(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A solution of material A of concentration C_{0}kg/m^{3}flows into a tank at the flow rate F m^{3}/min. The tank has fixed volume V m^{3}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

C_{0}(t) = C_{init}for t<0 and

C_{0}(t) = C_{0}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) = C_{0}F - CF - kCV

dC/dt + (F/V+k)C = FC_{0}/V

b)

Let C_{P}(t)=A (constant)

0=(F/V+k)A=FC_{0}/V

=> A=FC_{0}/(F+Vk)

:. C(t) = FC_{0}/(F+Vk) + me^{-(F/V+k)t}for arbitrary m

Initially C(0)=FC_{0}/(F+Vk) + m =C_{init}

m=C_{init}-FC_{0}/(F+Vk)

:. C(t)=FC_{0}/(F+Vk) + (C_{init}-FC_{0}/(F+Vk))e^{-(F/V+k)t}, t>0

As t->infinity, C(t)->FC_{0}/(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 C_{0}? Or does the loss of material A into the atmosphere change that?

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# Homework Help: Modelling DE mixture problem

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