Heat Transfer Su Yongkang School of Mechanical Engineering # 1 HEAT TRANSFER CHAPTER 6 Introduction to convection

Heat Transfer Su Yongkang School of Mechanical Engineering # 2 CH6 – INTRODUCTION Where we’ve been …… Basic Conduction Heat Transfer Finished Fourier’s law: Where we’re going: Begin study of convective heat transfer. Newton’s law of cooling:

Heat Transfer Su Yongkang School of Mechanical Engineering # 3 Convective transfer problem

Heat Transfer Su Yongkang School of Mechanical Engineering # 4 CH6 INTRODUCTION KEY POINTS THIS CHAPTER What are the key variables when analyzing convection heat transfer? Review boundary layer concept and significance General idea of relationship between velocity and thermal profiles in a boundary layer. Effect of laminar versus turbulent flow on heat transfer potential Boundary layer similarity This chapter will be taught in two lectures: the first includes text book sections §6.1 to 6.4 the other includes text book sections §6.5 to 6.10

Heat Transfer Su Yongkang School of Mechanical Engineering # 5 Convection overview Consider a flat plate of length L, in air flow with velocity u  and temperature T  Local heat flux is: where h is the local heat transfer coefficient Total heat transfer rate: average heat transfer coefficient Determination of ‘h’ will rely on analytical as well as empirical data

Heat Transfer Su Yongkang School of Mechanical Engineering # 6 Convection overview (Cont’d) Same principal applies to any arbitrary shape, not just a flat plate Average convection heat transfer coefficient: So, we need to know how h varies with x, the distance from the leading edge…….. What do you think key parameters that might influence h?

Heat Transfer Su Yongkang School of Mechanical Engineering # 7 Key parameters Transfer potential: forced flow or free flow Phase change: boiling and condensation Flow conditions: laminar or turbulent flow Geometries: shape, size, position and roughness. Properties: density, viscosity, thermal conductivity, specific heat, and so on.

Heat Transfer Su Yongkang School of Mechanical Engineering # 8 Example Given Experimental results for measured local heat transfer coefficient h for flow over a flat plate with a rough surface where: a = coefficient x = distance from leading edge –Find expression for average heat transfer coefficient, and the relation of average heat transfer coefficient to the local coefficient

Heat Transfer Su Yongkang School of Mechanical Engineering # 9 The Convection Boundary Layers Velocity Boundary Layer For fluid flow over a flat plate, which disturbs the fluid flow: –As y  : where u is velocity in x-direction –As y  0: (no-slip condition) –The boundary layer thickness is defined as the value at which: –The boundary layer thickness  varies with x Shear Stress Local friction coefficient Dynamic viscosity

Heat Transfer Su Yongkang School of Mechanical Engineering # 10 The Convection Boundary Layers Thermal Boundary Layer A hot or cold plate alters the temperature distribution in the air –As y  : –As y  0: –The thermal boundary layer thickness is defined as the value at which: –The thermal boundary layer thickness,  t also varies (increases) with x

Heat Transfer Su Yongkang School of Mechanical Engineering # 11 The Convection Boundary Layers Thermal Boundary Layer (Cont’d) Heat Flux Heat flux analogous to shear stress in velocity boundary layer Heat flux proportional to the temperature gradient at the surface, AND since u(y=0) =0, energy transfer to/from fluid occurs by conduction only! Since thermal boundary layer gets larger along x direction, the temperature gradient changes with x, and therefore fluid thermal conductivity wall temperature gradient ____________________

Heat Transfer Su Yongkang School of Mechanical Engineering # 12 The Convection Boundary Layers Thermal Boundary Layer (Cont’d) Heat Flux (Cont’d) Using Newton’s law of cooling: We obtain While  increases with increasing x, temperature gradients in the boundary layer must decrease with increasing x. Accordingly, and h decrease with increasing x.

Heat Transfer Su Yongkang School of Mechanical Engineering # 13 The Convection Boundary Layers Laminar Versus Turbulent Flow Characterization of laminar flow Low amount of “mixing” of fluid within the boundary layer (smooth flow) Characterization of turbulent flow High amount of mixing of fluid within the boundary layer (irregular flow) High amount of mixing means increased surface friction as well as convection transfer rates (heat and mass)

Heat Transfer Su Yongkang School of Mechanical Engineering # 14 Convection Heat Transfer Variations Along Flat Plate Consider flat plate with: and all laminar flow Thermal boundary layer defined by Compare temperature gradients at points 1 and 2 to evaluate the heat flux rate (and hence the heat transfer coefficient) 12

Heat Transfer Su Yongkang School of Mechanical Engineering # 15 Convection Heat Transfer Variations Along Flat Plate (Cont’d) Consider flat plate with: at x 1 at x 2 To determine the location that transition begins, we define the Reynolds number, And the critical Reynolds number,

Heat Transfer Su Yongkang School of Mechanical Engineering # 16 Convective Transfer Equations Topic of the Day Project Teams

Heat Transfer Su Yongkang School of Mechanical Engineering # 17 Convective Transport Equations Where we’ve been …… Last section: Overview of the topic of convective transport of heat. Where we’re going: Convection transfer detailed equations Heat and mass transfer analogy Eventually get to applications in external and internal flow.

Heat Transfer Su Yongkang School of Mechanical Engineering # 18 Convective Transport Equations KEY POINTS THIS SECTION Detailed development of boundary layer equations for velocity, temperature and species concentrations What approximations can be made in the boundary layers?

Heat Transfer Su Yongkang School of Mechanical Engineering # 19 Recall the convection overview Local heat flux is: where h is the local heat transfer coefficient

Heat Transfer Su Yongkang School of Mechanical Engineering # 20 Develop convection transfer equations Consider steady, 2-D flow of a viscous, incompressible fluid with constant properties. Key point to remember: At each point in the fluid, conservation of mass, energy and momentum must be satisfied. Appendix E contains detailed development of the full boundary layer equations, for example: Conservation of mass (continuity)

Heat Transfer Su Yongkang School of Mechanical Engineering # 21 The magnitude of variables in the thermal boundary layer variables x (main flow direction) yuvt magnitude 111 Thermal boundary layer Velocity boundary layer Boundary Layer Approximations

Heat Transfer Su Yongkang School of Mechanical Engineering # 22 Continue convection transfer equations Quick overview of fluid equations Conservation of mass (continuity): OR x-momentum equation: y-momentum equation: So:

Heat Transfer Su Yongkang School of Mechanical Engineering # 23 Continue convection transfer equations Energy equation: NOTE: is usually small unless u is high (as in sonic flows) or is high (such as flow of oils). Result is 4 equations and 4 unknowns: Unknowns are: u, v, P, and T Since: P(x) can be obtained from free stream flow.

Heat Transfer Su Yongkang School of Mechanical Engineering # 24 Review boundary layer concept. General relationship between velocity and thermal boundary layers. Convective heat transfer is dependent on the temperature gradient at the fluid/solid interface Boundary layer grows with distance from the leading edge, and this decreases the local heat transfer rates. Turbulent boundary layers have much greater potential for heat and mass transfer due to velocity fluctuations. Convection transfer equations (4). Go back and review fluids course notes! SUMMARY