Conduction Shape Factors

Prequisite Knowledge
Heat Diffusion Equation

Problem
Pressurized steam at 400 K flows through a long, thin-walled pipe of 0.4-m diameter. The pipe is enclosed in a concrete casing that is of square cross section and 1.2 m on a side. The axis of the pipe is centered in the casing, and the outer surfaces of the casing are maintained at 300 K. What is the heat loss per unit length of the pipe?

Cross Section View

T1 is at pipe wall
T2 is at outer surface of concrete casing


Solution
Heat loss per unit length would be in terms of W/m.

Assumptions:
• steady-state conditions
• negligible steam side convection resistance, pipe wall resistance and contact resistance (T1 = 400 K)
• constant properties

The thermal conductivity for concrete must be found.
Referring to the table of thermal conductivity of some common materials from Engineering ToolBox, there are two values: one for light concrete, another for stone concrete. The value for stone concrete will be used, which is 1.7.

The heat rate can be expressed as:

q = SkΔT1−2 = Sk(T1 − T2)

The proper shape factor is needed for the solution and is found from the list of shape factors. The shape factor must be for a circular cylinder of length L centered in a square solid of equal length and is:

S = 
2πL
ln (1.08w/D)

Then,

q' = 
q
L

The full equation is:

q' = 
2πk(T1 − T2)
ln (1.08w/D)

Substituting known values:

q' = 
2π(1.7 W/mK)*(400 K − 300 K)
ln [1.08(1.2 m)/0.4 m]

q' = 908.61 W/m