3. A reservoir contains 600 litres of pure water. Brine (salty water) that contains 0.2 kg/L of salt is added at a rate of 2 L/min. Brine from a second source with 0.05 kg/L of salt is added at a rate of 3 L/min. Assume that the reservoir is instantaneously well-mixed. The reservoir is drained at a rate of 5 L/min. Let Q(t) be the amount of salt (in kg) at time t (in min).
(a) Set up and solve the differential equation for Q(t).

Question

3. A reservoir contains 600 litres of pure water. Brine (salty water) that contains 0.2 kg/L of salt is added at a rate of 2 L/min. Brine from a second source with 0.05 kg/L of salt is added at a rate of 3 L/min. Assume that the reservoir is instantaneously well-mixed. The reservoir is drained at a rate of 5 L/min. Let Q(t) be the amount of salt (in kg) at time t (in min).
(a) Set up and solve the differential equation for Q(t).

australopithecus · Answer

so I know the general formula is Rin - Rout = dy/dt

australopithecus · Answer

but yeah I'm stuck

dumbcow · Answer

well you know the volume remains constant at 600
combined sources flowing in is 5L/min  and flows out at 5L/min

Rate of salt going in is (.2*2 = .4) + (.05*3 = .15)
--> 0.55 kg/min
Rate of salt going out is 5*concentration at time t
--> 5*(Q/600) = Q/120 kg/min

thus
$$\frac{dQ}{dt} = 0.55 - \frac{Q}{120}$$

australopithecus · Answer

thanks,  that makes sense :) so the Rin is normally a constant and Rout is always the outflow*concentration at time t/volume

dumbcow · Answer

yep

australopithecus · Answer

if there was an initial concentration in the chamber like 0.2kg/L how would I represent that in the equation

dumbcow · Answer

to solve this DE you need to find the integrating factor

...hmm, i don;t think it would change the rate equation just the function Q(t) because you would apply different initial conditions

australopithecus · Answer

I know how to solve these problems I just need help setting up the equations

australopithecus · Answer

so it would be the same if it was 0.2kg/L  in the tank instead of pure water initially?

dumbcow · Answer

yes, because dq/dt is rate that salt is changing over time
concentration is ratio of salt per volume so its an initial condition and does not affect the rate the salt changes once you start pumping in/out water

dumbcow · Answer

obviously the value of Q in the equation will be different but you would still use same rate equation

australopithecus · Answer

ok thanks

dumbcow · Answer

no prob

australopithecus · Answer

I got the equation

Q = 66e^(x/120) - 66, does this seem reasonable? I dont think I trust it

dumbcow · Answer

http://www.wolframalpha.com/input/?i=y%27+%3D+.55+-+%28y%2F120%29%2C+y%280%29+%3D+0 forgot a negative