CHAPTER 6 Introduction to convection HEAT TRANSFER CHAPTER 6 Introduction to convection # 1

Boundary Layer Similarity Parameters The boundary layer equations (velocity, mass, energy continuity) represent low speed, forced convection flow. Advection terms on the left side and diffusion terms on the right side of each equation, such as: Advection Diffusion Non-dimensionalize the equations by setting: # 2

Boundary Layer Similarity Parameters (Cont’d) The boundary layer equations can be rewritten in terms of the non-dimensional variables Continuity x-momentum energy With boundary conditions # 3

Boundary Layer Similarity Parameters (Cont’d) From the non-dimensionalized boundary layer equations, dimensionless groups can be seen Reynolds # Prandtl # Substituting gives the boundary layer equations: Continuity: x-momentum: Energy: # 4

Back to the convection heat transfer problem… Solutions to the boundary layer equations are of the form: Rewrite the convective heat transfer coefficient Define the Nusselt number as: # 5

Nusselt number for a prescribed geometry (For a prescribed geometry, is known)  Many convection problems are solved using Nusselt number correlations incorporating Reynolds and Prandtl numbers The Nusselt number is to the thermal boundary layer what the friction coefficient is to the velocity boundary layer. # 6

Heat transfer coefficient, simple example Given: Air at 20ºC flowing over heated flat plate at 100ºC. Experimental measurements of temperatures at various distances from the surface are as shown Find: convective heat transfer coefficient, h # 7

Heat transfer coefficient, simple example Solution: Recall that h is computed by From Table A-4 in Appendix, at a mean fluid temperature (average of free-stream and surface temperatures) the air conductivity, k is  0.028 W/m-K Temperature gradient at the plate surface from experimental data is -66.7 K/mm = -66,700 K/m So, convective heat transfer coefficient is: # 8

Example: Experimental results for heat transfer over a flat plate with an extremely rough surface were found to be correlated by an expression of the form where is the local value of the Nusselt number at a position x measured from the leading edge of the plate. Obtain an expression for the ratio of the average heat transfer coefficient to the local coefficient. # 9

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SUMMARY General boundary layer equations Nusselt number for heat transfer coefficient in the thermal boundary layer Many convection problems are solved using Nusselt number correlations incorporating Reynolds and Prandtl numbers. # 11

Momentum and Heat Transfer Analogy # 12

Momentum and Heat Transfer Analogy Where we’ve been …… Development of convective transport of heat transfer equations. Where we’re going: Momentum and heat transfer (Reynolds) analogy. # 13

Momentum and Heat Transfer Analogy KEY POINTS THIS SECTION Physical significance of the dimensionless parameters (Reynolds, Prandtl numbers) Describe how the convective heat transfer equations can be related (Heat Transfer Analogy) Review of the general convection equations, prepare for application to external and internal flows. # 14

Review boundary layer equations for heat transfer For heat transfer (conservation of energy) usually small, except for high speed or highly viscous flow Momentum equation Begin to see where the momentum and heat transfer analogy is going ….. # 15

These two equations are of precisely the same form. If and , we obtain: These two equations are of precisely the same form. We know that if , . The boundary conditions for these two equations are: The boundary conditions are equivalent. Therefore, the boundary layer velocity and temperature profiles must be of the same functional form. # 16

Momentum and Heat Transfer Analogy (continued) For the solution, the function f must be the same. As we know, We conclude that # 17

Momentum and Heat Transfer Analogy (continued) Define the Stanton number St, The analogy takes the form The restrictions: the validity of the boundary layer approximations, and . The modified Reynolds, or Chilton-Colburn, analogy has the form Reynolds analogy Colburn j factor For laminar flow, it’s only appropriate when # 18

Boundary Layer Similarity Parameters Recall the non-dimensional parameters Reynolds # - Ratio of the inertia to the viscous forces of a fluid flow Prandtl # Ratio of the momentum to the thermal diffusivity in a fluid flow. For laminar boundary layers, Where n is a positive exponent. # 19

Example: Forced air at ℃ and is used to cool electronic elements on a circuit board. One such element is a chip, 4 mm by 4 mm, located 120 mm from the leading edge of the board. Experiments have revealed that flow over the board is disturbed by the elements and that convection heat transfer is correlated by an expression of the form Estimate the surface temperature of the chip if it is dissipating 30 mW. # 20

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Convective Transport Equations Summary General boundary layer equations Nusselt number for heat transfer coefficient in the thermal boundary layer Empirical evaluation of Nusselt number involves correlations incorporating Re and Pr Local heat flux is: where h is the local heat transfer coefficient # 22

Convective Transport Equations Summary (Cont’d) Reynolds # Prandtl # Reynolds analogy # 23

Nothing Is Impossible To A Willing Heart # 24