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: In winter, the outer surface of the wall will have a temperature closer to the surrounding air temperature.

Step-by-step solution

Introduce convection heat transfer coefficients

A convection heat transfer coefficient is the ratio of heat transferred between a solid surface and a fluid to the temperature difference between the surface and the fluid. In this problem, we are given the information that the convection heat transfer coefficient at the outer surface of the wall is three times that of the inner surface due to winds. Let h_in denote the convection heat transfer coefficient of the inner surface and h_out denote the convection heat transfer coefficient of the outer surface. So, we have: h_out = 3 * h_in

Define temperature variables

Let's use the following notation for the temperatures: T_in = temperature at the inner surface of the wall, T_out = temperature at the outer surface of the wall, T_room = temperature of the room (inside), and T_air = temperature of the air outside.

Apply the convection heat transfer equation

For steady heat transfer through the wall, the heat transfer rate along the wall should be constant. We can use the convection heat transfer equation given by: Q_cond = Q_conv where, Q_cond - heat transfer rate by conduction through the wall, Q_conv - heat transfer rate by convection from the inner surface to the room and the outer surface to the air outside. Using the convection heat transfer equation, we can write: Q_conv_in = h_in * A * (T_room - T_in) Q_conv_out = h_out * A * (T_out - T_air) Considering steady heat transfer, Q_conv_in = Q_conv_out.

Calculate temperature differences

Using the information given, we can find the temperature difference between the inner and outer surfaces of the wall and the surrounding air temperatures. Since h_out = 3 * h_in and they are equal: h_in * A * (T_room - T_in) = 3 * h_in * A * (T_out - T_air) By canceling out h_in and A, we get: (T_room - T_in) = 3 * (T_out - T_air)

Determine which surface is closer to surrounding air temperature

Considering the relationship derived in Step 4, we can now determine which surface has a temperature closer to the surrounding air temperature. In winter, T_room > T_air, so (T_room - T_in) > (T_out - T_air). Since the temperature difference between the inner surface and the room is greater than the temperature difference between the outer surface and the air outside, the temperature of the outer surface is closer to the surrounding air temperature. Thus, the temperature at the outer surface of the wall is closer to the surrounding air temperature.