Question

A computer chip inside an orbiting satellite heats up during operation. Since there is no air in the satellite the only cooling will be by radiation. The chip is 1.50 cm wide by 2.50 cm long with a negligible thickness; it has an emissivity of 0.600; and dissipates 0.420 Watts during usage (assume that the chip radiates heat evenly from both the top and bottom surfaces). Assuming the inside of the satellite is held at a constant temperature of -100°C; calculate the surface temperature of the chip in °C

Answer 1

89.967°C

Step-by-step explanation:

According to Stefan-Boltzmann's law,

E = μΩA(Ts⁴ - Ti⁴)

Where,

Emissitivity, μ = 0.6,

Stefan-boltzmann's constant, Ω = 5.67 x 10^-8 W/m²K⁴

Surface area, A = 1.5 x 2.5 x 10-⁴ = 3.75 x 10-⁴ m²

Satellite temperature, Ti = -100°C = 173K

Power dissipated, E = 0.42 W

Since we're evaluating the temperature for both the top and bottom surfaces,

E = 2 x μΩA(Ts⁴ - Ti⁴)

0.42 = 2 x 0.6 x 3.75 x 10-⁴ x 5.67 x 10^-8 x (Ts⁴ - 173⁴)

0.0165 x 10^-12 = Ts⁴ - 8.957 x 10^8

Ts⁴ = 173.57 x 10^8

Taking 4th root of both sides,

Ts = 362.97 K

Ts = (362.97 - 273)°C

Ts = 89.967°C