compartmental analysis: 100-gal tank of water, 5 gal/min of

J

jsaxman

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can sombody help me with this problem:

A tank contains 100 gallons of water. five gallons of brine per minute flow into the tank, and each gallon of brine contains 1 pound of salt. five gallons of water flow out of the tank per minute. assume the tank is well stirred.
find a differential equation for the number of pounds of salt in the tank assuming the tank initailly contains no salt, solve the differential equation. how much salt is in the tank after one hour? at what time will there be 499 lbs of salt in the tank?

jsaxman
 
Re: compartmental analysis problem

\(\displaystyle \frac{dy}{dt}=\text{rate in-rate out}\)

rate in: \(\displaystyle \left(1 \frac{lb}{gal}\right)\left(5 \frac{gal}{min}\right)=5\frac{lb}{min}\)

rate out: \(\displaystyle \left(\frac{y(t)}{100}\frac{lb}{gal}\right)\left(5\frac{gal}{min}\right)=\frac{y(t)}{20}\frac{lb}{min}\)

\(\displaystyle \frac{dy}{dt}=5-\frac{y}{20}\)

\(\displaystyle \frac{dy}{dt}+\frac{y}{20}=5\)

We have the I.C. y(0)=0

Find the integrating factor: \(\displaystyle e^{\int \frac{1}{20}dt}=e^{\frac{t}{20}}\)

Continue?. Let me know what you get.
 
Re: compartmental analysis problem

It can also be separable-variable type:

\(\displaystyle \frac{dy}{100-y} \, = \, \frac{dt}{20}\)
 
Re: compartmental analysis problem

How would you solve this question if the initial conditions weren't y(0) = 0; if instead there was an initial level of salt, say 10?
 
Re: compartmental analysis problem

charlesmci said:
How would you solve this question if the initial conditions weren't y(0) = 0; if instead there was an initial level of salt, say 10?
Click to expand...

The same way as shown.

Why do you think a different IC will change the DE?
 
Re: compartmental analysis problem

Oh yeah sorry was trying to apply that working to a different question but turns out it doesn't correlate that well.
Thanks anyway.
 
A tank contains 100 gallons of water. five gallons of brine per minute flow into the tank, and each gallon of brine contains 1 pound of salt. five gallons of water flow out of the tank per minute. assume the tank is well stirred.
find a differential equation for the number of pounds of salt in the tank assuming the tank initailly contains no salt, solve the differential equation. how much salt is in the tank after one hour? at what time will there be 499 lbs of salt in the tank?
Click to expand...

If the tank always has 100 gallons in it, and the max concentration is 1 lb salt per gallon, the maximum amount of salt achievable is 100 lb. Are you sure the initial volume is not 500 or 1000 gallons or…?
 
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