# [Physics] Make an equation dimensionless

dimensional analysisfluid dynamics

Suppose the concentration $c(x,y,t)$ inside a reactor evolves the following
$$\frac{\partial c}{\partial t}=D\frac{\partial^2 c}{\partial x^2}-kc ,$$
where $D$ is the diffusion coefficient and $k$ is the reaction rate.
If the characteristic length of the reactor is $L$, how does one make the equation dimensionless and identify important dimensionless parameters?

I imagine that the "new" unit of length must be $L$, however, I don't know how to continue (by the way, I don't fully understand what exactly this implies).

The problem seems to contain three dimension parameters, $D$, $k$ and $L$. The first thing would be to determine what the typical scale parameters are. You need a length scale $x_0$ and a time scale $t_0$. The options for the length scale are $x_0=L$, $x_0=D/k/L$ or $x_0=\sqrt{D/k}$ and those for the time scale are $t_0=1/k$, $t_0=L^2/D$ or $t_0=L/\sqrt{kD}$. To determine which one to choose, one needs to think a little about the physics.
Next, one convert the function (concentration) into a function of dimensionless variable $$c(x,t) \to c_0(u,v) = c_0\left(\frac{x}{x_0},\frac{t}{t_0}\right) .$$ Then we plug this back into the equation. The result is $$\frac{1}{t_0} \partial_v c_0(u,v) = \frac{D}{x_0^2}\partial_u^2 c_0(u,v) - kc_0(u,v) .$$ Based on this result, it seems that the appropriate choice for the scale parameters would be $x_0=\sqrt{D/k}$ and $t_0=1/k$, because then the dimensionless of the equation becomes completely independent of any parameters: $$\partial_v c_0(u,v) = \partial_u^2 c_0(u,v) - c_0(u,v) .$$