Time to cool molten metal pool?

[mas4444]

Searched through the forums and feel like I have a pretty unique problem that has stumped me for weeks. Have come up with a number of methods...all providing differing solutions.

Here goes:

We have a furnace that we are using to heat up aluminum to 2000 degrees C in a ceramic crucible. Below the crucible is a tantalum sheet large enough to contain a spill should the crucible break. We are trying to calculate a conservative equilibrium temperature should the full volume of aluminum spill onto the tantalum "tray." Conservatively, we assume that the spill will form a axially symmetric (cylindrical) pool on the tray. To keep things simple (and very conservative) we neglect convection to the surrounding air and only account for conduction in the cooling of the aluminum.

Does anyone have a method that they think best applies to this situation?

I have tried lumped capacitance and transient conduction in a semi-infinite solid, but having read in my old textbooks it seems like neither of these situations is very similar to this. I fear I am going to have to use some sort of iterative process as neither surface is being held at a constant temperature. I am fine assuming that the bodies are sufficiently thin to assume a constant temperature across both metals, just do not know what method really applies. Thanks for thinking!

 

[sailoday28]

Isn't this also a transient radial flow problem, with the spill falling to the center of the thin radial cylinder?

 

[IRstuff]

Radiation could be a big part of the solution.

What's under the tray? That'll have a big impact as well.

You can iterate by solving the immediate steady state problem for heat loss. Then apply that to the thermal mass to get a new temperature. Iterate.

TTFN

FAQ731-376

 

[zekeman]

Are you saying that the process will continue to heat in which case heat would be continually pouring into the crucible and you hope that the conductance through the sheet to ground will absorb the heat?


The conduction problem is the easier ( but not easy) part if you know the the initial temperatures of the structure underneath the sheet.
The radiation, which is very significant is more difficult to assess.


Also what do you mean by time to cool?

 

[dvd]

why 2000C for aluminum? What do you do with all of the vapor at that temperature?

 

[corus]

The equilibrium temperature, or steady state solution, will be that everything will be at room temperature, presuming that heat is lost to the room ambient temperature.

Why you neglect convection (and radiation) from the surfaces is a mystery, however if you're simply assuming that the tray is at one temperature and the molten metal is at another with no heat loss to the outside then the mean temperature will be the sum of Density*Specific Heat*Volume*Initial Temperature iof each, divided by the sum of Density*Specific Heat*Volume of each material. Strictly speaking you need to calculate the integral of the terms, but a mean value for the temperature dependent properties wil give you a rough answer.

corus

 

[mas4444]

In our current setup the radiation and convection are very difficult to determine accurately. Regardless, we are looking for a very conservative solution to time to reach equilibrium. This of course makes use of the assumption that the only form of heat transfer is by conduction between the aluminum and the tantalum tray. Maybe I should have posed the question like this (to avoid all of the "well why are you doing this" questions):

I take a hot puck of aluminum and lay it on a "cold" sheet of tantalum. Assuming this system is completely insulated on all sides and all energy exchanged is from the aluminum puck to the tantalum sheet, how long will it take for the two metals to come to equilibrium with each other. I can calculate the equilibrium temperature of interest by using a simple energy balance between the two. I am simply looking to find how long this energy exchange will take.

Again, I know that this is a VERY approximate solution to my situation, but I am not looking for an exact time to cool. I am looking for a ballpark figure. Without getting into exact numbers, does anybody have any input on what the governing rate equations might be for this situation. I appreciate your input!

 

[IRstuff]

In such a constrained problem, it's just the thermal conductivity.

TTFN

FAQ731-376

 

[zekeman]

mas4444,

You will have to give us an idea of the thickness of the tantalum sheet and the thickness of the aluminum pool.

 

[zekeman]

And we need the mass of the alumnum pool and the overall dimensions of the sheet of tantalum, since it appears that the heat will be getting out radially. A closed form solution can be developed involving Bessel functions after a few assumptions.

 

[corus]

The time taken to reach equilibrium, or steady state, is infinite. You'd have to be more specific as to what temperature would be acceptable as reaching 'almost' steady state.

corus

 

[zekeman]

Is the OP still around? Haven't heard from him lately.

 

[IRstuff]

He logged on today, but apparently had nothing to say...

TTFN

FAQ731-376

 

[mas4444]

I am still here. I am trying to avoid getting too bogged down in an exact solution. All I am looking for are governing equations for this type of situation. Again, take a hot brick and set it on a thin sheet of metal, insulate it from all thermal interaction with the environment. How long will it take for these to come to a thermal equilibrium with each other. Substitute m for an actual mass, cp for specific heat, etc. I understand that the literal solution is INFINITY, so ok how about 90% of the way to equilibrium for instance?

Maybe I am too simplistic, but it seems like the model described above is the most conservative assumption you can make. Any other interactions will simply decrease the time to equilibrium. I am only looking for a bounding solution.

 

[corus]

The problem is that if you say that the two objects are thin, or infinitely small, then the time to reach the equilibrium temperature will be zero. The best way to calculate this is to assume that you have heat flow in one dimension only and use finite difference or finite element methods to calculate the temperature change across a known thickness. The exact solution involves the summation of an infinite series and fourier analysis. With a 1D solution you can also add in heat transfer to the ambient to give a better solution.

corus

 

[zekeman]

4444,

For somebody looking for a handle ("ballpark solution" on an extremely difficult problem, your reticence amazes me.

Did it ever occur to you that sometimes you can't get a solution in closed form meaning that it may not be possible to give you a general solution in terms of the parameters you want. This is NOT F=Ma.

Now you must know that the next best thing to do is present actual estimated geometry of the given problem with temperature data so that somebody can give you the free advice you need.

To repeat, all you have to now do is give us the geometry of the tantalum sheet size the volume of the aluminum spill, the equilibrium temperature and let us handle the rest.

If we fail to give you satisfactory answers, what have you lost?

 

[vpl]

Perhaps the solution that mas4444 needs at this point is to hire a consultant. That way he can provide the specifics without fear of releasing proprietary information.

Patricia Lougheed

******

Please see FAQ731-376: Eng-Tips.com Forum Policies for tips on how to make the best use of the Eng-Tips Forums.

 

[zekeman]

My final answer is:

About 1 hour.

I leave it to the OP to figure out how I got there.

 

[chicopee]

From your original description, it would be hard to describe the cooling process since we don't know about the effects of the oven on the spilled Aluminum and how the tantalum sheet is supported. We can assume the spilled aluminum to be a thin sheet and the tantalum sheet to be insulated underneath and removed from the oven immediately after the breakage of the crucible to the open. You can then develop you heat transfer equation by assuming a constant temperature thru its thickness at any time. Internal heat loss= heat transfer by convection + radiation. I don't think that you'll have a direct solution but you can do a time step analysis on a spread sheet.