GATE Mechanical Engineering (ME) 2019 Shift 2 Solved Paper

Question : 26 of 65

Marks: +1, -0

One-dimensional steady state heat conduction takes place through a solid whose cross-sectional area varies linearly in the direction of heat transfer. Assume there is no heat generation in the solid and the thermal conductivity of the material is constant and independent of temperature. The temperature distribution in the solid is

Quadratic
Exponential
Logarithmic
Linear

Validate

Solution:

Let the given body cross-sectional area varies linearly, and heat is flowing from left to right.

If there is

∫ \;{d}/{d x}(-K A \;{d T}/{d X})=∫ 0

(-K A \;{d T}/{a X})=C

-K(A_{0}+A_{c} x) \;{d T}/{d X}=C

d T=\;{-C d X}/{K(A_{0}+A_{c} x)}

∫↙{1}↖{2} d T=∫↙{1}↖{2} \;{-C d X}/{K(A_{0}+A_{c} x)}

T_{2}-T_{1}=-\;{C}/{K} \ln (\;{A_{c} x_{2}+A_{0}}/{A_{c} x_{1}+A_{0}})

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