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### meeg342_lecture16_2012

Course: MEEG 342, Spring 2009
School: Maryland
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Word Count: 847

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Convection: General 4/6/2012 Free Considerations and Results for Vertical and Horizontal Plates Chapter 9 Sections 9.1 through 9.6.2, 9.9 General Considerations Lecture 16-2 General Considerations Free convection refers to fluid motion induced by buoyancy forces. Buoyancy forces may arise in a fluid for which there are density gradients and a body force that is proportional to density. In heat transfer,...

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Convection: General 4/6/2012 Free Considerations and Results for Vertical and Horizontal Plates Chapter 9 Sections 9.1 through 9.6.2, 9.9 General Considerations Lecture 16-2 General Considerations Free convection refers to fluid motion induced by buoyancy forces. Buoyancy forces may arise in a fluid for which there are density gradients and a body force that is proportional to density. In heat transfer, density gradients are due to temperature gradients and the body force is gravitational. Stable and Unstable Temperature Gradients 1 4/6/2012 General Considerations (cont) Lecture 16-3 Free Boundary Flows Occur in an extensive (in principle, infinite), quiescent (motionless at locations far from the source of buoyancy) fluid. Plumes and Buoyant Jets: Free Convection Boundary Layers Boundary layer flow on a heated or cooled surface Ts T induced by buoyancy forces. General Considerations (cont) Lecture 16-4 Pertinent Dimensionless Parameters Grashof Number: GrL g Ts T L3 Buoyancy Force Viscous Force 2 L characteristic length of surface thermal expansion coefficient (a thermodynamic property of the fluid) 1 T p Liquids: Tables A.5, A.6 Perfect Gas: =1/T K Rayleigh Number: RaL GrL Pr g Ts T L3 2 4/6/2012 Lecture 16-5 "Heat and Mass Transfer: A Practical Approach, 3/e" By Yunus A. engel The flow regime on a vertical plate becomes turbulent at Grashof numbers greater than 109 IF Natural convection is small volume expansion coefficient Lecture 16-6 "Heat and Mass Transfer: A Practical Approach, 3/e" By Yunus A. engel 3 4/6/2012 General Considerations (cont) Lecture 16-7 Mixed Convection A condition for which forced and free convection effects are comparable. Exists if Gr L / Re 2 o 1 L - Free convection GrL / Re 2 1 L - Forced convection GrL / Re2 1 L Heat Transfer Correlations for Mixed Convection: n n Nu n Nu FC Nu NC assisting and transverse flows - opposing flows n3 Vertical Plates Lecture 16-8 Vertical Plates Free Convection Boundary Layer Development on a Heated Plate: Ascending flow with the maximum velocity occurring in the boundary layer and zero velocity at both the surface and outer edge. How do conditions differ from those associated with forced convection? How do conditions differ for a cooled plate Ts T ? 4 4/6/2012 Vertical Plates (cont) Lecture 16-9 Form of the x-Momentum Equation for Laminar Flow 2 u u u g T T u x y y 2 Net Momentum Fluxes Buoyancy Force ( Inertia Forces) Viscous Force Temperature dependence requires that solution for u (x,y) be obtained concurrently with solution of the boundary layer energy equation for T (x,y). 2 u T T T x y y 2 The solutions are said to be coupled. Vertical Plates (cont) Lecture 16-10 Nusselt Numbers Nu x and Nu L : 1/ 4 Gr Nu x hx x k 4 g Pr dT d 1/ 4 0 Gr x 4 g Pr 0.75 Pr1/ 0.609 2 1.221 Pr 1/ 2 1.238 Pr 1/ 4 0 Pr L h 1 o h dx Nu L 4 NuL L 3 Transition to Turbulence Amplification of disturbances depends on relative magnitudes of buoyancy and viscous forces. Transition occurs at a critical Rayleigh Number. Rax , c Grx , c Pr g Ts T x3 109 5 4/6/2012 Lecture 16-11 Horizontal Plates Horizontal Plates Buoyancy force is normal, instead of parallel, to the plate. Flow and heat transfer depend on whether the plate is heated or cooled and whether it is facing upward or downward. Hot Surface Facing Upward or Cold Surface Facing Downward Ts T Ts T Nu L 0.54 Ra1/ 4 L Nu L 0.15 Ra1/ 3 L 10 10 4 RaL 107 7 RaL 1011 How does h depend on L when Nu L Ra1 / 3 ? L RaL GrL Pr g Ts T L3 Lecture 16-12 Hot Surface Facing Downward or Cold Surface Facing Upward Horizontal Plates (cont) Ts T Nu L 0.27 Ra1/ 4 L Ts T 10 5 RaL 1010 Why do these flow conditions yield smaller heat transfer rates than those for a heated upper surface or cooled lower surface? RaL GrL Pr g Ts T L3 6 4/6/2012 RaL GrL Pr g Ts T L3 Lecture 16-13 Lecture 16-14 Problem 9.18 Problem 9.18 Why is the frost at the bottom ? 7 4/6/2012 Lecture 16-15 "Heat and Mass Transfer: A Practical Approach, 3/e" By Yunus A. engel Lecture 16-16 8 4/6/2012 Lecture 16-17 Lecture 16-18 9 4/6/2012 Lecture 16-19 Cylinders Lecture 16-20 The Long Horizontal Cylinder Boundary Layer Development and Variation of the Local Nusselt Number for a Heated Cylinder: The Average Nusselt Number: 0.387 Ra1/ 6 D Nu D 0.60 9 /16 8 / 27 1 0.559 / Pr 2 RaD 1012 How do conditions change for a cooled cylinder? 10 4/6/2012 Lecture 16-21 Spheres Spheres The Average Nusselt Number: Nu D 2 0.589 Ra1/ 4 D 1 0.469 / Pr 9 /16 4/9 Enclosures Lecture 16-22 Enclosures Rectangular Cavities Characterized by opposing walls of different temperatures, with the remaining walls well insulated. RaL g T1 T2 L3 q h T1 T2 Horizontal Cavity 0, 180deg Vertical Cavity 90 deg 11 4/6/2012 Enclosures (cont) Lecture 16-23 Horizontal Cavities Heating from Below 0 RaL RaL , c 1708 : Fluid layer is thermally stable. Nu L hL 1 k 1708 Ra L 5 104 : Thermal instability yields a regular convection pattern in the form of roll cells. 3 105 RaL 7 109 : Buoyancy driven flow is turbulent Nu L 0.069 Ra1/ 3 Pr 0.074 L Enclosures (cont) Lecture 16-24 Heating from Above 180 deg Fluid layer is unconditionally stable. Nu L 1 Vertical Cavities Nu L hL 1 k Heated from below: For Ra > 1708 For Ra <1708, Nu L 0.069 Ra1/ 3 Pr 0.074 L 3 RaL 10 : Nu L 1 RaL 103 : A primary cellular flow is established, as the core becomes progressively more quiescent, and secondary (corner) cells develop with increasing RaL . Correlations for Nu L Eqs. (9.50) - (9.53). 12
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Maryland - MEEG - 342
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UC Davis - ECH - 51
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UC Davis - ECH - 51
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UC Davis - ECH - 51
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UC Davis - ECH - 51
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UC Davis - ECH - 51
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UC Davis - ECH - 51
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