HT2D was developed to study the heat transfer in two dimensions with prescribed temperatures at the boundaries. In each boundary there are four functions that can describe the evolution of temperature over time. Code: http://earc96.vprc.net/

1. Heat Transfer (2D) - HT2D
Emanuel Camacho
Objective
HT2D was developed to study the heat transfer in a material that is isolated both up and down
and has prescribed temperatures at the edges.
Mathematical Introduction
The heat equation describes the distribution of heat over time and for a three dimensional
space plus the time, the heat equation is:
∂T
∂t
= α
∂2
T
∂x2
+
∂2
T
∂y2
+
∂2
T
∂z2
where α is the thermal diffusivity.
Since this program was just programmed for two directions in a non steady state, the heat
equation is now:
∂T
∂t
= α
∂2
T
∂x2
+
∂2
T
∂y2
In this program, the finite difference method was used to resolve these types of problems. After
some mathematical manipulation which included expanding the Taylor series around T(x, y, t), we
can conclude that:
∂T
∂t
=
Tn+1
i,j − Tn
i,j
∆t
∂2
T
∂x2
=
Tn
i+1,j − 2Tn
i,j + Tn
i−1,j
(∆x)2
&
∂2
T
∂y2
=
Tn
i,j+1 − 2Tn
i,j + Tn
i,j+1
(∆y)2
So,
Tn+1
i,j − Tn
i,j
∆t
= α
Tn
i+1,j − 2Tn
i,j + Tn
i−1,j
(∆x)2
+
Tn
i,j+1 − 2Tn
i,j + Tn
i,j−1
(∆y)2
Tn+1
i,j = Tn
i,j + α∆t
Tn
i+1,j − 2Tn
i,j + Tn
i−1,j
(∆x)2
+
Tn
i,j+1 − 2Tn
i,j + Tn
i,j−1
(∆y)2
The equation above is stable if:
α∆t
min {∆x2, ∆y2}
≤
1
2