Question 102: Thermodynamics & KTG

Question

Two identical plates P and Q , radiating as perfect black bodies, are kept in vacuum at constant absolute temperatures $\mathrm{T}_{\mathrm{P}}$ and $\mathrm{T}_{\mathrm{Q}}$, respectively, with $\mathrm{T}_{\mathrm{Q}}<\mathrm{T}_{\mathrm{P}}$, as shown in Fig. 1. The radiated power transferred per unit area from P to Q is $W_0$. Subsequently, two more plates, identical to P and Q , are introduced between P and Q, as shown in Fig. 2. Assume that heat transfer takes place only between adjacent plates. If the power transferred per unit area in the direction from $P$ to $Q$ (Fig. 2) in the steady state is $W_S$, then the ratio $\frac{W_0}{W_S}$ is ________.

Accepted Answer

Since the plates are blackbodies, they emit radiation according to the Stefan-Boltzmann law. The power emitted per unit area by a blackbody is given by.
$$ P=\sigma T^4 . $$
Where, $\sigma=$ Stefan - Boltzmann constant
$$ \sigma=5.67 	imes 10^{-8} \mathrm{W} / \mathrm{m}^2 \mathrm{k}^4 $$
When two blackbodies face each other, the net power transfer per unit area from $P$ to $Q=$ power emitted by $P$ toward $Q$ - power emitted by $Q$ toward $P$
So,
$$ W_0=\sigma T_p^4-\dot{\sigma} T_a^4=\sigma\left(T_p^4-T_a^4ight)\,\,\,\,\,\,\,\,\,\,\,\,...(i) $$
After introducing two additional plates: assuming temperature $T_1$ and $T_2$ for these plates.
In the steady state, the power transfer per unit area through each interface (From $P$ to plate $1,1 ightarrow 2,2 ightarrow Q$ ) must be same, because there is no accumulation of energy in the intermediate plates.
So
$$ \begin{aligned} & \sigma\left(T_p^4-T_1^4ight)=W_S \,\,\,\,\,\,\,\,\,\,\,\,...(ii)\ & \sigma\left(T_1^4-T_2^4ight)=W_S \,\,\,\,\,\,\,\,\,\,\,\,...(iii)\end{aligned} $$
and $\sigma\left(T_2^4-T_a^4ight)=W_S\,\,\,\,...(iv)$
by adding $e q^n$ (2), (3) and (4),
$$ \begin{aligned} & \sigma\left(T_p^4-T _1^4+T_1^4-T_2^4+T_2^4-T_Q^4ight)=3 W_S \ \Rightarrow & \sigma\left(T_p^4-T_Q^4ight)=3 W_S \ \Rightarrow & W_0=3 W_S \ & \Rightarrow \frac{W_0}{W_S}=3 \end{aligned} $$

Thermodynamics & KTG

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