HT1D was developed to study the heat transfer in a beam that is isolated both up and down and has prescribed temperatures at the tips.

1. Heat Transfer (1D) - HT1D
Emanuel Camacho
Objective
HT1D was developed to study the heat transfer in a beam that is isolated both up and down
and has prescribed temperatures at the tips.
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
Since this program was just programmed for one direction in a non steady state, the heat
equation is now:
∂T
∂t
= α
∂2
T
∂x2
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, t), we
can conclude that:
∂T
∂t
=
Tn+1
i − Tn
i
∆t
∂2
T
∂x2
=
Tn
i+1 − 2Tn
i + Tn
i−1
(∆x)2
So,
Tn+1
i − Tn
i
∆t
= α
Tn
i+1 − 2Tn
i + Tn
i−1
(∆x)2
Tn+1
i = Tn
i + α
∆t
(∆x)2
Tn
i+1 − 2Tn
i + Tn
i−1