Find the quantity of salt in the tank?

[Math10]

Homework Statement

A 200 gallon tank initially contains 100 gallons of water with 20 pounds of salt. A salt solution with 1/4 pound of salt per gallon is added to the tank at 4 gal/min, and the resulting mixture is drained out at 2 gal/min. Find the quantity of salt in the tank as it's about to overflow.

Homework Equations

None.

The Attempt at a Solution

(1/4)lbs/gal*(4)gal/min=1lbs/min
(Q)lbs/100 gal*(2)gal/min=Qlbs/50 min
dQ/dt=1-Q/50=(50-Q)/50
50/(50-Q)dQ=dt
-50ln abs(50-Q)=t+C
ln abs(50-Q)= -t/50+C
50-Q=Ce^(-t/50)
Q=50-Ce^(-t/50)
Q(0)=0
C=50
Q(t)=50(1-e^(-t/50))
Q(50)=31.606
But the answer is Q(50)=47.5
Where did I make a mistake?

 

[LCKurtz]

Homework Statement

A 200 gallon tank initially contains 100 gallons of water with 20 pounds of salt. A salt solution with 1/4 pound of salt per gallon is added to the tank at 4 gal/min, and the resulting mixture is drained out at 2 gal/min. Find the quantity of salt in the tank as it's about to overflow.

Homework Equations

None.

The Attempt at a Solution

(1/4)lbs/gal*(4)gal/min=1lbs/min
(Q)lbs/100 gal*(2)gal/min=Qlbs/50 min
dQ/dt=1-Q/50=(50-Q)/50
50/(50-Q)dQ=dt
-50ln abs(50-Q)=t+C
ln abs(50-Q)= -t/50+C
50-Q=Ce^(-t/50)
Q=50-Ce^(-t/50)
Q(0)=0
C=50
Q(t)=50(1-e^(-t/50))
Q(50)=31.606
But the answer is Q(50)=47.5
Where did I make a mistake?

I didn't check all your work, but I notice you used Q(0)=0. But the problem says the tank started with 20lbs of salt.

 

[Math10]

So I used Q(0)=20,
C=30
Q=50-30e^(-t/50)
Now what?

 

[DrClaude]

(Q)lbs/100 gal*(2)gal/min=Qlbs/50 min

This line I don't understand. Why are you dividing by 100 gal?

 

[Math10]

So what do I do? Can you guys correct me?

 

[haruspex]

This line I don't understand. Why are you dividing by 100 gal?

It's attempting to calculate the instantaneous rate at which salt is leaving the tank (in lb/min).
Math10, you've overlooked that fact that the volume is increasing too. It's only 100 gal at the start.
We shouldn't have to guess what you are doing. Explain as you go along.

 

[Chestermiller]

You need to do 2 material balances: (1) The volume of water in the tank and (2) the amount of salt in the tank.

Let V (gal) represent the volume of water in the tank at time t, and let C (lb/gal) represent the concentration of salt in the tank at time t. You need to solve for both of these. The balances should be of the form

(rate of input) - (rate of output) = (rate of accumulation).

Let q_in represent the rate of water addition to the tank, and let q_out represent the rate of water output from the tank. Let C_in represent the concentration of salt in the inlet stream. Assume that the tank is always well-mixed, so that the concentration of salt in the tank is always uniform.

To start with, show us your balance on the volume of water in the tank.

Chet