4. Steady-State Conduction through a Pipe Wall
Firstly, it is useful to remember (from the mathematics module MAS261) that cylindrical coordinates define a position in 3D by the radius, angle and height (if vertical, or length if horizontal) of the position.
Finding the temperature in a pipe wall of constant length as a function of radial position:
The conduction equation for cylindrical coordinates is given by:
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:
No heat generated, so qgen = 0;
System is at steady state, so dT/dt = 0;
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:
Substituting c and d into the integrated form of the conduction equation and simplifying results in the following equation, which shows temperature as a function of the radial position in the pipe wall:
Finding the thermal resistance of the pipe wall:
By including the area of heat transfer through the pipe wall, the rate of heat transfer can be found:
The thermal conduction resistance (the denominator of the above equation) can therefore be found:
Combining Newton's law of cooling for the fluids on either side of the pipe wall with this knowledge of the thermal conductive resistance of the pipe wall allows analysis of the following general scenario where a fluid is flowing through a pipe:
