Cost-Optimal Design of Dry Cooling Towers Through Mathematical Programming Techniques

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1 J. D. Buys Snior Lcturr, Dpartmnt of Mathmatics. D. G. Krogr rofssor, Dpartmnt of Mchanical Enginring. Mm. ASME Univrsity of Stllnbosch, Stllnbosch, South Africa Cost-Optimal Dsign of Dry Cooling owrs hrough Mathmatical rogramming chniqus h Constraind Variabl Mtric Algorithm is chosn to minimiz th objctiv function (cost) in th dsign of a natural draft dry cooling towr. An xisting cooling systm dsign that has spcific prformanc charactristics undr prscribd oprating conditions is slctd as a rfrnc unit. By changing dsign variabls, but not xcding prscribd constraints, a mor cost-ffctiv dsign is achivd. h influnc of various paramtrs, and th snsitivity of th objctiv function to ths paramtrs, ar valuatd. Introduction In this study th application of modrn minimization tchniqus to obtain cost-optimal dsigns of larg natural draft dry cooling towrs and systms incorporating ths towrs is illustratd. Various optimization mthods applid to th dsign of air-coold hat xchangrs hav bn summarizd by Hddrich t al. (1982). W hav chosn th Constraind Variabl Mtric Algorithm (owll, 1978) (also known as th Squntial Quadratic rogramming Mthod, or Han's Mthod), to solv th problm of minimizing th objctiv function (cost), subjct to prscribd constraints. A comparativ study of 27 computr cods for constraind minimization by Hock and Schittkowski (1983), has ratd cods basd on this mthod as th bst currntly availabl. h mthod is rlativly complicatd and rquirs a substantial amount of computation, as a quadratic minimization problm has to b solvd at ach itration of th minimization procss. Howvr, w hav found that th FORRAN routin VMCWD (owll, 1983) solvs our problm rliably and fficintly. h computational ovrhad of th numrical algorithm is compnsatd for by fwr itrations and hnc fwr valuations of th problm functions. his is important in our problm, sinc ach computation of th cost and th constraint quations involvs a rlativly larg numbr of numrical oprations. Contributd by th Hat ransfr Division for publication in th JOURNAL OF HEA RANSFER. Manuscript rcivd by th Hat ransfr Division March 2, Kywords: Environmntal Hat ransfr, Finnd Surfacs, Hat Exchangrs. 322/Vol. 111, MAY 1989 Copyright 1989 by ASME Optimizing th Cost Suppos that a dry cooling towr with circular finnd tub hat xchangrs, of th typ rfrrd to in Appndix A, is to b dsignd, for a spcifid cooling rat Q ct, whn coupld to a condnsr having a spcifid thrmal conductanc UA con. For a minimum-cost dsign, valus hav to b assignd to all dsign variabls in ordr to solv th following minimization problm: Minimiz C (Appndix C, quation (CI)) subjct to th constraints: (a) Hat balanc quations (Al) and (A2). (b) Draft quation (A39). (c) Constant condnsr thrmal conductanc UA C0 (Appndix B). (d) Gomtric constraints, for xampl, > dp For practical rasons, furthr constraints such as t > f L (avoid fouling) and W b < W bu (limit bundl width for transportation) may b rquird. h minimization problm as formulatd abov is an xampl of th gnral nonlinar constraind mathmatical programming problm, which may b writtn as follows (s Lunbrgr, 1984, for xampl): Minimiz F(X) subjct to th constraints: gj(x) 0, i 1,..., m, g,(x) a; 0, i m + 1,..., m X (x u x 2 x ) is a vctor of variabls. A numbr of numrical tchniqus for solving problms of this typ hav bn dvlopd and ar finding thir way into nginring dsign. S Hddrich t al. (1982) for a discussion of som applications of minimization in hat xchangr dsign and also for additional rfrncs. h Constraind Variabl Mtric Mthod dos not rquir a fasibl starting point and in gnral no fasibl point is availabl bfor th itration has convrgd to th minimum valu of th cost function. During ach itration th partial drivativs of th cost function and all constraint functions ar rquird; in this cas th drivativs ar calculatd numrically with a finit diffrnc approximation. Variabls and constraints of th minimization problm should b carfully scald to nsur that all quantitis usd by th minimization algorithm hav approximatly qual magnituds. S Gill t al. (1981) for a discussion of th importanc and tchniqu of propr scaling. In addition to th dsign variabls that ar to b chosn to minimiz th total annual cost C, th minimization problm contains paramtrs such as al, s, L C wf, and othrs, which ar fixd at prscribd valus du to physical constraints or practical considrations. It is of som intrst to invstigat th "snsitivity" of th optimal valu of C with rspct to ths paramtrs, sinc this givs an indication of th rlativ importanc of th paramtrs and may hlp in dciding which of th paramtrs hav to b spcifid accuratly and thos for which a rough stimat will suffic. h snsitivity of C is dtrmind as follows: Suppos a is a paramtr of th modl, thn th mimimum valu of C, say C*, is a function of a: C* C* (a) By th snsitivity of C with rspct to a, w man th drivativ dc*/da, valuatd at th prscribd valu of a. If th Lagrangian function of th gnral optimization problm is dfind as ransactions of th ASME

2 m L(X, A) F(X) - \ igi (X) il A (X,,..., \ m ) is th vctor of Lagrangian multiplirs, and th minimum valu of Fas a function of a paramtr a is dnotd by F*(a), thn it is shown by Fiacco (1983) that da da A*) X* is th optimal solution and A* th associatd Lagrang multiplirs. Sinc th Constraind Variabl Mtric Mthod computs th Lagrang multiplirs A, in th cours of solving th constraind minimization problm, it is a simpl mattr to comput df*/da onc th optimal solution is known. A convnint scal-invariant masur of th snsitivity of th optimal valu with rspct to a may thn b writtn as AF* Aa df* --i-f* da A Cas Study In ordr to illustrat th application of minimization tchniqus to th cooling towr dsign, an xisting dry cooling systm dsign is considrd (dtails ar listd in Appndix B), and th possibility of an improvd dsign is invstigatd. Using xisting and stimatd valus for all variabls and paramtrs, th work point of th cooling towr is dtrmind by solving th hat balanc quations (Al) and (A2) and th draft quation (A39) by varying m, a0, and wo. h rsulting valu of Q c, MW is thn assumd as th targt cooling rat of th towr, and th computd valu of UA C x 10 7 W/K, as th prscribd condnsr thrmal conductanc. h minimization problm is thn solvd as dscribd prviously, in ordr to invstigat whthr th dsign of this cooling towr and th hat xchangrs might b improvd, i.., whthr a towr with th sam cooling rat but with a lowr annual cost can b dsignd. h following ar usd as dsign variabls, which can b varid, subjct to th constraints, to minimiz C: tf, df, d h 6, H 5, H 3, d 5 and m w. In ordr not to xcd th limitations of th finnd tub corrlations, an quilatral tub layout is chosn ( t 0.866,). Both th valus of th tub wall thicknss {d 0 - tf,)/ mm and th fin root thicknss (d r - d 0 )/ mm ar rtaind. Sinc th variation of d 3 rsults in a towr of unaccptably low hight, its original dsign valu of m is rtaind. Bcaus of gomtric considrations, it is only possibl to covr ffctivly 52.4 prcnt of th cooling towr inlt cross-sctional ara with hat xchangr bundls, as is th cas in th original dsign. h optimum numbr of bundls is thus a function of this ara and th bundl dimnsions. Although f is a variabl, it is not listd sinc its optimum valu is always found to b qual to its lowr bound valu f L 2.35 mm, xcpt whn th fin thicknss is rquird to b 0.5 mm, in which cas th optimum fin pitch is 3.22 mm. Similarly th optimum valu of W b is always found to b qual to its uppr bound valu, which is chosn as W bu 3 m in this cas. h rsults of th calculations ar prsntd in abls 1,2, and 3. h first lin of abl 1 shows th valus of th dsign variabls for th xisting towr dsign, and th scond lin th optimal valus. Subsqunt lins show th valus obtaind whn on of th dsign variabls is rgardd as a fixd paramtr and th cost is optimizd with rspct to th othr variabls. h cost of th "optimum" towr is found to b vry much lss than that basd on th original dsign. It is howvr notd that th aluminum fin would hav to b only mm thick, which is impractical considring th larg fin diamtr. By incrasing th fin thicknss to a mor practical valu, th cost of th towr incrass corrspondingly. Changs in othr paramtrs within practical limits hav only a vry small influnc on th cost of th towr. abl 2 shows th snsitivity of th optimal cost with rspct to som prscribd paramtrs. For xampl, if th cost of th fin matrial Cj m is incrasd by 1 prcnt, th optimal cost will incras by approximatly prcnt. A C c D C d Eu F D F, f G g H h i K k L m fn n r ara, m 2 cost, $/annum drag cofficint spcific hat, J/kg K diamtr, m Eulr numbr ffctivnss drag forc, N LMD corrction factor friction factor mass vlocity, kg/s m 2 gravitational acclration, m/s 2 hight, m hat transfr cofficint, W/m 2 K intrst rat on capital loss cofficint thrmal conductivity, W/m K lngth, m numbr of constraints flow rat, kg/s numbr of dsign variabls or numbr of yars or numbr pitch, m or powr, W randtl numbr Q R R t U V W X z a V 6 a prssur, N/m 2 hat transfr rat, W gas (air) constant, J/kg K Rynolds numbr tmpratur, C or K thicknss, m ovrall hat transfr coffi cint, W/m 2 K vlocity, m/s width, m variabl hight paramtr of modl surfac roughnss fficincy smi-apx angl, dg dnsity, kg/m 3 contraction ratio annual oprating hours Subscripts a air b bundl c contraction con condnsr ct cooling towr tc ct d f fr h i L I m 0 r s t tb tr ts U w wf wp cooling towr contraction cooling towr xpansion downstram or diagonal ffctiv or lctricity or quality fin frontal hat xchangr inlt or insid lowr bound longitudinal matrial outlt or outsid pump root stram or shll tub or total or transvrsal tubs pr bundl tub rows towr supports uppr bound watr wighting factor watr passs Journal of Hat ransfr MAY 1989, Vol. 111/323

3 abl 1 Dimnsions of optimizd cooling systm abl 2 Snsitivity of paramtrs aramtr l f d f d. 6 C) p B 5 (n) H 3 (m) d 5 (m) (kg/s) I (S/annum) aramtr Sns i t iv i ty aramtr Snsi t ivi ty Existing dsign Optimum C tm towr C s L t O.I C fm ' f fl <", abl 3 Snsitivity ol corrlations Corrlat ion Sns it ivi ty Corr lation Sns i t ivi ty d ha (A.5) K ct (A.28) Kh [A.31) K ctc (A.27) Kh (A.29) K ct (A.23) Eu (A.30) K,s (A.22) p t H 5 H 3 d 5 K fz,0mm f-2.5.~,3.0mm ' Conclusions In a multidimnsional dsign spac, an intuitiv approach to optimizing a systm bcoms a futil xrcis. For a limitd numbr of variabls, analytic approachs may indicat crtain trnds (Moor, 1972, 1973), but ths mthods ar inadquat in th cas of mor dtaild systm optimizations. An ffctiv procdur for achiving a cost-optimizd dsign for a dry cooling systm is dmonstratd. h rsultant finnd tub has dimnsions corrsponding closly to commrcially availabl tubs with d mm (1.5 in.) and df 76.2 mm (3 in.). h optimum fin thicknss is howvr impractically thin and a mor costly thickr fin would hav to b spcifid. Although it may b argud that th optimum finnd tub»nd not always ncssarily b on that is commrcially availabl, or may b difficult to manufactur with xisting quipmnt, th gomtry proposd dos pos a challng and givs dirction to manufacturrs of finnd tubs. Furthrmor, in th cas of larg towrs, significant lngths of tubs ar rquird (up to 2000 km pr towr), justifying th stting up of suitabl machinry for th manufactur throf. h optimum cooling towr is considrably lowr than th towr proposd in th xisting dsign, whil th inlt is considrably highr. It should howvr b notd that th optimum towr hight incrass as th fin thicknss incrass. For a fixd fin pitch, an incras in fin thicknss causs th flow rsistanc to incras and this rquirs a gratr draft ffct to achiv th sam cooling rat. abl 3 shows th snsitivity of th optimal cost with rspct to som of th mpirical corrlations which wr mployd in th calculations. hs wr calculatd as follows. o invstigat th snsitivity with rspct to K h, for xampl, K h was rplacd by ctk h, th valu of a was quatd to unity and th snsitivity of th cost was calculatd with rspct to a. From abl 3 it may b dducd, for xampl, that th towr support loss cofficint K ts has a ngligibl ffct on th optimal valu of C and may wll hav bn dltd from th calculations. Rfrncs Briggs, D. E., and Young, E. H., 1963, "Convction Hat ransfr and rssur Drop of Air Flowing Across riangular itch Banks of Finnd ubs," Chm. Eng. rog. Symp. Sr., Vol. 59, No. 41, pp Fiacco, A. V., 1983, Introduction to Snsitivity and Stability Analysis in Nonlinar rogramming, Acadmic rss, Nw York. Gldnhuys, J. D., and Krogr, D. G., 1986, "Arodynamic Inlt Losss in Natural Draft Cooling owrs," roc, 5th IAHR Cooling owr Workshop, Montry. Gill,. E., Murray, W., and Wright, M. H., 1981, ractical Optimization, Acadmic rss, Nw York. Gnilinski, V., 1975, Forsch. Ing.-Ws., Vol. 41, No. 1. Haaland, S. E., 1983, "Simpl and Explicit Formulas for th Friction Factor in urbulnt ip Flow," ASME Journal of Fluids Enginring, Vol. 105, No. 3, pp /Vol. 111, MAY 1989 ransactions of th ASME

4 Hddrich, C.., Kllhr, M. D., and Vandrplaats, G, N., 1982, "Dsign and Optimization of Air-Coold Hat Exchangrs," ASME JOURNAL or HEA RANSFER, Vol. 104, pp Hock, W., and Schittkowski, K., 1983, "A Comparativ formanc Evaluation of 27 Nonlinar rogramming Cods," Computing, Vol. 30, pp Kays, W. M., 1950, "Loss Cofficints for Abrupt Changs in Flow Cross Sction With Low Rynolds Numbr Flow in Singl and Multipl ub Systms,'' rans. ASME, Vol. 72, No. 8, pp Kotz, J. C. B., Bllstdt, M. O., and Krogr, D. G., 1986, "rssur Drop and Hat ransfr Charactristics of Inclind Finnd ub Hat Exchangr Bundls," roc. 8th Int. Hat ransfr Conf., San Francisco, CA. Lunbrgr, D. G., 1984, Linar and Nonlinar rogramming, Addison-Wsly, Nw York. Moor, F. K., 1972, "On th Minimum Siz of Natural-Draft Dry Cooling owrs for Larg owr lant," ASME apr No. 72-WA/H-60. Moor, F. K., 1973, "h Minimization of Air Hat-Exchang Surfac Aras of Dry Cooling owrs for Larg owr lants," Hat ransfr Digst, Vol. 6, pp owll, M. J. D., 1978, "A Fast Algorithm for Nonlinarly Constraind Optimization Calculations," rocdings of th Dund Confrnc, 1977, Lctur Nots in Mathmatics, Vol. 630, G. A. Watson, d., Springr-Vrlag, Brlin, pp owll, M. J. D., 1983, "VMCWD: A Fortran Subroutin for Constraind Optimization," SIGMA Bulltin, No. 32. Robinson, K. K., and Briggs, D. E., 1965, "rssur Drop of Air Flowing Across riangular itch Banks of Finnd ubs,'' AJChE Eighth National Hat ransfr Confrnc, pp Schmidt,. E., 1945, "La roduction Calorifiqu ds Surfacs Munis Dailtts," Annx du Bulltin d I'Institut Intrnational du Froid, Annx G-5. 0^ i( W \!,,,i; <D. / *-Hof xchangr owr support --' h,,hi oc Fig. A1 Natural draft dry cooling towr AENDIX Considr th xampl of a hyprbolic natural draft dry cooling towr as shown in Fig. Al. h hat xchangr bundls assmbld in th form of A frams or V arrays, as shown in Fig. A2{b), ar locatd horizontally at th inlt cross sction to th towr. h dnsity of th hatd air insid th towr is lss than that of th atmosphr outsid th towr, with th rsult that th prssur insid th towr at H 3 is lss than th xtrnal prssur at th sam lvation. his prssur diffrntial causs air to flow through th towr at a rat that is dpndnt on th various flow rsistancs ncountrd, th cooling towr dimnsions, and th hat xchangr charactristics. h hat transfr charactristics of th lattr ar xprssd by th following quations: and Qct mcfipa ( a 4 a3 ) ( w wo ) (Al) UAF,[( wi - aa ) - { wo - a3 )] \ict i r/^ I* \ II \i (AA) lnk^ - a4 )/( m - a} )] UA (l/h a A a + l/h A w )~ l (A3) A For round bimtallic, xtrudd finnd tubs, as shown in Fig. A2(a), th ffctiv air-sid thrmal conductanc may b xprssd as "a^a 1 L ln(<vtf,) \n(d/d 0 ) ~ I + (A4) h^jaa 2vk,L n 2-wkfL,, According to Briggs and Young (1963) th air-sid hat transfr cofficint for such tubs may b approximatd by h a ^ k a R 81 r -» m-t f ) dfd r iizh (A5) in th cas of a staggrd arrangmnt, whil th ffctivnss of th xtndd surfac is dfind as f l-a f (l-n f )/A a (A6) According to Schmidt (1945), th fin fficincy may b approximatd by 7) f tanh <j>/<j> (A7) Fig. A2 Finnd tub and A-fram hat xchangr In (i) h watr-sid hat transfr cofficint is (Gnilinski, 1975) K kj w (R w - looojrjl + WL,) 061 ] 8d,[l </,/8) - 5 (r 67 w - 1)] (A8) th pip friction factor is, according to Haaland (1983), givn by Journal of Hat ransfr MAY 1989, Vol. 111/325

5 ( Y'" 6.9 f w (A9) R V 3.7 d, ) h powr rquird to pump watr through th hat xchangr tubs is w ird 2 Ap G w L t n, b n b /4p w (A10) Ap w z r and G w 4m»n wp 2p d, -Kdj n tb n b h air flow rat through th towr is dtrmind by th draft quation, which is drivd as follows: In th atmosphr xtrnal to th towr th variation of prssur with lvation in a gravity fild is givn by dp a -asdz For a prfct gas th following rlation holds: (All) p a p a /R a (A12) Sinc th tmpratur distribution in th atmosphr nar ground lvl changs continuously, an arbitrary rfrnc condition is slctd for this analysis. According to th Intrnational Standard Atmosphr (ISA), th man tmpratur laps rat in th troposphr is K/m, rsulting in th following tmpratur distribution in th atmosphr: a al z (A13) Substitut quations (A 12) and (A 13) into quation (All) and intgrat from 1 to 6: ai - a6 ai [1 - ( H 5 / al )!» «**] (A14) H 6 H i is th towr hight, g 9.8 m/s 2, and R J/kg K. h air acclrats from stagnant ambint conditions at 1 and ntrs through th towr supports at 2 to sction 3 bfor flowing through th hat xchangr bundls. A total prssur balanc btwn 1 and 4 yilds al (a4 a^li \ (K ls + K ct + K ctc + K ha + K ct ) h ( ^ - ) ( f 2 ) 2 + aish^ (A15) rh a is th air mass flow rat through th towr and all loss cofficints ar basd on th frontal ara of th hat xchangr and th man dnsity through it. In dtrmining th dnsity aftr th hat xchangr, th spcifid prssur at ground lvl is mployd in th prfct gas rlation, i.., Furthrmor, for all practical purposs at a\/r a t (A 16) ai al al/ral (A 17) h man dnsity of th air flowing through th hat xchangr follows from l/ am 0.5 (l/p + I/ft*) 0.5R( al + a4 )/ai (A18) h loss cofficint through th towr supports K ts is basd on th drag cofficint for such bodis, i.., C m 2F Dts /p al v 2 a2a ts (A 19) With quation (A 19), th ffctiv prssur drop across th towr supports is givn by &ats n ls F DIS / d 3 H p al yj 2 C m L ls d ts n ls /ir rf 3 // 3 (A20) 326/Vol. 111, MAY 1989 L ts is th support lngth and d, s is its ffctiv diamtr or width, and n ls is th numbr of towr supports. h corrsponding loss cofficint basd on th conditions at th towr supports 2 is K ts 2Ap als /p al vl 2 C dts L ts d ls n ls /ir d } H 3 (A21) For substitution in quation (A 15), this loss cofficint is rquird to b basd on conditions at th hat xchangr, i.., K ts 2Ap als /p am {m a /A f f _ wto L ts d ts n ts Aj r ( am \ (A22) (x d } Htf V al ) It is assumd that th air dnsity and vlocity distribution through th supports ar uniform. Du to sparation at th lowr dg of th cooling towr shll and distortd inlt flow pattrns, a cooling towr loss cofficint K Q(i basd on th towr cross-sctional ara 3 can b dfind to tak ths ffcts into considration. h cooling towr loss cofficint basd on conditions at th hat xchangr is ah K r. K rl (A23) al \Aj K ct3 may b approximatd by (Gldnhuys and Krogr, 1986) K ca (d 3 /H 3 ) d 3 /H (A24) Dpnding on th hat xchangr bundl arrangmnt in th cooling towr bas, only a portion of th availabl ara is ffctivly covrd du to th rctangular shap of th bundls. his rduction in ffctiv flow ara rsults in contraction and subsqunt xpansion losss. hs losss may b approximatd by corrsponding loss cofficints basd on th ffctiv rducd flow araa } (Kays, 1950) 12 K clc3 1-2/ff c + I/a 2. o c is th jt contraction ratio and K*a (1 - A 3 /A 3? (A25) (A26) h ffctiv ara A 3 corrsponds to th frontal ara of th hat xchangr bundls if thy ar installd horizontally. In th cas of V arrays, A 3 corrsponds to th projctd frontal ara of th bundls. Basd on th conditions at th hat xchangr, th abov xprssions bcom and K clc (1-2/o c + Vol) (p a34 / al ) (A /r /A 3 y (A27) K ct (1 - A 3 /A 3 f (p a3,/ a4 ) (A /r /A 3 y (A28) Whr th cooling air flows normally through a hat xchangr, th loss cofficint is radily xprssd in trms of th Eulr numbr, i.., Kh - Eu + al ~ ai, (A29) al + al according to Robinson and Briggs (1965) n tr /rf r \»»' /,\0-515 Eu i^u) U) If th flow approachs and lavs th hat xchangr obliquly, as is th cas for an array of V bundls, th following rlation holds (Kotz t al., 1986): ali ( _ al \Si sin 6 6 m O.OO K r - 5 a34 v K d xp ( ai x lo x 1O" ) (A31) ransactions of th ASME (A30)

6 and K c 0.05 Aftr th hat xchangr, th flow through th towr is ssntially isntropic. h chang in total prssur btwn sctions 4 and 5 in th towr is thus (a p a4 4,) - (p a5 + Q.5p a5 vl 5 ) fl «g(tf 5 - H 3 ) (A32) it is assumd that H 4 ~ H 3 for a rlativly thin hat xchangr. h tmpratur at th towr outlt is as a4 + [0.5(^4 - t&) + g(h 3 - H 5 ))/c pa (A33) From th prfct gas rlation it thus follows that for p a5 p a6 th dnsity at th outlt of th towr is as JR\,A + 0.5(^4 - V 2 a5)/c pa + g(h 3 - H 5 )/c pa ] (A34) p a6 is obtaind from quation (A 14). In most practical towrs, th chang in kintic nrgy btwn sctions 4 and 5 is normally approximatly an ordr of magnitud smallr than th corrsponding chang in potntial nrgy. Equation (A34) may thus b simplifid: as - a 6 /Rl a4 + g(h 3 - H 5 )/C pa ] p a6 /R [ a (H 5 - H 3 )] (A35) h man dnsity in th towr btwn 4 and 5 is ats 0.5(p a4 + p as ) (A36) h chang in prssur through th cooling towr is obtaind by adding quations (A 15) and (A32): al ~ as (K ts + K ct + K clc + K Aw + K cl ) h - ( ) 2ft,34 \AfrJ + ai gh 3 + a45g(h 5 - H 3 ) + a5vh/2 (A37) From continuity it follows that v a s injasas (A3 8) Sinc p a[ - p a6 p al - p a5, quations (A14) and (A37) can b quatd to giv with quation (A3 8) p ai [1 - ( // 5 /r al ) 5-25 ] (K ts + K ct 1 /««\ + K clc + K h0 + K cl ) h 2p a34 m \A fr ) [p al H 3 + a45 (H 5 - H 3 )] + (A39) UaS his xprssion is known as th draft quation for natural draft cooling towrs. 2 AENDIX B A cooling systm, as shown schmatically in Fig. Al is dsignd to rjct Q cl MW hat at an ambint air prssur of p a 84,600 N/m 2 and a tmpratur of al 15.6 C. h cooling towr inlt watr tmpratur is m C and th flow rat is fn 4390 kg/s. h proposd cooling towr is H m high and has an inlt hight of H m. Its outlt diamtr is d 5 58 m and th inlt diamtr is d m. hr ar a total of n ts - 60 towr supports, ach L ls m long with an ffctiv diamtr of d, s 0.5 m. h drag cofficint of ach support is C DIS 2. Extrudd bimtallic finnd tubs as shown in Fig. A2(«) ar mployd. h stl cor tub has an insid diamtr of d- x mm and an outsid diamtr of d mm. h aluminum fins hav a diamtr of d f 57.2 mm with a root diamtr of d r 21.6 mm, a man thicknss of tj 0.5 mm and a pitch of j 2.8 mm. h dnsity of th stl cor tub is p, 7850 kg/m 3 and its thrmal conductivity is k, 50 W/m K. h aluminum fins hav a dnsity of p f 2101 kg/ m 3 and a thrmal conductivity of k f 204 W/m K. h finnd tubs ar assmbld in th form of hat xchangr bundls, ach L, 15 m long and W b m wid. Each bundl contains n lb 154 finnd tubs arrangd on a triangular pitch with, 58 mm and t 50.2 mm. hr ar n lr 4 tub rows with n t/r 39 tubs pr row and n wp 2 watr passs. h bundls ar arrangd in th form of A frams or V arrays as shown in Fig. A2(b). h smi-apx angl of ach A fram is dg and thr ar a total of n b 142 bundls arrangd in this mannr. For this particular dsign th ovrall thrmal conductanc of th surfac stam condnsr is givn as UA C X 10 7 W/K. his valu is rtaind as a constant in th optimization procss. In viw of th fact that th cost of th condnsr is usually small whn compard to that of th dry cooling towr, this assumption is not unrasonabl. AENDIX h annual oprating cost of a natural draft cooling towr may b approximatd by th quation i(l+0" c c + [C c, + C h + C p ] $/annum (C1) pump (i + 0"-i th first trm rprsnts annual pumping costs and th scond trm th annual capital rpaymnt ovr n yars at an intrst rat /'. h annual cost of th watr pumping powr is ^ pump 4.33 C -F w i\-2) w is givn by quation (A10) and th constant 4.33 is a factor to account for additional circuit prssur drops and pump and motor fficincis. If th cooling towr shll is approximatd by a conical frustrum, its construction cost may b xprssd as C ct \C S (rfj + d s ) [(Hi - H 3 f + (d 3 - di?\ (C3) h cost of th hat xchangrs is modld by C h n tb n b L, [C w/ (C, + C f ) + C fix ] C (C4) th cost of th tubs and th fins pr unit lngth of tub ar, rspctivly, and Q x p, C lm (dg - df )/4 (C5) C f it jc fm [(4 - dj)t f + (fi - di) ( f - tjft/af (C6) A wighting factor C f and fixd cost factor C fix tak into considration labor and rlatd costs. h capitalizd cost of th watr pump and piping is approximatd by C p 0.02 (C ct + C h ) (CI) It is mphasizd that th cost quations givn abov ar an attmpt to modl th annual cost of a spcific cooling towr and ar not mant to b gnrally applicabl. For th cas study discussd in th main body of th papr, th following valus wr assumd: Intrst on capital / 0.1, rpaymnt priod n 30 yars, lctricity cost C 5 c/kwh, towr shll construction cost C s $100/m 2, wighting factor C f 2 and C fix $2/m, pric of stl for tubs C, m $0.8/kg, and pric of aluminum for fins C fm $4.2/kg. h towr is oprational for 6570 hours/annum and to simplify th analysis it is assumd that stam and ambint air tmpraturs rmain unchangd. Journal of Hat ransfr MAY 1989, Vol. 111/327