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unscientific
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Homework Statement
Part (a): Under what circumstances do these equations work?
Part (b): Find Temperature as a function of x, in steady state.
Part (c): Show heat loss is proportional to r^1.5
Part (d):Assume A = 0, show solution works, and find γ.
Homework Equations
The Attempt at a Solution
Part(a)
dQ = T dS works only for reversible processes.
dW = -p dV works only for reversible processes.
Part (b)
In steady state, ##\frac{\partial T}{\partial t} = 0##.
DE becomes:
[tex]k\frac{\partial^2T}{\partial x^2} = \frac{A}{r}(T-T_0)[/tex]
Solving, one obtains:
[tex]T = (T_1-T_0)e^{-\sqrt{\frac{A}{rk}}x} + T_0[/tex]
Part (c)
[tex]P \propto \pi r^2\int R dx \propto r^{\frac{3}{2}}[/tex]
Part (d)
The DE now becomes:
[tex]\rho C_v \frac{\partial T}{\partial t} = k\frac{\partial ^2 T}{\partial x^2} [/tex]
I have subtituted ##T = Bt^{-\frac{1}{2}}e^{-\frac{x^2}{\gamma t}}## and it is a solution, but I can't seem to find an expression for ##\gamma##, as it cancels out on both sides.