Thermodynamics and ice melt flows
by Dan Hughes
I recently ran across the paper by Isenko et al.  listed below. The second paragraph of the introduction states:
“According to the conservation of energy, the loss of potential energy for a volume of water is sufficient to warm it by 0.2 C for each 100 m of lowering.”
The described process corresponds to isentropic compression of liquid water by increasing the pressure by about 1 MPa, through a change in elevation of 100.0 m. Note that the temperature change is given independent of any other information relating to flow velocity, kinetic energy, viscosity, dissipation, or any details of the flow channel that might affect conversion to thermal energy by viscous dissipation of kinetic energy. Especially note that for the case of flows in horizontal channels, for which the potential energy change is zero, apparently there would not be any temperature changes. The same can be said relative to flows upward against gravity.
The calculation by the authors is related the same concept that is the subject of this previous post. That is, the total potential energy at the top of a column of water is converted to thermal energy content by the action of viscous dissipation. As in the subject papers of the previous post, the temperature increase is too high.
When the process is considered to be compression of subcooled liquid water isolated from interactions with its surroundings, the temperature increase is estimated to be about 0.01 K per 100 m.
In general, textbooks recommend that temperature increase due to viscous dissipation can be neglected for all but a few special situations. The recommendation is particularly valid whenever thermal interactions between the fluid and channel walls, i.e. heat transfer, is the focus of the application.
The temperature increase for compression of subcooled liquid water is estimated in the attached PDF FILE. [WorkPost03]
Evgeni Isenko, Renji Naruse, and Bulat Mavlyudov, “Water temperature in englacial and supraglacial channels: Change along the flow and contribution to ice melting on the channel wall,” Cold Regions Science and Technology, Vol. 42, pp. 53– 62, 2005.
Herbert B. Callen, Thermodynamics: An Introduction to the Physical Theories of Equilibrium Thermostatistics and Irreversible Thermodynamics, John Wiley & Sons, Incorporated, New York, (1960).
W. Bridgman, “A Complete Collection of Thermodynamic Formulas,” Physical Review, Vol. 3, No. 4, pp. 273–281, (1914). doi:10.1103/PhysRev.3.273.
W. Bridgman, The Thermodynamics of Electrical Phenomena in Metals and a Condensed Collection of Thermodynamic Formulas, Dover Publications, Inc. New York. (1961).
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