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Best fit value of eddy thermal diffusivity

  1. Apr 20, 2013 #1
    1. The problem statement, all variables and given/known data

    A mathematical model for temperature T as a function of depth y (in m) and time t (in days) is:

    (T(y,t)-T0)/(Tsurf(t)-T0)=e^(-y2/4αt) (2)


    where Tsurf(t) is the water temperature of the lake surface at time t, α is a property called the “eddy thermal diffusivity” and T0 is the lake temperature at time zero. Time zero must be chosen to be on a day when the lake temperature is more or less uniform.

    Fit equation (2) to the data for July 19th to obtain the best fit value of α.

    20-Dec 18-Apr 16-May 19-Jul
    y (m) T(C) T(C) T(C) T(C)
    0 10.8 19.1 22.2 28.4
    1 10.7 18.7 21.8 27.9
    2 10.5 18 21.4 28
    3 10.5 17.4 21.2 27.9
    4 10.5 17 21.1 27.4
    5 10.5 16.4 20.7 26.2
    6 10.5 16 19.3 23.6
    7 10.5 15.2 17.1 21.4
    8 10.5 14.7 15.6 19.3
    9 10.5 13.7 14.6 17.9
    10 10.5 12.9 14.1 16.8
    11 10.5 12.1 13.2 15.9
    12 10.5 11.6 12.7 15
    13 10.5 11.1 12.1 14.1
    14 10.5 10.7 11.6 13.2
    15 10.4 10.4 11.3 12.4
    20 10.3 9.3 9.9 10.6
    25 10.3 8.9 9.4 9.8
    30 10.1 8.7 9.1 9.3
    35 10.3 8.7 8.8 9.1

    2. Relevant equations
    T0=10.5
    t=211 since Dec. 20th is t=0


    3. The attempt at a solution
    I solved for α which gives the equation -y^2/[844ln(T - 10.5)/17.9]

    so I started going down the list using the values from July 19th which gave me a different value for every value of T but once i got to T=9.8 I couldn't get any value for alpha because it would be the ln of a negative number... my question is how exactly should I look for the best fit value of α? is it the average value of the ones I was able to calculate? Or maybe Im missing something? Any help would greatly be appreciated.
     
    Last edited: Apr 20, 2013
  2. jcsd
  3. Apr 20, 2013 #2
    Sorry, here is a better look at the Temperatures from July 19th

    19-Jul
    T(C)
    28.4
    27.9
    28
    27.9
    27.4
    26.2
    23.6
    21.4
    19.3
    17.9
    16.8
    15.9
    15
    14.1
    13.2
    12.4
    10.6
    9.8
    9.3
    9.1
     
  4. Apr 20, 2013 #3
    You probably want to do a non-linear least squares fit to the data, minimizing the error with respect to the eddy diffusivity.
     
  5. Apr 21, 2013 #4
    Im not quite sure how I would go about doing that; but thanks for the reply.
     
  6. Apr 21, 2013 #5
    Make a plot of T(y,t)-T(0) vs y2 on a semi-log plot, including only the points for which T(y,t) > T(0). The semi-log parameter should be T(y,t) - T(0). You should get something close to a straight line. The slope of this line should be -1/(4αt). Draw your best straight line in, and then calculate the slope. The points where T(y,t) < T(0) are omitted because, within experimental uncertainty, they are essentially equal to T(0).
     
  7. Apr 21, 2013 #6
    so once I have the slope just solve for α?
     
  8. Apr 22, 2013 #7
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