Question

It is stated that the temperature in a plane wall with constant thermal conductivity and no heat generation varies linearly during steady one- dimensional heat conduction. Will this still be the case when the wall loses heat by radiation from its surfaces?

Short answer

Answer: No, the temperature in a plane wall will not vary linearly when heat is lost by radiation from its surfaces. The temperature variation becomes more complex due to the non-linear contribution of radiation heat loss in the net heat transfer equation.

Step-by-step solution

Understand the heat conduction equation

For steady one-dimensional heat conduction, we use the heat conduction equation, which states that the rate of heat transfer through the wall is equal to the product of the wall's thermal conductivity, its cross-sectional area perpendicular to the direction of heat flow, and the temperature gradient. Mathematically, this equation can be written as: q = -k * A * (dT/dx) where q is the rate of heat transfer, k is the thermal conductivity, A is the cross-sectional area, and (dT/dx) is the temperature gradient in the x-direction.

Incorporate heat loss by radiation

When the wall loses heat by radiation, we must take into account the rate of heat loss due to radiation in addition to the rate of heat transfer due to conduction. The heat loss by radiation can be calculated using Stefan-Boltzmann's law, which states that the rate of heat transfer by radiation is proportional to the difference in the fourth power of the absolute temperatures of the two surfaces. Mathematically, this equation can be written as: q_rad = σ * A * (T1^4 - T2^4) where q_rad is the rate of heat transfer by radiation, σ is the Stefan-Boltzmann constant, T1 and T2 are the absolute temperatures of the two surfaces of the wall.

Combine the conduction and radiation equations

Since both conduction and radiation are contributing to the overall heat transfer, we can combine the two equations to get the net heat transfer in the wall. The net heat transfer can be written as: q_net = q + q_rad or, q_net = -k * A * (dT/dx) + σ * A * (T1^4 - T2^4)

Analyze the impact on the temperature distribution

Now let's analyze the given statement that the temperature varies linearly during steady one-dimensional heat conduction. From the heat conduction equation, we can deduce that if the temperature gradient (dT/dx) remains constant, the temperature will vary linearly with respect to the position. However, when we add the contribution of radiation heat loss, the net heat transfer equation becomes a more complex function of temperature. Due to the presence of the T1^4 - T2^4 term, the temperature variation will no longer be linear if the wall loses heat by radiation from its surfaces. In conclusion, the temperature in a plane wall will not vary linearly when the wall loses heat by radiation from its surfaces, as the addition of radiation heat loss introduces a more complex dependency on temperature in the net heat transfer equation.