# Comparison of convection time - theoretical value vs computed

This is a follow up to my previous post here,

I'm solving for convection in 1D

$$\frac{\partial C}{\partial t} = - v\frac{\partial C}{\partial x}$$

The discretization of the above equation is presented using a backward difference formula, $$\frac{d C}{dt} = -v \frac{C_{i} - C_{i-1}}{\Delta x}$$

The boundary condition is imposed at the left end of the domain (i.e inlet)

$$\frac{d C}{dt}_{i=1} = -v \frac{C_{i} - C_{0}}{\Delta x}$$

and $$C_0$$ is specified for observing the transient change in C over a time span [0 tend].

Similar to the exponential kinetics

$$C(t) = C_{\infty} (1-\exp(-\frac{t}{\tau}))$$

presented for describing mass transport with diffusion mechanism, I'd to know if a similar expression can be derived for describing convection.

• The exact solution is simply translation of the initial data to the right at speed $v$. So yes, you can easily determine exactly how long it will take a signal to reach any given point. – David Ketcheson Feb 17 '20 at 5:36
• @DavidKetcheson In fact, any given function in the form of $C(x,t) = f(x-vt)$ satisfies that equation. Still, you are confused cause you are trying to evaluate your governing equation at the inlet node, which is not correct. – Alone Programmer Feb 17 '20 at 13:42
• @AloneProgrammer I'd to know if an expression similar to exponential kinetics can be derived for describing convection – Natasha Feb 26 '20 at 17:20
• @Natasha The answer is no, unless you provide more information. – Alone Programmer Mar 4 '20 at 14:44
• @AloneProgrammer Thanks a lot for the response. By more information do you mean boundary conditions? It's Dirichlet at the inlet. Please let me know if you are looking for other information. – Natasha Mar 5 '20 at 1:27