Question

Radiative heat flux \( \dot{q} \) at a hot surface at a temperature \( T_s \) can be expressed as \[ \dot{q} = A f(T_s, T_\infty) (T_s - T_\infty) \] where \( A \) is a constant and \( T_\infty \) is the temperature of the surroundings (temperatures are expressed in K). The function \( f(T_s, T_\infty) \) is given by ______.

• A. \( (T_s + T_\infty)^2 (T_s - T_\infty) \)
• B. \( (T_s^2 + T_\infty^2)(T_s + T_\infty) \)
• C. \( (T_s^2)(T_\infty^2)(T_s + T_\infty) \)
• D. \( (T_s - T_\infty)^2(T_s + T_\infty) \)

Hint:

In radiative heat transfer, the heat flux depends on the fourth power of the absolute temperatures. When the temperatures are expressed as \( T_s^4 - T_\infty^4 \), you can approximate this expression for the given form as \( (T_s^2 + T_\infty^2)(T_s + T_\infty) \).

Answer 1

The Correct Option is B

The radiative heat flux \( \dot{q} \) is expressed by the Stefan-Boltzmann law, which is typically given as: \[ \dot{q} = A \sigma \left( T_s^4 - T_\infty^4 \right) \] Where \( \sigma \) is the Stefan-Boltzmann constant, and \( T_s \) and \( T_\infty \) are the absolute temperatures of the hot surface and the surroundings, respectively. Now, in the given problem, we have the form: \[ \dot{q} = A f(T_s, T_\infty) (T_s - T_\infty) \] This implies that the function \( f(T_s, T_\infty) \) should represent the difference of the fourth powers of the temperatures. Upon expanding and factoring, the form of \( f(T_s, T_\infty) \) is found to be: \[ f(T_s, T_\infty) = (T_s^2 + T_\infty^2)(T_s + T_\infty) \] Thus, the correct option is Option B.
Step 1: Analyzing each option - Option (A): \( (T_s + T_\infty)^2 (T_s - T_\infty) \) - This does not match the form of the required function, and therefore is incorrect.
- Option (B): \( (T_s^2 + T_\infty^2)(T_s + T_\infty) \) - Correct: This matches the expected form for the function \( f(T_s, T_\infty) \), and correctly accounts for the temperatures raised to the power of four.
- Option (C): \( (T_s^2)(T_\infty^2)(T_s + T_\infty) \) - This is an incorrect form and does not correctly represent the temperature dependence for radiative heat flux.
- Option (D): \( (T_s - T_\infty)^2 (T_s + T_\infty) \) - This does not match the required form for \( f(T_s, T_\infty) \), making it incorrect.
Step 2: Conclusion The correct function \( f(T_s, T_\infty) \) is given by Option B, \( (T_s^2 + T_\infty^2)(T_s + T_\infty) \), which is consistent with the form derived from the Stefan-Boltzmann law.