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Differential equations to determine water level

  1. Mar 24, 2012 #1
    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 cant 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 im kinda stuck?Help, please.
     
  2. jcsd
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