The Dominant Thermal Resistance Approach for Heat Transfer to Supercritical-Pressure Fluids

Transcription

1 The Dominant Thermal Resistance Approach for Heat Transfer to Supercritical-Pressure Fluids Donald M. McEligot 1,2, Eckart Laurien 3, Shuisheng He 4 and Wei Wang 4,5 1. Nuclear Engineering Division, U. Idaho, Idaho Falls, Idaho U.S.A. 2. Professor Emeritus, U. Arizona, Tucson, Ariz U.S.A. 3. Institut für Kernergetik und Energiesysteme (IKE), Uni. Stuttgart, D Stuttgart, Germany 4. Mechanical Engineering Dept., U. Sheffield, Sheffield S1 3JD, England 5. No at Daresbury Laboratory, Science and Technology Facilities Council, Warrington, WA4 4AD England page 1

2 sco 2 Recuperative Recompression Brayton Cycle 'Advanced Design' of the MIT-Study 200 bar 77 bar critical point region of interest page 2

3 Outline Introduction Test Case : Direct Numerical Simulation (Wang, He) Dominant Thermal Resistance Approach Results Conclusions and Outlook page 3

4 Physical Properties of Supercritical Water pressure : 23.5 MPa case 1 2 page 4

5 Why Develop an Approximate Method? Approach to provide approximate predictions and improved analyses ith varying fluid properties Possibly a useful basis for extending constant property correlations to variable properties A reasonable, sensible, simple analysis ill (may) provide better predictions than empirical correlations for fluids ith significant property variation Improved treatment for all functions in CFD page 5

6 Outline Introduction Test Case : Direct Numerical Simulation (Wang, He) Dominant Thermal Resistance Approach Results Conclusions and Outlook page 6

7 DNS of Supercritical Pipe or Channel Flos (Water or CO 2 ) DNS : Direct Numerical Simulation Computational Fluid Dynamics (CFD) ithout turbulence model, all scales resolved Rynolds Number Re must belo (typically: Re < 6000) authors case HT mode published year Bae, Yoo, Choi (Korea) upard vertical pipe all heating Phys Fluids 2006 used in the present ork Nemati et al. (Delft) Chu and Laurien (Stuttgart) Wang and He (Sheffield, UK) vertical pipe hith/no buoyancy horizontal pipe plane channel hith/no buoyancy all heating IJHMT 2015 all heating constant T at the alls J. Supercritical Fluids 2016 NURETH Pandey and Laurien (Stuttgart) vertical pipe all heating or cooling this conference 2018 page 7

8 Description of the Wang-DNS (no accelation, no buoyancy) p = 23.5 MPa, G = kg/m2s, T c = 367 C Geometry case: 1, T h = 367 C case 2, T h = 380 C Turbulence structures visualized by the second-largest eigenvalue of the stress tensor um 2 Reb 5730 Reb 5670 W. Wang & S. He, NURETH-16, 2015 b page 8

9 case 1 Forced-Convection Correlations for Nu Dittus Boelter (DB), Gnielinski (VG), Mokry and Pioro (Mok) q 2 Nu ( T T ) k Nu b b pseudo-critical temperature K at the all page 9

10 Forced-Convection Correlations for Nu Dittus Boelter (DB), Gnielinski (VG), Mokry and Pioro (Mok) q 2 Nu ( T T ) k b b case 2 30 Nu pseudo-critical temperature K at the all page 10

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12 Importance of the Turbulent Core DNS data from Bae, Yoo and Choi, Phys. Fluids % y laminar sublayer r/ R page 12

13 Outline Introduction Test Case : Direct Numerical Simulation (Wang, He) Dominant Thermal Resistance Approach Results Conclusions and Outlook page 13

14 Dominant Thermal Resistance Approach assumptions Steady state, Quasi-established velocity and temperature profiles Constant shear layer and heat flux layer approximations Negligible buoyancy, negligible acceleration Turbulent core - high turbulence, high, high c p ----> T T T T b centerline laminar conducting sublayer sublayer page 14

15 WangF653-Profs-T.qpc left all Wang-DNS Mean Temperature Profile Case 1 region of interest 1.01 T T ref cooled side 2 heated side conducting sub-layer y cs h c pc y all distance right all page 15

16 Wang-DNS Mean Velocity Profile Case 1 WangF653-Profs.qpc region of interest u U cooled side heated side viscous sub-layer y vs h 0 2 c pc y all distance page 16

17 Integration of the Thermal Energy Equation in the laminar, conducting sub-layer: the region of dominant thermal resistance T q 0 cu p q( y) const. q x y Near the all e have Fourier s la: Define then q( y) k( T) T T y Integrate: ( T) k( T) dt a property q y ( T) ( T) At y = y cs T ref ( Tb) ( T) q y cs y T ( y) q( y) d y k( T) dt 0 May be a good approximation If one has a good estimate of y cs T page 17

18 Ho to get good Estimates for y cs using the universal all units Prandtl [1910] approach y cs+ y vs + To-layer approximation y vs here y + = y ( /) 1/2 / or y + = (y/d h ) Re Dh (C f /2) 1/2 DO WE HAVE A GOOD ESTIMATE OF C f? page 18

19 Forced-Convection Correlations for all friction used in Gnielinski (VG), Dre, Koo, McAdams (DKM), M.F. Taylor (MFT), Blasius (Blas) case 1 c f c f 0.5 u 2 m c f T K page 19

20 Forced-Convection Correlations for Nu used in Gnielinski (VG), Dre, Koo, McAdams (DKM), M.F. Taylor (MFT), Blasius (Blas) c f u Case 2 m c f T K page 20

21 Integration of the Momentum Equation in the laminar, conducting sub-layer: the region of dominant flo resistance U ( y) ( y) const. y Near the all e have Neton s la and Fourier s la: dy ( T ) du dy du ( T ) Define Integrate U Solve for 0 b ( T) du T T and ref q kt ( ) ( T ) T T b q dy dy dt a property kt ( ) ( T) k( T) dt kt ( ) dt q dt Ub q ( T ) ( T ) b du Ub ( Tb) ( T) q kt ( ) dt ( T) q page 21

22 Integration of the Momentum Equation (contd.) in the laminar, conducting sub-layer: the region of dominant flo resistance Substitute into i.e. into to give q y ( T ) ( T ) cs cs ycs q ( Tcs) ( T) ith ycs q ( Tcs) ( T) Ub q ( T ) ( T ) and solve for b y cs y cs q U ( T ) ( T ) b b 2 ( y cs) ( T ) ( Tb ) 2 page 22

23 Outline Introduction Test Case : Direct Numerical Simulation (Wang, He) Dominant Thermal Resistance Approach Results Conclusions and Outlook page 23

24 case 1 Nu Nu Result Heat Transfer q 2 ( T T ) k b b present y y vs y y cs 11.6 upper index + in all units T K page 24

25 Nu Result Heat Transfer q 2 ( T T ) k b b case 2 Nu present T K page 25

26 c f Result Friction 0.5 u 2 m case 3 c f present T K page 26

27 Nu Result Heat Transfer q 2 ( T T ) k b b case 3 T h = 655 K Nu present T K page 27

28 Concluding Remarks Demonstrated a closed-form, approximate, coupled analysis for Nu for ScPF (ith negligible buoyancy and acceleration) Some reasonable agreement ith DNS of Wang+He Nu is sensitive to choice of y vs + Useful approach to provide approximate predictions and improved analyses Improved treatment for all functions in CFD Can provide a first estimate for interative processes in "more sophisticated" analyses page 28

29 Outlook Extend to significant buoyancy and acceleration Revise analysis to treat differing y cs+ and y vs + Add thermal resistance for turbulent core? page 29

30 DNS data: 80 bar, CO 2, D = 2 mm upard donard T [K] bulk bulk q W = 61 kw/m 2 K h [kj/kg] 1 h [kj/kg] T [K] q W = 61 kw/m 2 K 8 8 page 30

31 Backup Slides page 31

32 State-of-the-Art Correlations for Narro Channels (2 mm) G = 60 kg/m2s, q = -30 kw/m2 (cooled all) Forced convection Mixed convection (/ gravity influence) index 0 : constant properties : all b : bulk Nu : Nusselt number Gr : Grashof number Re : Reynolds number + upard, - donard flo almost the same result for upard and donard flo page 32

33 DNS-Results of a Heated PipeFlo at Re = 5400 upards: thermally stable buoyancy first damps later induces turbulence (recovery) g donards: thermally unstable buoyancy induces turbulence Visualization of turbulence structures using the λ 2 criterium g page 33

34 Terminology of Single-Pipe or Channel Experiments at Super-Critical Pressure 100 pseudo-critical point (p pc, T pc ) pseudo-critical line heating ρ = 100 kg/m 3 p [bar] 73.1 critical point cooling 50 saturation line bulk temperature (T b ) of a heated/cooled pipe/channel T [ C] page 34

35 sco 2 Recuperative Recompression Brayton Cycle 'Advanced Design' of the MIT-Study: Net Efficiency ~47 % HighTemp Lo Temp 77 bar Recuperator Recuperator % p T 4 Input HX 200 bar Reject HX Alternator % MPa C 1 7,7 32, , , , ,0 6 7,7 534,3 7 7,7 165,8 8 7,7 68,9 Turbine Re-Compressor Main Compressor 200 bar 77 bar region of interest critical point page 35

36 Example: To-Layer Model for Turbulent Boundary Layers Prandtl 1910, Kays and Craford 1980, constant properties T T laminar sub-layer T l R turbulent core layer y D 2 T b T R The temperature in all units c u p T y T T y cp u R q Can be considered a non-dimensional thermal resistance. Expand to T y c u R R p lam turb And compare to the Kays and Craford relation Prt T R Pr y ln R ln y T lam lam b The total resistance is the sum of to individual resistances fore the to layers page 36

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