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GATE ME 2022 Set 1 — Question 64

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NAT+2 / -0Medium1D Steady ConductionConductionHeat TransferBlack & Grey SurfacesRadiationStefan-Boltzmann & Wien's Laws

Heat Transfer → Conduction → 1D Steady Conduction

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Question

Consider a solid slab (thermal conductivity, k=10 W⋅m−1⋅K−1k = 10 \text{ W}\cdot\text{m}^{-1}\cdot\text{K}^{-1}) with thickness 0.2 m0.2 \text{ m} and of infinite extent in the other two directions as shown in the figure. Surface 2, at 300 K300 \text{ K}, is exposed to a fluid flow at a free stream temperature (T∞T_\infty) of 293 K293 \text{ K}, with a convective heat transfer coefficient (hh) of 100 W⋅m−2⋅K−1100 \text{ W}\cdot\text{m}^{-2}\cdot\text{K}^{-1}. Surface 2 is opaque, diffuse and gray with an emissivity (ϵ\epsilon) of 0.50.5 and exchanges heat by radiation with very large surroundings at 0 K0 \text{ K}. Radiative heat transfer inside the solid slab is neglected. The Stefan-Boltzmann constant is 5.67×10−8 W⋅m−2⋅K−45.67 \times 10^{-8} \text{ W}\cdot\text{m}^{-2}\cdot\text{K}^{-4}. The temperature T1T_1 of Surface 1 of the slab, under steady-state conditions, is _________ K (round off to the nearest integer).
Schematic of a solid slab with Surface 1 at temperature $T_1$ and Surface 2 at temperature $T_2 = 300 \text{ K}$, showing thickness $0.2 \text{ m}$, thermal conductivity $k = 10 \text{ W/m-K}$, and fluid flow at $T_\infty = 293 \text{ K}$ with $h = 100 \text{ W/m}^2\text{-K}$ and emissivity $\epsilon = 0.5$
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