Question

Consider steady heat transfer through the wall of a room in winter. The convection heat transfer coefficient at the outer surface of the wall is three times that of the inner surface as a result of the winds. On which surface of the wall do you think the temperature will be closer to the surrounding air temperature? Explain.

Short answer

Answer: The outer surface of the wall will have a temperature closer to the surrounding air temperature compared to the inner surface. This is because the higher convection heat transfer coefficient caused by winds on the outer surface allows for more efficient heat transfer, thus maintaining a smaller temperature difference between the surface and the air.

Step-by-step solution

Understand the problem

We need to compare the temperatures at the inner and outer surfaces of the wall, taking into consideration the convection heat transfer coefficients at both surfaces.

Convection heat transfer coefficients

Let's denote the convection heat transfer coefficient at the outer surface as h_out and the one at the inner surface as h_in. According to the problem statement, we have h_out = 3 * h_in.

Heat conduction through the wall

We can express the heat conduction through the wall using Fourier's Law of heat conduction, given by: q = -k * (dT/dx) Where q is the heat flux, k is the thermal conductivity of the wall, dT/dx is the temperature gradient across the wall.

Heat convection at the surfaces

Heat convection at both surfaces can be expressed using Newton's Law of cooling, as follows: For the outer surface: q = h_out * (T_wall,out - T_air,out) For the inner surface: q = h_in * (T_wall,in - T_air,in)

Steady heat transfer condition

Given that heat transfer is steady, the heat flux through the wall must be equal to heat convection at both surfaces. Therefore, h_out * (T_wall,out - T_air,out) = h_in * (T_wall,in - T_air,in)

Solve for the temperature difference

From the previous step, we can rewrite the equation in terms of temperature difference: 3 * (T_wall,out - T_air,out) = (T_wall,in - T_air,in)

Determine which surface is closer to the air temperature

Since the outer surface has a higher convection heat transfer coefficient (h_out = 3 * h_in), the temperature difference between the outer surface and the surrounding air temperature will be smaller than the temperature difference between the inner surface and the surrounding air temperature. Therefore, the temperature at the outer surface of the wall will be closer to the surrounding air temperature compared to the inner surface. This is because the higher convection heat transfer coefficient caused by winds on the outer surface allows for more efficient heat transfer, thus maintaining a smaller temperature difference between the surface and the air.

Key concepts

Convection Heat Transfer Coefficient

Heat transfer between a solid surface and the adjacent fluid that is in motion is described by the convection heat transfer coefficient, represented by the symbol 'h'. This coefficient measures the rate at which heat is transferred per unit area and per degree of temperature difference between the surface and the fluid. Essentially, a high 'h' means heat can more efficiently move from the surface to the fluid or vice versa.

For example, when the wind increases outside the room mentioned in our original exercise, the convection heat transfer coefficient at the wall's outer surface increases. This leads to a higher rate of heat loss from the wall to the cold outside air, which conveys the idea that with a higher 'h_out', the wall is more effective at