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timman_24
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Homework Statement
I need to set up the mathematical formulation of the following heat conduction scenarios:
a) A slab in 0\le x \le L is initially at a temperature f(x). For times t>0 the boundary at x=o is kept insulated and the boundary at x=L dissipates heat by convection into the medium at zero temperature.
b)A semi-infinite region 0\le x \le \infty is initially at temperature f(x). For times t>0, heat is generated in the medium at a constant rate of g (constant), while the boundary at x=0 is kept at zero temperature.
c) A solid cylinder 0\le r \le b is initially at a temperature f(r). For times t>0 heat is generated in the medium at a rate of g(r), while the boundary at r=b dissipates heat by convection into a medium at zero temperature.
d) A solid sphere 0\le r \le b is initially at temperature f(r). For times t>0, heat is generated in the medium at a rate of g(r), while the boundary at r=b is kept at a uniform temperature T'.
2. The attempt at a solution
Problem a)
Solution:
\frac{1}{\alpha}\frac{\partial T}{\partial t}=\frac{\partial ^2 T}{\partial x^2 }
IC: T(x,0)=f(x)
BC:
1.) Insulated Boundary:q\prime(0,t)=0
2.) Convection Boundary:k\frac{\partial T}{\partial x}|_L =-hT(L,t)
Problem b)
Solution: \frac{1}{\alpha}\frac{\partial T}{\partial t}=\frac{\partial ^2 T}{\partial x^2 }+\frac{1}{k}g
IC: T(x,0)=f(x)
BC:
1.) T(0,t)=0
2.) Is there another boundary for the x=\infty?
Problem c)
Solution: \frac{1}{\alpha}\frac{\partial T}{\partial t}=\frac{\partial ^2 T}{\partial r^2 }+\frac{1}{k}g(r)
IC: T(r,0)=f(r)
BC:
k\frac{\partial T}{\partial r}=-hT
Problem d)
Solution: \frac{1}{\alpha}\frac{\partial T}{\partial t}=\frac{\partial ^2 T}{\partial r^2 }+\frac{1}{k}g(r)
IC: T(r,o)=f(r)
BC: T(b,t)=T'
3. Discussion
I do not know if I need another boundary condition on question b for the infinite side, I don't know what to do there. Also, c and d should be rather easy to express due to symmetry, but it almost seems too easy. Is there anything I am missing in my mathematical representations of the physical systems described? Any guidance would be very appreciated.