal weights al*, , . . , 6,,* are bounded from below by the nonnegativity relations (Inequality 14) and from above by Equation 12. I n addition, they must also satisfy the orthogonality conditions of Equation 13. Therefore by computing all nonnegative solutions of Equations 12 and 13, \\e can find for each \\eight 6 j a range which is invariant and independent of the coefficients p 3 . This can be accomplished conveniently by solving the follo\ving 2 no linear programs:
+
1. F o r j = 1, , , , , no, maximize 6, subject to Equations and Inequalities 12 to 14. Similarly, 2. For j = 1, , , , no, minimize 6 j subject to Equations and Inequalities 12 to 14. ,
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T h e solutions to these programs yield feasible upper and lower bounds on the variables 61, . . ., ,6, associated with the terms of the posynomial objective function. I n many engineering problems the exponents ai, usually represent technological interdependence of the variables. Accordingly, we refer to the matrix of coefficients appearing in Equation 13 as the technological matrix. We can say, therefore, that for a given technological matrix one can find the invariant range of weights of the various terms in the objective function without knowing the coefficients PJ, which usually represent unit costs or fixed design parameters.
where the pressure drop through straight tubes is APT =
32 fLW2 T2pgD15N2
Assuming that the flow through the tubes is fully developed turbulent flow, the friction factor can be written as (77) f = -
0.046 (Re)o.2
where
Condenser Design Relations Geometric programming will now be applied to the simplified design of a vapor condenser with fixed heat load. Consider then a horizontal condenser in which a fluid having a given flow rate W is heated without phase change from temperature Tbl to To*by condensing saturated steam. Optimal design involves minimizing the annual cost of the condenser, consisting of three terms:
The inside film coefficient is calculated by the well-known Dittus-Boelter correlation (77) :
ht = 0.023
(k)
(Re)0.8(Pr)0.4
The outside coefficient for film-type condensation is given by the modified Nusselt equation (77) :
1. Cost of steam. 2. Fixed charges on the condenser. 3. Cost of pumping the fluid through the condenser tubes. Assuming that the cost of steam can be expressed as a linear function of its saturation temperature ( 7 4 , we can write:
C,
=
+
~ o Q wTSQ ($/year)
(18)
T h e fixed charges on the condenser are expressed as follows:
C, = CHp,Ao
($/year)
T h e number of tubes in a vertical tier of N u horizontal rows can be approximately related to the total number of tubes in the condenser by
(19)
N u N '/2(N)'I2
(33)
Combining Equations 18, 22, 24, 25, 26, 30, 31, 32, and 33 and letting
and the pumping cost is given by: cpu =
CEAPWPF
($/year)
Pt)
(20) (35)
T h e objective now is to select values for T,, Ao, and A P which minimize the total annual cost, given by:
c = c,
+ c, + cp,
(21)
T o simplify the derivation of the necessary mathematical model for geometric programming, certain assumptions are made which may or may not be used in actual design problems. Similarly, the particular correlations chosen for the present analysis may not be the best ones for every practical design problem. I n general, the design engineer has to decide on the simplifying assumptions and the particular correlations he will use. Selection of design parameters is governed by the following design equations, representing quantitative relations between the parameters. The steam temperature can be written as:
Ts =
Tbm
+ ATmo + ATmt
(22)
Combining Equation 19 with 23 and letting (37) (38) From Equations 20, 26, 27, 28,29, and 30 and defining I34 =
(32) (O.O46)CEBpp(W)2J (p/4)".2 gt)P2(a)1'8
(39)
we finally derive
The heat transfer area, based on the (outer) steam side, is given by: A0 = ?rNDoL
(23)
Substitution of Equations 36, 38, and 40 into 21 gives the total annual cost function as
T h e rate of heat transfer is expressed as follows :
C q = hoAoATmo
(24)
= hfAiATmi
(25)
= WCpATb
(26)
T h e total pressure drop through the condenser is given by: A P = BAPT 258
(27)
l & E C PROCESS D E S I G N A N D DEVELOPMENT
aoQ
+
aiTbmQ
+
Since the first two terms of Equation 41 are constants, only the last four terms may vary and are subject to optimization. T h e sum of these last four terms-Le., the following variable cost function,
is a posynomial in the variables N , DO,Di, and L . However, the optimization problem in this form cannot be solved correctly, since DOand Di cannot take on arbitrary values. This difficulty can be overcome by adding a constraint which connects the outside diameter Do to the inside diameter D T
Do - Di 2 2 1
(43)
which can be rewritten in the correct form for geometric programming as 21
lk-+Do
Di Do
(45) Without imposing some penalty o n using tubes of very small wall thickness, it is reasonable to assume that in a n optimal design Inequality 45 will become a n equation rather than inequality. T h e case where 1 approaches zero (DO-+ Di) has special interest. I n fact, let
Dt
=
=
Re
2 Remin
(51)
or by letting
(44)
where the right member is a posynomial. Defining p5 = 21 and P 6 = 1, one can rewrite Equation 44 as
Do
D, N , and L , the values of D and N should be substituted into Equation 30 to see if the Reynolds number obtained is high enough to justify the use of the correlations. Leaving now the case of the “ideal” condenser, we turn our attention to more practical problems. T h e first step in obtaining a n optimal design for a practical condenser is to minimize Equation 42, subject to Inequality 45, and keeping in mind the turbulence constraint on R e :
D
we obtain
No constraints are imposed yet on the numerical values of Di, DO,L, and N . I n fact, constraints of the type of Inequality 53 can be added to the problem in view of the results of a n optimal solution for a previous problem without this constraint. For example, assume that a n optimal solution yields a n impractically low value for DO. Then a n additional constraint is added to the problem in the form of
(46)
and by substituting Equation 46 into 42 one obtains:
DO2 DOm i n
(54)
or
Pe
I n this case m = 3, no = 4 and by Equation 16 this geometric program has zero degrees of difficulty. T h e optimal solution is readily obtained by solving the following system of four linear equations in four unknowns: 61
+
62
+ + 63
84 = 1
7 ---61-oo.262+63-1.864=o 6 -61
4 --613
+ 0.8 + 62
83
- 4 . 8 64
8 2 + 6 3 +
Pe
UminP
= -
4w
the velocity constraint becomes
Such inequality constraints can be easily found by the design engineer and applied in cases where they are necessary. Numerical Solution
(49)
3.3%
4
II ( P J s ~ * ) ~ ~ * j=1
(57)
Urnin
After change of variables and definition of
and Crnin’ =
2
0
43.3%
copu#* =
u
64=0
eoF+*= 53.3%
(55)
When the velocity of the fluid inside the tubes is too low, we add the constraint
(48)
6 3 * = 8 / 1 5 , and = which yields 61* = 2/6, 6 2 * = Recalling the original meaning of the terms appearing in Equation 47, one concludes that the distribution of the annual cost of a n optimal “ideal” condenser (less the cost ofsteam is as follows: a t temperature Torn)
cos#*=
12Do
(50)
T h e values of 6, * obtained and the corresponding distribution of the annual cost are invariants-Le., independent of the numerical values of the coefficients P,. Using Equations 15, 47, and 50 and the values of a3*, one can calculate the corresponding design parameters. Since no constraints whatsoever were imposed on the numerical values of the design parameters, certain values of P3 may yield impractical values of these parameters. T h e only “hidden” constraint in the derivation of the preceding relations was the assumption of turbulent flow inside the tubes. Therefore, after obtaining numerical values for the design parameters,
I t was indicated above that in geometric programming one maximizes the product function V(6) subject to linear constraints rather than minimizes the posynomial subject to nonlinear constraints. Rewriting the product function, we maximize with respect to 6 n,
4
Since the natural logarithm is a strictly increasing function of its argument, we can maximize the natural logarithm of Equation 60 instead of Equation 60 itself-Le., we maximize nn
b
nb
and since maximizing a function f ( x ) is equivalent to minimizing - f ( x ) , we can minimize VOL. 6
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I n order to demonstrate how certain modifications of the original relations developed above can be made, assume that the effect of fouling inside the tubes should be taken into account. Based on experimental results, let the fouling resistance, defined on the inside surface, be given by ( 2 ):
no
C6,=1
RF
]=1 np
aij6j = 0 ;
i
=
I, . . ., m
j=1
(64)
=
0.0002 f 2.63
RF
2 0; j
=
1,
..., np
(65)
Inspection of the function appearing in Equation 62 reveals a n interesting correspondence between geometric programs and chemical thermodynamics, Let 6, represent the number of moles of the chemical species j in phase k of a heterogeneous chemical system a t a constant pressure P and absolute temperature T . Similarly, y k then represents the total number of moles in phase k and Gi/yic the mole fraction of species j in phase k. Also let 1n(l/Pr) = (Fjo/RT)
+ In P
(66)
where Fjo is the standard free energy of species j and R is the gas constant. The function -lnV(6) then becomes the total Gibbs free energy of a n ideal heterogeneous chemical system (9). The equilibrium composition of such a system is found by minimizing its total free energy, subject to mass balance constraints of the type of Equations 63 to 65. Thus, the solution of a geometric program is equivalent to finding the equilibrium composition of a n abstract heterogeneous chemical system ; conversely, chemical equilibrium problems are closely related to geometric programs ( 7 , 72). We take advantage of the above stated equivalence by solving a n optimal condenser design problem with the aid of a numerical method developed for the chemical equilibrium problem by Clasen ( 4 ) .
(Tom- 60)
(68)
and in our example
and 6j
x
0.0005682
=
(69)
Equation 22 is modified as follows:
Ts
=
Thm f ATmo f A T m t f A r m ,
(70)
and a n additional relation exists-namely,
Defining
Equation 42 becomes
where the last term a t the right-hand side represents the additional cost due to fouling. Substituting numerical values into Equations 34, 35, 37, 39, 45, and 72, one obtains:
Example
Consider the case of a sea water desalting plant using the single-effect multistage flash distillation method. Sea water a t a rate of 500,000 pounds per hour is to be heated from 195' to 205' F. by condensing steam. Assume that steam is available in a continuous range of temperature with cy1 = $/B.t.u.-' F. Unit cost of the condenser is C, = 596/sq. foot. For annular charges assume 10% maintenance and depreciation-i.e., p c = 0.1. T h e cost of electricity is C, = lo-* $/kw.-hr. and the plant factor is PF
=
0.9 X 24 X 365 = 7884 hr./yr.
(67)
Let the over-all pump efficiency, 7, be 0.8and for simplicity assume that sea water has the same physical properties as pure water. At the average bulk temperature, Thmof 200' F.: 60.13 lb./cu. ft. p = 0.28 CP. k = 0.393 B.t.u./hr.-sq. ft.-' F./ft. C, = 1.01 B.t.u./lb.-' F. p
First the problem of minimizing Equation 73, subject t o the constraint represented by Inequality 45, was solved by geometric programming, using Clasen's computer algorithm for the chemical equilibrium problem. The optimal solution was found to be DO* = 1.3 inches. As a practical consideration, a constraint expressing the fact that the maximum practical value of DOcannot exceed 1 inch, was formulated-i.e., or
=
Assume that the average outside film temperature is T , = 210' F. and 59.88 lb./cu. ft. 0.26 CP. k , = 0.393 B.t.u./hr.-sq. X = 960 B.t.u./lb. p, =
I.(, =
ft.-O
F./ft.
Let the minimum allowable tube wall thickness be that of BSYG 18-i.e., 1 = 0.049 inch. Also let B = 1 . 2 . 260
(74)
l&EC PROCESS D E S I G N A N D D E V E L O P M E N T
where 1 --
(77)
011 = 12
(78)
811 and since DO
=
l/12
=
Do max
foot,
At this point some simplifications of the problem are possible. Since the optimal value of D Ofor a problem without Inequality 76 was greater than DOmax it is clear that the optimal value of DOfor the same problem, augmented by Inequality 76 will be
Do* = Do ma=
(79)
Since no penalty was imposed for selecting the smallest allowable wall thickness, in any optimal solution Inequality 45 will be a n equality. Therefore, from 45 and 79 the optimal values of D jand D O can be calculated and substituted into Equation 73, yielding new values of the Pj and leaving L and N as the only unknowns. I n practice, however, this kind of simplifications is negligible and next the augmented problem was solved. Inequality 45 as a n equality cannot be used to eliminate DO or D ifrom Equation 73, since this would result in transforming Equation 73 into a n expression which is not a posynomial, and geometric programming would not be applicable. As a matter of fact, frequently a function which is not a posynomial can be transformed into the desired form by a change of variables and addition of constraints. T h e mathematical programming problem now is to minimize Equation 73 subject to Inequalities 45 and 76. T h e equivalent maximization problem, as was shown in previous sections, is 6
+ + 66)ln(6s + + ln(PlO/b) + (80) subject to the constraints + + + + = 1 Maximize
[6j
j=l
In(Pj/6j)]
(65
66)
67
68
1x3 P I 1
:
-
z6
61
62
61
- 0.2 62
+
+
-61
63
-
-
62
67
-
1 . 864
- 87
63
+
- 4.8 6 4
+ +
66
64
63
+ + + +
=o
6,
+ 6 g = o
- 6 6 - 6 6
0.8 62 4 - 61 3
64
83
Comparing the optimal cost distribution of a practical condenser with the “ideal” case, we observe that the latter provides a rough estimate of the relative importance of the various cost items in the practical case. T h e “ideal” distribution was obtained by solving a square system of linear equations, while the practical distribution was found by the constrained maximization of the corresponding product function. Thus, one can assume that in certain practicaI cases the design engineer or economist can get a n estimate of the optimal cost distribution by possibly reducing the geometric program under consideration to a zero degrees of difficulty case using reasonable simplifying assumptions. This estimate can be obtained in early phases of the engineering or economic analysis, nithout evaluating the coefficients Pi. However, for the particular coefficients of this example, the “optimal” design variables of the “ideal” case were absolutely impractical. More complete information on the cost distribution can be generated by finding the invariant ranges of the M eights. T h e invariant ranges of the weights of the terms in the variable cost function (Equation 73) were found by solving a set of linear programs as \%asoutlined above. Thus, the invariant range of the steam cost contribution is found by solving the following two problems: 1. Minimize 61 89 a7 subject to Constraints 81 and 82. 2. Maximize 61 62 6; subject to the same constraints. T h e contributions of other terms are found similarly. Let
- 87
=o
- 67
=o
=
61
6’y =
63
6,u
+ + 62
87
(86)
= 64
Then the solution of the six linear programs of minimizing and maximizing 6,, b H , and 6,, yielded the following invariant ranges of the weights:
42.86% _< CSp_< 50.00% 35.71% T h e solution to the maximization problem yields the following optimal variables: 61* = 0.1115 62*
=
0.1869
63*
=
0.4585
64*
=
0.0601
6 ~ *= 0 , 0 3 5 0 66*
=
0.3217
a7*
=
0.1830
(83)
0.00%
5 C,# 5 5 C,,* 5
54.59%
(87)
14.2970
T h e optimal values of the four design variables are found by substituting the first four terms of Equation 73 into 15, using the first four values of Equation 83 and 84, and taking the logarithm of both sides:
- -76
log N
- log D O - !log 3
L
-
- 0 . 2 log N
-
log L
+ 0.8 log Dj =
log N
+ log Do +
log L
-
6 s * = 0 0096
and the minimum variable annual cost is
Cmln4*= 883.77 $/yr.
(84)
For the “ideal” condenser the corresponding cost would be C#* = 526.10 $/yr., but the numerical values of the design variables are such that the “hidden” turbulent flow constraint is violated. Recalling the original meaning of the terms in Equation 73, one gets from Equation 83 the distribution of the minimum variable condenser cost: Cs4* = 48.1%
(43l/3%)
c,$* = 45.8%
(53*/3%)
c,,p*
( 3l/3%)
=
6.1%
T h e figures in parentheses correspond to the “ideal” case.
(85)
log 63*Crni,% P3
+
- 1 . 8 log N
log L
- 4.8 log Dj = log 64*Crni,# P4
Solution of the above system of four equations in four unknowns yields : Do* = 1 inch
Di*= 0.902 inch (89)
AT*
=
112.2
L*
=
27.78 feet
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Simplified heat transfer design relations for heat exchangers in which a saturated vapor is condensed a t a constant temperature are the same for single-pass as well as for multipass exchangers ( 3 ) . Xeglecting the difference in pressure drop between a single-pass and a two-pass exchanger and rounding off the values of Equation 89 to the nearest integer, we finally conclude that the optimal condenser will be a twopass type, containing 224 one-inch o.d., 18 BWG, 14-feet-long tubes. T h e other design variables are found by direct substitutions and straightforward calculations. Thus
T8* = 210.68' F.
U*
=
580.7 B.t.u./hr.-sq. ft.-" F.
(90)
Re* = 113,600 u*
= 4.73 ft./sec.
where the over-all heat transfer coefficient, U*, was obtained from
number of terms over the number of variables. Addition of inequality constraints to the optimization problem causes a considerable increase in the computational effort required by conventional methods. O n the other hand, in geometric programming the difficulty of solving such problems is proportional to the excess of the total number of terms, including both terms of the objective function and terms in the constraints, over the number of variables. I n fairly complex optimal design problems such inequality constraints are usually unavoidable. Aside from the solution of the optimization problem, geometric programming also provides considerable insight into the distribution of the minimum total cost among the various cost items. I t was shown that in the case of zero degrees of difficulty this distribution is invariant, while in the general case an invariant range of weights of the cost items can be obtained which is independent of unit costs or given process conditions. Acknowledgment
All preceding calculations were based on a value of Tbmwhich was calculated as the arithmetic mean bulk temperature,
Since in Equation 91 a better expression for the temperature difference would be the logarithmic mean,
ATlm
=
In
Tb2 - Tbl Ts - Tbl T8
-
(93)
Tb2
we compare the temperature difference (Ts*- Tbm) with Equation 93 for our optimal solution:
Ts* - Tom
=
10.68
(94)
and Equation 93 yields
ATZ,* = 10.0
(95)
Although there is a 6.8y0 difference between the temperature differences in Equations 94 and 95, recalculation of the optimal design using the value of Tbm = 200.68' F. yielded negligible improvements. Actual computation time of such a condenser design problem on an IBM 7090 computer was less than 10 seconds. Conclusions
Application of geometric programming to chemical process equipment design was demonstrated by finding the optimal design of a horizontal tube condenser. I n case of a geometric programming problem in m variables and m 1 terms (including constraints), a unique solution is obtained by solving a system of linear equations. I n cases where the number of terms exceeds the number of variables by two, a simple procedure can be used to solve the problem (78). If the number of terms exceeds the number of variables by more than two, the problem is reduced to the minimization of a convex function subject to linear constraints, solvable numerically by Clasen's algorithm for minimizing the Gibbs free energy function. Consider now an unconstrained minimization problem of a posynomial in rn variables and no terms. T h e difficulty of solving such a problem by conventional methods of calculus is proportional to the number of variables, while in geometric programming it is essentially proportional to the excess of the
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262
I & E C PROCESS D E S I G N A N D D E V E L O P M E N 1
The authors thank the RAND Corp. for supplying the computer program. Nomenclature
inside heat transfer surface, sq. ft. outside heat transfer surface, sq. ft. pressure drop factor annual cost, $/year cost of electricity, S/(ft.) (lb.) fixed charges on condenser, $/year unit cost of condenser surface, $/sq. ft. specific heat, B.t.u./(lb.)(" F.) pumping cost, $/year cost of steam, $/year inside tube diameter, ft. outside tube diameter, ft. Fanning friction factor specific gravity, ft./(hr.)2 inside film coefficient, a.t.u./(sq. ft.) (hr.) (" F.) outside (ccndensing) film coefficient, B.t.u./(sq. ft.) (hr.)(" F.) thermal conductivity, B.t.u./(ft.) (hr.) (' F.) tube ivall thickness, ft. tube length, ft. No. of variables in posynomial KO. of terms in objective function total No. of terms in objective function and constraints No. of tubes in condenser No. of tubes in a vertical tier rate of depreciation, l/year plant factor-effective operating time, hr./year positive terms Prandtl number condenser heat load, B.t.u./year rate of heat transfer, B.t.u./hour Reynolds number Fouling resistance, (sq. ft.)(hr.) (" F.)/B.t.u. mean bulk temperature, ' F. steam temperature, F. over-all heat transfer coefficient, B.t.u./(sq. ft.) (hr.)(" F.) velocity inside tubes, ft./hr. product function flow rate inside tubes, lb./hr. GREEKLETTERS = coefficient in steam cost equation, $/B.t.u. = coefficient in steam cost equation, $/(B.t.u.)(' F.) = positive coefficient = sum of weights of kth constraint = total pressure drop, lb./sq. ft. = pressure drop in a straight tube, lb./sq. ft.
ATb = temperature rise of fluid in condenser, F. ATrni = mean temp. drop through inside tube film, F. AT,, = mean temp. drop through inside tube fouling, F. AT,, = mean temp. drop through condensing film, O F. 6, = nonnegative weight B = over-all pump efficiency X = latent heat of evaporation, B.t.u./lb. p = viscosity, lb./(ft.) (hr.) p = density, lb./cu. ft. SUBSCRIPTS b = bulk = physical property evaluated a t outside film temp. f max = maximum min = minimum 0 = “ideal” condenser case SUPERSCRIPTS * = optimal = corresponds to variable cost # literature Cited
(1) Avriel, M., Ph.D. dissertation in chemical engineering, Stanford University, 1966. (2) Bechtel Corp., private communication to the authors. (3) Bennett, C. O., Myers, J. E., “Momentum, Heat and Mass Transfer,” McGraw-Hill, New York, 1962.
(4) Clasen, R. J., “The Numerical Solution of the Chemical Equilibrium Problem,” RAND Corp. Research Memorandum RM-4345-PR (1965). ( 5 ) Duffin, R. J., J . SZAM 10, 119 (1962). (6) Duffin, R. J., Operations Res. 10, 668 (1962). (7) Duffin, R. J., Peterson, E. L., Zener, C., “Geometric Programming,” Wiley, New York, in press. (8) Ferron, J. R., Chem. Eng. Progr. Symp. Ser. 60 (50), 60 (1964). (9) Gibbs, J. W., “The Scientific Papers of J. Willard Gibbs,” Vol. I, Dover Publications, New York, 1961. (10) Hardy, G. H., Littlewood, J. E., Polya, G., “Inequalities,” University Press, Cambridge, 1964. (11) McAdams, W. H., “Heat Transmission,” 3rd ed., McGrawHill, New York, 1954. (12) Passy, U., Ph.D. dissertation in chemical engineering, Stanford University, 1966. (13) Sherwood, T. K., “A Course in Process Design,” M.I.T. Press, Cambridge, Mass., 1963. (14) U. S. Atomic Energy Commission, “Engineering and Economic Feasibility Study Phases I and I1 for a Combination Nuclear Power and Desalting Plant,” TID-22330 (1965). (15) Wilde, D. J., Znd. Eng. Chem. 57, 19 (1965). (16) Zener, C . , IEEE Trans. Mil.Elec. MIL-8,63 (1964). (17) Zener, C . , Proc. Natl. Acad. Sci.48, 537 (1961). (18) Zener, C., Duffin, R. J., Westinghouse Engr. 24, 154 (1961). RECEIVED for review September 2, 1966 ACCEPTED December 9, 1966 58th Annual Meeting, American Institute of Chemical Engineers, Philadelphia, Pa., December 1965. Research supported in part by the Office of Saline Water Grant 14-01-0001-699.
COM M UN ICATl ON
SIZING O F STORAGE T A N K S FOR OFF=GRADE MATERIAL In some processes it is advisable to include in the design a storage facility for off-grade batches which are then added in small amounts to the regular production. A method for determining the required size of such storage tanks is given and it is shown that such problems can b e solved employing queuing theory. DESIGSING blending facilities for materials with close specifications (such as pigments, drugs, and some polymers), the problem often arises how to take into account the occasional appearance of an off-grade batch. Standard design methods for blending facilities (7, 2) are based on the inherent variability of the process, and the size of the blender is determined by the desired ratio of output variability to input variability. I n this context a n off-grade batch (or production period) is not a batch 1% hose composition is outside the specifications, but a batch M hose deviation from the average composition is larger than some multiple of the standard deviation. Such batches might occur during startup or shutdown periods of the plant, or occasionally if the process gets out of control. I t has been suggested ( 7 ) that in processes in which there is reasonable finite probability of the occurrence of off-grade batches it might be advisable to include a n off-grade storage tank in the design of the blending facility. Often, this material can, after suitable tests, be added in small amounts to the regular product. T h e first thing to consider in sizing a n off-grade storage tank is the probability of the occurrence of an off-grade batch. This should preferably be ascertained from previous plant performance. I n a continuous process there are obviously no identifiable batches, but the maximum time of off-grade production is often determined by the nature of the quality control methods used. I n a batch process any single batch has a certain probability of becoming off-grade. Let us now assume, from a previous experience, that on the average a fraction [ of all batches turns
IN
out to be off-grade. If the probability that a batch will become off-grade is independent of what happened to the previous batch, then f is also the probability that any batch will be off-grade. Now we have assumed that by blending a n offgrade batch with a suitable number of acceptable batches we obtain a product that is still within specifications. If we spread the off-grade material completely evenly over all the other batches, each off-grade batch would be blended with (1/4 1) acceptable batches. However, it might not be necessary to achieve such a uniform spreading of the material and it is sufficient to spread the off-grade batch over n acceptable batches. The fraction of off-grade material in any material to which a n off-grade batch has been added is 1/ (n l ) , and let us designate this fraction as p. I t is immediately apparent that p must be larger than or equal to 5, as n must be smaller than (l/[ - l ) , or we would have to discard some of the off-grade batches. T h e ratio
-
+
P = €/I =
number of blend-off batches average number of batches containing one off-grade batch which is less than or equal to unity, is a quantitative index designating how well the off-grade material has been spread out over the regular production. For any specific case the required minimum value of p can be determined from the specification of the material. Having determined p, 6, and p, one can now calculate the size of the off-grade storage tank necessary. Obviously, if the VOL
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