# Homework Help: Modelling DE mixture problem

1. Aug 14, 2010

### EnSlavingBlair

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

A solution of material A of concentration C0 kg/m3 flows into a tank at the flow rate F m3/min. The tank has fixed volume V m3 and is continuously stirred. The solution on the tank is of concentration C and flows out the top of the tank at the flow rate F. The material A in solution in the tank reacts with air and breaks down such that the amount present in the tank decreases at the rate kCV km/min, where k is constant. At time t=0, the inlet concentration undergoes a step change given by

C0(t) = Cinit for t<0 and
C0(t) = C0 for t>0

a) Write the balance equation for material A in the tank.

b) Find the solution C(t) of the model. What is the time constant and the limiting concentration in the tank?

2. Relevant equations

Rate of change of mass = (rate in) - (rate out)

3. The attempt at a solution

a)
d/dt(CV) = C0F - CF - kCV

dC/dt + (F/V+k)C = FC0/V

b)
Let CP(t)=A (constant)

0=(F/V+k)A=FC0/V

=> A=FC0/(F+Vk)

:. C(t) = FC0/(F+Vk) + me-(F/V+k)t for arbitrary m

Initially C(0)=FC0/(F+Vk) + m =Cinit

m=Cinit-FC0/(F+Vk)

:. C(t)=FC0/(F+Vk) + (Cinit-FC0/(F+Vk))e-(F/V+k)t, t>0

As t->infinity, C(t)->FC0/(F+Vk)

My problem is the very last part, or at least, that's where I noticed I had a problem. Shouldn't the concentration approach C0? Or does the loss of material A into the atmosphere change that?