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Heating and Cooling

  1. Apr 4, 2008 #1
    1. The problem statement, all variables and given/known data

    A solar hot-water-heating system consists of a hot water tank and a solar panel. The tank is well insulated and has a time constant of 64 hr. The solar panel generates 2000 Btu/hr during the day, and the tank has a heat capacity of 2 degrees F per thousand Btu. If the water in the tank is initially 110 F and the room temperature outside the tank is 80 degrees F, what will be the temperature in the tank after 12 hr of sunlight.

    2. Relevant equations

    [tex] T(t) = e^{-kt} (\int e^{kt}[KM(t) + H(t) + U(t)]dt +C)[/tex]

    M is the outside temperature, H is other things that affect temperature in the tank(0 in this case), and U is the solar panel. K comes from the time constant, and should be the inverse of the time constant I believe. T is temperature, t is time.

    3. The attempt at a solution

    [tex] T(t) = e^{-\frac{1}{64}t} (\int e^{\frac{1}{64}t}[ \frac{1}{64} (80) + 4t]dt[/tex]

    After integrating I keep getting

    [tex] -16304 + 256t + Ce^{- \frac{1}{64} t}[/tex]

    I calculate C to be 16414 setting t equal to 0 and using the initial conditions. However, when plugging in 12 to my answer for t, I get in the 300s, whereas the book says the answer is 148.6. Could anyone please show me where I'm going wrong. I would greatly appreciate it.

    Thanks for your time and help.
     
  2. jcsd
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