Rate of heat loss? Speed of heat.

Gerenuk

I have some extended aluminum part which is at low temperatures. It is connected by a thin alu rod to a cooling device held at a constant temperature.
How can I estimate the rate of cooling through this thin rod given the material specific constants?

I assume the heat exchange with the surrounding atmosphere is small, but wouldn't mind including it in my calculations.

The Green function is a particle diffusion function. So can I associate the particle/heat speed somehow with a velocity that would give me the "speed of heat"? Then for 1D or 3D I'd calculate the "distance travelled by heat" keeping in mind some diffusion correction depending on the dimension?!

John Creighto

It would seem to me the most important parameters are the specific heat of the aluminum part, and the thermal conductivity of the metal rod. It sounds like you are trying to kill a fly with a sledgehammer.

gel

You can't associate any speed to diffusions such as Brownian motion, because of the way they scale under rescaling time.

John Creighto

I'm not sure what you mean.

John Creighto

If we're talking about solids then the particles only vibrate. It is the phonons in the solid which transmit heat. I presume you could do something like radiative transfer but instead of light in a gas consider phonons in a solid, where the scattering would obay something like Beer's law.

gel

I mean that if X(t) is a Brownian motion then [itex]a^{-1}X(a^2t)[/itex] is also a BM. However, if there was some speed you could naturally associate with X, it should multiply by [itex]a[/itex] under the same transformation.

Equivalently, if f(t,x) is a solution of the heat equation, then so is [itex]f(a^2t,ax)[/itex], but velocities/speeds are not invariant under rescaling space & time like this, because velocity has units of length over time.

Edit:
that is, there is no fixed speed.
I suppose you could define a speed v such that [itex]df/dt+v\cdot\nabla f=0[/itex], but it would depend on f.

Cyrus

What does brownian motion have to do with this problem? This is basic heat transfer via conduction and convection. Just set up a thermal resistance network and solve. Jeesh.

gel

because probabilities for BM satisfy the heat equation. As the OP mentioned particle diffusions, I thought he must be refering to this.

Cyrus

I think hes trying to use that because it mentions particle speed, to somehow relate it to heat speed, when all he has to do is look at the heat transfer rate.

I could be wrong though. I read it as a solid aluminum bar being cooled on one end. No particle motions are necessary.

John Creighto

I'm not very good at vector aspects of physics. I did however, read though that the laplacian somehow represents diffusion. I'm not sure why but doesn't the wave equation also depend on the laplacian? I know nothing about how Brownian motion relates to the heat equation but I thought the poster might somehow be interested in transitive properties of heat transfer.

John Creighto

Isn't Brownian motion just a random walk? Was is that relevant to heat transfer given that energy is exchanged between particles?

gel

Yes, but the wave equation also has second order derivatives in time, unlike the heat equation which only has a first order time derivative, which makes the solutions very different.
If you have [itex] d^2f/dt^2=k\nabla^2 f[/itex], as in the wave equation, then k has units of velocity^2. In the heat equation you have [itex]df/dt=k\nabla^2 f[/itex] and k has units of length^2/time.

gel

Well you calculate heat flow with a monte carlo simulation. Don't know if anyone does this in physics, but you could. If you start a BM off randomly distributed according to the temperature at the start, then its probability density evolves in the same say as temperature does.

Integral

To the first order this is a simple problem in conduction, You need the thermal properties, cross sectional areas and the lengths of your various components. If you set up the more complex methods mentioned, which could be done, your solution will be very nearly the conduction equation.

Unless there is a large temperature difference to the atmosphere or you are blowing air across the surfaces, convective and radiative losses will be small to negligible.

Gerenuk

Good that at least someone understands the question. I can see that the Green function has a [itex]r^2/t[/itex] dependence, so it's diffusion. Then I wondered: "Do Brownian particles have a velocity associated with them"? If so, it should be possible to find a velocity for heat by analogy. Of course this velocity would have to be corrected to include the diffusion character.

Gerenuk

The difficulty is that that shape of the aluminium block on the rod is complicated or maybe to first approximation a plate.

I tried to solve the problem and assumed that the heat spreads instantly in this alu block and that along the rod there is always a constant gradient. I get an exponential solution for the temperature of the alu block.
[tex]T=A+Be^{-kt}[/tex]

Are these approximations OK?

GT1

You don't need to work hard - what you describe is a fin with a constant base temperature. the fin conducts heat and loses (or gains) heat to the environment by convection.
You can find the solution of the differential equation for this problem in any heat transfer book. all you need to do is to put your numbers and you have the answer.