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# Fluid Mechanics Help?

Freshwater flows steadily into an open 55-gallon drum initially filled with seawater. The freshwater mixes thoroughly with the sea water and the mixture overflows out of the drum. If the freshwater flowrate is 10 gal/min, estimate the time in seconds required to decrease the difference between the density of the mixture and the density of the freshwater by 50%.

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This is really not a fluid mechanics problem but rather a simple first order D.E. problem...

Q(t) = the quantity of salt in tank at time t

Q(0) = the quantity of salt in tank at time zero

The salt concentration is C = Q(t) / Volume(t)

dQ/dt is the instantaneous rate of change of the salt concentration, thus

dQ/dt = (Cin)(Flow in) - (Cout)(Flow out)

dQ/dt = (Cin)(Flow in) - (Qout(t)/Volume(t)) (Flow out)

Since freshwater has zero salt, Cin = 0 and the first term drops away

For simplicity, let's let Q equal the salt in tank solution at time t

The volume in the tank is fixed at 55 gallons and the flow in, 10 GPM, equals the flow out.

dQ/dt = - ( Q lbs / 55 gal )( 10 gal/min )

dQ/dt = -10Q / 55

Move Q to the LHS and then integrate

∫ 1/Q dQ = -10/55 ∫ dt

ln Q = -10t/55 + C

Then exponentiate

Q(t) = [ e^C ] [ e^(-10t/55) ]

At t = 0 then Q(0) = e^C

So we can say...

Q(t) = Q(0)e^(-10t/55)

Let's let Q(0) = 1 and then we need to find Q(t) = 1/2

1/2 = e^(-10t/55)

t = (-55/10)ln(1/2)

t = 3.8 minutes

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• Complex yet you can do this. Homework is to make you think. Get busy.

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