Question

How do you determine what percentage of chlorine is in the pool after 1 hour if a pool whose volume is 10000 gallons contains water the is 0.01%chlorine and starting at t=0 city water containing .001% chlorine is pumped into the pool at a rate of 5 gal/min while the pool water flows out at the same rate?

Answer 1

The concentration will be:

• `approx 0.009 "%"`

Explanation:

Volume flow rate for the pool:

• `dot V = dot V_("in") - dot V_("out") = 0`

Mass flow rate, with the idea that:

• `"mass" = "concentration" times "Volume"`:

`dot m = dot m_("in") - dot m_("out")`

`implies color(red)( d/dt(c V ))= c_("in") dot V_("in") - c_("out") dot V_("out")`

[Where `c` and `V` are real-time concentration and volume of pool]

`V dot c = 5(0.001"%") - 5c`

`2000 dot c = 0.005 - 5c`

This is now just a separable DE, with IV:

• `c(0) = 0.01 "%"`

#implies c(t) = 0.001 + 0.009 e^(-0.0025 t) #

After 1 hour:

`implies c(60) approx 0.009`

ie `0.009 "%"`, a slight drop as would be expected