1D Heat Transfer Model

The one-dimensional transient heat conduction equation without heat generating sources is given by:

$$\rho c_p \frac{\partial T}{\partial t} = \frac{\partial}{\partial x}\left(k\frac{\partial T}{\partial x}\right)$$

where $\rho$ is the density, $c_p$ heat capacity, $k$ thermal conductivity, $T$ temperature, $x$ distance, and $t$ time. If $\rho$, $c_p$ and $k$ are constant then the equation can be simplified to:

$$\frac{\partial T}{\partial t} = \kappa\frac{\partial^2 T}{\partial x^2}$$

The numerical solution to this equation is:

$$ T_i^{n+1} = T_i^n + \kappa \Delta t\left( \frac{T_{i+1}^{n}-2T_{i}^{n}+T^{n}_{i-1}}{\Delta x^2} \right) $$

Further details of its derivation can be found here.

An example of this in practice is where a thin body with thermal conductivity $\kappa$ (e.g. rod or laminate) is at a starting temperature of $T_s$, is heated at both ends at a temperature $T_b$ and insulated along its length $L$.

Below is an implementation of the numerical solution that determines the time for the body to reach the desired temperature.