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 boundry 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?