Home › Previous year questions › GATE 2021 PI › Question 64GATE 2021 PI – Question 64 General Engineering · Thermal and Fluids Engineering · 2 marks · Numerical answer
A sphere has radius 1 m, centered at the origin, and thermal conductivity K = 50 W/(m·K). Its temperature field is $T=100-70x+51y-80z-10x^2-20y^2-20z^2$. Under Fourier’s law, find net heat leaving per unit time in W to one decimal.
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Show answer and explanation Correct answer: 20943.8 to 20944.1
Explanation Heat flux is $\mathbf q=-K\nabla T$. The divergence theorem gives outward heat rate $\int_V\nabla\cdot\mathbf q\,dV=-K\int_V\nabla^2T\,dV$. Here ∇²T = −20−40−40 = −100, constant. Hence rate = 50×100×4π/3 = **20944.0 W**.
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