convective heating of a sphere 3

[laLuz]

thread997-113741
I have the same exact problem of the referenced post above and did not see anyone post any direct answer to the problem so I'm reposting. I am trying to heat a sphere by forced convection and am tracking the temperature within the sphere (lumped capacitance not valid). I am using finite differencing method in Matlab to solve the problem. It is a 1D problem with radial temperature gradient and for some reason the sphere is heating slowly and after a certain period of time the temperature does not change at all. I am using a convective boundary condition, -k(dT/dr) = h(Tfluid-Tsurface) as well as the insulated boundary condition at the center of sphere: (dT/dr) = 0. I have a feeling that this has something to do with the way that I discretize the convective boundary condition. I have tried several methods and none seem to work. For example, I've tried:
-(k(Tsurface-T(node2))/dr) = h(T(fluid) - Tsurface). I then rearrange to solve for Tsurface.
I've been working on this for a few months and getting no where. Any help would be GREATLY appreciated.

 

[IRstuff]

IS this for school?

TTFN
faq731-376

Need help writing a question or understanding a reply? forum1529

 

[laLuz]

Hi,
No this is not for school. It is a research problem.
Thanks.

 

[ione]

For Fourier number Fo > 0.2 you can use the following approximate expression

[T(r,t) - Tinf]/[Ti - Tinf] = A1*exp(-λ1^2*Fo)*[sin (λ1*r/ro)]/(λ1*r/ro)

where r0 is the radius of the sphere and A1 and λ1 are tabulated values which are a function of Biot number (see attachment).

 

[prex]

You should state exactly what you do for us to help. For example I don't see how you treat the time variable and how you discretize the Fourier equation in spherical coordinates.
The correct discretization of the boundary equation should be k(T₂r₂²-T₁r₁²)/(r₁-r₂)=h(T_f-T₁)r₁², but this, for a small spatial step, is equivalent to yours.
And of course if the steady final temperature equals T_f, that is normal, otherwise you have a big problem in the discretization.

prex
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[laLuz]

Hi Prex,
I use a constant spatial step size by defining a set number of grid points and then divide the radius of the particle by (grid points-1) to get deltaR. I then use this deltaR value to determine the time step size using the stability criteria that the Fourier number must be less than 1/2. This ensures convergence. I should note that I am solving this explicitly and am wondering if it needs to be solved implicitly. Any help would be appreciated and thanks for the reply ione.

 

[prex]

OK, but again: you told us nothing about the details of the discretized equations and what you get from them. So I don't know what to comment.
About explicit or implicit, this cannot change the result, if both are convergent. By the way, if you mean explicit in time, than that's absolutely normal.

prex
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[racookpe1978]

But you are specifying the delta T/dr at the center of the sphere = 0, then also specified delta T/dr at the edge of the sphere = heat transfer across the boundary layer.

But if you are not always changing (allowing to change) the delta T/dtime throughout the sphere - then the equations will "stop" and NOT change temperatures any more with respect to time.

 

[laLuz]

Hi, I attached discretization of the energy equation for the sphere and the boundary conditions. I hope this is what you were looking for prex.

racookpe1978, I believe that I am always changing deltaT/dtime (please see attached) but I must be doing something wrong because the equations are doing exactly as you say. The temperatures at and near the surface initially increase quickly, start to slow down and then eventually the sphere stops heating completely. The nodes close to the center of the sphere do not change in temperature at all so the heat is not reaching the inner region. Any help on this would be greatly appreciated.
Thanks

 

[prex]

Again you didn't show us what exactly you did: the mistake must reside in how you actually wrote the discretized equations in Matlab.
Anyway, you'll find the solution in the Excel file in annex. It is a correct solution, as the estimated time to reach half the temperature difference is correctly checked in the time behavior. You also see the graph of the average temperature versus time.
Note that I used a fixed time step. Both solutions, in space and in time, are fully explicit (no matrix inversion required). The solution in space could be turned into implicit by using the iterations in Excel, but this doesn't change much.
Hope you'll be able to decode and understand it.

prex
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[laLuz]

Wow, this helped me out tremendously. I had one term switched in the energy equation. Thank you very, very much prex. This was a big headache for me so I'm definitely going to make a donation to the forum.

 

[ione]

Dear Mr. Presciuttini,

You've been really too kind with laLuz: I'll give you the star you deserve on his/her behalf.

 

[yukipilas]

Hello Everbody,

Sorry for bombarding your Thread wih my own issue. I have a similar problem to as of the convective heating of the sphere (

http://www.eng-tips.com/viewthread.cfm?qid=364897)

which i have been dealing with since a couple weeks now with no success, as you can see on my post the replies have not been very helpful. My problem deals with the progressive heating of a cylinder.

I checked the file that prex posted and seems to go on the direction that i will also like to go to, I am still missing fundamental knowledge on using FEM for this, without trolling your thread i would like to invite you to visit my thread:

http://www.eng-tips.com/viewthread.cfm?qid=364897

I appreciate your help,

kindest regards,