FOUNDATIONS OF CONVECTION

The formulation of convection problems, to be outlined by including the effect of fluid motion into conduction, presents no real difficulties. Let us proceed to the formulation of these problems by the help of some conduction problems formulated in the preceding chapters. Recall the original conduction problem, 

Further extension to the unsteady case presents no difficulty but need not be considered here. 

Suppose we now wish to determine the temperature of a fluid in motion from Eq. (5.4). First we need to know the local velocity of the fluid. It can be shown, in a manner similar to the development leading to Eq. (5.4), but starting with Newton s law of motion, 

For a constant-property fluid, Eq. (5.5) is decoupled from Eq. (5.4) and is the only equation needed for velocity. Then, Eq. (5.4) in terms of the velocity obtained from Eq. (5.5) gives the steady temperature of an incompressible fluid in motion. Keep in mind, in addition to fluid temperatures, that convection studies are ultimately and more importantly concerned with heat transfer through a solid-fluid interface. In terms of a heat transfer coefficient h, this heat transfer is 

1 Clearly, qc may also be expressed by means of conduction in the solid, which leads to the definition of the Biot number [recall Eq. (3.1)]. Note the fundamental difference in the use of Eqs, (3.1) and (5.8). In conduction problems, k and Too are given, and Eq. (3.1) is employed as a boundary condition. Because of their complexity, however, convection problems are usually solved in terms of simpler boundary conditions unrelated to h (such as specified temperature or heat flux), and Eq. (5.8) is utilized for the evaluation of h. 

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The temperature dependence of thermal conductivity for liquids, metal alloys, and nonconducting solids is more complicated than those mentioned above. Because of these complexities, the temperature dependence of thermal conductivity for a number of materials, as illustrated in Fig. 1,11, does not show a uniform trend. Typical ranges for the thermal conductivity of these materials are given in Table 1.1, We now proceed to a discussion of the foundations of convective and radiative heat transfer. 

In Chapter 5, we learned the foundations of convection. Integrating the governing equations for laminar boundary layers, we obtained expressions for the heat transfer associated with forced convection over a horizontal plate and natural convection about a vertical plate. We also found analytically, as well as by the analogy between heat and momentum, that the thermal and momentum characteristics of laminar flow over a flat plate are related by... 

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