GATE ME 2022 Set 1 — Question 60
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Heat Transfer → Conduction → 1D Steady Conduction
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Question
Consider a one-dimensional steady heat conduction process through a solid slab of thickness 0.1 m. The higher temperature side A has a surface temperature of 80 °C, and the heat transfer rate per unit area to low temperature side B is 4.5 kW/m². The thermal conductivity of the slab is 15 W/m.K. The rate of entropy generation per unit area during the heat transfer process is __________ W/m².K (round off to 2 decimal places).
Correct answer
1.12 to 1.24
Solution
Let the slab thickness be , thermal conductivity be , heat flux be , and the temperatures of the hot and cold surfaces be and respectively.Given data:
For one-dimensional steady-state heat conduction, Fourier's law is:
Rearranging to solve for :
Substituting the given values:
Step 2: Calculate the rate of entropy generation per unit area.
The entropy balance for the slab (a steady-state control volume) is:
Here, entropy is transferred into the slab with heat at temperature and out of the slab with heat at temperature . The rate of heat transfer is .
So, the entropy balance equation becomes:
Solving for the rate of entropy generation, :
We need the rate of entropy generation per unit area, which is :
Substituting the values:
Rounding off to two decimal places, the rate of entropy generation per unit area is 1.18 W/m².K.
This value is within the specified range of 1.12 to 1.24.
- Slab thickness,
- Temperature of side A,
- Heat transfer rate per unit area (heat flux),
- Thermal conductivity,
For one-dimensional steady-state heat conduction, Fourier's law is:
Rearranging to solve for :
Substituting the given values:
Step 2: Calculate the rate of entropy generation per unit area.
The entropy balance for the slab (a steady-state control volume) is:
Here, entropy is transferred into the slab with heat at temperature and out of the slab with heat at temperature . The rate of heat transfer is .
So, the entropy balance equation becomes:
Solving for the rate of entropy generation, :
We need the rate of entropy generation per unit area, which is :
Substituting the values:
Rounding off to two decimal places, the rate of entropy generation per unit area is 1.18 W/m².K.
This value is within the specified range of 1.12 to 1.24.
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