Fourier and Heat Conduction ????

[t_n_so...@hotmail.com]

In Boyce-DiPrima, on page 582, they do a non-homogenuous heat
conduction problem as an example.

At t=0, the temperature dist is:

U(x,0)=60-2x, 0<x<30

and at t=infinity, the temperature dist. is:

U(x,t=1000000000) = 20+x, 0<x<30.

However, there is an *increase* in the amount of energy here!!! We can
see this using a "common sense approach."

At t=0, the *average* temperature is halfway between the linear
distribution of 0 to 60 = 30.

At t=infinity, the *average* temperature is halfway between the linear
dist. of 20 and 50 = 35.

What has happened here? Did the ghost of Fourier decide to take his
revenge on all those evil mathematicians (like LaGrange) who debunked
him?

TNS

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sout...@aurora.mindsprings.org

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[t_n_so...@hotmail.com]

[Brooks Moses]

t_n_so...@hotmail.com wrote:
> In Boyce-DiPrima, on page 582, they do a non-homogenuous heat
> conduction problem as an example.
>
> At t=0, the temperature dist is:
>
> U(x,0)=60-2x, 0<x<30
>
> and at t=infinity, the temperature dist. is:
>
> U(x,t=1000000000) = 20+x, 0<x<30.
>
> However, there is an *increase* in the amount of energy here!!! We can
> see this using a "common sense approach."

[...]

> What has happened here? Did the ghost of Fourier decide to take his
> revenge on all those evil mathematicians (like LaGrange) who debunked
> him?

It would help greatly to know the boundary conditions....

However, since the final result is non-uniform, there must necessarily
be at least two boundary conditions which have either a specified
non-zero heat flux or a specified temperature, or some combination
thereof. If the temperature is specified at a point, then there is the
possibility of heat flux into that point to maintain this temperature.
This heat flux is of course an energy flux, and thus the energy in the
bar need not remain constant.

Hope this helps,
- Brooks

[t_n_so...@hotmail.com]

In article <7lukqr$276$1...@nnrp1.deja.com>,

t_n_so...@hotmail.com wrote:
> In Boyce-DiPrima, on page 582, they do a non-homogenuous heat
> conduction problem as an example.
>
> At t=0, the temperature dist is:
>
> U(x,0)=60-2x, 0<x<30
>
> and at t=infinity, the temperature dist. is:
>
> U(x,t=1000000000) = 20+x, 0<x<30.
>
> However, there is an *increase* in the amount of energy here!!! We
can
> see this using a "common sense approach."
>

> At t=0, the *average* temperature is halfway between the linear
> distribution of 0 to 60 = 30.
>
> At t=infinity, the *average* temperature is halfway between the linear
> dist. of 20 and 50 = 35.
>

> What has happened here? Did the ghost of Fourier decide to take his
> revenge on all those evil mathematicians (like LaGrange) who debunked
> him?
>

[t_n_so...@hotmail.com]

[t_n_so...@hotmail.com]

[t_n_so...@hotmail.com]