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

rate of heat loss = 114.34(T - t)w/m

Explanation: Please find the attached file for the solution

Some figures are missing in the question. The complete question is;

"Pressurized steam at 450 K flows through a long, thin-walled pipe of 0.5-m diameter. The pipe is enclosed in a concrete casing that is of square cross section and 1.5 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 pipe?"

Rate of heat loss per unit length of pipe = 1122 W/m

Explanation:

Heat analysis is given by the expression;

q = SkΔT(1-2)

Now for this problem, we will work under the following assumptions;

- steady state conditions

- negligible steam side convection resistance

- pipe wall resistance, contact resistance, and constant properties.

For concrete at 300K, the constant k = 1.4 W/m.K

From the heat analysis equation earlier, S is alternatively given an expression;

S = 2πL/(In(1.08w/D)) where w is width and D is Diameter

And thus,

q = [2πkL(T1-T2)] /(In(1.08w/D))

Now, rate of heat transfer per unit length is given by the expression;

q' = q/L

So,

q' = [2πk(T1-T2)] /(In(1.08w/D)

q' = [2π(1.4)(450-300)] /(In(1.08 x 1.5)/0.5)) = 1319.47/In(3.24) = 1122 W/m

wavelength divides in half and speed stays the same.