Calculating Temperature of Insulated Room Base - Thermal Conduction Homework

[thereddevils]

Homework Statement

The base of the room in a house consists of wooden board which is 15 mm thick and of area 200 m^2. The thermal conductivity of the board is 0.15 W/m/K. The temperature of the room is maintained at 0 degree celsius and the base is at 20 degree celsius. If the owner now decides to insulate the room by covering the board with a layer of insulating material of thickness of 10 cm and thermal conductivity of 0.030 W/m/K, calculate the temperature of the board and insulating material interface.

Homework Equations

$$\frac{dQ}{dt}=kA\frac{d\theta}{dx}$$

The Attempt at a Solution

Is it the temperature of the board or insulating material interface the question is asking? Or is it the temperature between them?

Can i assume that the rate of heat flow across the board and insulating material is constant such that,

$$0.030(\frac{\theta-0}{0.1})=0.015(\frac{20-\theta}{0.015})$$

but the temperature calculated here is the temperature of the insulating material.

The answer given is 20 degree celsius.

 

[Mapes]

The question is asking about the temperature between the board and insulating material. And your approach is correct, except you have a typo in your equation.

 

[thereddevils]

The question is asking about the temperature between the board and insulating material. And your approach is correct, except you have a typo in your equation.

Thanks Mapes, i got 19.42 degree celsius from my above working(after correcting the typo as well) but the answer given is 20.

Also, isn't that what i calculated the temperature of the insulating material, and not the temperature between the insulating material and the board? My approach assumes the rate of heat flow across the two materials are constant. Is this assumption valid? I missed out something, it says at the end of the question that the board and insulating material are in good contact. Does this statement make that assumption valid? Why?

 

[Mapes]

The temperature increases from 0°C to 19.42°C across the insulation, and from 19.42°C to 20°C across the board. The heat flow (in W) is constant. Because the board and insulation are in good contact, we don't need to worry about any air gap that would influence heat transfer.