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Heat Transfer and Cold Beer

May 6, 2013

For many years I was engaged in a type of crystal growth called liquid phase epitaxy. The process was simple. Wafers from a crystal were dipped into a molten solution of oxides dissolved in lead oxide. After just a few minutes, an epitaxial crystal layer was grown on this seed crystal wafer, the wafer was removed, and the excess lead oxide solution was dissolved in a dilute acid.

The difficulty in this process lay in the fact that the lead oxide solution needed to be at about 1000°C; and the crucible and heat shielding in the furnace needed to be made from platinum, one of the few materials impervious to the corrosive effects of lead oxide vapors.

The furnaces consumed about 7 kilowatts of electrical power, and there was unavoidable heat flow into the laboratory. The heat transfer was by several mechanisms; namely, conduction, convection, evaporation and radiation.

In a room environment, heat transfer is generally by conduction, as when your hand touches the surface of a cold beverage can. In this case, the transfer of heat is simply proportional to the temperature difference between your hand and the aluminum can.

If you heat water in a pan, you've entered the convection regime, which is somewhat harder to model. Essentially, the cooler liquid at the top of the pan sinks to the bottom along the sides, and the warmer liquid rises through the center.

convection in a vertical pot

Liquid Convection

With a heat source at the bottom and heat loss at the top and sides, the warm liquid rises at the center and cooler liquid flows to the bottom at the sides.
(Image by Oni Lukos, modified, via Wikimedia Commons.)


When the water in your pan reaches higher temperatures, evaporating water (steam) will transfer heat from the liquid; and, to your hand if it comes too close. This heat transfer process is easy to model, since a quantity of water will always carry a latent heat of vaporization, also called the enthalpy of vaporization.

Fortunately for most heat transfer processes, heat loss from radiation is negligible, since the temperature is too low. In its simplest form, as expressed by the Stefan–Boltzmann law, radiation loss is proportional to the fourth power of the temperature difference between an object and its surroundings.
∂j/∂T = σT4,
where the value of the proportionality constant σ, the Stefan–Boltzmann constant, is 5.670 x 10-8 joule·meter-2·second-1·kelvin-4. The constant is very small, so radiation loss doesn't matter below a few hundred degrees Celsius. The loss, of course, depends on the exposed surface area of the object, as the dimensions of the constant show.

Predicting heat transfer seems all too easy. We have the equations, so we just plug in our numbers, right? As one of my physics professors said, everything is just F = ma; the problem, however, is deciding what F, m and a are for your particular problem.

The Mpemba effect is the name given to a supposed phenomenon, noted from antiquity, involving the freezing of water into ice. I wrote about the Mpemba effect in an earlier article (October 19, 2006, Mpemba Effect). Writes Aristotle in his Meteorology (Book I, Part 12),
"The fact that the water has previously been warmed contributes to its freezing quickly: for so it cools sooner. Hence many people, when they want to cool hot water quickly, begin by putting it in the sun. So the inhabitants of Pontus when they encamp on the ice to fish (they cut a hole in the ice and then fish) pour warm water round their reeds that it may freeze the quicker, for they use the ice like lead to fix the reeds."[1]
The example given by Aristotle of pouring hot water over reeds in freezing weather indicates to me that this might be an effect caused by evaporative cooling, assisted by the flow of cooling air over the reeds by convection.

Plato and Aristotle from Raphael's The School of Athens

Plato (c.428 BC-c.348 BC), left, and Aristotle (384 BC-322 BC), from Raphael's The School of Athens.

Plato is holding his Timaeus, and Aristotle is holding his Nicomachean Ethics.

Aristotle was a student of Plato.

(Via Wikimedia Commons.)


Since people only started thinking thermodynamically in the last two centuries, the idea that heating water helped in its freezing persisted to the time of Francis Bacon (1561-1626). As Bacon wrote in his Novum Organum (Book 2, Chapter 50),
"Nor should we omit the means of preparing bodies to receive cold. Among others I may mention that water slightly warm is more easily frozen than quite cold."[2]
In the 1960s, a Tanzanian high school student, Erasto B. Mpemba, noticed the effect while making ice cream from hot mixes. Mpemba published the results of experiments with his teacher, Denis G. Osborne, in 1969, causing renewed interest in the "Mpemba effect."[3] There has been much published about whether the effect is real; and, if it is real, why.[4] Controversy still persists, since there isn't a formal definition of the Mpemba effect. The end-point might be when 0°C is reached; or, when first ice appears.

One reason why the Mpemba effect might be true is a difference in the heat transfer of the initial states. Ice is insulating, and a container of warm water will melt through any frost layer between it and the freezer surface on which it's placed. This ensures an intimate contact with an important heat sink. Good experiments should levitate the water cell with wires to prevent this. This sounds like a good experiment for the International Space Station, an idea suggested by others.[5] A recent study finds that supercooling is an important factor.[6]

Thermodynamic processes, such as the evaporation of water mentioned above, are reversible. While it takes a certain quantity of heat to evaporate a quantity of water, that heat is liberated when the vapor condenses. The enthalpy of condensation of water has the same large absolute magnitude as the enthalpy of vaporization, 40.68 kilojoule per mole (2,260 kj/kg), but with a negative sign to denote that heat is being released. University of Washington scientists have found that condensation of water on the exterior of beverage cans is responsible for a large portion of their warming.[7-8]

This research vindicates the use of beer koozies/coosies (the cool variant of the tea cosy), which prevents condensation of humidity on the surface of beverage containers, and this effort started with the desire by Dale R. Durran, a professor of atmospheric sciences, to devise an interesting example for his students of the importance of the heat generation by condensation.[8]

As usual in science, a back-of-the-envelope calculation was employed, and that showed a measurable effect.[8] Durran calculated that the heat released by a layer of condensate just four thousandths of an inch thickness on a can would heat its contents by 9 degrees Fahrenheit.[8] My own calculation based on the dimensions of a standard 12 ounce US beverage can (6.6 cm, or 2.6 inches, in diameter and 12.1 cm, or 4.75 inches in height) modeled as a cylinder filled with water using an enthalpy of condensation of 2,260 j/g, gives 3.88°C, or 7 degrees Fahrenheit. Alcohol has a lower heat capacity than water, so alcoholic beverages will tend to warm faster.

Durran was able to interest one of his colleagues, Dargan M. W. Frierson, in a collaboration to do an experimental validation of this result. In true Ernest Rutherford "string and sealing wax" style,[9] they conducted experiments in a bathroom, using a hot shower to adjust humidity.[8] With validation, some better experiments were in order. Said Durran,
"You can't write an article for Physics Today where the data has come from a setup on the top of the toilet tank in one of the author's bathrooms."[8]
These experiments were conducted using an antique apparatus formerly used to simulate cloud formation, along with some funding from the National Science Foundation.[8] This work resulted in an article in the April issue of Physics Today[7] which gives the following information:[8]
• On a typical summer's day in New Orleans, the heat released by condensation will warm a drink by 6° Fahrenheit in five minutes.
• In Dhahran, Saudi Arabia, on the hottest and most humid day, condensation will warm a can's contents from near-freezing to 48° Fahrenheit in just five minutes.

A beer can under test

The rare time that alcoholic beverages are allowed in a laboratory.

Condensation on a beer can is measured on an electronic balance.

The cap at the top prevents air from entering the opening at the top of the can.

(University of Washington Image.)


References:

  1. Aristotle, "Meteorology," E. W. Webster, Trans., MIT Classics Web Site.
  2. Francis Bacon, "Novum Organum." Latin text, "Etiam praeparationes corporum ad excipiendum frigus non sunt omittendae; veluti quod aqua parum tepida facilius conglacietur quam omnino frigida, et huiusmodi."
  3. E. B. Mpemba and D.G. Osborne, "Cool?," Physics Education, vol.4, no. 3 (May, 1969), pp. 172-175.
  4. Monwhea Jeng, "Can hot water freeze faster than cold water?"
  5. A Search for the Mpemba Effect: When Hot Water Freezes Faster Than Cold Water, ZapperZ's physics blog on the world of Physics and Physicists, March 17, 2010.
  6. James D. Brownridge, "A search for the Mpemba effect: When hot water freezes faster then cold water," arXiv Preprint Server, March 16, 2010. Also appears as J.D. Brownridge, "A search for the Mpemba effect: When hot water freezes faster then cold water," Amer. Jour. Phys., vol. 79, no. 1 (January, 2011), p.78ff.
  7. Dale R. Durran and Dargan M. W. Frierson, "Condensation, atmospheric motion, and cold beer," Physics Today, vol. 66, no. 4 (April 2013), pp. 74ff.
  8. Hannah Hickey, "Keeping beverages cool in summer: It's not just the heat, it's the humidity," University of Washington Press Release, April 25, 2013.
  9. Mark Oliphant, "The beginning: Chadwick and the Neutron," Bulletin of the Atomic Scientists Dec 1982 pp.14-18.

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