Differential Equation - Brine Solution Entering Tank

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SUMMARY

The discussion centers on solving a differential equation related to a brine solution entering a tank containing 80 gallons of pure water. The equation governing the amount of salt in the tank over time is given as S(t) = 160 - 160e^(-t/40). The key question is determining when the concentration of salt in the outflow reaches 1 lb/gal. The correct formulation of the differential equation is dS/dt = Ri*Ci - (S*Ro)/(Vo + (Ri - Ro)t), where Ri is the inflow rate, Ro is the outflow rate, Ci is the concentration of salt in, and Vo is the initial volume of the tank.

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Homework Statement



A tank contains 80 gallons of pure water. A brine solution with 2 lb/gal of salt enters at 2 gal/min, and the well-stirred mixture leaves at the same rate. Find (a) the amount of salt in the tank at any time and (b) the time at which the brine leaving will contain 1 lb/gal of salt.

Homework Equations



I'm just wondering about (b) really. I know we set S=80 below to solve it, but why?

The Attempt at a Solution



The differential equation that gives (a) is

S=160 - 160*e^(-t/40)

where S is the amount salt in the tank at any time t.
 
Last edited:
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If you correctly modeled a diff. eq for this problem and, also correctly solved it to come up with the sol

S=160 - 160*e^(-t/40), then part b)is not a problem at all. what it is asking u is that when will S(t)=1, and not 80 as you are saying!
remember S(t) is the amount of salt that the tank contains at any time.
The diff eq for this problem is

dS/dt=Ri*Ci- (S*Ro)/(Vo+(Ri-Ro)t) , where

S--- is the amount of salt in the tank,
Ri rate in
Ro rate out
Ci concentration in
Vo the initial volume

EDIT: You haven't actually showed us what u have done at all, remember one of the forums main policy is that you must first show your work, for after the people here to give you hints!
 
Last edited:
Oh, I'm sorry about that. I'll be sure to put up my work soon. Are you sure that what it's asking though? My notes say that I should get somewhere around 28 minutes.
 
A tank contains 80 gallons of pure water. A brine solution with 2 lb/gal of salt enters at 2 gal/min, and the well-stirred mixture leaves at the same rate. Find (a) the amount of salt in the tank at any time and (b) the time at which the brine leaving will contain 1 lb/gal of salt.

here it is :

dS/dt=2*2- (S*2)/(80+(2-2)*t)
dS/dt=4-2S/80, just solve this diff eq, if you haven't gone like this.
 
and for the part b) it is just asking you at what time t=? will S(t)=1, like i said.
NOTE: Next time show your work if you want to receive any help!
 
Last edited:

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