# Need feedback if this is possible (first order removal based model)

## Main Question or Discussion Point

Please pardon my lack of understanding here. I am a scientist with just limited experience in calculus and de (long time ago too), but I am willing to learn.

I'm in the process of working through a theoretical model for a wastewater treatment plant that removes constituents assuming first order kinetics:

$\frac{d[c]}{dt} = -k[c]$

For convenience, here is the integrated form:

$[c]=[c_{o}]e^{-kt}$

I would like determine the effects of evaporation on removal of the constituent as it moves through the system. Based on simple chemistry, evaporation will have two effects on the concentration of C.

First, as water is evaporated and C is left behind, the concentration of C will increase due to simple enrichment.

But, here is where it gets more tricky, as water is evaporated, the retention time of C will increase within a fixed control volume receiving a constant flow since hydraulic retention time can be written as follows:

$t=\frac{V}{Q}$

As you can see from the above equation, as the flowrate is decreased, the time for the first order reaction increases and removal is increased.

Now I'm trying to find a way to come up with a set of models to determine the different effects of evaporation on removal of C.

1. normal removal (done, lol):

$[c]=[c_{o}]e^{-kt}$

2. enrichment (no removal)

3. removal and enrichment

4. removal and the lengthened t caused by evaporation

5. removal and enrichment and lengthened t caused by evaporation

I'm having a hell of a time with this... One of the problems is that everything involving the rate equation is based on time observations (lagrangian I believe would be the term), but since I am interested in [C] at a distance from the inflow and time is no longer related to distance as water is removed (t=V/(Q-Evaporation*Area) like it would be in a problem were Q is conserved, I'm not sure how to solve this.

What I have done is create some discretized forms of the equations based on simple substitutions into the integrated first order equation (is this even a legitimate mathematical operation?).

Here's what I have.... All equations are based on a simple 1-d discretized model with 100 cell blocks (i from 1-100) 1m apart and a unit surface area of 1m^2.

I first calculated the flow rate leaving each cell assuming that flow through each cell was decreased by Evaporation*Area.

For i from 1 to 100
$Q_{i}=Q_{i-1}-E*A$

Then my enrichment term (concentration leaving each cell assuming no removal other than simple enrichment) is simply going to be the flow entering the cell divided by the flow leaving:

2. $C_{i}=\frac{Q_{i-1}}{Q_{i}}C_{i-1}$

To add the combined effects of enrichment and removal (no modified t due to evaporation), I just combined the terms together:

3. $C_{i}=\frac{Q_{i-1}}{Q_{i}}C_{i-1}e^{-kt}$

To calculate the removal enhancement caused by t increasing due to evaporation, but no enrichment, I used the equation below to calculate $t_{i}$ for each cell i.

$t_{i}=\frac{2V}{Q_{i}+Q_{i-1}}$

And then substituted into the integrated rate law equation:

4. $[c]=[c_{o}]e^{-kt_{i}}$

And finally a combination of enrichment and modified t:

5. $C_{i}=\frac{Q_{i-1}}{Q_{i}}C_{i-1}e^{-kt_{i}}$

Are these valid operations?

I'm really at a loss here. I assume there is some other way to do this, but I need some help. If this is in the wrong section please let me know.

Thanks