Question

Two large black plane surfaces are maintained at constant temperature $T_1$ and $T_2 (T_1 > T_2)$. Two thin black plates are placed between the two surfaces and in parallel to these. After some time, steady conditions are obtained. What is the ratio of heat transfer rate between plate-1 & plate-3 to the ratio of original (when plate-3 & plate-4 was not present) heat tranfer rate between plate-1 & plate-2 ($\eta$) in steady state?

• A. $\eta = \frac{1}{2}$
• B. $\eta = \frac{1}{3}$
• C. $\eta = 1$
• D. $\eta = 0$

Answer 1

Answer: (B)

$\eta = \frac{1}{3}$

Answer 2

Let $A$ be the area of the plates and $\sigma$ be the Stefan-Boltzmann constant. All surfaces are black bodies, so their emissivity $e=1$.

Original Situation: The original heat transfer rate per unit area between plate-1 and plate-2 is given by Stefan-Boltzmann's law: $H_{orig} = \sigma (T_1^4 - T_2^4)$

New Situation: Two thin black plates (plate-3 and plate-4) are placed between plate-1 and plate-2. The arrangement is plate-1, plate-3, plate-4, plate-2. Let their temperatures in steady state be $T_1, T_3, T_4, T_2$ respectively. Since $T_1 > T_2$, it follows that $T_1 > T_3 > T_4 > T_2$ in steady state.

In steady state, the net heat transfer rate per unit area between any two adjacent plates is the same. Let this rate be $H_{new}$. The heat transfer rate per unit area from plate-1 to plate-3 is: $H_{new} = \sigma (T_1^4 - T_3^4)$ (1)

The heat transfer rate per unit area from plate-3 to plate-4 is: $H_{new} = \sigma (T_3^4 - T_4^4)$ (2)

The heat transfer rate per unit area from plate-4 to plate-2 is: $H_{new} = \sigma (T_4^4 - T_2^4)$ (3)

Adding equations (1), (2), and (3): $H_{new} + H_{new} + H_{new} = \sigma (T_1^4 - T_3^4) + \sigma (T_3^4 - T_4^4) + \sigma (T_4^4 - T_2^4)$ $3H_{new} = \sigma (T_1^4 - T_2^4)$

Comparing this with the original heat transfer rate: $3H_{new} = H_{orig}$ So, $H_{new} = \frac{H_{orig}}{3}$

The question asks for the ratio $\eta$ of the heat transfer rate between plate-1 & plate-3 to the original heat transfer rate between plate-1 & plate-2. The heat transfer rate between plate-1 and plate-3 in the new configuration is $H_{new}$. The original heat transfer rate between plate-1 and plate-2 is $H_{orig}$.

Therefore, the ratio $\eta$ is: $\eta = \frac{\text{Heat transfer rate between plate-1 \& plate-3}}{\text{Original heat transfer rate between plate-1 \& plate-2}} = \frac{H_{new}}{H_{orig}}$

Substituting $H_{new} = \frac{H_{orig}}{3}$: $\eta = \frac{H_{orig}/3}{H_{orig}} = \frac{1}{3}$