Fourth Interntionl Wter echnology Conference IWC 99, Alexndri, Egypt 9 Abstrct Optimum esign of Horizontl Condenser A.M. Abdllh nd Aly Krmeldin Atomic Energy Authority-Nucler eserch Center ectors eprtment - P.O. 1759 Ciro, Egypt In this study rigorous constrined therml hydrulic model hs been derived for clculting the economic design of vpor surfce condenser used in power plnts nd sewter deslintion plnts which use distilltion methods. According to the economicl dt obtined in the present study, it hs been shown tht the totl condenser cost components: stem genertion, brine wter pumping, nd fixed chrge contribution nerly s %, %, nd 11% respectively ccording to the fuel cost. However the clculted cost is closed to the known ordinry vlues when the clcultions re bsed on the minimum vrible condenser cost [1] (About 55% for stem cost, 10% for pumping cost, nd 5% for the fixed cost). As for the effect of some technicl prmeters, it hs been shown tht: (i) totl condenser cost is nerly independent of the tube dimeter (ii) totl cost depends wekly on the brine velocity, nd more strongly on the brine flow rte. he plnt fouling condition is of prime importnce nd should not exceed the design vlues; otherwise it cn led to increse of 1 % of the totl condenser cost if the fouling fctor incresed to 0.05 m o C/W. From het trnsfer point of view, the effect of the different tube mteril used in deslintion, hs slck effect on the totl condenser cost (0.055 0.0%). In short, the generl model derived in this study is cpble to del with ny horizontl condenser. Introduction he optimum design of engineering equipment (e.g. evportor, condenser, het exchnger, etc) usully involves the clcultion of the importnt design vribles, which minimize the totl cost of given equipment. However, some constrints re usully imposed on the equipment governing equtions so s to get logicl or cceptble numericl vlues for the clculted design vribles. Vrious optimiztion methods re vilble, for exmple LGrnge s multiplier, liner, dynmic, nd geometric progrmming. he choice of suitble method depends on the mthemticl form of the function to be optimized nd type of
50 Fourth Interntionl Wter echnology Conference IWC 99, Alexndri, Egypt imposed constrints (e.g. equlities nd inequlities). wo efficient optimiztion methods, which re pplicble to nonliner nd multivrible cost functions, re the clculus of vrition method nd geometric progrmming. In this respect it hs been demonstrted tht geometric progrmming my result in considerble sving of effort in comprison to the clculus method when the degree of difficulty-equls the number of terms in the objective function nd constrints minus the number of vribles minus one-is zero. But for degree of difficulty greter thn zero, it my demnd more effort thn optimiztion by clculus method. It hs been shown tht the degree of difficulty in the underlying problem is three, which precludes the use of geometric progrmming [1] s n optimiztion method. In the present study ccordingly, we use the method of clculus of vrition to determine the economic design of horizontl surfce condenser used in conventionl nd nucler power plnts nd in sewter deslintion plnts which use distilltion methods. Formultion of the Problem In the present section the problem of simplified design of vpor condenser with fixed het lod like brine heter used in multi stge flsh deslintion will be studied. Considering brine heter in which sewter mss flows t rte of W (kg/s) nd to be heted without phse chnge from bi to bo ( o C) by stem condenstion. Optiml design of such brine heter involves minimiztion of the nnul totl cost, which includes the following three min items: Fixed chrges on the brine heter, Cost of circulting sewter in the brine heter, Cost of stem supply. he optimiztion of the brine heter nnul cost cn be hndled through the determintion of the brine heter cost lgorithm. Brine heter cost lgorithm he lgorithm cn be determined be superimposing the before mentioned three nnul costs. he fixed chrges on the brine heter C F ($/y) cn be expressed s follows: C C C A π C C LN LN (1) F d d 7
Fourth Interntionl Wter echnology Conference IWC 99, Alexndri, Egypt 51 Where C, C d, nd A re the cost per unite re, deprecition nd mintennce, nd condenser tube surfce re respectively. he Cost of circulting sewter brine to the brine heter C P ($/y) cn be determined s function of the tube friction pressure drop P t ( Arcy eqution for tube friction), pressure fctor B (typiclly the tubes friction losses 0-70% of brine recycle pump), friction fctor f, pump efficiency η p, (not including the motor efficiency 0.7-0.5 [] typiclly 0. [1], mss flow rte W, nd power fctor P F 0.5 [], s follows: C Pu Where C B PWP E t F () ρη ( V ) fl W P 1 ρ & V () t π ρn i i W f 0.0 & e 0. e πµ N () i Substituting from Equtions (&) in Eqution () nd referring inner tube dimeter to the outer, the cost of circulting sewter cn be written in the form:. 1. C Pu LN (5) he cost of the vilble stem C S ($/y) cn be determined through the therml lod Q nd worth of supplied stem S. So, the totl brine heter nnul cost C ($/y) cn be expressed s: C C QS () S fuel he worth of the supplied stem S cn be correlted to the supplied stem condition, which is function of stem temperture S for sturted stem (by direct referring the cost to the vrible fuel cost nd correlting the given dt in []) s follows: S c + c 1 S (7) Agin, the stem temperture S cn be clculted from the het trnsfer reltions between the stem nd the sewter het blnce. he totl resistnce to het flow is the summtion of resistnces through the inside film i, [5] internl tubes fouling fi, wll thickness w, externl tubes fouling fo, noncondensible gses nc, nd the out tube side condenste film o [5] represented s: + + + + + () i fi w fo nc o
5 Fourth Interntionl Wter echnology Conference IWC 99, Alexndri, Egypt Where, + + b (9) f fi fo nc Where b o m K 0.000[5] 0.0005[ W 1 ] 0. 1 1 k W c µ m pm m 1 i h π Nµ k i i i i m m ln( / ) i b π k W W b 1. N 0. (10) (11) O gρ k h 1 0.75 fg ( ) µ n S WO 1 1 (1) n b N (1) For clcultion simplicity considering the lyout of the brine heter tubes hs squre form (b 1), so the het flux cn be represented s: Q A WC So tht ( S PM ( A O I ) FI I M WI FI FI S O nc S M (1) WC ( ) PM O I ) (15) WO O A Substituting for N nd ( S WO ) in Equtions (1 nd 15) in Eqution (1) gives: O 1 πgρ k h 1 1 9 (0.75) fg L N b L N 5 ( ) µ WC PM O I From Eqution (1) S WC ( ) PM O I + (17) M A Substituting from Eqution () for in Eqution (17), Eqution (17) for S in Eqution (7), nd Eqution (7) for S in Eqution (), nd rerrnging on cn get: 0. C S + + + 1 0. + 9 (1) 5 (1) 11 LN LN LN 9 L N he totl nnul cost cn be expressed in terms of C S, C Pu nd C F s: C C + C + C (19) S Pu F
Fourth Interntionl Wter echnology Conference IWC 99, Alexndri, Egypt 5 Substituting from Equtions (1,, nd 1) for C S, C Pu, nd C F respectively in Eqution (19), the totl nnul cost cn be obtined s follows: 0. L 5 + + + + + LN 1 0. 11 LN LN LN L N. N 1. + (0) 9 C 7 From technicl point of view, this lgorithm is subjected to some constrints concerning tube mteril, dimeter, wll thickness, length, erosive velocity, mnufcturing processes,..etc. he tube dimeter nd wll thickness used in deslintion rticle re [,7]: 0.015 0.05( m) (1) nd 0.9 up to 1. [mm] respectively. he limiting brine velocity [7] is: 1.- [m/s] for Cu-Ni 90/10, - for Cu-Ni 70/0 1.5-.5 for Aluminum Brss (polyphosphte brine chemicl tretment plnts). he number of brine heter tubes cn be expressed in terms of tube dimeter. So, from Eqution (): W N πρ V M () Cse Study In the present study 1 (t/dy) MSF plnt t Brindisi power sttion (Itly) [] with the following dt is tken s working exmple: W 11 (kg/s), sewter concentrtion,000 (ppm), B 100 ( o C), nd tubes mteril Cu-Ni 90/10 9. (W/m. o K), therml conductivity). ke the wll thickness 1 (mm), dimeter rtio 1.0559, the brine input/output temperture 91/9 ( o C), nd the verge outside film temperture 100( o C). Brine properties [9] ρ 9 (Kg/m ), C PM 111.5 (J/kg.K), k M 0.75 (W/m. o K), µ M 0.000 (kg/m.s) Outside film properties ρ 95 (Kg/m ), C p 195 (J/kg.K), h fg 5.9 (KJ/kg) k 0.1 (W/m.K), µ 0.00079 (kg/m.s) he corresponding constnts of the brine heter cost lgorithm in eqution [0] re:
5 Fourth Interntionl Wter echnology Conference IWC 99, Alexndri, Egypt 1 9.791 * C fuel * W 7.179 * C fuel * W.17 * C fuel * W 1..7011 * C fuel * W 5 1.7779 * C fuel * W (7 / ).7E- * C fuel * W. 7 9.510 * C fuel * W (for C 5 $/m [10]) 0.0017 * W / V cr Algorithm Optimiztion he method of LGrnge s multipliers is useful nd powerful nd could very well be the best choice for mny prcticl optimiztion problems. he brine heter cost lgorithm to which LGrnge s multipliers optimiztion directly pplied to the totl nnul cost represented by Eqution (0) is subjected to the constrint: 1. V ( m / s) () he optiml vlue of totl cost lgorithm C is found by the solution of the following sclr non-liner equtions [] C λ ( V V ) 0 () cr V V cr 0 (5) he grde opertor on C nd (V - V cr ) in eqution (), is ctive on, L, nd N so, the following equtions cn be obtined by direct differentition of Eqution () with respect to, N, nd L respectively in Eqution () s follows: N 0.. λ + LN + 7 N 5 : + 0. 0. 11 5. 1. 9 LN LN 0. 1. LN L N 115 9L N LN 1. 0. : + L + 0.. 7 9 LN LN 0. 5 L : + N 0 () 0. 7 11. 1. 7 L N L N L N 9 L N N Multiplying Equtions (,7&) by L, N L, nd L LN 0 λ N 0 () (7) respectively, subtrct Eqution () two times Eqution (7), nd substituting for N in Eqution () resulting in: 1 9 1. 1. 1 1. L 5 + + + + L 0. 11 1 1. 1. 7 9 9 L 1 1. 9 L 5 + + L 0. 11 1 1. 1. 7 9 L 0 0 (9) (0)
Fourth Interntionl Wter echnology Conference IWC 99, Alexndri, Egypt 55 From the bove two equtions, one cn obtin the following nlyticl expression for L s: 1. 1. 11 + + 0. L (1) 0. + 1. 7 A trnscendentl eqution in cn be obtined by substituting for L either in Eqution (9) or (0) nd solving for. It should be noted tht the solution for must be subjected to the constrint in Inequlity (0). So, pplying different vlues of nd V in Inequlities (1) nd () respectively, one cn get the corresponding vlues of L, N, A, C S, C P, C F, nd C t the different vlues of C fuel, W, V, nd. he obtined results re represented in Figs. (1,,, nd ). esults nd iscussion As mentioned before the model derived for estimting the minimum cost of horizontl surfce condenser nd hence the corresponding optimum design vribles, s tube dimeter, tube length nd number of tubes, hs been pplied to 1 [t/dy] MSF plnt t Brindisi power sttion (Itly) [7]. Fig (1A) shows the vrition of totl cost nd specific tube surfce re with the tube dimeter t n rbitrry brine velocity of 1.9 (m/s), brine mss flow rte of 11 (kg/s) nd fuel cost of 1. ($/GJ) [11]. It cn be seen tht there is wek dependence of the totl tube cost nd the specific condenser re on the tube dimeter. Fig (1B) shows the dependence of both tube length nd number of condenser tubes on tube dimeter. In this figure, s the tube dimeter increses, the number of condenser tubes decreses nd the tube length increses. his simple figure cn be considered s design chrt for clculting the tube dimeter, length, nd number of tubes of horizontl condenser. he vrition of specific re nd totl cost with the brine velocity is depicted in Fig (A). It cn be seen tht the totl condenser cost reches minimum vlue t brine velocity of bout 1. (m/s). As the brine velocity increse, the condenser specific re decreses. Fig (B) shows tht both the condenser tubes length nd number of tubes decrese with the increse of brine velocity. Figs (A&B) show tht the specific re nd tube length re independent of the brine mss flow rte, while the totl cost nd number of tubes vry linerly with it.
5 Fourth Interntionl Wter echnology Conference IWC 99, Alexndri, Egypt Fig (A) shows tht the totl condenser cost increses linerly with the fuel cost, while the specific re increses less steeply with it. When compring this figure with Fig (A), it cn be immeditely note tht the fuel cost is the key prmeter which controls the optimiztion process. his, becuse it contributes in both the cost of consumed stem (more thn 0% of the totl condenser cost) nd the pumping of sewter, lthough the ltter is much less thn the former. Accordingly, it is very importnt to form the stem cost component nd mking use of ll methods nd mens to reduce this cost tking into considertion innovtion of new technology. In comprison with the previous work of Avriel-Wilde [1], they mde used of geometric progrmming optimiztion in solving the condenser problem. hey concluded tht the designer or economist my resonbly simplify the problem to zero degree of difficulty to obtin solution of the problem. So, further ssumptions were mde to obtin solution of the problem. hey used the minimum vrible condenser cost, ignoring the fixed term in the condenser lgorithm cost. However, for comprison purpose, the following results cn be obtined when compring the minimum vrible cost percentge of the totl cost [1] (t 0.05 m, L.9 m, nd N11 tube) In the present study: C F %.7, C Pu %.0, nd C S %5.55 While in Avriel-Wilde work: C F %5., C Pu %.1, nd C S %.1 hese results re lmost closed to ech other with smll differences. hese differences my be ttributed to elimintion of the wll therml resistnce in Avriel- Wilde work, nd the different electricity nd heting prices. In the present study, wide condenser lgorithm cost ws covered. his enbles the designer to get cceptble shortcut optimized cost estimtion in ccordnce with demnd nd mrket conditions. For the purpose for nlyzing the effect of individul terms of the cost lgorithm, the following nlysis of the effect of fouling nd tube mteril is presented. he optimized cost lgorithm t W11 (kg/s), C fuel 1. ($/GJ), V1.9 (m/s), nd 0.019 (m) gives optimized L. (m) nd N0 tube. Fouling t plnt condition fter 0 dys norml opertion since tube clening representing less thn 0.7 % of the totl condenser cost ( t fouling fctor f 0.0005 (m o K/W)). If the fouling fctor incresed to f 0.05 (m K/W), this cn led to increse to the condenser cost to 1 % of the totl condenser cost. So, it should be
Fourth Interntionl Wter echnology Conference IWC 99, Alexndri, Egypt 57 pointed out tht the plnt fouling condition is of prime importnce nd should not exceed the design vlues. From het trnsfer point of view, when studying the effect of the different tube mteril on the condenser totl cost (typiclly therml conductivity of mterils used in deslintion is 1.7 up to.7 (W/m.K) [10]), it cn be shown tht it hs slck effect on the totl condenser cost (0.055 0.0%). Nomenclture A Condenser het trnsfer re (m ) 1,,,, 5,, 7, Constnt of Eqution (0) B Pressure drop fctor b 1 b, b, b otl fouling Fctor (m o K/W) Constnts c 1, c Constnts of Eqution (7) C Cost per unit re ($/m ) C d C E C F C fuel eprecition nd mintennce (1/y) Cost of electricity ($/kw-hr) Cost of fixed chrge ($/y) Cost of fuel ($/GJ) C p specific het (J/kg o K) C Pu C S C Cost of sewter circultion ($/y) Cost of stem ($/y) otl nnul cost of condenser ($/y), i Outer nd inner tube dimeters (m) f Fnning friction fctor g Grvittionl ccelertion (m/s ) h Het trnsfer coefficient (W/ m o K) h fg Evportion specific enthlpy (kj/kg) k Het trnsfer conduction coefficient (W/ m o K) L ube length (m) N Number of tubes P F Q Power fctor herml lod (GJ/y)
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Fourth Interntionl Wter echnology Conference IWC 99, Alexndri, Egypt 59
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Fourth Interntionl Wter echnology Conference IWC 99, Alexndri, Egypt herml resistnce (m K/W) S Stem cost coefficient emperture ( o C) V Brine velocity (m/s) W Brine mss flow rte (kg/s) Greek Letters P Pressure drop (N/m ) η Pump efficiency µ ynmic viscosity (kg/ms) ρ ensity (kg/m ) Subscripts F Fixed chrge fi Inner foul fo Outer foul I, i Inner M, m Men nc Noncondensible O, o Outer Pu Pump W, w Wll WI, wi Inner wll WO, wo Outer wll S, s Stem otl eferences [1] Avriel M. nd Wilde., Optiml condenser design by geometric progrmming, I&Ec process design nd evelopment, V N O, 197. [] IAEA, Computer mnul series, N O 1, Methodology for the economic evlution of co-genertion /deslintion options: user mnul, IAEA, 1997. [] Hmmod.P., Eissenberg.M., Emmermnn.K., Jones J.E.,Jr, Sephton H.H., Stndiford F.S., ider W.J., nd en.w., Sewter deslintion plnt for southern Cliforni, eslintion,v99, 199.
Fourth Interntionl Wter echnology Conference IWC 99, Alexndri, Egypt [] Stoecker W.F., esign of therml systems, McGrw-Hill, rd ed., 199. [5] Henning S., Wngnich K. nd Wngnich K, Comprison of different equtions for the clcultion of het trnsfer coefficients in MSF Multi-Stge Flsh evportors, IA/UAE V, 1995. [] Porteous A., eslintion technology, Applied Science Pub., 19. [7] Hunbury W.., Some thoughts on the limittions on incresing the unit size of conventionl cross-tube MSF distilltion plnts, eslintion, V9, 199. [] Zennoni., esini I. And Lucenti., esign construction of x10 /dy MSF plnts t Brindisi s power sttion (Itly): ntiscle dditive tretment for 100 C top brine temperture, eslintion, V, 197. [9] Loir N., Mesurements nd control in wter deslintion, Elsevier, 19. [10] Hornburg C.., odd B., nd uthill A.H., Het trnsfer tubing selection for MSF deslintion plnts, IA/UAE V5, 1995. [11] Nshr A.M., Kmlddin B.A., McGregor I.., Khn S., nd El-Hres H., Overview of the design fetures, performnce nd economic of the MSF plnts operted by Abu hbi wter nd electricity deprtment, IA/UAE, V, 1995.