It is desired to transport liquid metal through a pipe embedded in a wall at a point where the temperature is 650 K. A 1.2‐m‐thick wall constructed of a material having a thermal conductivity varying with temperature according to k = 0.0073 (1 + 0.0054 T), where T is in K and k is in W/m ⋅ K, has its inside surface maintained at 925 K. The outside surface is exposed to air at 300 K with a convective heat‐transfer coefficient of 23 W/m2 ⋅ K. How far from the hot surface should the pipe be located? What is the heat flux for the wall?

q = 16.366 W/m^2  

L = 0.646m

Explanation:

Given:

- The inside surface temperature T_i = 925 K

- The pipe surface temperature T_1 = 650 K

- The ambient temperature T_a = 300 K

- The outermost surface temperature = T_rw

- The thermal conductivity coefficient :

- Total thickness of the wall = 1.2 m

                         k = 0.0073*( 1 + 0.0054*T)

- The convection heat transfer coefficient h = 23 W / m^2K

Find:

How far from the hot surface should the pipe be located? What is the heat flux for the wall?

Solution:

- The conduction through the wall is given by:

- Solve the above quadratic equation we get:

                 T_rw = 300.71 K

                  q = 16.366 W/m^2      

-  The pipe must be located where the surface temperature of 650 K can be maintained. Hence, we have: