Best fit value of eddy thermal diffusivity

[rico22]

Homework Statement

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

(T(y,t)-T0)/(T_surf(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

Homework Equations

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

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 I am missing something? Any help would greatly be appreciated.

 

[rico22]

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

 

[Chestermiller]

You probably want to do a non-linear least squares fit to the data, minimizing the error with respect to the eddy diffusivity.

 

[rico22]

You probably want to do a non-linear least squares fit to the data, minimizing the error with respect to the eddy diffusivity.

Im not quite sure how I would go about doing that; but thanks for the reply.

 

[Chestermiller]

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).

 

[rico22]

so once I have the slope just solve for α?

 

[Chestermiller]

Yes.