# Math Help - differential equations- single compartment model

1. ## differential equations- single compartment model

Suppose that a tank holds 1000 liters of water, and 2 kg of salt is poured into the tank.

a)
computer the concentration of salt in g/liter.
b) assume now that you want to reduce the salt concentration. One method would be to remove a certain amount of salt water from the tank and then replace it by pure water. How much salt water do you have to replace by pure water to obtain a salt concentration of 1 g/L?
c) Another method to reduce salt concentration would be to hook up an overflow pipe and pump pure water into the tank. This way salt concentration would be gradually reduced.

Assume that you have two pumps, one that pumps water at 1 L/s, and other at 2 L/s. For each, Find out how long it would take to reduce the salt concentration from the orig. concentration to 1 g/L and how much pure water is needed in each case. (Note that the rate at which water enters that tank is equal to the rate at which water leaves the tank)

I don't know how to use the equation... do I need to use the single compartment model? $\frac {d}{dt} (CV) = qC1-qC?$

2. Originally Posted by juicysharpie
Suppose that a tank holds 1000 liters of water, and 2 kg of salt is poured into the tank.

a) computer the concentration of salt in g/liter.
b) assume now that you want to reduce the salt concentration. One method would be to remove a certain amount of salt water from the tank and then replace it by pure water. How much salt water do you have to replace by pure water to obtain a salt concentration of 1 g/L?
c) Another method to reduce salt concentration would be to hook up an overflow pipe and pump pure water into the tank. This way salt concentration would be gradually reduced.

Assume that you have two pumps, one that pumps water at 1 L/s, and other at 2 L/s. For each, Find out how long it would take to reduce the salt concentration from the orig. concentration to 1 g/L and how much pure water is needed in each case. (Note that the rate at which water enters that tank is equal to the rate at which water leaves the tank)

I don't know how to use the equation... do I need to use the single compartment model? $\frac {d}{dt} (CV) = qC1-qC?$
Let the concentration at $t$ be $c(t)$, then in a time interval $\delta t$: $\rho\ \delta t\ c(t)$ grams of solute are removed where $\rho$ is the pumping rate, so the concentration becomes:

$
c(t+\delta t)=\frac{V c(t) - \rho\ \delta t\ c(t)}{V}
$

$
\frac{c(t+\delta t)-c(t)}{\delta t}=-\frac{\rho c(t)}{V}
$

or in the limit as $\delta t \to$ zero:

$\frac{dc}{dt}=-\frac{\rho c(t)}{V}$

where $V$ is the tank volume.

CB