Show that the temperature distribution in a sphere of radius ro, made of a homogeneous material in which energy is released at a uniform rate per unit volume Gq, is
GIVEN
- A homogeneous sphere with energy generation
- Radius = ro
FIND
- Show that the temperature distribution is as shown above.
ASSUMPTIONS
- Steady state conditions persist
- The thermal conductivity of the sphere material is constant
- Conduction occurs in the radial direction only
SKETCH
Let k = the thermal conductivity of the material
To= the surface temperature of the sphere
can be simplified to the following equation by the assumptions of steady state and radial
conduction only
With the following boundary conditions
Integrating the differential equation once
From the first boundary condition
Integrating once again
Applying the second boundary condition
Therefore, the temperature distribution in the sphere is
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