(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I need to set up the mathematical formulation of the following heat conduction scenarios:

a) A slab in [itex]0\le x \le L[/itex] 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 [itex]0\le x \le \infty[/itex] 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 [itex]0\le r \le b[/itex] 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 [itex]0\le r \le b[/itex] 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:

[itex]\frac{1}{\alpha}\frac{\partial T}{\partial t}=\frac{\partial ^2 T}{\partial x^2 }[/itex]

IC: T(x,0)=f(x)

BC:

1.) Insulated Boundary:[itex]q\prime(0,t)=0[/itex]

2.) Convection Boundary:[itex]k\frac{\partial T}{\partial x}|_L =-hT(L,t)[/itex]

Problem b)

Solution: [itex]\frac{1}{\alpha}\frac{\partial T}{\partial t}=\frac{\partial ^2 T}{\partial x^2 }+\frac{1}{k}g[/itex]

IC: T(x,0)=f(x)

BC:

1.) T(0,t)=0

2.) Is there another boundry for the [itex]x=\infty[/itex]?

Problem c)

Solution: [itex]\frac{1}{\alpha}\frac{\partial T}{\partial t}=\frac{\partial ^2 T}{\partial r^2 }+\frac{1}{k}g(r)[/itex]

IC: T(r,0)=f(r)

BC:

[itex]k\frac{\partial T}{\partial r}=-hT[/itex]

Problem d)

Solution: [itex]\frac{1}{\alpha}\frac{\partial T}{\partial t}=\frac{\partial ^2 T}{\partial r^2 }+\frac{1}{k}g(r)[/itex]

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.

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# Heat Conduction Initial Problem Set Up

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