One of the first mathematical models that students encounter is that of the cooling of a cup of coffee. A related, but more complicated, problem is how the temperature in a thermos full of ice-cold water changes as a function of both time and position in the thermos. We use the approach developed by Fourier for the heating of an insulated rod to establish a model for a thermos. We verify the model by comparing it to data recorded with a calculator-based laboratory.

1.
Newton’s law of cooling is neither a law nor due to Newton, but we will conform to common usage.
2.
E. D. Rainville, P. E. Rainville, and R. E. Bedient, Elementary Differential Equations, 8th ed. (Prentice–Hall, Upper Saddle River, NJ, 1997).
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4.
For more information about the Texas Instruments CBL and new CBL 2, see 〈http://education.ti.com/〉. The CBL 2 is compatible with the TI-73, TI-82, TI-83, TI-83 Plus, TI-83 Plus Silver Edition, TI-86, TI-89, TI-92, TI-92 Plus, and Voyage 200. The TI-85 is compatible with the CBL and can also be used with CBL 2. However, there is not a built-in user program available for the TI-85. Sensors can be obtained from Vernier 〈http://www.vernier.com/〉 or Pasco 〈http://www.pasco.com/〉. Vernier also offers LabPro, a data collection interface similar to the CBL and CBL 2, which can be connected to a TI programmable calculator, a Windows or Macintosh personal computer, or a Palm OS handheld device. Pasco also provides data collection software and PASPORT, their data collection interface that can be used with either a Windows or Macintosh personal computer to collect, graph, analyze, and print data collected from their sensors.
5.
The software can be freely downloaded from 〈http://education.ti.com/〉.
6.
L. Antinone, A. Bellman, C. Brueningsen, W. Krawiec, and J. Randall, CBL System Experiment Workbook (Texas Instruments Incorporated, Austin, TX, 1997).
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D. L. Powers, Boundary Value Problems, 3rd ed. (Saunders College, New York, 1987).
9.
See EPAPS Document No. E-AJPIAS-71-020306 for sample Mathematica code.
A direct link to this document may be found in the online article’s HTML reference section. The document may also be reached via the EPAPS homepage (http://www.aip.org/pubservs/epaps.html) or from ftp.aip.org in the directory /epaps. See the EPAPS homepage for more information.
10.
J. P.
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F. J. Bayley, J. M. Owen, and A. B. Turner, Heat Transfer (Barnes and Noble, New York, 1972).
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J. E. Scherschel, “Modeling Heat Flow in a Thermos,” senior honors thesis, Ball State University, 2000, pp. 24–27.
13.
R. A. Serway, Physics for Scientists and Engineers with Modern Physics, 3rd ed. (Saunders College, Philadelphia, 1990).
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H. J. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd ed. (Clarendon, Oxford, 1959).
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H. M.
McGee
,
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The virtual cook: Modeling heat transfer in the kitchen
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W. Mendenhall, Introduction to Probability and Statistics, 6th ed. (Duxbury, Boston, 1983).
17.
J. Stewart, Calculus: Early Transcendentals, 4th ed. (Brooks/Cole, Pacific Grove, 1999).
18.
Reference 8 provides an introduction to Stürm–Liouville theory. For a more in-depth study of these ideas, see Ref. 19.
19.
G. P. Tolstov, Fourier Series (Prentice-Hall, Englewood Cliffs, NJ, 1962).
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