if anyone is interested, the general solution to these problems is as follows:
you have a tank with volume,
initially with
kg of salt present. You pump a solution of salt of concentration
in at a rate of
. The tank is continuously stirred and at any time
there is
kg of salt present. The mixture is pumped out at a rate of
.
Now you differential equation is:
}=V_{in} C_{in} - V_{out} \frac{Q}{V_0+t\Delta V})
the second one is just an abbreviated form of the first one, so i don't have too many fractions everywhere...
Now, to get the general solution we use the
Integrating Factor.
Basically, let
=e^{\int \frac{V_{out}}{V_0+t \Delta V} dt} = \left ( V_0+ t \Delta V \right )^{\frac{V_{out}}{\Delta V}} )
then using the formula on that wiki page,
} \int V_{in} C_{in} \left (V_0+t \Delta V \right ) ^{\frac{V_{out}}{\Delta V}} dt)
then crank the handle to get:
}{V_{out} + \Delta V} + \frac{k}{( V_0 + t\Delta V )^{\frac{V_{out}}{\Delta V}}})
where
} \right ) V_0^{\frac{V_{out}}{\Delta V}} )
pretty sure all of that is right, if anyone wants to check for any errors that'd be awesome.
This should only be used to check answers, the methods used are out of the VCE scope and they will never ask you to solve one of these without prompting the solution first and you showing that it is indeed a solution. also its pretty cool i reckon haha
and of course it reduces to the general solution when
which is the ones you have to know how to solve, actually it might be a bit more complex given the
in the exponent, but using some identities it should reduce haha
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