Mass transfer coefficient from a numerical model

Summary
A numerical model has been made to obtain the relation between Sh and Re for different velocity profiles. How can Sh be expressed though, since the mass transport is not able to be expressed, or can it be left out? Without it, a negative power exponential relation is found.
Dear all,

For an assignment, I am trying to find the relationship between the Sherwood number and the Reynolds number in a channel for different laminar velocity profiles, where there is a concentration of a species at both the top and bottom wall which is transported to the fluid. For this, I solve a convection-diffusion problem. Now what it basically comes down to is find the relation between the mass transfer coefficient (the relevant parameter in the Sherwood number) and the average velocity of the fluid (the relevant parameter in the Reynolds number). However, the mass transfer coefficient depends on the mass transport and the concentration gradient, the driving force. The concentration gradient can be obtained by taking the difference between the concentration of the wall and the first layer of the liquid in steady state. My question is now, how do I determine the mass transport of the species exactly? Or would it be sufficient to just take: k (the mass transfer coefficient) = proportional to 1/average concentration gradient? This is what I took for now, but if I plot this, I get a negative power relation between average velocity and k. I would expect to find a positive power relation between 0 and 1. Could anybody help me with this? Thanks in advance!
 
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The mass transfer coefficient is based on the difference between the concentration at the wall and the mixing cup average concentration: $$D\left(\frac{\partial C}{\partial z}\right)_{wall}=k(C_{wall}-\bar{C})$$For laminar flow, this is also going to be a function of distance along the channel.
 

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