1 problem found
In the manufacture of Grandma's Home Made Ice-cream, chemicals \(A\) and \(B\) pour at constant rates \(a\) and \(b-a\) litres per second (\(0 < a < b\)) into a mixing vat which mixes the chemicals rapidly and empties at a rate \(b\) litres per second into a second mixing vat. At time \(t=0\) the first vat contains \(K\) litres of chemical \(B\) only. Show that the volume \(V(t)\) (in litres) of the chemical \(A\) in the first vat is governed by the differential equation \[ \dot{V}(t)=-\frac{bV(t)}{K}+a, \] and that \[ V(t)=\frac{aK}{b}(1-\mathrm{e}^{-bt/K}) \] for \(t\geqslant0.\) The second vat also mixes chemicals rapidly and empties at the rate of \(b\) litres per second. If at time \(t=0\) it contains \(L\) litres of chemical \(C\) only (where \(L\neq K\)), how many litres of chemical \(A\) will it contain at a later time \(t\)?
Solution: The total volume in the first vat at time \(t\) is always \(K\), since \(b\) litres per second are coming in and \(b\) litres per second are going out. \begin{align*} &&\frac{\d V}{\d t} &= \underbrace{a}_{\text{incoming chemical }A} - \underbrace{b}_{\text{outgoing volume}} \cdot \underbrace{\frac{V(t)}{K}}_{\text{fraction of outgoing which is }A} \\ &&&= a - b \frac{V}{K} \\ \Rightarrow && \int \frac{1}{a-b\frac{V}{K}}\d V &= \int \d t \\ && - \frac{K}{b} \ln |a - b \frac{V}{K}| &= t +C\\ (t,V) = (0,0): && -\frac{K}{b} \ln a &= C \\ \Rightarrow && 1-\frac{b}{a} \frac{V}{K} &= e^{-bt/K} \\ \Rightarrow && V &= \frac{aK}{b} (1 - e^{-bt/K}) \end{align*} \begin{align*} &&\frac{\d W}{\d t} &= \underbrace{b}_{\text{incoming volume}} \cdot \underbrace{\frac{a}{b} (1 - e^{-bt/K})}_{\text{incoming fraction }A} - \underbrace{b}_{\text{outgoing volume}} \cdot \underbrace{ \frac{W(t)}{L}}_{\text{fraction of outgoing which is }A} \\ &&&= a (1 - e^{-bt/K}) - b \frac{W}{L} \\ \Rightarrow && \frac{\d W}{\d t} + \frac{b}{L} W &= a (1-e^{-bt/K}) \\ && \frac{\d}{\d t} \left ( e^{b/L t} W\right) &= ae^{b/L t}(1-e^{-bt/K}) \\ \Rightarrow && W &= e^{-bt/L} \left ( \frac{aL}{b}e^{b/Lt} - \frac{a}{\frac{b}{L} - \frac{b}{K}}e^{b/L t - b/K} \right) + Ce^{-bt/L} \\ &&&= \frac{aL}{b} \left (1 - \frac{K}{K-L}e^{-b/Kt} \right)+ Ce^{-bt/L} \\ (t,W) = (0,0): && 0 &= \frac{aL}{b} \frac{-L}{K-L} + C \\ \Rightarrow && C &= \frac{aL^2}{b(K-L)} \\ \Rightarrow && W &= \frac{aL}{b} \left (1 - \frac{K}{K-L} e^{-bt/K} + \frac{L}{K-L} e^{-bt/L} \right) \end{align*}