Transport phenomena: Gauss error function and transient heat transfer

The Gauss error function (Equation 1) appears when you want to solve transient heat transfer problems.

$$ erf(x) = \frac{2}{\sqrt{\pi}} \int{e^{-x^2}} dx \tag{1} $$

The complementary error function is:

$$ erfc(x) = 1 - erf(x) \tag{2} $$

An example of its use is in solving the heat equation for a semi-infinite solid.

$$ \frac{\partial^2{T}}{\partial{x^2}} = \frac{1}{\alpha} \frac{\partial{T}}{\partial{t}} \tag{3} $$

T is the temperature, x the length (of a plate or soil depth for example), $\alpha$ is the thermal diffusivity and t the time. The solution to this equation involves using a similar variable $ \eta \equiv \frac{x}{{4 \alpha t}^{1/2}} $. Derivatives of $\eta$ are calculated and replaced in Equation 3. After changing the variable, the partial differential equation is transformed into an ordinary differential equation that, when integrated, asks for the solution of the integral of Equation 1. The solution of Equation 3 is:

$$ \frac{T(t)-T_s}{T_i-T_s} \equiv erf (\eta) \tag{4} $$

Where T(t) is the temperature in time t, $T_s$ the surface temperature (constant and different from the initial temperature) and $ T_i $ the initial temperature.

The development, boundary conditions used and table with the values ​​of the erf(x) and erfc(x) functions are presented in detail by Incropera ¹.

References Link to heading