COMPUTATIONAL NUCLEAR THERMAL HYDRAULICS

Transcription

1 COMPUTTIONL NUCLER THERML HYDRULICS Cho, Hyoung Kyu Dpartmnt of Nuclar Enginring Soul National Univrsity

2 CHPTER4. THE FINITE VOLUME METHOD FOR DIFFUSION PROBLEMS 2

3 Tabl of Contnts Chaptr 1 Chaptr 2 Chaptr 3 Chaptr 4 Chaptr 5 Chaptr 6 Chaptr 7 Chaptr 8 Chaptr 9 Chaptr 10 Chaptr 11 Chaptr 12 Chaptr 13 Introduction Consrvation las of fluid motion and thir boundary conditions Turbulnc and its modlling Th finit volum mthod for diffusion problms Th finit volum mthod for convction diffusion problms Solution algorithms for prssur vlocity coupling in stady flos Solution of systms of discrtisd quations Th finit volum mthod for unstady flos Implmntation of boundary conditions Uncrtainty in CFD modlling Mthods for daling ith complx gomtris CFD modlling of combustion Numrical calculation of radiativ hat transfr

4 Contnts Introduction FVM for 1D stady stat diffusion Workd xampls: 1D stady stat diffusion FVM for 2D diffusion problms FVM for 3D diffusion problms Summary

5 Introduction Gnral transport quation Dlting th transint and convction trms, Intgral form From this 1D stady stat diffusion quation, th discrtizd qs. ar introducd. Th mthod is xtndd to 2D and 3D diffusion problms.

6 Contnts Introduction FVM for 1D stady stat diffusion Workd xampls: 1D stady stat diffusion FVM for 2D diffusion problms FVM for 3D diffusion problms Summary

7 1D diffusion quation FVM for 1D stady stat diffusion Stp 1: grid gnration Control volum for FVM Usual convntion of CFD mthods

8 Stp 2: discrtization FVM for 1D stady stat diffusion n d d dx V SdV d dx d dx SV 0 n d d dx f n x d dx f d dx d dx fds S k f k S k n x 1, n 1 x

9 FVM for 1D stady stat diffusion Stp 2: discrtization Taylor sris approximations 0 V S dx d dx d? dx d x x x x dx d ) ( ) ( WP W P PE P E x dx d x dx d?

10 Stp 2: discrtization FVM for 1D stady stat diffusion d dx d dx S V 0 d dx?? Intrfac proprtis (for uniform grid) W 2 P P 2 E W W 1 W P 1 E E EP W x x P WP E x x x PE PE P

11 FVM for 1D stady stat diffusion Stp 2: discrtization 0 V S dx d dx d PE P E x dx d WPW P x dx d p S u S p V S 0 p p u WP W P PE P E S S x x

12 Stp 2: discrtization FVM for 1D stady stat diffusion

13 FVM for 1D stady stat diffusion Stp 3: Solution of quations St up th discrtizd quation at ach of th nodal points Modify th quation for th control volums that ar adjacnt to th domain boundaris. Driv th systm of linar algbraic quations Matrix solution tchniqus

14 Contnts Introduction FVM for 1D stady stat diffusion Workd xampls: 1D stady stat diffusion FVM for 2D diffusion problms FVM for 3D diffusion problms Summary

15 1D Hat conduction quation Workd xampls: 1D stady stat diffusion Exampl 4.1 Considr th problm of sourc fr hat conduction in an insulatd rod Whos nds ar maintaind at constant tmpraturs of 100 C and 500 C rspctivly. Calculat th stady stat tmpratur in th rod. Thrmal conductivity k = 1000 W/mK Cross sctional ara = m 2

16 Exampl 4.1 Workd xampls: 1D stady stat diffusion Fiv control volums (clls) Cll cntr nods: 5 Cll facs: 6 (4 intrnal facs, 2 boundary facs) Constant thrmal conductivity Constant nod spacing Constant cross sctional ara No sourc trm

17 Exampl 4.1 Workd xampls: 1D stady stat diffusion Fiv control volums (clls)

18 Workd xampls: 1D stady stat diffusion Exampl 4.1 For boundary #1 0 WP W P PE P E x T T k x T T k

19 Workd xampls: 1D stady stat diffusion Exampl 4.1 For boundary #5 0 WP W P PE P E x T T k x T T k

20 Exampl 4.1 Workd xampls: 1D stady stat diffusion Systms of algbraic quations Constant thrmal conductivity (1000) Constant nod spacing (0.1) Constant cross sctional ara ( ) For cll #1 For cll #2 ~#4 For cll #5

21 Exampl 4.1 Workd xampls: 1D stady stat diffusion Systms of algbraic quations For cll #1 For cll #2 ~#4 For cll #5

22 Exampl 4.1 Workd xampls: 1D stady stat diffusion Systms of algbraic quations matrix form Linar solvr Gaussian limination LU dcomposition TDM ILU CGM BICG BICGSTB Etc.

23 Exampl 4.1 Workd xampls: 1D stady stat diffusion

24 Exampl 4.2 Workd xampls: 1D stady stat diffusion problm that includs sourcs othr than thos arising from boundary conditions. larg plat of thicknss: L = 2 cm Constant thrmal conductivity: k = 0.5 W/m.K Uniform hat gnration: q = 1000 kw/m 3 Th facs and B ar at tmpraturs: 100 C and 200 C rspctivly. 1D problm Dimnsions in y and z ar so larg.

25 Exampl 4.2 Workd xampls: 1D stady stat diffusion Intgral form and discrtization

26 Exampl 4.2 Gnral form Workd xampls: 1D stady stat diffusion For boundaris

27 Exampl 4.2 For boundaris Workd xampls: 1D stady stat diffusion

28 Exampl 4.2 Workd xampls: 1D stady stat diffusion Fiv discrtizd quations matrix form T 100 C T 200 B C Error=1.5~2.8 %

29 Exampl 4.3 Workd xampls: 1D stady stat diffusion Cooling of a circular fin by mans of convctiv hat transfr along its lngth. Convction givs ris to a tmpratur dpndnt hat loss or sink trm cylindrical fin ith uniform cross sctional ara. Th bas is at a tmpratur of 100 C (TB) Th nd is insulatd. Th fin is xposd to an ambint tmpratur of 20 C. Govrning q. nalytical solution h: th convctiv hat transfr cofficint P: th primtr k: th thrmal conductivity of th matrial T : th ambint tmpratur

30 Exampl 4.3 Workd xampls: 1D stady stat diffusion Cooling of a circular fin by mans of convctiv hat transfr along its lngth. Convction givs ris to a tmpratur dpndnt hat loss or sink trm cylindrical fin ith uniform cross sctional ara. Th bas is at a tmpratur of 100 C (TB) Th nd is insulatd. Th fin is xposd to an ambint tmpratur of 20 C. Govrning q. nalytical solution d dx k dt dx hpx x T 0 T h: th convctiv hat transfr cofficint P: th primtr k: th thrmal conductivity of th matrial T : th ambint tmpratur

31 Data and mshs Workd xampls: 1D stady stat diffusion Uniform grid, dividd into fiv control volums Modifid govrning quation and its intgral form

32 Discrtization Workd xampls: 1D stady stat diffusion For msh 2~4

33 For msh 2~4 Workd xampls: 1D stady stat diffusion For msh 1 For msh 5 q k Tnd T x / 2 5 0

34 Workd xampls: 1D stady stat diffusion Matrix form of th algbraic quations n 2 hp k 25

35 Workd xampls: 1D stady stat diffusion Matrix form of th algbraic quations HW#1 Solv Exampl 3 by yourslf Driv th discrtizd quation Writ a cod. Find th analytical solution Compar th calculation rsult ith th solution With 5, 10, 20 mshs Error: max. 6.3 % ith 5 mshs Error: max. 2.1 % ith 10 mshs

36 Homork ssignmnt HW #2 1D conduction quation for nuclar ful rod Chck th hat transfr ara carfully! Using FVM Constant conductivity, k For pllt For cladding Gap conductanc 1) Xnon conductivity 2) Constant gap conductanc valu 3) Gap conductanc modl Flo condition HTC: Dittus Boltr Fluid vlocity Chck th unit of ach paramtr!

37 HW #2 Gap conductanc Homork ssignmnt Chck th unit of ach paramtr! h g [ W / m 2 C] k gas ff 1/ f T 3 fo 1/ c 1 Gap conductanc q in q " g, fo in h g in T fo T gap q out q " g, ci out h g in out out T gap T ci Non linar dpndncy Start calculation ith th gap conductanc graph. hg valu from th graph With th calculatd tmpratur, updat hg. Rpat th calculation Updat hg Rpat until th solution convrgs. Nuclar Systms, vol. I, p. 418

38 Homork ssignmnt HW #2 What you nd to rport Discrtizd quations Calculation conditions Tmpratur profils for thr cass Gap conductanc