### Problem Statement

Determine the overall heat transfer by conduction per unit area occurring across a furnace wall made of fire clay. Furnace wall has a thickness of 12" or a foot. The wall is insulated from outside. Thermal conductivity values for the wall and insulation materials are 0.1 W/m·K and 0.01 W/m·K, respectively. The furnace operates at 650^{0}C. Average ambient temperature outside the furnace wall is 30^{0}C and allowable temperature on the outer side of insulation is 80^{0}C. If the air side heat transfer coefficient is 0.4 W/m^{2}·K, calculate the minimum insulation thickness requirement.

### Solution

The sample problem can be solved by following the steps given here. First the maximum possible heat transfer rate from furnace wall to the atmosphere is calculated. Then based on this maximum possible rate, minimum requirement of insulation thickness can be estimated.

### Step1

Refer to EnggCyclopedia's article about heat transfer coefficients, for relation between heat transfer rate and the individual heat transfer coefficients between wall and air.

Q/A = h_{A}×(T_{2}-T_{A}) – Air side heat transfer rate in W/m^{2}

Hence, Q/A = 0.4×(80-30) = 20.0 W/m^{2}

This is the maximum limit of heat transfer rate through the furnace wall and insulation.

### Step2

The conductive heat transfer through a flat wall is described in EnggCyclopedia's article on conduction.

For a flat wall,

Q/A = k1×(T_{1}-T_{i})/L_{1} = k2×(T_{i}-T_{2})/L_{2}

Hence,

(T_{1}-T_{i}) = (Q/A)×(L_{1}/k1)... (1)

and (T_{i}-T_{2}) = (Q/A)×(L_{2}/k2)... (2)

(1) + (2) gives,

(T_{1}-T_{2}) = (Q/A)×(L_{1}/k1 + L_{2}/k2)... (3)

From this equation we can say that for composite walls with layers of different materials, the overall heat transfer rate can be represented as,

(T_{1}-T_{2}) ÷ (Q/A) = (L_{1}/k1 + L_{2}/k2) = heat transfer resistance

The inverse of heat transfer resistance represents conductive heat transfer coefficient, given by,

Conductive heat transfer coefficient =1 / (L_{1}/k1 + L_{2}/k2) = k1k2/(L_{1}k2+L_{2}k1)

### Step3

Maximum allowable heat transfer rate represents minimum insulation thickness requirement. Hence, Q/A = 20.0W/m^{2 }

So, in equation (3), all the variables are known except for L_{2}. Hence this equation can be solved to determine L_{2}.

The solution to equation (3) is L_{2} = 0.25 m or 10 inches.

This is the minimum insulation thickness requirement for furnace wall.