Natural Circulation Systems

Transcription

1 Natural Crulatn Systems Natural rulatn lays an mrtant rle n the ln term ln f reatr systems durn adent ndtns, and n the strae f sent fuels. Cnsder the fllwn smle reresentatn f a reatr lant system eratn under steady-state natural rulatn. Whle the daram suests a One-Thruh steam eneratr, there s nthn whh requres a One-Thruh desn. Steam Generatrs Reatr Ht Les Cld Les Fure : Smle Reresentatn f a Natural Crulatn L We are nterested n the ndtns neessary fr natural rulatn, and the mantude f the natural rulatn flw rate as a funtn f system arameters. We make the fllwn assumtns reardn the flw: a) The lant s a snle hase lqud. Densty varatns are nly mrtant n the flud weht terms (Busnessq Arxmatn) b) The lqud s nmressble wth reset t ressure, but thermally exandable,.e. the flud densty s a funtn f temerature alne, = ( T ) ) The flw ath s msed f nstant area flw sements, wth mmentum hanes due t area hanes aunted fr by arrate lss effents. The steady-state ntnuty equatn ves dga ( x ) = () at any nt wthn the l. Sne we have assumed the flw l an be nstruted f nstant area n, wthn any nstant rss setnal area sement Interate the steady state mmentum equatn arund the flw l dg = ()

2 A d G Ax dp f G K D G e = + δ ( ) sn θ () x and examne the nterals ndvdually. Mmentum Flux Term We rewrte the nteral f the mmentum flux as the sum f nterals ver eah lenth f nstant rss setnal area. Sne we have assumed the densty s nstant everywhere but the bdy fre term d G A x = Ax Mass nservatn ves dg L () dg = G dg = (5) suh that A x d G A x = (6) Pressure Gradent By defntn, the nteral f the ressure hane arund the lsed l s er dp = (7) Frtn and Frms Lsses f D e G K G f D G K G + δ ( ) = + δ( ) e L fl G K G = + D e (8) Aan mass nservatn requres GA x = m where m s the ttal system mass flw rate. The frtn and frms lsses an then be wrtten as e e x fl G K G fl m K m + = + D D A A x (9)

3 suh that e f D G K G fl m K m + δ ( ) = + De Ax Ax () and the nterated mmentum equatn has the frm e x x fl m K m = D A A snθ () T evaluate the nteral f the bdy fre term requres knwlede f the densty dstrbutn arund the l. We have assumed a state equatn f the frm The flud temerature may be btaned frm the smle enery equatn = ( T ) () mc dt = q () () nterated arund the l. Assumn fr nw that the temerature and densty dstrbutns are knwn, we break the nteral f the bdy fre term u nt the sum f nterals ver the heated and nn heated lenths. θ θ T sn = θ θ T sn + ( ) sn + sn + ( ) snθ () snθ = ( H H) (5a) ( T) snθ = SG ( H H) (5b) where SG s the averae lant densty n the steam eneratr snθ = ( H H) (5) ( T) snθ = Rx ( H H) (5d) where Rx s the averae lant densty n the reatr

4 vn fr the nteral f the weht fre term snθ = ( H H) + SG ( H H) + ( H H) + Rx ( H H ) (6) The averae denstes n the reatr and steam eneratr an be wrtten as a wehted averae between the nlet and ext denstes,.e. = ε + ( ε ) ε Rx [,] (7a) Rx Rx Rx = ε + ( ε ) ε SG [,] (7b) SG SG SG Substtutn and smlfyn ves sn θ = [ H + εsg ( H H) H εrx ( H H) ] + ε ε [ H SG ( H H ) + H + Rx ( H H )] (8) Nte, H H + ε ( H H ) (9) SG SG s an averae elevatn wthn the steam eneratr at whh heat s remved frm the lant. Smlarly H H + ε ( H H ) () Rx Rx s an averae elevatn wthn the re at whh heat s added t the lant. The elevatns H SG and H Rx are ften alled the thermal enters f the steam eneratr and reatr resetvely. The abve nteral an then be wrtten as snθ = [ HSG HRx ] + [ HRx HSG ] = ( ) [ HSG HRx ] () suh that the nterated mmentum equatn s fl m K m = ( ) [ ] D A A H SG H Rx () r m e x x fl D A K A + e x x = ( ) [ ] H SG H Rx () The nterated mmentum equatn reresents a balane between frtnal fres (LHS) and buyany fres (RHS). The l wll reah a steady state when these fres are equal.

5 Nte: Sne >, the mmentum equatn has real slutns fr m nly when HSG > HRx. Ths mles natural rulatn an nly devel f the steam eneratrs are laed at a hher elevatn than the reatr. The mantude f the mass flw rate s als dtated by ths elevatn dfferene. In eneral, t ntate natural rulatn, fur requrements must be met: () There exsts a heat sure avalable t the flwn flud. () There exsts a marable heat snk. () The thermal enter f the heat snk be hher than the thermal enter f the sure. () There exsts a mlete flw ath. The smle flw l ndated here bvusly satsfes these requrements. Slutn fr the mass flw rate stll requres the denstes and be sefed. Sne densty s a funtn f the flud temerature, ths requres the flud temerature dstrbutn arund the l be knwn. As nted revusly, the flud temerature dstrbutn may be btaned by nteratn the enery equatn arund the flw l. Anther examle f a flw system whh satsfes the requrements fr natural rulatn s that f a sent fuel strae l. Examle: Cnsder a sent fuel assembly n a fuel strae l. We are nterested n the steady state mass flw rate thruh the assembly as a result f natural nvetn. If we assume n mmunatn between assembles wthn the l, and a nstant flw area wthn the assembly, then the steady-state flud nservatn equatns are Pl Pl Assembly H Fure : Natural Crulatn n a Sent Fuel Pl Cntnuty Equatn

6 dga x = () Enery Equatn We assume snle hase flw, suh that the smle enery balane mc dt = q () () determnes the flud temerature wthn the assembly Mmentum Equatn v P f v v = + Kδ( ) sn ϑ () De Equatn f State = ( T ) () The Cntnuty Equatn states that the mass flw rate s nstant, and n setns f nstant rss setnal area, the mass flux s nstant. If we assume the l temerature des nt hane wth tme, then the flud temerature n the assembly s btaned by nteratn the enery equatn dt = mc q () (5) frm the hannel nlet at nt t sme arbtrary nt aln the lenth f the assembly dt = q ( ) (6) mc T () T() = q ( ) mc (7) Interate the mmentum equatn frm nt n the l t nt at the nlet t the assembly. Neletn frtn and frms lsses n the l ves ϑ v P = sn (8) where s the bulk l densty. Evaluatn the nterals ( v v ) ( P P) ( H H) = (9) 5

7 r ( P P) ( ) ( H H) ( ) v v v v = + = H () Nte, the same result uld have been btaned by alyn Bernull s equatn between these same nts. We next nterate the mmentum equatn frm nt t nt wthn the assembly δ v P f v v = + K ( ) D () e where we have taken advantae f the fat that snθ = n ths ren. Assumn the Busnessq arxmatn s aan vald v = () fh = ( P P) + De v K () By the free et ndtn P = P suh that fh = ( P P) + De v K () r fh = + De ( P P ) K v (5) Equatn and 5 ( v v ) H fh v = + K De (6) Nte, n the remann nteral, densty s a funtn f temerature, and temerature a funtn f stn aln the assembly. T smlfy ths nteratn, exand the flud state equatn n a Taylr s seres abut a knwn referene temerature T 6

8 ( T) = ( T ) ( T T ) d + +L (7) dt T Assumn the temerature varatns are suffently small, we an nelet the hher rder terms n the exansn suh that the flud state equatn s arxmately ( T) = ( T ) ( T T ) d + (8) dt r T { } ( T) = ( T ) β( T T ) (9) where β d () dt T s the vlume exansvty. If we take the referene temerature as the bulk l temerature then ( ) = { } ( T) = β( T T ) () and the densty nteral bemes { β } β T T H = ( ) = ( T T ) () Frm the enery equatn, we have T T () T = q ( ) mc () where the hannel nlet temerature s assumed t be the bulk l temerature. The densty nteral an then be wrtten as β = H q mc ( ) () suh that the nterated mmentum equatn s fnally H ( v v ) H fh v = + K De H β H q ( ) (5) mc r 7

9 H fh v β ( ) v v + + K = q ( ) (6) De mc The terms n the rht hand sde f Equatn 6 reresent the net buyany fres atn t drve the flud. The terms n the left hand sde reresent the net frtn and frms lsses sn the flw. As n the lsed l rblem, a steady state wll be reahed when the buyany fres are balaned by the frtnal fres. In addtn, mass nservatn ves m= va = va suh that mass flw an be determned n rnle frm Equatn 6 fr any funtnal frm f the lnear heat rate. Cnsder the smle ase f a unfrm lnear heat rate. The nterals n the rht hand sde ve H q = q = H ( ) Q H (7) where Q s the ttal heat eneratn rate f the assembly. If we further assume that v fh + + De << v, then v β K = mc Q H (8) r n terms f the mass flw rate fh + + De m β K = A mc Q H (9) A m β QH fh = + + K () C De Equatn mles the mass flw rate thruh the assembly shuld nrease as the ube rt f the heat nut. Nte, n ths examle, the l tself ats as the heat snk, wth the assumtn that the l temerature s mantaned by sme external heat exhaner. If the heat exhaner were t fal, the l wuld eventually heat t the bln nt unless lsses t the envrnment were suffent t brn the l t a new steady state. 8