Free Convection: General Considerations and Results for Vertical and Horizontal Plates Chapter 9 Sections 9.1 through 9.6.2, 9.9 General Considerations General Considerations вЂў Free convection refers to fluid motion induced by buoyancy forces. вЂў Buoyancy forces may arise in a fluid for which there are density gradients and a body force that is proportional to density. вЂў In heat transfer, density gradients are due to temperature gradients and the body force is gravitational. вЂў Stable and Unstable Temperature Gradients General Considerations (cont) вЂў Free Boundary Flows пѓ� Occur in an extensive (in principle, infinite), quiescent (motionless at locations far from the source of buoyancy) fluid. пѓ� Plumes and Buoyant Jets: вЂў Free Convection Boundary Layers пѓ� Boundary layer flow on a heated or cooled surface пЂЁ T s п‚№ Tп‚Ґ пЂ© induced by buoyancy forces. General Considerations (cont) вЂў Pertinent Dimensionless Parameters пѓ� Grashof Number: g пЃў пЂЁ T s пЂ Tп‚Ґ пЂ© L 3 G rL пЂЅ пЃ® B u o yan cy F o rce 2 V isco u s F o rce L п‚® characteristic length of surface пЃў п‚® thermal expansion coeffi cien t (a thermo dynamic property of the flui d) пѓ¦ п‚¶пЃІ пѓ¶ пЃў пЂЅпЂ1пѓ§ пѓ· пЃІ пѓЁ п‚¶T пѓё p Liquids: пЃў п‚® T ables A.5, A.6 P erfect G as: пЃў =1/ T пЂЁ K пЂ© пѓ� Rayleigh Number: g пЃў пЂЁ T s пЂ Tп‚Ґ пЂ© L 3 R a L пЂЅ G rL P r пЂЅ пЃ®пЃЎ General Considerations (cont) вЂў Mixed Convection пѓ� A condition for which forced and free convection effects are comparable. пѓ� Exists if пЂЁGr L / ReL пЂ© 2 0 пЂЁ1 пЂ© - Free convection п‚® пЂЁ G rL / R e L пЂ© 1 2 - Forced convection п‚® пЂЁ G rL / R e L пЂ© 1 2 пѓ� Heat Transfer Correlations for Mixed Convection: N u п‚» N u FC п‚± N u NC n n n пЂ« п‚® assisting and transverse flow s - п‚® opposing flow s nп‚»3 Vertical Plates Vertical Plates вЂў Free Convection Boundary Layer Development on a Heated Plate: пѓ� Ascending flow with the maximum velocity occurring in the boundary layer and zero velocity at both the surface and outer edge. пѓ� How do conditions differ from those associated with forced convection? пѓ� How do conditions differ for a cooled plate пЂЁ T s пЂј Tп‚Ґ пЂ© ? Vertical Plates (cont) вЂў Form of the x-Momentum Equation for Laminar Flow п‚¶u п‚¶u п‚¶ u пЂ«пЃµ пЂЅ g пЃў пЂЁ T пЂ Tп‚Ґ пЂ© пЂ« пЃ® 2 п‚¶x п‚¶y п‚¶y 2 u Net Momentum Fluxes Buoyancy Force ( Inertia Forces) Viscous Force пѓ� Temperature dependence requires that solution for u (x,y) be obtained concurrently with solution of the boundary layer energy equation for T (x,y). п‚¶T п‚¶T п‚¶ T u пЂ«пЃµ пЂЅпЃЎ 2 п‚¶x п‚¶y п‚¶y 2 вЂ“ The solutions are said to be coupled. Vertical Plates (cont) вЂў Similarity Solution пѓ� Based on existence of a similarity variable, пЃЁ through which the x-momentum equation may be transformed from a partial differential equation with twoindependent variables ( x and y) to an ordinary differential equation expressed exclusively in terms of пЃЁ . y пѓ¦ Gr пѓ¶ пЃЁ п‚є пѓ§ xпѓ· xпѓЁ 4 пѓё 1/ 4 пѓ� Transformed momentum and energy equations: пЂЄ f п‚ўп‚ўп‚ў пЂ« 3 ff п‚ўп‚ў пЂ 2 пЂЁ f п‚ў пЂ© пЂ« T пЂЅ 0 2 T пЂЄ п‚ўп‚ў пЂ« 3 P r fT f п‚ў пЂЁпЃЁ пЂ© п‚є df dпЃЁ пЂЄп‚ў пЂЅ пЂЅ0 пЂ1 / 2 x G rx пЂЁ пЂ©u 2пЃ® T пЂЄ п‚є T пЂ Tп‚Ґ T s пЂ Tп‚Ґ Vertical Plates (cont) пѓ� Numerical integration of the equations yields the following results for f п‚ў пЂЁпЃЁ пЂ© and T : пЂЄ пѓ� Velocity boundary layer thickness пЂЁ пЃ¤ пЂ© п‚® пЃЁ п‚» 5 for P r пЂѕ 0.6 пѓ� пѓ¦ G rx пѓ¶ P r пЂѕ 0.6 : пЃ¤ пЂЅ 5 x пѓ§ пѓ· пѓЁ 4 пѓё пЂ1 / 4 пЂЅ 7.07 x пЂЁ G rx пЂ© 1/ 4 п‚µ x 1/ 4 Vertical Plates (cont) пѓ� Nusselt Numbers пЂЁ N u x an d N u L пЂ© : пѓ¦ G rx пѓ¶ hx Nux пЂЅ пЂЅ пЂпѓ§ пѓ· k пѓЁ 4 пѓё g пЂЁ Pr пЂ© пЂЅ hпЂЅ 1/ 4 пЂЄ dT dпЃЁ пЃЁ пЂЅ0 0.75 P r пЂЁ 0.609 пЂ« 1.221 P r пѓ¦ G rx пѓ¶ пЂЅпѓ§ пѓ· пѓЁ 4 пѓё 1/ 2 пЂ« 1.238 P r пЂ© вЂў Transition to Turbulence пѓ� Amplification of disturbances depends on relative magnitudes of buoyancy and viscous forces. пѓ� Transition occurs at a critical Rayleigh Number. g пЃў пЂЁ T s пЂ Tп‚Ґ пЂ© x пЃ®пЃЎ g пЂЁ Pr пЂ© 1/ 2 1 L 4 пѓІo h dx п‚® N u L пЂЅ N u L L 3 R a x , c пЂЅ G rx , c P r пЂЅ 1/ 4 3 п‚» 10 9 1/ 4 пЂЁ0 пЂј Pr пЂј п‚Ґ пЂ© Vertical Plates (cont) вЂў Empirical Heat Transfer Correlations пѓ� Laminar Flow пЂЁ R a L пЂј 10 9 пЂ© : 1/ 4 N u L пЂЅ 0.68 пЂ« 0.670 R a L пѓ©1 пЂ« пЂЁ 0.492 / P r пЂ© 9 / 16 пѓ№ пѓ« пѓ» 4/9 пѓ� All Conditions: Nu L пѓ¬ пѓј 1/ 6 0 .3 8 7 R a L пѓЇ пѓЇ пЂЅ пѓ 0 .8 2 5 пЂ« 4/9 пѓЅ 9 / 16 пѓ©1 пЂ« пЂЁ 0 .4 9 2 / P r пЂ© пѓ№ пѓЇ пѓЇ пѓ« пѓ» пѓѕ пѓ® 2 Horizontal Plates Horizontal Plates вЂў Buoyancy force is normal, instead of parallel, to the plate. вЂў Flow and heat transfer depend on whether the plate is heated or cooled and whether it is facing upward or downward. вЂў Heated Surface Facing Upward or Cooled Surface Facing Downward T s пЂѕ Tп‚Ґ T s пЂј Tп‚Ґ N u L пЂЅ 0.54 R a L пЂЁ10 4 пЂј R a L пЂј 10 7 N u L пЂЅ 0.15 R a L пЂЁ10 7 пЂј R a L пЂј 10 11 1/ 4 1/ 3 How does h depend on L when N u L п‚µ R a 1L / 3 ? пЂ© пЂ© Horizontal Plates (cont) вЂў Heated Surface Facing Downward or Cooled Surface Facing Upward T s пЂѕ Tп‚Ґ T s пЂј Tп‚Ґ N u L пЂЅ 0.27 R a L 1/ 4 пЂЁ10 5 пЂј R a L пЂј 10 10 пЂ© пѓ� Why do these flow conditions yield smaller heat transfer rates than those for a heated upper surface or cooled lower surface? Problem: Solar Receiver Problem 9.31: Convection and radiation losses from the surface of a central solar receiver. q conv Too = 300 K q S = 10 5 W /m 2 q rad KNOWN: Dimensions and emissivity of cylindrical solar receiver. Incident solar flux. Temperature of ambient air. FIND: (a) Heat loss and collection efficiency for a prescribed receiver temperature, (b) Effect of receiver temperature on heat losses and collector efficiency. ASSUMPTIONS: (1) Steady-state, (2) Ambient air is quiescent, (3) Incident solar flux is uniformly distributed over receiver surface, (4) All of the incident solar flux is absorbed by the receiver, (5) Negligible irradiation from the surroundings, (6) Uniform receiver surface temperature, (7) Curvature of cylinder has a negligible effect on boundary layer development, (8) Constant properties Problem: Solar Receiver (cont) PROPERTIES: Table A-4, air (Tf = 550 K): k = 0.0439 W/mпѓ—K, пЃ® = 45.6 п‚ґ 10-6 m2/s, пЃЎ = 66.7 п‚ґ 10-6 m2/s, Pr = 0.683, пЃў = 1.82 п‚ґ 10-3 K-1. ANALYSIS: (a) The total heat loss is q пЂЅ q rad пЂ« q co n v пЂЅ A s пЃҐпЃі Ts пЂ« h A s пЂЁ Ts пЂ Tп‚Ґ 4 пЂ© With RaL = gпЃў (Ts - Tп‚Ґ)L3/пЃ®пЃЎ = 9.8 m/s2 (1.82 п‚ґ 10-3 K-1) 500K (12m)3/(45.6 п‚ґ 66.7 п‚ґ 10-12 m4/s2) = 5.07 п‚ґ 1012, the Churchill and Chu correlation yields пѓ¬ k пѓЇ пѓЇ h пЂЅ пѓ 0.825 пЂ« L пѓЇ пѓЇпѓ® 2 пѓј 1/ 6 пѓЇпѓЇ 0.387 R a L 0.0439 W / m пѓ— K 2 пЂЅ пЃ» 0.825 пЂ« 42.4 пЃЅ пЂЅ 6.83 W / m 2 пѓ— K пѓЅ 8 / 27 12 m пѓЇ пѓ©1 пЂ« пЂЁ 0.492 / P r пЂ© 9 / 16 пѓ№ пѓЄпѓ« пѓєпѓ» пѓЇпѓѕ Hence, with As = пЃ°DL = 264 m2 2 q пЂЅ 264 m п‚ґ 0.2 п‚ґ 5.67 п‚ґ 10 пЂ8 6 2 W / m пѓ—K 4 пЂЁ 800 K пЂ© 5 4 пЂ« 264 m п‚ґ 6.83 W / m пѓ— K пЂЁ 500 K пЂ© 2 6 q пЂЅ q rad пЂ« q conv пЂЅ 1.23 п‚ґ 10 W пЂ« 9.01 п‚ґ 10 W пЂЅ 2.13 п‚ґ 10 W 2 Problem: Solar Receiver (cont) With 7 A s q п‚ўп‚ўs пЂЅ 2.64 п‚ґ 10 W , пѓ¦ A q п‚ўп‚ў пЂ q пЃЁ пЂЅпѓ§ s s пѓЁ A s q п‚ўп‚ўs the collector efficiency is пЂЁ 7 2.64 п‚ґ 10 пЂ 2.13 п‚ґ 10 пѓ¶ пѓ· 100 пЂЅ 7 2.64 п‚ґ 10 W пѓё 6 пЂ© W пЂЁ100 пЂ© пЂЅ 91.9% 5E6 100 4E6 95 Collector efficiency, % Heat rate, W (b) As shown below, because of its dependence on temperature to the fourth power, q rad increases more significantly with increasing T s than does qconv, and the effect on the efficiency is pronounced 3E6 2E6 1E6 0 600 700 800 900 Receiver temperature, K Convection Radiation Total 90 85 80 1000 75 600 700 800 900 1000 Receiver temperature, K COMMENTS: The collector efficiency is also reduced by the inability to have a perfectly absorbing receiver. Partial reflection of the incident solar flux will reduce the efficiency by at least several percent.