# Recirculating DE Mixture Problem

1. Apr 29, 2010

### The0wn4g3

1. The problem statement, all variables and given/known data

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

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

3. 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.

2. Apr 29, 2010

### The0wn4g3

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.

3. May 1, 2010

### The0wn4g3

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