HEAT EXCHANGERS (1) Overall Heat Transfer Coefficient
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1 HEAT EXCHANGERS Heat exangers transfer eat fr ne wrkng flud t anter. Fr nstane, stea generatrs, feedwater eaters, reeaters and ndensers are all exaples f eat exangers fund n nulear pwer systes. Te eat transfer rate arss a eat exanger s usually expressed n te fr were: Q & eat transfer rate U verall eat transfer effent A eat exanger area Δ average teperature dfferene between te fluds T Q & UAΔ ( Te verall eat transfer effent s a funtn f te flw geetry, flud prpertes and ateral pstn f te eat exanger. Te average teperature dfferene between te fluds s n general a funtn f te flud prpertes and flw geetry as well. Heat exanger desgn requres nsderatn f ea f tese fatrs. T Overall Heat Transfer Ceffent Te verall eat transfer effent represents te ttal resstane t eat transfer fr ne flud t anter. Te funtnal fr f U r te prdut UA, ay be derved fr any partular geetry by perfrng a standard ndutn analyss n te syste f nterest. T llustrate ts, nsder frst a planar wall f tkness L, subjet t nvetn n bt sdes. TH T TC z+δ z T L z Fgure : Planar Wall Heat Exanger Te eat transfer rate fr te t flud t te wall fr a surfae area defned by te lengt segent Newtn s Law f Clng as Δ z s gven by su tat [ T ] q A ( s H T TH q T As (3 4
2 Te teperature drp arss te wall s fund by slutn f te ndutn equatn d dx k dx 0 (4 subjet t te bundary ndtn k dx 0 q A s (5 Integratng Equatn 4, x 0 d dx k dx dx 0 ( k k 0 (7 dx dx 0 and applyng te bundary ndtn gves q k + 0 (8 dx A s Integrate Equatn 8 ver te wall tkness t btan te teperature drp arss te wall L L q dx dx dx 0 kas (9 ql T T (0 ka s Te teperature drp fr te wall t te ld flud s als fund fr Newtn s Law f Clng, slar t Equatn 3 abve q T TC ( A Addng Equatns 3, 0, and gves te teperature drp fr te t t ld flud as q L T H TC + + ( As k r were s q UA [ T TC ] (3 s H 5
3 L U + + (4 k s te Overall Heat Transfer Ceffent. In atual eat exanger desgn, te planar wall s seld used. A re n desgn nvlves eat transfer arss a tube wall as llustrated n Fgure. TH T TC z+δ z T z r r r 0 Fgure. Crular Tube Heat Exanger In ts ase, te eat transfer rate fr te t flud t te wall wtn te lengt segent Δ z s su tat Fr te tube wall, te ndutn equatn s [ T ] q π r Δz H T (5 T H q T ( π r Δz r d dr rk dr 0 (7 subjet t te bundary ndtn πrkδz q (8 dx r Integratng Equatn 7, r d rk dr 0 (9 r dr dr
4 r k kr 0 (0 dr dr r and applyng te bundary ndtn gves q r k + 0 dr πδz ( Integrate Equatn ver te wall tkness t btan te teperature drp arss te wall r r dr + dr T T r r q dr 0 πkδz r q r πkδz r ( (3 Te teperature drp fr te wall t te ld flud s q T TC (4 πr Δz Addng Equatns, 3, and 4 gves te teperature drp fr te t t ld flud as T H T C q Δz πr r + πk r + πr (5 r q UA[ T H TC ] ( were UA πr Δz r + + Δ πk z r πr Δz (7 Sne te eat transfer area n te nterr f te tubes s dfferent fr tat n te exterr n ylndral geetry, te prdut UA s nrally used t desrbe eat exanger perfrane. Equatns 3 and represent te eat transfer arss a sall lengt segent Δ z were te t and ld flud teperatures an be nsdered nstant. In realty, te t and ld flud teperatures ange ntnuusly alng te lengt f te eat exanger as llustrated n Fgure 3 belw fr te Duble Ppe Heat Exanger gven n Fgure 4. 7
5 T T q q T T da T T T da T A Parallel Flw A Cunter Flw Fgure 3: Flud Teperature Prfles n a Duble Ppe Heat Exanger If we let Δ z be a dfferental lengt, ten te eat transfer arss te dfferental lengt segent s q dq & UdA( T H TC (8 were da s te dfferental area assated wt lengt dz. Te ttal eat transferred arss te eat exanger s ten btaned by ntegratng Equatn 8 ver te lengt f te eat exanger d Q & UdA( T C H T (9 Q & UAΔ (30 T were ΔT ( TH TC da (3 A s te average teperature dfferene between te t and ld fluds ver te lengt f te eat exanger. Sne te area n Equatn 30 s nw te ttal area f te eat exanger nstead f tat assated wt a sall lengt segent f a sngle tube, te expressn gven fr UA n Equatn 7 s dfed t reflet te atual area f te eat exanger UA A n r + kl + r π A (3 were: n nuber f tubes 8
6 nvetve eat transfer effent n te tube nterr A nterr surfae area f ne tube nvetve eat transfer effent n te tube exterr A exterr surfae area f ne tube r tube nner radus r tube uter radus k tube teral ndutvty L tube lengt Te value f Δ T depends n te eat exanger desgn, wt analyt expressns nly avalable n speal ases. One su ase s tat f te sple duble ppe eat exanger llustrated belw. Fgure 4: Duble Ppe Heat Exanger Lg Mean Teperature Dfferene We ws t alulate te ttal eat transfer n te duble ppe eat exanger trug an expressn f te type Q& UAΔ T were Δ T s se apprprate average teperature dfferene between te t and ld fluds. If we assue parallel flw, te steady state eat transferred trug a dfferental area da s ( C & p ( C & p d Q& (33 were te subsrpts and dente te t and ld fluds respetvely. We ave already seen tat we an wrte te eat transfer arss ts dfferental area n ters f te Overall Heat Transfer Ceffent as dq & UdA( T T (34 Fr Equatn 33, dq& and ( C & p dq& su tat ( C & p d T T dq& ( + ( ( (35 C & p C & p 9
7 Substtute fr d Q & fr Equatn 34 d( T T UdA ( T T + ( ( C & C & p p (3 Assung all ters n te rgt and sde f Equatn 3 are nstant, we an ntegrate fr pnt ( t pnt ( alng te lengt f te eat exanger d( T T ( T T ( T T UdA UA + ( C & ( C & p p + ( ( C & C & p p (37 (38 We furter take advantage f te fat su tat ( T T (39 p p ( UA + ( ( T T C & C & ( C & ( T T ( C ( T T Q& & (40 p p and ( C & Q& ( C & Q& T T T T (4 p p Substtutng nt Equatn 39 ( T T ( T T UA Q& [( T T + ( T T ] (4 and rearrangng gves were ( T T ( T T UAΔT [( T T /( T T ] Q& UA (43 ( T T ( T T [( T T /( T T ] Δ T (44 s alled te Lg Mean Teperature Dfferene (LMTD. Te LMTD represents te effetve, average teperature dfferene between te tw eat transfer fluds ver te lengt f te eat exanger and tug derved ere fr parallel flw, Equatn 44 s als vald fr unter flw eat exangers. Te LMTD gven n Equatn (44 s nly strtly vald fr sngle pase fluds n duble ppe eat exangers. In a eat exanger ter tan a duble-ppe 0
8 type, te eat transfer s alulated by usng a rretn fatr appled t te LMTD fr a unterflw duble-ppe eat exanger wt te sae t and ld flud teperatures. Equatn ( s ten dfed su tat Q & UAFΔ (45 T Crretn fatrs fr a sngle sell pass, and ultple tube passes are prvded n Fgure 5 belw (fr Hlan, Heat Transfer 3rd Edtn. Fgure 5: Crretn Fatrs fr a sngle sell pass, and even nuber f tube passes (Hlan, Heat Transfer 3rd Ed. Stea Generatrs and Cndensers In stea generatrs and ndensers, te sell sde teperature an be assued nstant at te uratn teperature. Ts splfes te dervatn f te LMTD as fllws. Assue te nstant teperature s tat f te ld flud. dq & UdA T T (4 ( and dq& su tat ( C & p d T T dq& ( ( (47 C & p Substtute fr d Q & fr Equatn 4
9 d( T T ( T T UdA ( C & p (48 Assung all ters n te rgt and sde f Equatn 48 are nstant, we an ntegrate fr pnt ( t pnt ( alng te lengt f te eat exanger d( T T ( T T UdA ( C & p (49 ( T T ( UA ( T T C & (50 p Reall ( C ( T T Q & & (5 p su tat ( T T ( T T UA Q& [( T T ] (5 w ay be rearranged t gve ( T T UAΔT [( T T /( T T ] Q& UA (53 were ΔT ( T T [( T T /( T T ] ( T T ( T T [( T T /( T T ] (54 Nte, tat ts s dental t Equatn 44 wt T T T. A slar expressn an be develped fr ndensers wt te t flud teperature replaed by te uratn teperature. In One Trug Stea Generatr desgns, te sendary water usually enters subled, bls, and n se ases leaves supereated. Fr re prese eat transfer area alulatns, an verall eat transfer effent and LMTD suld be deterned fr ea regn. U-tube stea generatr desgns atually ntan ne sell sde pass and tw tube-sde passes. Ts wuld nrally requre te use f Equatn 45 wt an apprprate value f te rretn fatr F. Sne te sell sde teperature s nstant wever, te F fatr bees.0 and te LMTD an be used wtut rretn. Ts als lds fr ndensers. U-tube desgns nrpratng ntegral preeaters are re plated and requre ultple LMTDs. Te verall eat transfer effent an be alulated fr Equatn 3 wt te tube sde eat transfer effent alulated fr te Dttus-Belter r se ter apprprate sngle-pase rrelatn. Te sell sde blng eat transfer effent an be puted fr te Jens-Lttes rrelatn. It suld be nted, tat n ts dervatn t was assued tat te UA value was nstant. We ave seen n ur dsussn f blng eat transfer tat te blng eat transfer effent s a strng funtn f te wall
10 teperature. Te sae s true fr ndenn. T aurately predt te perfrane f eat exangers subjet t eter blng r ndenn generally requres Equatn 49 t be ntegrated nuerally. Fulng Fatrs After a perd f te, te eat transfer surfaes f a eat exanger ay bee ated wt depsts fr te eat transfer fluds r rrsn. In eter ase, te addtnal resstane t eat transfer due t tese aterals dereases te perfrane f te eat exanger and ust be aunted fr. Ts s dne trug te use f experentally deterned fulng fatrs. Tese fulng fatrs are ten used t dfy te verall eat transfer effent ardng t R f / U / U were: R f fulng fatr U verall eat transfer effent fr te "drty" eat exanger U verall eat transfer effent fr te lean eat exanger Reended values fr fulng fatrs are gven n Table fr se n fluds. Type f Flud Fulng Fatr (r-ft -F/Btu Sea water belw 5 F Sea water abve 5 F 0.00 Treated bler feedwater abve 5 F 0.00 Fuel l Quenng l All vaprs Stea, nn-l bearng Industral ar 0.00 Refrgeratng lqud 0.00 Table : Fulng Fatr fr Cn Fluds (Hlan, Heat Transfer 3rd Ed. 3
11 Exaple: Estate te tube bundle lengt requred fr a U-Tube stea generatr avng te fllwng aratersts. Prary Clant Flw Rate x 0 lb/r Tube Sde Inlet Teperature 595 F Tube Sde Outlet Teperature 540 F Sendary (sell Sde Pressure 830 psa Nuber f Tubes 859 Tube Outer Daeter 0.75 nes Tube Wall Tkness nes Tube Teral Cndutvty 0 Btu/r-ft-F SOLUTION Te ttal eat transfer rate trug te stea generatr s gven by su tat Q& C & ΔT UAΔ p T Q& UA Δ T In addtn, we ave UA A n r + KL + r π A UA n r + πr L πkl r + πr L UA r πnl r + + K r r fr w L an be deterned. Flud Prpertes Te prary sde flud prpertes are taken at te average flud teperature and a pressure f 000 psa. Tave F C p.3 Btu/lb-F k 0.39 Btu/r-ft-F μ 0.5 lb/r-ft Lg Mean Teperature Dfferene At a sendary sde pressure f 830 psa, te uratn teperature s 5.4 F. 4
12 ΔT ΔT ( T T ( T T [( T T /( T T ] ( T T [( T T /( T T ] ( [( /( ] Δ T 38. 3F Cnvetve Heat Transfer Ceffent Te tube sde nvetve eat transfer effent an be btaned fr te Dttus-Belter rrelatn D k G 0.03 D μ 0.8 C pμ k 0.3 G & A x & nπ D 0 4 (859( π (0.54 / lb r - ft (0.39 ( ( ( (.3( Btu r - ft - F Blng Heat Transfer Ceffent Te blng eat transfer effent s puted usng te Jens-Lttes rrelatn su tat q exp(4p / 900 ( T 4 w T exp(4p / ( T 3 4 w T. 0 Te alulatn f te blng eat transfer effent s plated by te nnlnearty n tube wall teperature w s unknwn. We an estate te blng eat transfer effent by akng se reasnable estate f te wall teperature wever, and fr ts exaple assue gvng fr te blng eat transfer effent Tube Bundle Lengt 0 Tw T + ( 0.5 ΔT 54F exp(4 830 / 900 Te UA value s fund fr te ttal eat transfer rate and te LMTD ( ,9Btu r - ft - F 5
13 Q& UA ΔT C & pδt ΔT ( 0 (.3( UA Btu r - F Te bundle lengt s ten btaned fr te UA value trug te relatnsp. UA r πnl r + + K r r UA L πn r r + K r + r L 44 ft ( (50(0.54 / (4,9(0.75 / 4 π
14 Exaple: (Fr Hlan, Heat Transfer 3rd Edtn Water wt a ass flw rate f 30,000 lb/r s eated fr 00 t 30 F n a tube and sell eat exanger. A sngle pass f water at 5,000 lb/r and 00 F s used n te sell sde as te eatng flud. Te verall eat transfer effent s 50 Btu/r-ft -F and te average water velty n te 0.75 n I.D. tubes s. ft/se. Beause f spae ltatns, te eat exanger ust be n lnger tan 8 feet. Cnsstent wt ts restrtn, alulate te nuber f tube passes, te nuber f tubes per pass, and te lengt f te tubes. SOLUTION Cnsder frst a sngle tube pass. Te ext teperature n te sell sde s alulated fr ( C & p ΔT ( C p ΔT Q& & ( C & p ( C & ( 30,000 ( 5,000 ΔT (30 ΔT 0F p Te LMTD s gven by Δ T T ext F ( T T ( T T [( T T /( T T ] w fr ts exaple (assung a unterflw eat exanger gves Te ttal eat transfer area s btaned fr te equalty ( ( [( /( ] Δ T 53. F Q& & ( C p ΔT UAΔT ( C p ΔT ( 30,000 (30 A & UΔT (50( ft Te nuber f tubes s btaned fr te ttal tube sde ass flw rate and te average velty n te tubes trug te relatnsp & ρ va ρvnπd / 4 x 4& 4 30,000 n tubes ρvπd ((. 300 π (.75 / A Te ttal eat transfer area s A nπdl L nπd lengt fr a sngle pass eat exanger gvng fr te tube lengt ( and terefre te eat exanger 7. 3 L 9. ft ( 37 π(. 75 / 7
15 Ts lengt s greater tan te allwable 8 feet fr te eat exanger lengt, plyng re tan ne tube pass s requred. Nte, wle ts redues te lengt f te eat exanger t wll nt redue te tube lengt. Exane next tw tube passes. Te ttal eat transfer rate s nw gven by Q& UAFΔ T Te rretn fatr F s gven n ters f te rats T R t T t Fr ts prble T 00 T 40 t 00 t 30 t t P T t R P Fr te grap f rretn fatrs, F 0.88 w nreases te ttal eat transfer area t 03. ( C p ΔT ( 30,000 (30 A & UFΔT 7.3 ft (50(.88(53. Te ttal nuber f tubes reans unanged at 37. Te ttal eat transfer area s nw gven by A A nπd L L nπd were te s due t te tw tube passes trug te eat exanger. Te requred lengt fr te tw pass eat exanger s ten w sfes te desgn requreent L 55. ft ( ( 37 π(. 75/ Nte: Ts prble s sewat dealst, as te verall eat transfer effent s assued t be nstant. In realty, as yu nrease te nuber f passes, yu redue te ass flux n te sell sde and rrespndngly te sell sde nvetve eat transfer effent. 8
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