2D Heat Tranfer
This MATLAB GUI presents 2D heat transfer in a plate.
The general equation for 2D is given by
ߩܿ௣
߲ܶ
߲ଶܶ ߲ଶܶ
= ߣ ቆ ଶ + ଶቇ
߲‫ݐ‬
߲‫ݔ‬
߲‫ݕ‬
We try to solve it numerically:
Time derivation :
߲ܶ ܶ௜௧ାଵ − ܶ௜௧
=
∆‫ݐ‬
߲‫ݐ‬
- x direction derivation
௧
௧
௧
− 2ܶ௜,௝
+ ܶ௜ାଵ,௝
߲ ଶ ܶ ܶ௜ିଵ,௝
=
ሺ∆‫ݔ‬ሻଶ
߲‫ ݔ‬ଶ
- y direction derivation
௧
௧
௧
− 2ܶ௜,௝
+ ܶ௜,௝ାଵ
߲ ଶ ܶ ܶ௜,௝ିଵ
=
ሺ∆‫ݕ‬ሻଶ
߲‫ ݕ‬ଶ
And the equation is as follows
ߩܿ௣
௧ାଵ
௧
ܶ௜,௝
− ܶ௜,௝
∆‫ݔ‬
= ߣቆ
௧
௧
௧
௧
௧
௧
ܶ௜ିଵ,௝
− 2ܶ௜,௝
+ ܶ௜ାଵ,௝
ܶ௜,௝ିଵ
− 2ܶ௜,௝
+ ܶ௜,௝ାଵ
+
ቇ
ሺ∆‫ݔ‬ሻଶ
ሺ∆‫ݕ‬ሻଶ
We assume ∆‫ ݕ∆ = ݔ‬:
௧ାଵ
=
ܶ௜,௝
௧
௧
௧
௧
௧
ܶ௜ିଵ,௝
+ ܶ௜ାଵ,௝
+ ሺ‫ ܯ‬− 4ሻܶ௜,௝
+ ܶ௜,௝ାଵ
+ ܶ௜,௝ିଵ
‫ܯ‬
where ‫ ܯ‬is Modul and is given by
ߩܿ௣ ሺ∆‫ݔ‬ሻଶ
‫=ܯ‬
ߣ∆‫ݐ‬
The stability condition have to be satisfied:
‫≥ܯ‬4
Using the boundary conditions
q = −λ
Ti ,t j −1 − Ti t, j
∆x
so
Ti ,t j −1 − Ti ,t j = 0
Ti ,t j −1 = Ti ,t j
Using the GUI
=0
USING the GUI
The material properties, initial temperature, temperatures on the edges, and some additional
parameters are required to be set up. (There are some initial values in the program).