1. Free Convection: General Considerations and Results for Vertical and Horizontal Plates 1

2. General Considerations Free convection refers to fluid motion induced by buoyancy forces. In heat transfer, density gradients are due to temperature gradients and the body force is gravitational. Stable and Unstable Temperature Gradients 2 General Considerations Buoyancy forces may arise in a fluid for which there are density gradients and a body force that is proportional to density.

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 induced by buoyancy forces. 3

4. Pertinent Dimensionless Parameters  Grashof Number:  Rayleigh Number : 4

5. Mixed Convection  A condition for which forced and free convection effects are comparable.  Exists if  Heat Transfer Correlations for Mixed Convection: 5

6. 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 6 STEADY STATE

7. Form of the x-Momentum Equation for Laminar Flow (constant properties) Net Momentum Fluxes ( Inertia Forces) Buoyancy ForceViscous 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). – The solutions are said to be coupled. 7 Form of the Continuity Equation (  =cte)

8. 7C Non-dimensional equations

9.  Based on existence of a similarity variable, through which the x-momentum equation may be transformed from a partial differential equation with two- independent variables ( x and y) to an ordinary differential equation expressed exclusively in terms of.  Transformed momentum and energy equations: 8 Similarity Solution

10.  Numerical integration of the equations yields the following results for  Velocity boundary layer thickness  9

11.  Nusselt Numbers Transition to Turbulence  Amplification of disturbances depends on relative magnitudes of buoyancy and viscous forces.  Transition occurs at a critical Rayleigh Number. 10

12. Empirical Heat Transfer Correlations  Laminar Flow  All Conditions: 11 CHURCHILL E CHU  What if the vertical plate is subjected to a constant heat flux (temperature is varying)? In this case, the same correlations are applied, but the non-dimensional parameters (Nusselt and Rayleigh numbers) are defined in terms of the temperature difference at the midpoint of the plate:

13. 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. Heated Surface Facing Upward or Cooled Surface Facing Downward How does depend on L when 12 IT IS INDEPENDENT OF L !

14. Heated Surface Facing Downward or Cooled Surface Facing Upward  Why do these flow conditions yield smaller heat transfer rates than those for a heated upper surface or cooled lower surface? 13 BECAUSE THE PLATE IMPEDES THE ASCENDING/DESCENDING NATURAL CONVECTION FLOW THAT HAS TO MOVE HORIZONTALLY. THIS MAKES CONVECTION HEAT TRANSFER INEFFECTIVE.

15. x- and y-Momentum Equations for Laminar Flow (constant properties) 7A Continuity Equation  (  = constant) 

16. 7B Energy Equation for Laminar Flow (constant properties)

17. 8A BOUNDARY CONDITIONS

18. 9A

19. 10A Nusselt number