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Got an interesting problem here, I am thinking I did it correctly but I wanted to see your opinions. Here is the problem..

*A 0.2-m-thick plane wall is constructed of concrete. At steady state, the energy transfer by conduction through the wall is 0.15 kW/m2. The inside temperature of the wall is 24oC.*

a. If the temperature distribution through the wall is linear, what is the temperature on the outside surface of the wall?

b. A layer of insulation is added to the inside of the wall. Assuming the same temperatures at the surfaces, determine the thickness of the insulation that will reduce the energy transfer through the composite wall by a factor of 2.

The insulation has a thermal conductivity of 0.08 W/m.K. For concrete , k = 1.4 W/m.K.

a. If the temperature distribution through the wall is linear, what is the temperature on the outside surface of the wall?

b. A layer of insulation is added to the inside of the wall. Assuming the same temperatures at the surfaces, determine the thickness of the insulation that will reduce the energy transfer through the composite wall by a factor of 2.

The insulation has a thermal conductivity of 0.08 W/m.K. For concrete , k = 1.4 W/m.K.

**My Solution**

Since it is stated that we can assume a linear relationship, the heat transfer through the wall will be proportional to the temperature differential between the inside and outside of the wall and the thickness of the wall. (Which will be denoted L) Therefore..

[tex]\frac{dQ}{dt} = kL(T_{outside}-T_{inside})[/tex]

Plugging in the values we obtain..(Using Kelvin for temperature)

[tex]\frac{dQ}{dt} = kL(T_{outside}-T_{inside}) \rightarrow 0.15 = (1.4)(0.2)(T_{outside}-299.15) \rightarrow T_{outside} = 299.7K[/tex]

For the second part, if we look at the wall on a cross section (Going to do my best to make a wall out of text here...)

---L-2x

| |---| |

| |---| |

| |---| |

| |---| |

x ----- x

Therefore the above equation for heat transfer turns into..

[tex]\frac{dQ}{dt} = 2k_{concrete}L(T_{outside}-T_{inside})+k_{insulation}(L-2x)(T_{outside}-T_{inside})[/tex]

Plugging in all the values I get an x value of 4.6cm which means the thickness of the insulation is L-2x or 10.9cm.

This seems reasonable to me, do you guys think my logic is correct?

Thanks!