# Heat transfer through a multi-layered wall

1. May 26, 2015

### Lieberkuhn

1. The problem statement, all variables and given/known data
A furnace wall is made up of 3 layers:

1. 4mm layer of material with thermal conductivity of 52 W/m.K
2. 2mm layer of material with thermal conductivity of 20 W/m.K
3. 1mm layer of material with thermal conductivity of 3 W/m.K

There is an air gap between layers 1&2 with a thermal resistance of 0.16 K/W.

The temperature at the inner surface of layer 1 is 873.15 K
The ambient temperature outside layer 3 is 343.15 K

The heat transfer coefficient from outside surface to surroundings is 17 W/m^2 .K

Find the following:
1. Rate of heat loss per square metre of outside surface (heat flux)
2. Temperature at each interface of wall, including outside surface temperature.

2. Relevant equations

h=q/ΔT

h = Heat transfer coefficient (in W/m^2 .K)
q = Heat flux (in W/m^2)
ΔT = Change in temperature (in K)

q=(kΔT)/L
q = Heat flux (in W/m^2)
k = Thermal conductivity (in W/mK)
ΔT = Change in temperature (in K)
L = Thickness of material (in m)

R=L/kA
R = Thermal resistance (in K/W)
L = Thickness of material (in m)
k = Thermal conductivity (in W/mK)
A = Area (in m^2)

q=ΔT/[(L1/k1)+(L2/k2)+(L3/k3)]
q = Heat flux (in W/m^2)
ΔT = Change in temperature between outer surfaces (in K)
L1 = Thickness of layer 1 (in m)
L2 = Thickness of layer 2 (in m)
L3 = Thickness of layer 3 (in m)
k1 = Thermal conductivity of layer 1 (in W/mK)
k2 = Thermal conductivity of layer 2 (in W/mK)
k3 = Thermal conductivity of layer 3 (in W/mK)

3. The attempt at a solution
Found the following equation but do not know how to remove A (area) to find q (heat flux).

3147.064q=28111200-8486.4A

Any useful hints would be much appreciated!

2. May 26, 2015

### CWatters

Problem asks for heat loss per square meter.

3. May 26, 2015

### Staff: Mentor

Let's see your calculations in detail. Your units on the air gap resistance are incorrect. Your equation for the heat transfer omits the air gap resistance and the outside heat transfer coefficient.

Chet