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unscientific
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Homework Statement
In general, a sphere with conductivity ##\kappa##, heat capacity per unit volume ##C## and radius ##R## obeys the differential equation at time t:
[tex] C\frac{\partial T}{\partial t} = \kappa \frac{\partial^2 T}{\partial r^2} + \frac{2\kappa}{r}\frac{\partial T}{\partial r} [/tex]
Part (a): A sphere with a cavity of radius ##a## generates heat at a rate Q. Heat is lost from the outer surface of the sphere to the surroundings, with surrounding temperature ##T_s##, given by Newton’s Law of Cooling with constant ##\alpha## per unit area. Find the temperature ##T_a## at the surface of the cavity at thermal equilibrium.
Part (b): A second sphere without a cavity generates heat uniformly at ##q## per unit volume. Like the first sphere, heat is lost to the surroundings at its surface. Find temperature at the center of the sphere ##T_0## at equilibrium.
Homework Equations
The Attempt at a Solution
Part (a)
Heat generated in cavity = Heat loss at surface
[tex]Q = \alpha (T_s - T_R)(4\pi R^2) [/tex]
[tex]T_R = T_s - \frac{Q}{\alpha (4\pi R^2)} [/tex]
We will use this to solve for the constants in the differential equation later on.
At steady state, ##C\frac{\partial T}{\partial t} = \kappa \frac{\partial^2 T}{\partial r^2} + \frac{2\kappa}{r}\frac{\partial T}{\partial r} = 0##.
[tex]\frac{\partial^2 T}{\partial r^2} + \frac{2}{r}\frac{\partial T}{\partial r} = 0 [/tex]
Solving, we get:
[tex]T = A + \frac{b}{r}[/tex]
We need one more equation with the one for ##T_R## to solve for constants ##A## and ##B##.
Using ##\int \vec J \cdot d\vec S = k \int \nabla \vec T \cdot d\vec S##:
[tex]Q = -\kappa \frac{\partial T}{\partial r}(4\pi r^2)[/tex]
[tex]Q = 4\pi \kappa b[/tex]
[tex]b = \frac{Q}{4\pi \kappa}[/tex]
Solving for A:
[tex]A = T_s - \frac{Q}{4\pi \alpha R^2} - \frac{Q}{4\pi \kappa R} [/tex]
Together, temperature at surface of cavity is:
[tex]T_a = T_s - \frac{Q}{4\pi \alpha R^2} + \frac{Q}{4\pi \kappa}\left( \frac{1}{a} - \frac{1}{R} \right) [/tex]
Part(b)
I'm not sure how to approach this, as ##T = A + \frac{b}{r}## doesn't work at ##r=0##..
I have found the temperature at surface though:
[tex]Q = \frac{4}{3}\pi R^3 q = \alpha (T_s - T_R)(4\pi R^2)[/tex]
[tex]T_R = T_s - \frac{Rq}{3\alpha}[/tex]
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