4. Steady-State Conduction through a Pipe Wall

        where r is radial position, phi is the angle, and z is the position along the length of the cylinder. This             equation is derived in Principles of Heat and Mass Transfer by Incropera et al. 8th edition, but                     derivation of it is not examinable.

Assuming:

• 1. No heat generated, so qgen = 0;

  2. System is at steady state, so dT/dt = 0;

  3. Temperature is independent of angular position and the pipe is of constant length;

The conduction equation can be simplified to:

And simplified again:

Integrating both sides with respect to r results in:

        where c is a constant of integration.

Integrating again with respect to r results in:

        where d is another constant of integration.

The following boundary conditions can be assumed for the case of a pipe:

        where r1 and r2 are the inner and outer radii of the pipe respectively, and Ts1 and Ts2 are the                         temperatures of the inner and outer surfaces of the pipe wall.

The boundary conditions are substituted into the integrated equation, providing the following equations:

Solving this pair of simultaneous equations for c and d results in: