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 celcius and the base is at 20 degree celcius. 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 celcius.

 

[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 celcius 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.

 

[rwisz]

Can someone please explain the reasoning behind this result?

I would first clarify the question and make sure I understand what is being asked. It seems that the question is asking for the temperature at the interface between the wooden board and the insulating material. To calculate this, we can use the equation for thermal conduction, where the rate of heat flow is equal to the thermal conductivity (k) multiplied by the surface area (A) and the temperature gradient (dθ/dx). We can assume that the rate of heat flow is constant, so we can set the two equations for the wooden board and the insulating material equal to each other and solve for the temperature at the interface.

However, the given answer of 20 degrees Celsius does not seem to be correct. To double check, we can plug in the values given in the problem and solve for the temperature at the interface. Using the equation mentioned above, we get:

(0.030 W/m/K)(200 m^2)(dθ/dx) = (0.15 W/m/K)(200 m^2)(dθ/dx)

Solving for dθ/dx, we get dθ/dx = 0, which means there is no temperature difference between the two materials. This does not seem to be a realistic result, so I would suggest double checking the calculations and/or seeking clarification from the teacher or professor.