Project ForgeHeat Transfer
Conduction
GATE 2012
Hard
The one-dimensional unsteady state heat conduction equation in a hollow cylinder with a constant heat source q is ∂T/∂t=(1/r)∂/∂r(r∂T/∂r)+q. If A and B are arbitrary constants, then the steady state solution to the above equation is
- AT(r)=−qr²/2+A/r+B
- T(r)=−qr²/4+A ln r+B
- CT(r)=A ln r+B
- DT(r)=qr²/4+A ln r+B
Explanation
At steady state, 0=(1/r)d/dr(r dT/dr)+q ⟹ d/dr(r dT/dr)=−qr. Integrating once: r dT/dr=−qr²/2+A, so dT/dr=−qr/2+A/r. Integrating again: T(r)=−qr²/4+A ln r+B.
Source: GATE CH 2012 (2012)
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