Steady, one-dimensional conduction of heat

As a simple, but practically important application, the conduction of heat independent of time, so called steady conduction, in a flat plate, in a hollow cylinder and in a hollow sphere will be considered in this section. The assumption is made that heat flows in only one direction, perpendicular to the plate surface, and radially in the cylinder and sphere, Fig. 1.3. The temperature field is then only dependent on one geometrical coordinate. This is known as one-dimensional heat conduction. 

The position coordinate in all three cases is designated by r. The surfaces r = const are isothermal surfaces and therefore 1 = d(r). We assume that d has the constant values d = dwl, when r = r1 and d = dW2, when r = r2. These two surface temperatures shall be given. A relationship between the heat flow Q through the flat or curved walls, and the temperature difference dwl — dW2, must be found. For illustration we assume dwl dW2 without loss of generality. Therefore heat flows in the direction of increasing r. The heat flow Q has a certain value, which on the inner and outer surfaces, and on each isotherm r = const is the same, as in steady conditions no energy can be stored in the wall. 

In the flat wall A is not dependent on r A = Ax= A2. If the thermal conductivity is constant, then the temperature gradient dd/dr will also be constant. The steady 

From the mean value theorem for integration comes 

The heat flow is directly proportional to the difference in temperature between the two surfaces. The driving force of temperature difference is analogous to the potential difference (voltage) in an electric circuit and so XmAm/5 is the thermal conductance and its inverse 

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