Calculate the temperature drop from the center line to the surface for a 2.5 cm diameter rod having a thermal conductivity of 26 W/(m K) if the rate of heat removal from its surface is 1.6 MW/ m2 .
In a cylindrical fuel rod of a nuclear reactor, heat is generated internally according to the
equation
GIVEN
• A cylindrical rod with internal generation and heat removal from its surface
• Outside diameter (Do) = 2.5 cm=0.025 m
• Rate of heat generation is as given above
• Thermal conductivity (k) = 26 W/(m K)
• Heat removal rate (q/A) = 1.6 MW/m2
FIND
• The temperature drop from the center line to the surface (?T)
ASSUMPTIONS
• The heat flow has reached steady state
• The thermal conductivity of the fuel rod is constant
• One dimensional conduction in the radial direction
SKETCH
The equation for one dimensional conduction in cylindrical coordinates is given
With the boundary conditions
Integrating once
From the first boundary condition: C1 = 0, therefore
Integrating again
Evaluate this expression at the surface of the cylinder and at the centerline of the sphere and subtracting the results gives us the temperature drop in the cylinder
The rate of heat generation at the centerline (q1) can be evaluated using the conservation of energy. The total rate of heat transfer from the cylinder must equal the total rate of heat generation within the cylinder
Therefore, the temperature drop within the cylinder is
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