Cleaning dirty pool question?

Question

The theoretical problem is the following: I have a dirty swimming pool of V (liter) with a starting contamination of Czero (parts / liter). The actual number of parts is Pzero=Czero.V. The is a pump which pumps Q (liter / sec) and filters the water completely.

The clean water goes back in the pool, so the volume V is constant, only the concentration C will decrease... the question is what is the concentration in function of the time??

Answer 1

P = C*V

dP/dt = C*dV/dt since C is a constant

But dV/dt = -Q since Q is sucking water out

dP/dt = -C*Q ...... and we know that Q is a constant so

P = -C*Qt + k ..... k is a constant of integration

at t = 0 we have P = C so:

P = C - C*Q*t = C(1 - Q*t)

Concentration as a function of time is therefore:

P = C(1 - Qt)