Time to cool molten metal pool?
Time to cool molten metal pool?
(OP)
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!
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!





RE: Time to cool molten metal pool?
RE: Time to cool molten metal pool?
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
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RE: Time to cool molten metal pool?
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?
RE: Time to cool molten metal pool?
RE: Time to cool molten metal pool?
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
RE: Time to cool molten metal pool?
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!
RE: Time to cool molten metal pool?
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RE: Time to cool molten metal pool?
You will have to give us an idea of the thickness of the tantalum sheet and the thickness of the aluminum pool.
RE: Time to cool molten metal pool?
RE: Time to cool molten metal pool?
corus
RE: Time to cool molten metal pool?
RE: Time to cool molten metal pool?
TTFN
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RE: Time to cool molten metal pool?
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.
RE: Time to cool molten metal pool?
corus
RE: Time to cool molten metal pool?
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?
RE: Time to cool molten metal pool?
Patricia Lougheed
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RE: Time to cool molten metal pool?
About 1 hour.
I leave it to the OP to figure out how I got there.
RE: Time to cool molten metal pool?