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I'm having problems with third type heat transfer boundary condition

Y L
Posted 4 years ago
Attachments:
Untitled-3.nb
POSTED BY: Y L
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I did some more work on this problem. See the attached notebook.

Attachments:
diffbarnum2sides-a.nb
POSTED BY: Hans Dolhaine
Y L
Y L
Posted 4 years ago

Hans, I'm so grateful for your help!! Learned a lot from reading the notebooks you send me!
POSTED BY: Y L

In the 2nd part of the attached notebook ( after the 1st line of stars ) you will find an example (although with different values compared to your problem) how to find that series which is a solution to your problem using the method of Laplace-Transforms. I did it only for 0 <= x <= 5, as the problems is symmetric with respect to x.

If you have any question concerning the method please feel free to contact me.

Regards Hans

Attachments:
diffbarnum2sides.nb
POSTED BY: Hans Dolhaine

Super. As I said: NDSolve is full of mysteries. I have Version 7. And I think there is a theoretical solution to your problem, albeit in form of an infinite series. You may want to have a look in these books

Frieder Häfner, Dietrich Sames, Hans-Dieter Voigt, Wärme- und Stofftransport Springer Verlag, Berlin 1992

https://www.amazon.de/s?k=frieder+h%C3%A4fner+w%C3%A4rme&__mk_de_DE=%C3%85M%C3%85%C5%BD%C3%95%C3%91&ref=nb_sb_noss

Horatio Scott Carslaw, John Conrad Jaeger, Conduction of Heat in Solids, 2nd Edition Oxford Science Publications, Oxford University Press 1959

https://www.amazon.de/Conduction-Solids-Oxford-Science-Publications/dp/0198533683/ref=sr_1_5?__mk_de_DE=%C3%85M%C3%85%C5%BD%C3%95%C3%91&dchild=1&keywords=carslaw+jaeger&qid=1632414324&sr=8-5

Both contain lots of solved problems.
POSTED BY: Hans Dolhaine

I don't know what is happening. I attach a notebook with my code, look at the plot (before running the notebook). It contains D[u[t, x], x] /. x -> 5, and gives a reasonalbe result.

Attachments:
diffbarnum2sides.nb
POSTED BY: Hans Dolhaine
Y L
Y L
Posted 4 years ago

Hi Hans! I'm not sure what happened wither, probably different versions of Mathematica? I ran your code and the same wierd result replaced your original one, but when I increase the MinPoints from 100 to 200 it worked! Thank you!
POSTED BY: Y L

NDSolve is full of mysteries. I learnt that sometimes forcing something to be zero for a (very) short time can help. You could try this.

h = 13/(1000^2);(*units are converted to mm*)k = 0.3/1000;
c = 1030;
rou = 1380/(1000^3);
heq = rou*c*D[u[t, x], t] == k*D[u[t, x], {x, 2}];
ic = u[0, x] == 275;
bcs1 = (D[u[t, x], x] /. x -> 5) == -h*(u[t, 5] - 25) (1 - Exp[-30 t]);
bcs2 = (D[u[t, x], x] /. x -> -5) == 
   h*(u[t, -5] - 25) (1 - Exp[-30 t]);
sol = NDSolve[{heq, ic, bcs1, bcs2}, u[t, x], {t, 0, 20}, {x, -5, 5},
   Method -> {"MethodOfLines", 
     "SpatialDiscretization" -> {"TensorProductGrid", 
       "MinPoints" -> 100}}];
F[t_, x_] = u[t, x] /. sol[[1]];
Plot3D[F[t, x], {x, -5, 5}, {t, 0, 15}, PlotRange -> All, 
 AxesLabel -> Automatic]
POSTED BY: Hans Dolhaine
Y L
Y L
Posted 4 years ago

Hi Hans! Thank you so much for your help! But unfortunately, it still didn't work. So strangely, by implementing boundary conditions like this: 

bcs1 = D[u[t, 5], x] == -h*(u[t, 5] - 25), 
bcs2 = D[u[t, -5], x] == -h*(u[t, -5] - 25)

the result looks reasonable. It looks like this: enter image description here

with the temperature at left/right side gradually decreasing. BUT I also learned from somewhere else that it's incorrect to implement boundary conditions like this: 

D[u[t, 5], x] == -h*(u[t, 5] - 25)

because in D[u[t, 5], x], the u[t, 5] is no longer a function of x, so instead the derivative in boundary condition should be like D[u[t, x], x] /. x -> 5. With the boundary condition like D[u[t, x], x] /. x -> 5, I tried what you suggested, but unfortunately, the result still looks erroneous enter image description here may I ask if you can give me some more help on this? I'm really appreciated for your help!
POSTED BY: Y L
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