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 (all temperatures 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:

Always remember that radiative heat transfer involves $T^4$ dependence. Expanding and factoring $(T_s^4 - T_\infty^4)$ directly leads to the required function.

Answer 1

The Correct Option is B

Step 1: Recall Stefan–Boltzmann law.
Radiative heat flux between a surface at temperature $T_s$ and surroundings at $T_\infty$ is \[ \dot{q} = \sigma \, (T_s^4 - T_\infty^4) \] Step 2: Factorize the difference of fourth powers.
\[ T_s^4 - T_\infty^4 = (T_s - T_\infty)(T_s + T_\infty)(T_s^2 + T_\infty^2) \] Step 3: Compare with the given expression.
We are given: \[ \dot{q} = A \, f(T_s, T_\infty) \, (T_s - T_\infty) \] Thus, \[ f(T_s, T_\infty) = (T_s + T_\infty)(T_s^2 + T_\infty^2) \] Step 4: Select the correct option.
This matches option (B). \[ \boxed{f(T_s, T_\infty) = (T_s^2 + T_\infty^2)(T_s + T_\infty)} \]