M ODELLING OF THERMAL DISPERSION IN HEATED PIPES Marie D ROUIN 1 , 2 Olivier G RÉGOIRE 1 Olivier S IMONIN 2 Augustin C HANOINE 1 1 CEA Saclay, DEN/DANS/DM2S/SFME/LETR, 91191 Gif-sur-Yvette, France 2 IMFT, UMR CNRS/INP/UPS, Allée du Professeur Camille Soula, 31400 Toulouse, France Scaling Up and Modeling for Transport and Flow in Porous Media Dubrovnik, Croatia, 13-16 October 2008
1. Introduction 2. Averaged equation 3. Dispersion coefficients 4. Numerical results 5. Conclusion 1.1. Context C ONTEXT : HEAT EXCHANGERS , NUCLEAR REACTORS 3 x 6 plaques cintrées Entrefer : 1,84 mm Hauteur active : 600 mm Combustible U-7Mo à 8 gU/cm 3 Gainage : AlFeNi (cf. RHF ) Assemblage combustible Fast Breeder Reactor Caisson monobloc en alliage d'aluminium 48 alvéoles (version à 600 kW/L) (FBR) 2 réservées pour des dispositifs spécifiques 48 pour les éléments combustibles Jules Horowitz Reactor (JHR) Pressurized Water Reactor (PWR) Marie D ROUIN (LETR, IMFT) 2 / 19 :
ra y on quart de assem blage om bustible o eur 1. Introduction 2. Averaged equation 3. Dispersion coefficients 4. Numerical results 5. Conclusion 1.2. Up-scaling U P - SCALING Elementary cell Rod Bundle 1/4 core 1 . 26 cm 21 . 3 cm 170 . 4 cm f Subchannel fine scale simulation Macroscale simulation Marie D ROUIN (LETR, IMFT) 3 / 19 :
xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx � ξ � f + δξ + � ξ ′ � f + δξ ′ xxx xxx xxx xxx xxx xxx Doubly averaged xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx Macroscopic scale δ T f xxx xxx equation xxx xxx � u � f , � T f � f xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx 4 / 19 xxx xxx xxx xxx xxx xxx xxx xxx B xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx A xxx δ u xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx Pedras and De Lemos ( IJHMT , 2001), Quintard and Whitaker (Transport Porous Med., 1994) xxx xxx xxx xxx spatial average xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx 1.3. Averaging procedure Statistically averaged microscopic scale xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx 5. Conclusion xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx Statistically averaged xxx xxx xxx xxx xxx xxx xxx xxx (RANS model) xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx equation u , T f xxx xxx xxx xxx ξ + ξ ′ xxx xxx xxx xxx xxx xxx 4. Numerical results xxx xxx xxx xxx : xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx A VERAGING PROCEDURE xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx 3. Dispersion coefficients statistical average Instantaneous microscopic scale xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx Marie D ROUIN (LETR, IMFT) xxx xxx xxx xxx xxx xxx 2. Averaged equation xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx u , T f xxx xxx xxx xxx Microscopic equation xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx (DNS) xxx xxx xxx xxx xxx xxx xxx xxx ξ xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx 1. Introduction xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx xxx
1. Introduction 2. Averaged equation 3. Dispersion coefficients 4. Numerical results 5. Conclusion 1.4. Hypothesis H YPOTHESIS Properties of the flow Incompressible flows; Constant fluid properties; Laminar to high Reynolds number ( Re ∼ 10 6 ) flows; REV Velocity no-slip condition at the wall. Properties of the media Stratified (flow along the z -axis); Spatially periodic; The porosity is constant; ⇒ Heat exchanger study is reduced to a unit cell study. Marie D ROUIN (LETR, IMFT) 5 / 19 :
1. Introduction 2. Averaged equation 3. Dispersion coefficients 4. Numerical results 5. Conclusion 2.1. Statistically averaged temperature equation S TATISTICALLY AVERAGED TEMPERATURE EQUATION Microscopic temperature balance equation � � ∂ t + ∂ ( T f u i ) ∂ T f = ∂ ∂ T f + 2 α f Pr ∂ u i ∂ u i Q α f + , ( ρ C p ) f ∂ x i ∂ x i ∂ x i ∂ x j ∂ x j Φ ∂ T f Boundary condition on the wall: α f . n i = ( ρ C p ) f ∂ x i Satistically averaged temperature equation � � Q ∂ T f ∂ t + ∂ = ∂ ∂ T f − ∂ � � u ′ i T ′ ¯ + u i T f α f , f ∂ x i ∂ x i ∂ x i ∂ x i ( ρ C p ) f ���� turbulent heat flux Φ ∂ T f Boundary condition on the wall: α f . n i = ( ρ C p ) f ∂ x i Marie D ROUIN (LETR, IMFT) 6 / 19 :
1. Introduction 2. Averaged equation 3. Dispersion coefficients 4. Numerical results 5. Conclusion 2.1. Statistically averaged temperature equation S TATISTICALLY AVERAGED TEMPERATURE EQUATION Microscopic temperature balance equation � � ∂ t + ∂ ( T f u i ) ∂ T f = ∂ ∂ T f + 2 α f Pr ∂ u i ∂ u i Q α f + , ( ρ C p ) f ∂ x i ∂ x i ∂ x i ∂ x j ∂ x j Φ ∂ T f Boundary condition on the wall: α f . n i = ( ρ C p ) f ∂ x i Satistically averaged temperature equation � � Q ∂ T f ∂ t + ∂ = ∂ ∂ T f − ∂ � � u ′ i T ′ ¯ + u i T f α f , f ∂ x i ∂ x i ∂ x i ∂ x i ( ρ C p ) f ���� turbulent heat flux Φ ∂ T f Boundary condition on the wall: α f . n i = ( ρ C p ) f ∂ x i ∂ T f = ν t ∂ T f where: − u ′ i T ′ f = α t . ∂ x i Pr t ∂ x i Marie D ROUIN (LETR, IMFT) 6 / 19 :
1. Introduction 2. Averaged equation 3. Dispersion coefficients 4. Numerical results 5. Conclusion 2.2. Spatially averaged equation of the temperature S PATIALLY AVERAGED EQUATION OF THE TEMPERATURE Satistically and spatially averaged temperature equation � � ∂ � T f � f ∂ � T f � f + �Q� f ∂ u i � f � T f � f = − ∂ f � f + ∂ � u ′ i T ′ � ¯ + α f ∂ t ∂ x i ∂ x i ∂ x i ∂ x i ( ρ C p ) f � Φ δ ω � f + ∂ − ∂ + � α f δ T f n i δ ω � f � δ ¯ u i δ T f � f ( ρ C p ) f ∂ x i ∂ x i � ����������������� �� ����������������� � � ������������� �� ������������� � � ��� �� ��� � Tortuosity Thermal Wall heat transfer dispersion ∂ � T f � f def where: −� u ′ i T ′ f � f = α t φ . ∂ x i Marie D ROUIN (LETR, IMFT) 7 / 19 :
1. Introduction 2. Averaged equation 3. Dispersion coefficients 4. Numerical results 5. Conclusion 2.2. Spatially averaged equation of the temperature S PATIALLY AVERAGED EQUATION OF THE TEMPERATURE Satistically and spatially averaged temperature equation � � ∂ � T f � f ∂ � T f � f + �Q� f ∂ u i � f � T f � f = − ∂ f � f + ∂ � u ′ i T ′ � ¯ + α f ∂ t ∂ x i ∂ x i ∂ x i ∂ x i ( ρ C p ) f � Φ δ ω � f + ∂ − ∂ + � α f δ T f n i δ ω � f � δ ¯ u i δ T f � f ( ρ C p ) f ∂ x i ∂ x i � ����������������� �� ����������������� � � ������������� �� ������������� � � ��� �� ��� � Tortuosity Thermal Wall heat transfer dispersion ∂ � T f � f def where: −� u ′ i T ′ f � f = α t φ . ∂ x i For flows in flat plates, circular or annular pipes, the tortuosity contributions are zero. We focus on the analysis and modelization of the dispersion term. Marie D ROUIN (LETR, IMFT) 7 / 19 :
1. Introduction 2. Averaged equation 3. Dispersion coefficients 4. Numerical results 5. Conclusion 2.2. Spatially averaged equation of the temperature S PATIALLY AVERAGED EQUATION OF THE TEMPERATURE Satistically and spatially averaged temperature equation � � ∂ � T f � f ∂ � T f � f + �Q� f ∂ u i � f � T f � f = − ∂ f � f + ∂ � u ′ i T ′ � ¯ + α f ∂ t ∂ x i ∂ x i ∂ x i ∂ x i ( ρ C p ) f � Φ δ ω � f + ∂ − ∂ + � α f δ T f n i δ ω � f � δ ¯ u i δ T f � f ( ρ C p ) f ∂ x i ∂ x i � ����������������� �� ����������������� � � ������������� �� ������������� � � ��� �� ��� � Tortuosity Thermal Wall heat transfer dispersion ∂ � T f � f def where: −� u ′ i T ′ f � f = α t φ . ∂ x i For flows in flat plates, circular or annular pipes, the tortuosity contributions are zero. We focus on the analysis and modelization of the dispersion term. Marie D ROUIN (LETR, IMFT) 7 / 19 :
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