A spherical black body of radius R = 10 cm is at a temperature T 1 = 300 K . It is enclosed concentrically within a larger spherical black body shell of radius 2 R , which is maintained at a constan

Question

A spherical black body of radius R = 10 cm is at a temperature T 1 ​ = 300 K . It is enclosed concentrically within a larger spherical black body shell of radius 2 R , which is maintained at a constant temperature T 2 ​ = 400 K . What is the net rate of heat transfer (in Watts) from the outer shell to the inner sphere? Use Stefan-Boltzmann constant σ = 5.67 × 1 0 − 8 W m − 2 K − 4 and π ≈ 3.14 .

TestHub · Accepted Answer

For two concentric black bodies, the net rate of heat transfer from the hotter body (outer shell) to the colder body (inner sphere) is given by: $Q_{n e t} = σ A_{1} (T_{2}^{4} - T_{1}^{4})$

where $A_{1}$ is the surface area of the inner sphere, and $T_{2}$ and $T_{1}$ are the temperatures of the outer and inner spheres, respectively. The radius of the inner sphere is $R = 10 cm = 0.1 m$ . Its surface area is $A_{1} = 4 π R^{2} = 4 π (0.1)^{2} = 0.04 π m^{2}$ . Given $T_{1} = 300 K$ and $T_{2} = 400 K$ . Substituting the values: $Q_{n e t} = 5.67 \times 1 0^{- 8} \times (0.04 π) \times (40 0^{4} - 30 0^{4})$ $Q_{n e t} = 5.67 \times 1 0^{- 8} \times 0.04 π \times (256 \times 1 0^{8} - 81 \times 1 0^{8})$ $Q_{n e t} = 5.67 \times 1 0^{- 8} \times 0.04 π \times (175 \times 1 0^{8})$ $Q_{n e t} = 5.67 \times 0.04 \times 3.14 \times 175 \approx 124.63 W$

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