Solidification

[abanerj5]

Dear forum friends,

I am a metallurgist who is facing a very difficult problem in HT.

I have to find an analytical solution for the temperature profiles in a plate/slab (casting mould in a slab shape) filled with liquid with no turbulence, basically a one dimensional transient conduction problem of solidification of a liquid metal or alloy, with a moving boundary. It is also known as Stefan's problem. Does anyone have any idea about this topic? The solutions i found in text books are for semi-infinite slabs but my case is of a finite slab with the symmetry condition at the centerline. so the solution to the heat equation for one dimension has to satisfy an additional boundary condition of dT/db(at x=L)=0.

If anyone has any idea then please respond and I can describe it in detail. I am really confused and any help will be greatly appreciated.
I am exhausted reading these textbooks and it leads me no where.
Thanks
Arry

 

[zekeman]

I would think you almost have too use numerical methods for this.

If you have the semi-infinite solution, then very often , if you add a semi-infinite solution in the opposite direction at x= x+2b ,where b= slab thickness you satisfy dT/dx=0, but it will mess up if the temperature wave reaches x=0, then you have to add another mirrored solution. If your solution by this method gives you results before the boundaries are undone, you may have something.

 

[prex]

As the exact solution to the semi infinite slab is trascendental, it requires anyway a numerical solution. So why not simply starting from scratch with a numerical (finite differences) solution?
Here you can find a treatment of numerical solutions for your problem (see slide n.10).

prex
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[zekeman]

"..opposite direction at x= x+2b ,where b= slab thickness"

should read
opposite direction at x= 2b ,where the original slab is 0<x<b and the new "slab" is 0<x<2b which by symmetry assures
dT/dx=0 at x=b

 

[abanerj5]

Dear All,
Thank you for your help and I am sorry for not replying earler.
1. I didnt quite understand the solution when you said opposite direction with x=2b. Could you explain the whole thing in detail please? Just need a solution for the finite slab with symmetry condition at the centerline.
2. The reason I dont consider numerical solutions is because it doesnt take latent heat (source) term into consideration. Do u have a numerical solution where they will consider L(latent heat) along the interface? Please help me and I am sorry if I may be repeating myself I am not a Heat or Thermal guy and might not understand what you mean at first. I really appreciate the replies and your help guys.
Thank you