# Recirculating DE Mixture Problem

## Homework Statement

A rigid tank of volume "V" contains salt dissolved in water at a concentration "C". Fresh water flows into the tank at a rate of "F". A fraction "x" of the exit flow is recirculated back into the entrance flow.

Given:
V, Volume of tank
Fin, Flow rate in
Fo, Flow rate out
x, Fraction of volume flow-rate recirculated
Co, Initial concentration
C(t), Concentration in tank at time t

## Homework Equations

$$\frac{dC}{dt}=\text{concentration of salt entering}-\text{concentration of salt leaving}$$

$$C_{in}=xFC(t)$$

$$C_{out}=(1-x)FC(t)$$

so $$\frac{dC}{dt}=xFC(t)-(1-x)FC(t)$$

## The Attempt at a Solution

$$\frac{dC}{dt}=xFC(t)-(1-x)FC(t)$$

When I plot this using Euler's Method, I get this graph. This is obviously incorrect, because it does not reflect exponential decay. Can someone help me out? I'm really not sure what I've done wrong.

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I may have found my own mistake. I totally forgot to take into account to volume of the tank. I believe this wound change the Flow Out equation to:

$$C_{out}=(1-x)FC(t)$$

so the final equation would be:

$$\frac{dC}{dt}=xFC(t)-\frac{(1-x)FC(t)}{V}$$

is this correct?

With some randomly chosen values (V=10,000 ft^3/s F=10 ft^3 x=.2 Co=.8) I get this, which seems much more reasonable.

Anyone want to qualify my answer here? Should I re-post this somewhere else? Hello?