Differential equations to determine water level

[aquameatwad]

Hi, the problem i have is this:
How long will it take a water reservoir with an Average level of 300,000,000 m3 to drop to 90% of the average level, if there is a drought, taking into account average rainfall, evaporation and amount of water taken in and out?

Assumption are:
Reservoir is cylindrical.
Drought consists of zero rainfall
the only factors that affect reservoir level is stream inflow, rainfall, community outflow and evaporation.


the variables are:
Avg initial volume, V0=300,000,000 m3
Surface area of reservoir, S=10,000,000 m3
Avg in flow, K0=6,000,000 m3/Day
Avg evaporation rate, K1=.0012 m/sec
Avg out flow, K2=6,000,000 m3/Day
Height of water in reservoir, h0=30m
constant B=1/h0
constant a= 1day/m2
time, t=?
dependent variable, h=?

So my model is this dv/dt=IN-OUT
IN is a(K0/(t+1)) ,(stream inflow) since it will go down since there is a drought
OUT is K1(S), (evaporation) & K2-(B(h0-h)K2)K2 ,(Community outflow)

so the model is this
dv/dt=S(dh/dt)=[a(K0/(t+1))]-[K1(S)-(K2-(B(h0-h)K2)K2)]

which turns into
dh/dt=[K1-(K2/S)+(Bh0/S)] - [(BK/S)H] + [(2K0/S)(1/(t+1))]

so i guess in turn looks like the format:
dh/dt = C +(-Dh) + E/(t+1)

Now... is this even correct? cause i can't figure out how to solve E/(t+1). i know i could use variation of parameters if it was a 2nd order and find the Aux Eqn etc. But I am kinda stuck?Help, please.

 

[gtoguy97]

Yes, your model is correct. To solve for E/(t+1), you can use separation of variables. This involves separating the variables dh/dt and t and then integrating to find h as a function of t. After that, you can solve for t given the condition that h = 90%*V0.