For t BWR oprating paramtrs givn blow, comput and plot: a) T clad surfac tmpratur assuming t Jns-Lotts Corrlation b) T clad surfac tmpratur assuming t Tom Corrlation c) T clad surfac tmpratur assuming t Cn Corrlation On approac for andling t mixd boiling and fully dvlopd nuclat boiling rgims in flow boiling cannls is to assum a suprposition approac wr t wall at flux is t sum of singl pas forcd convction and nuclat boiling componnts from t point at wic t wall tmpratur xcds t saturation tmpratur, i.. q( z) [ T ( z) T ( z)] (z)[ T ( z) T ] FC w NB w sat wr FC is an appropriat singl pas forcd convction at transfr cofficint (.g. Wisman), and NB is an appropriat nuclat boiling at transfr cofficint (.g. Tom or Jns-Lotts). For Parts a) and b) you can assum tis suprposition approac olds. In t Cn corrlation, compar t tmpratur distributions obtaind wit t original Dittus-Boltr corrlation for o and tat obtaind by substituting t Wisman corrlation for o. BOILING WATER REACTOR PARAMETERS Prssur 040 psia Coolant Inlt Vlocity 7. ft/sc Cor Inlt Entalpy 527.9 Btu/lbm Cor Avrag at Flux 59,000 Btu/r-ft 2 Rod Pitc 0.640 incs Rod Diamtr 0.493 incs Ful igt 48 incs Fraction of nrgy dpositd in ful 0.97 Axial Pa to avrag ratio.4 T axial at flux may b tan to b ( - z) ( - z) () = q 0 sin q z
SOLUTION at Flux T at flux profil is in trms of two unnown paramtrs, t xtrapolation distanc and t amplitud q0. T xtrapolation distanc is dtrmind by t axial pa to avrag ratio. T amplitud sts t magnitud of t at flux. T axial pa to avrag ratio is dfind to b Extrapolation Distanc F z q ( z max ) q wr z max is t position of maximum at flux in a particular cannl, and q is t axially avragd at flux in t sam cannl. Not, tat sinc for any givn cannl q ( z max ) and q bot contain t amplitud q0, tis paramtr cancls and t axial pa to avrag ratio is only a function of sap. T position of maximum at flux is tat location suc tat d q dz z max 0 For tis at flux profil, t maximum at flux dos not occur at 2, nor is t function valuatd at t position of maximum at flux qual to on, suc tat q 0 q ( z max ). Dtrmination of zmax is furtr complicatd by t fact tat t solution for z max contains t xtrapolation distanc wic is as of yt unnown. W can avoid tis problm by dfining a nw variabl z x suc tat and maximizing wit rspct to x q ( x) q0 xsin( x) d q xsin( x) dx 0 0 x max 0 sin( xmax) xmax cos( xmax) wic is transcndntal in x max and must b solvd itrativly. Not, tat sinc z max [ 0, ] xmax (,0) for 0 2
Itrating on x max yilds t solution x max 2. 029. T axially avragd at flux is dfind to b q 0 q( z) dz 0 z z q0 dz sin q 0 ( ) ( ) q cos cos ( ) sin sin T axial paing factor can tn b writtn in trms of x max and q as F z x max sin( x max ) ( ) ( ) cos cos ( ) sin sin For 3.097 ft. 2, tis xprssion is transcndntal in and must b solvd itrativly. Itrating on givs at Flux Profil From t dfinition of avrag at flux, t magnitud of t at flux profil is givn by q q 0 ( ) ( ) cos cos ( ) sin sin For t data givn r 5 2 q 0.22327 0 Btu/r-ft T fluid proprtis assumd for tis problm ar f = 45.99 g = 2.3426 f = 0.292 g = 0.0460 f = 0.3289 C pf =.2986 = 0.008 T sat = 549.43 = 548.75 f 3
fg = 642.304 T coolant ntrs t cannl subcoold, suc tat t potntial xist for singl pas forcd convction ovr som portion of t cannl. T outr clad surfac tmpratur (in t absnc of boiling) is givn by q z Tco ( z) T ( z) ( ) c wr t fluid tmpratur can b obtaind dirctly from t ntalpy using a stat quation of t form T ( z) T [ ( z), P]. T ntalpy is obtaind from t simpl nrgy balanc z ( ) (0) q( z) Ddz o m f z 0 or qd 0 ( z) ( z) ( ) ( ) ( z) ( ) z ( ) (0) cos cos sin sin m f T cannl mass flow rat is m GA, wr t cross sctional flow ara is givn by x A S D and dnsity at t cannl inlt. G 2 2 2 2 2 3 2 x / 4 0.64 (0.493) / 4 0.287 in.59 0 ft. T mass flux is givn by fluid vlocity 6 ( v) inlt 47.7.3600.204 0 lbm/r-ft 2. T cannl mass flow rat is tn 6 3.204 0.590 829 lbm/r. m From t Wisman Corrlation Convctiv at Transfr Cofficint c CR D Pr 0.8 /3 wr C 0.042( S / D) 0.024 0.042(0.64/ 0.493) 0. 024 = 0.0305. If t Dittus-Boltr Corrlation is to b usd c 0.023 R Pr D 0.8 0.4 4
Equivalnt Diamtr 2 2 S D / 4 0. ft 4Ax 4 D 047 D D Rynolds Numbr 6 GD (.204 0 )(0.047) R 258,573 0.292 Prandtl Numbr C p.29860.292 Pr 0.865 0.3289 From wic t convctiv at transfr cofficint can b found Wisman Corrlation 0.8 /3 c CR Pr = D 0.3289 (0.0305)(258,573) 0.8 (0.865) /3 4,346. 0.047 Btu/r-ft 2 -F Dittus-Boltr Corrlation 0.8 0.4 c 0.023 R Pr = D 0.3289 (0.023)(258,573) 0.8 (0.865) 0.4 3,243 0.047 Btu/r-ft 2 -F Location wr t clad tmpratur xcds t saturation tmpratur T minimum critria for boiling is tat t wall tmpratur xcd t saturation tmpratur. If z sat is t position at wic t wall tmpratur racs t saturation tmpratur, tn z sat is t solution of T sat T ( z sat q ( z ) Assuming t Wisman Corrlation for t convctiv at transfr cofficint, t solution for z sat is itrativ. For t data givn r, t solution for z sat givs zsat.42 ft. T ngativ sign implis tat boiling is possibl ovr t ntir cannl. Wall Tmpratur Distributions If t wall tmpratur is assumd to b givn by q( z) [ T ( z) T ( z)] (z)[ T ( z) T ] wr t nuclat boiling at transfr cofficint is givn by FC w NB w sat c sat ) ( z) 0 ( T ( z) T ) m 6 NB w sat tis is a singl nonlinar quation in t wall tmpratur and may b solvd itrativly. 5
Onc t wall tmpratur xcds t saturation tmpratur, t wall tmpratur from t Cn corrlation is t solution of wr q ( z) lo[ Tco( z) T ( z)] 2 [ Tco( z) Tsat ] lo 0.042 S D G x D ( ) 0.024 0.8 C p / 3 D F if t Wisman Corrlation is assumd for t Liquid Only portion of t at transfr cofficint, and lo G( x) D C p 0.023 F D 0.8 0.4 if t Dittus-Boltr Corrlation is assumd. 0. 00. tt F 2.35 0736. 0. 23 00. tt tt tt x x 09. f g 05. 0. g f 0.75 f 99 sat 0.79 0.45 0.49 0.25 C pf g f c fg J 2 0. 0022 0.5 0.29 0.24 0.24 co f fg g Tsatv fg 0. T T S R 2 S 0.9622 0.5822tan 4 6.80 R 2 G xd. 25 F T singl pas liquid componnt of t Cn corrlation is quivalnt to tat for singl pas forcd convction prior to t fluid racing t saturation point. If t nuclat boiling cofficint ( nb ) is st to zro prior to t wall tmpratur xcding t saturation tmpratur, tn t Cn corrlation can b usd ovr t ntir cannl. At any axial location, t at flux, fluid tmpratur and ntalpy (quality) can b dtrmind. T Cn Corrlation is tn in trms of t singl unnown wall tmpratur T co. T dpndnc on wall tmpratur is nonlinar and T co must b solvd itrativly at ac spatial location. T fluid tmpratur profil and t wall tmpraturs computd from t diffrnt corrlations ar indicatd blow. T maximum wall tmpraturs ar 6
56.56 F if t Jns-Lotts corrlation is assumd for t nuclat boiling corrlation in t suprposition approac, 562.3 F if t Tom corrlation is assumd and 575.5 F according to t Cn corrlation. 580 570 Jns-Lotts Tom Cn (Wisman) Tmpratur (F) 560 550 540 530 0 2 4 6 8 0 2 Axial Position ft) A comparison btwn t wall tmpraturs computd using t Wisman and Dittus-Boltr Corrlations for t Liquid Only componnt is illustratd blow. T maximum wall tmpratur using Dittus-Boltr for t Liquid Only at transfr cofficint is 578.4 F. 7
590 580 Cn (Dittus-Boltr) Cn (Wisman) Tmpratur (F) 570 560 550 540 530 0 2 4 6 8 0 2 Axial Position ft) 8