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cost, amountingin itself to more than .Sl,OOO,OWJ, it is an appropriate example for an investigation of shape remaining number of sites suitable for once-through optimization. Before developing a complete program, coolingof power plants have resulted in an increaseduse including stress analysis, stability, and dynamical reof coolingtowers for condensercoolingof power stations. sponse,it seemedprudent to investigatethe magnitudeof In the past few years approximately40% of new plants possible savings achievable through shape optimization. have been constructed with coolingtowers, and by Ml0 Obviously,if there were to be no sign&ant potentialfor this figure may rise to 60%, depending upon the cost reductions, there would be no point in developinga interpretation and enforcement of environmentalprotec- sophisticated optimkation program. Moreover, several tion laws. Requirements for backfit of existing plants examples are needed to produce a valid conclusion. usingonce-throughcoolingcouldappreciablyincreasethe Therefore, it is necessary to have a set of independent demandfor coolingtowers. Many of these towers willbe results for a range of cooling towers, in order to mechanical draft, but many will be of the natural draft investigatethe potential benefits of conggmation optimtype, in which a chimney generates the flow of air ization. The only available published rest&s fulfill@ necessary to cool the water. More than 60 natural draft these criteria were those of Gould[2].In order to compare towers have been built or are under construction in the the results of the shapeoptimizationwith Gould’stowers, United States for Rower station service. These natural the followingpresentationfollows the problem definition draft towers are the evaporative type, but dry cooling as given by Gould[t]; only the shape of the tower is changed. towers have been built in Europe. A hyperboliccoolingtower followsthe shapefunction Several different shapes have been used for the veil, or structural shell of the tower. The first cooling tower, of hyperbolicshape,was built in Hollandin 1916.Cylindrical towers have been constructed, and many European towers have been composed of a frustum of a cone in which surmountedby a section of a torus, with another iJaring z = vertical distance measured from the base conical section at the top. These types have been S = height to throat (point of minimumradius) univetsally superceded by the hyperbolic tower, whichis k = slope factor relatively easy to analyze, exhibits primarily membrane P = throat radius. stresses under design loads[3], and does not have r = radius of shell at height t discontinuitieswhere one shape joins another. There is apparently, however, no proof that a hyperbolic shape is In the configurationoptimizationproblem, the height optimal for this service. and the radius of the tower (at every point) are the design variables.It is convenient, however, to specify a general function for the tower radius.The optimalconfigurationis FORMULATIONOF TBE SEW% here assumedto be a shellof revolutionwiththe following OFl’MEATION PB0BL.W Inasmuchas the structural cost of the veil of a natural equation: draft cooling tower is a major portion of the total tower -ON

Recent environmental legislationand the decline in the

r=i:a,z’

tFormerly U.N. Fellow, Center for Advanced Engineering Study, MIT, Cambridge,Massachusetts02139,U.S.A.

in which ai are shape parameters n = number of terms ‘inthe series.

321 CASVOL.

5 NO. 5&E

KENNETH F. REINSCHMIDT and R. NARAYANAN

322

The polynomial series was selected because it offers the possibility of generating a general shape and is fairly easily handled algebraically. Other functions, such as Fourier series, might perform as well or better; no investigation has been made of the best generic function, and lacking any other evidence this easily computed function was used. In actual computation it is more convenient to use the nondimensional variables R = r/H and 2 = z/H, where H is the total height, so that R

=QW _

and

If d(z) is the thickness of the shell at height z, and if c(z) is the construction cost per unit volume of the shell, as a function of height z. then the total cost of the shell may be approximated by cost = [“2+

+ ($!‘)

d(z)c(z) dz.

(5)

in which s = radius of the base of the tower M = specified performance coefficient. The geometrical constraints are

1.05+

1.17

VW

1.03=ks1.22

(6c)

15ftIa

(66)

S loOft

in which t = radius of the top of the tower.

WMMlZATlON

PROBUM

In the configuration optimization problem considered here, the design variables are the height H and the n + 1 coefficients ,¶ in equation (3). Inspection of the constraints (equations 5 and 6) and the objective function (equation 4) shows that this is a nonlinear programming problem. There are several methods available for solving such problems: the technique chosen here was that of iterated linear programming[4]. In this method, the objective function and constraints are expanded in Taylor Series and linearized by retaining only the first-order terms in these series. The optimization variables then become the incremental changes to the original design variables. New constraints, called move limits, are introduced to restrict these changes to the region in which the linear approximations are reasonably accurate. Because the first-order Taylor series are only approximate, the process must be iterated, generating new Taylor series expansions at each iteration, until it converges.

(4)

However, to agree with Gould’s de&&ion, the veil is assumed to be of constant unit thickness, d(z) = 1, and constant cost, c(z) = 1. The resulting objective function is thus proportional to both the volume, u, and the surface area of the shell. However, because equation 2 is used to represent the shape of the surface, the integral in equation 4 cannot be given in closed form. Instead, Simpson’s Rule is used to integrate equation 4 numerIcally. It was found that using 16 and 30 intervals in the numerical integration did not appreciably aifcct the result, and the numerical examples presented below were computed with 16 intervals of height HI16 each. In the paper by Gould[2] neither the structural nor the heat transfer behavior of the tower was considered explicitly. Instead, Gould minim&d the volume of a hyperbolic tower of unit thickness subject to a constraint on the ‘duty-performance coefficient’, M, defined by Chilton[l], and a set of geometric constraints intended to ‘. . . restrict the design variables to the practical range of hyperbolic cooling towers for which the dutyperformance coefficient has been developed’[21. The performance constraint is ns’d\/H 2 M

SOLIJlYON OF THE SHAPE

I

/

! t

I i

i

jolj JI;lse\rT

Fig. 1. Cooling tower shell, generalconfiguration. COME4RISON OF REWLTS

For purposes of comparison, a sample of 11 towers was taken from the set of 40 presented by Gould. The pertinent data for these I1 hyperbolic towers are given in Table 1, extracted from Gould’s Table 1[2]. The optimization procedure described above was begun using these examples as starting points. For each tower, a hyperbola matching Gould’s data was determined and a polynomial (equation 2) of the appropriate order was fitted to this hyperbola using the method of least squares. The fit was generally very good, as determined by comparing tower shape and the volume resulting from the numerical integration of the fitted polynomial to the exact parameters for the hyperbola given by Gould. The iterated linear programming cycle was then started with these initial values for the parameters. Move limits were introduced to restrict the change in any parameter in a single iteration to t 15% of its current value. It was found that the constraint on the throat radius (equation 6d) was never binding and was deleted as being redundant. However, in order to achieve results comparable to Gould’s, it was necessary to introduce a new constraint not specified by Gould: s 5 so

(7)

323

The optimum shape of cooling towers Table 1. Data for example towers [2] ExamPie

Gould Tower

Zt, feet

H. feet

k

254.55

147.37

258.60

1.12

813,347

148.589

16

170.00

261.54

200.00

215.04

1.22

787.803

139.184 222,336

2 3

24

144.50

222.31

160.56

422.36

1.04

797.703

4

38

98.W

217.70

103.16

358.43

1.05

705.214

158.939

5

39

150.00

230.77

157.89

290.05

1.07

712,332

160.826

6

40

130.00

236.36

136.84

310.01

1.07

772,570

162.826

7

1

120.00

266.67

141.18

220.98

1.22

830.237

126,183

a

10

150.w

272.73

157.89

195.92

1.22

817,689

125.134 120.951

9

15

120.00

266.67

133.33

209.49

1.22

808,362

10

25

120.00

266.67

126.32

196.28

1.22

782.466

115.384

11

26

140.00

254.55

155.56

198.27

1.22

716.549

116,583

Table 2. Optimization results, R = 6, base diameter restrained P'4

La,fwt

ZLfect

v,ft3

2s. fret

14o.w

2

in which sG= base radius of Gould’s solution. Without this constraint, the optimization tended to increase the base radius and reduce the height to such an extent that the resultingtower proportionswere quite dissimilarfrom Gould’s(see Table 3). Table 2 summarizesthe results for polynomials with n = 6. Ah examples were run for a uniform 20 iterations even though convergence on the volume generally occurred at a much fewer number of cycles. An IBM 1130 with 8 K core was used for all computations.The duty-performance coe5cients are in all cases greater to or equal to those given by Gould,and, due to the additional constraint (equation 7) the base diametersfor all towers are exactly those used by Gould. By virtue of equation 5, the tower height cannot vary unless the base diameter does. Even under this severe limitation,the optimizationprogram achieves savingsof 2-2295in volumecomparedto the hyperbolictowers. The explanationfor this is simple:the hyperbola (equation 1) has only 3 free parameters while the polynomial,with n = 6, has 7 free parameters. Therefore the polynomial

fxm-

M,R5'2

24,feet

1

k

v, f&3

XRcdvctlon V Conpwed ta Table 1

1

114.68

120.94

1.21

131.303

2

118.34

124.26

1.21

117.810

11.63X 15.31x

3

99.48

105.62

1.12

172,602

22.35x

4

98.47

103.47

1.17

144,350

9.18X

5

104.38

109.64

1.21

120.23a

20.26X

6

l%.%

112.29

1.21

159,097

14.571

7

120.66

126.70

1.21

123,746

1.93x

8

123.28

129.58

1.21

116.219

7.161

9

120.60

126.70

1.21

118,402

2.112

10

120.56

126.70

1.21

113.102

1.981

11

114.95

120.94

1.21

107.535

7.77X

configurationis able to achieve a closer approach to the constraints and, in many cases, a sign&ant reduction in volume. In most examples,the throat and top diameters have been substantiallyreduced from the initial values, while satisfyingthe geometric constraints (equation 6). If the additionalconstraint(q&on 7) is not imposed, the base diameters increase and tbe heights decrease dramatically,as shownin Table 3. The volumesavingsare substantial,with reductions of 18-5796compared to the hyperbolic towers. As the lower towers should be more economical to construct, the cost reductions should be even greater. Gf course, the duty-performancecoeflkient used to define the tower coolingperformance may not be adequate,and space limitationsmay precludetowers with very large base diameters, so the savingscomputed here may not be fully achieved in practice. Nevertheless, the’ saws indicated by Table 3 are sufkiently large to justify further investigationof the optimal con&u&on for cooling towers. In the polynomial expression for the cooling tower configuration(quation 2), the number of terms n is a matter of choice. To investigate the possible benefits from increasingII,six of the examplespresented in Table 2 were recomputed with n = 7. The results are shown in Table 4. Three of the six cases show no improvement, three showadditionalsavingsof approximately1%due to the use of seven terms as compared to six. To investigatethe sensitivityof the results to the values of the constraints,the limits given by Gould were varied and new optimization results obtained. In one case presented here, the constraintsgiven by Gould (equation 6) were modikd to 154+2.75

Table 3. Optimization results, n = 6, base diameter not constrained ExalllPlC

23,fnt

2t,faat

h.fnt

H.fwt

k

1

203.30

313.07

213.46

113.M)

1.21

99.955

32.73X

2

192.68

296.72

202.31

129.86

1.21 103.511

25.63X

3

206.43

348.50

237.60

65.20

1.21

94,197

67.63%

4

168.22

275.48

166.12

139.68

1.21

97,679

38.54X

5

182.62

291.81

191.75

113.03

1.21

91,179

43.31%

V,ft3

X Reduction In v cmpmd Table 1

to

6

188.83

298.71

198.27

120.81

1.21

98,500

39.441

7

218.56

336.60

229.51

87.15

1.21

92,962

26.33X

8

200.11

308.16

210.11

120.12

9

210.42

324.05

220.94

95.89

1.21 102,366

18.191

1.21

95,345

21.171

10

210.09

323.54

220.59

90.64

1.21

90,076

21.93X

11

203.26

313.06

213.44

86.80

1.21

83.800

28.12X

@a)

324

KENNETH F. REINSCHMIDT and R.

NARAYANAN

Table 4. Optimization results. n = 7 Examk

V.feet3

% Reduction in V hIPSred to Table

120.94

1.21

131,344

11.61%

124.26

1.21

116.530

16.28%

0.97%

Za.fect

Zt,feet

1

115.18

2

111.41

Pie

% Reduction in Y to Table 2

1 Compared 1

3

100.59

105.62

I.79

169,864

23.60%

1.25%

4

93.46

103.47

1.12

142,687

10.23%

1.05%

5

102.19

109.64

1.19

128,513

20.08%

3

6

106.02

112.30

1.18

139,396

14.39%

0

1.00+

value, and with the base diameter allowed to vary. The

144

@b) results are shown in Table 6. They are, in general, similar (84

1.035 k = 1.41.

Constraint equation 7 was retained, so that the base diameter and hence the height did not vary. The results are shown in Table 5. Clearly the savings attained by use of the more liberal constraints are substantially greater than those using the original constants. This result indicates that a more refined method of optimization is desirable, as the optimum configuration may be sensitive to the values of the arbitrary limits given in the geometric constraints. It also further demonstrates that the potential benefits from configuration optimization are sign&ant, even if the height and base diameter are not permitted to varyIn addition to the set of towers selected from Gould’s paper, it is of interest to compare the result using the shape optimizing procedure to the shape of a real tower. For this purpose, the hyperbolic tower erected at the Trojan 1 station of the Portland General Electric Company is selected as an example. The structural shell is 451 ft tall, and the tower is designed for a range of 37°F. an approach of 17”F, and a flow of 425,ooOgpm. Other data on the shell configuration are given in Table 6. A polynomial of the form of equation 2, with n = 6, was fitted to the shape of the Trojan tower by the method of least squares. The duty-performance coefficient, A4,was computed using the dimensions of the actual tower in equation 5. The shape optimization procedure described above was then performed using this configuration as a starting point and the constraints in equation 6, under two conditions: with the base diameter restrained at its initial

to those presented in Tables 2 and 3. The computed volume reduction is based on equation 4 as before, in which the thickness is constant, and is not based upon the actual volume of the Trojan tower. The results indicate that, using the assumptions listed above, a tower based upon a polynomial generator is about 1% lower in volume than a tower based upon the hyperbolic generator used for the Trojan tower, with the same height, base diameter, and duty-performance coefficient. The results given here do not necessarily imply that the design of the Trojan tower itself could be improved, as the method of analysis used is not sulilcient to support such a conclusion. To demonstrate the changes in conhguration graphically, the three towers in Table 6 are plotted in Fig. 2. The optimized shape with the base diameter fixed has reached the limiting values of k (1.211, s/a (2.21). and t/a (l*OS); the top of the tower does not Rare outward as in the hyperbolic; rather, it is practically vertical. The greatly reduced outlet diameter means that, for the same air flow rate, the air exit velocity will be greater than that for the hyperbolic. The optimized tower with the base diameter free to vary is obviously much squatter than the others. It is diicult to believe that this tower would perform identically to the original tower, even though they both have the same value for the duty-performance coelIicient. Of course, any shape intermediate between the two optimized configurations could be generated by the program, by varying the restraint on the base diameter; the squat configuration shown may be regarded as the limit of this process.

Table 5. Optimization results, revised constraints, n = 7 % Reduction in V Compared to Table 1

% Reduction in V CornpsredtO Table 2

ExamPie 1

87.58

129.46

1.18

126,107

15.13%

2

94.24

95.10

1.41

106,652

23.37%

8.06%

3

79.76

80.84

1.24

148,087

33.40%

11.05%

4

75.08

79.19

1.13

126,774

20.23%

11.05%

5

83.92

83.92

1.37

112,727

29.91%

9.65%

6

85.95

85.95

1.25

124.853

23.32X

a.75%

2a.feet

Zt.fset

V,f&3

k

3.50%

Table 6. Results of comparison with Trojan tower Za.feet

Zs.feet

Zt.feet

Trojan (actual)

232.00

367.00

250.00

: Reduction in V Compared to Trojan 451.00 1.04

Optrmrred (&se Restrained)

165.83

367.00

174.37

451.00 1.21

18.8%

Optimized (Base unrestrained)

319.42

441.90

335.39

138.94 1.21

47.7:

Tower

H.feet

k

The optimum shape of cooling towers

325

believe that a hyperbolic tower is necessarily more than a polynomial shape. Considering the significantvisualimpactof a large coolingtower, this area Trojan tower deserves further study, but it may be noted that the polynomial function, with a greater number of free parameters, offers much greater flexibilityin shape than does a hyperbola. There are, of course, many limitationsto the present study. The objective function used does not necessarily represent true cost, and more information on actual construction costs for large towers, includingthe effects of height and shape, is necessary. The thermodynamic behavior of the tower is represented only by the duty-performancecoefficient,and the possible effects of tower shape on air flowrate, head loss, air velocity, drift, and behavior of the plume are not considered. Most important, it is assumed that changes in shape will not cause major changesin the stresses or the stabilityof the 100- bosc fixed shell,which mightrequire significantincreasesin average shellthickness.This is consideredto be reasonablewhere the shell thickness is govetied by depth of cover and minimum thickness requirements, but the structural behavior must obviously be checked by. structural analysis. It should be noted, however, that these ft limitationsapply equally to Gould’stowers, and do not Fig. 2. Comparison of shape of Trojan tower with optimized invalidatethe conclusionsderived from comparisonwith con6guratilJns. his results. Further research is currently beinginitiatedto address these problems, by means of structural analysis coNu.usloNg methodsfor generalaxisymmetricshellsof revolutionand The study presented here is not intended as a final the developmentof programsto evaluatethe heat transfer statement on the con@ration of natural draft cooling and air flow conditions inside the tower. towers. Its purpose is to investigate the potential cost savings which might be achievabie through shape Acknowledgement-The authors thank tbe h4lT Center for optimization. Although the model used is very elemen- Advanced Engineering Studies for computer time on the IBM tary, it is believed that the results.indicatethat sign&ant 1130,and the autborities of S.E.R.C. and U.N.D.P. for the U.N. economiesare possibleand that further developmentof a Fellowship which supported tbe second author during this study. more refined system is justified. It is believedthat the shapesgeneratedby the use of the polynomial function would be no more difficult to 1. H. Cl&on, Performance of natural-draft water-cooling towers, Proc. Institution of Eiectricul Engineers, 99,440-456 (1952). construct than those generated from a pure hyperbola or any other function with variablecurvature.The ordinates 2. P. L. Gouid, Minimum weight design of hyperbolic cooling towers, 1. Stnrcf. I)iu. ASCE 95 (SR) 203-208 (1%9). of the shell surface can easily be generated by computer for use in field construction. Althoughthe aesthetics of 3. P. L. Gould and S-L. Lee, Bending of hyperbolic cooling towers, 1. Smrcf. Dia AXE 93 (STS) 125-146 (1%7). some of the more extreme configurations may be 4. K. F. Reinschmidt, C. A. CorneU and J. F. Brotchie, Iterative questionable, in many cases the polynomial shape is design and structural optimization, 1 Sfruct. l)io. ASCE 92 visually indistinguishablefrom a hyperbola. Moreover, (ST6) 281-318 (1966). the visual appearance of a real tower viewed at close 5. J. J. Der and R. Fidler A model study of the buckling behavior range from the ground is quite different from that of an of hyperbolic shells, Proc. Institution of Civil Engineers 41, elevation drawing, and there seems to be no reason to 105-118(KM). attractive

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