Heat Conduction Initial Problem Set Up

[timman_24]

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.

 

[timman_24]

Any thoughts?