# computing time scale and steady state concentration in microfluidic channels

I have been performing convection-diffusion transport studies on microfluidic channels like the following

The inlet concentration is specific and I obtain the time-dependent concentration profiles of the solute at the discretization nodes in the domain. Using this data, I would like to know how to compute the time scale required for the system to reach steady-state concentration at the channel outlet.

image source

EDIT1: The concentration profiles are obtained after performing numerical simulations. The equation that is solved is the 1D convection-diffusion equation

EDIT2: Yes, I consider an incompressible flow. As suggested below, I could compute Peclet number(Pe) to find out whether convective transport dominates or advective transport dominates. But I am not sure whether Pe number has to be determined for each branch. Pe for mass transport is given by, $$Pe = \frac{u L}{D}$$

i.e is L the effective length of the whole channel?. $$D$$ is the diffusion coefficient. What happens when convection-dispersion is studied? Can $$D$$ in the expression for Pe be replaced with the dispersion coefficient which is a function of diameter?

Is the diameter halved at every bifurcation point? I'm sorry if the image that I've used for illustration gives that impression. In the actual system, the diameters aren't halved at a junction point (which could be a bifurcation or trifurcation point). The ratio of diameters of the branches aren't constant always. The diameters range from 2-10 micrometers and lengths range from 50-150 micrometers. The diffusion coefficient is ~1E-10 $$m^2/s$$

EDIT3: From the explanations provided in the comments below and this post I understand the characteristic length scale is the smallest length scale.

For the characteristic length within the Peclet number you may use your discretization width, or possibly the width of your channel.

I would like to understand if the choice of the characteristic length(discretization length or the diameter channel) depends on the direction of diffusive transport or is it always the smallest length irrespective of whether diffusion occurs in the radial/axial direction.

• When you say you conducted convection-diffusion studies, do you mean the real world experiments, or numerical studies? For numerical approaches, what are the equations you want to solve for this setup? – MPIchael Sep 2 '20 at 11:48
• @MPIchael Please check my edit. – Natasha Sep 2 '20 at 12:14
• Do you have an incompressible flow $\nabla b = 0$? If it is water you are pumping through these channels, then that should be the case. If the advective transport is a lot faster than the diffusive one (high peclet number), then you might estimate the time by deviding the approx. length of the channel by the advective transport speed. – MPIchael Sep 2 '20 at 12:48
• If it is mostly diffusive transport, than the equilibration time might actually be infinite. Can you provide a bit more info about the parameters of the convection diffusion eq.? Are all the tubes the same diameter? When the tubes split, are the two smaller ones half the area? – MPIchael Sep 2 '20 at 12:49
• I don't think the Peclet Number is in any way depent on the direction. I think you may interpret the expression as: $Pe = \frac{|u| L }{|D|}$ – MPIchael Sep 10 '20 at 13:50