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EX4: Non-Constant Conductivity II

6: Non-Constant Conductivity II (Example 4)

Consider again radial flow in the cylindrical geometry studied in .

There the conductivity was taken to be constant. Now we make the more realistic assumption that conductivity varies with temperature. Specifically let



Equation 75

as in the previous discussion. [Note that the parameter in equation (5.1) is replaced here by .]

A cylindrical shell of infinite length and circular cross-section is under consideration, with inner radius and outer radius . The inner and outer cylindrical surfaces are assumed to have steady uniform temperatures and , respectively.

We wish to find how the steady state temperature varies within the shell. In practice it is important also to determine the rate at which heat flows from one boundary surface to the other. As in previous examples we shall formulate and solve an appropriate differential equation subject to the boundary conditions.

As in example 2, heat flows radially, at a steady rate (since the boundary temperatures, and , are unchanging). If , say, heat flows outward.

We consider a unit (one - meter) length of the infinite cylinder. Within this tube let denote a cylindrical surface of radius . At all points of the temperature is the same; thus depends only on : . Similarly , the rate of heat flow in watts per square meter (directed radially, perpendicular to ) is uniform over . The area of is , so heat flows through at the rate of watts.

Because this is a steady state process, the rate of heat flow through , has the same value for all (). Let denote this value. Thus



Equation 76

is a constant, independent of .

 
 
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