Original Problem of Conduction

Consider a flat plate of thickness l whose surfaces are kept at temperatures T and T. We wish to find the steady heat transfer through this plate. 

Under steady conditions, and in the absence of any mass flow and power input, the first law stated by Eq. (1.16) reduces to 

Interpretation of this result for the differential system shown in Fig. 1.7(a), gives 

That is, the heat flux is constant at any cross section of the plate. However, for the size of a heat transfer device, say for a heater providing this flux through the walls of a room to be heated, we need the specific value of this constant. Thermodynamics is silent to this need, The attempt to find an answer for this need is the origin of (conduction) heat transfer. Since the statement of our example specifies the temperatures of two surfaces, we need a relation between heat flow and temperature, Q = f(T), which is phenomenologically provided by heat transfer. Observations show that any relation of this nature is dependent on the medium it applies to and, consequently, is a particular law. The remainder of this section is devoted to particular laws of heat transfer. We begin with the particular law associated with our illustrative example. 

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As an illustration for the use of these steps, let us reconsider the original problem of conduction. This time, however, we wish to know the temperature distribution in, as well as the heat transfer through, the plate. 

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