The Second Law and Swamp Coolers

This post is part of a series, Nonsense and the Second Law of Thermodynamics. The previous post is entitled Heat Can Be Transferred From a Cold Body to a Hot Body: The Air Conditioner.

When I was an undergraduate, I had a physical chemistry professor who claimed that air conditioners that were completely inside a room could not possibly work.  Opening the refrigerator door on a hot day will not make your house cooler.

The refrigerator gives off more heat than it transfers from inside itself; a refrigerator is actually heating the house.  If the door is left open, the refrigerator works harder to try to maintain a cool temperature in accordance with its thermostat setting.  As the refrigerator works harder, it releases more heat into the house.

My professor, however, was not correct.  It is possible to have a cooling unit that does more cooling than it releases heat to the environment.  The trick with indoor coolers is that the process by which they operate is not cyclic.

These coolers are evaporative coolers and can also be used outside of a house to cool incoming air.  They work by a similar principle as closed-loop cooler, but the fluid being used is water.  The water needs to be resupplied as it evaporates.

Indoor evaporative coolers that do not have a source of outside air do not work as well as evaporative coolers with a source of dry outside air.  Eventually, the indoor-only coolers simply recirculate humidified air, at which point there is no cooling.

Swamp coolers have a reservoir of liquid water.  For the outdoor roof-mounted or window-mounted units, the reservoir is typically connected to the house's water supply.  A float valve similar, in principle,  to the float valve in a toilet ensures that the reservoir remains full.

A water pump pumps the water onto pads (these may be shredded aspen, foam, cardboard, or other formulations).  The excess water drains back into the reservoir.  A blower moves air through through the pads.  In the more effective setup dry outside air moves through the pads and comes into the house as cool moist air.

The blower exerts pressure into the house.  If doors and windows are left open, the cool air will displace the hot air in the house and cool the house down.

It takes heat to evaporate liquid water into water vapor.  The heat is supplied by the air.  If dry hot air enters an evaporative cooler, it will be cooled and humidified as water evaporates into it.

These coolers work best in arid climates; in humid climates the ability of the air to hold more water vapor (at a given temperature) is limited and thus the coolers are not very effective.

These coolers do not violate the second law of thermodynamics.  They do not involve a cyclic process.  if one were to consider the fact that somewhere, sometime, the water that has been evaporated will recondense, one can see that the cyclical process creates more heat than is absorbed from the hot reservoir (the house).

When the water condenses from vapor the heat of vaporization is released.  This heat balances out the heat that was absorbed by evaporating the water.

I am neglecting the fact that these processes may occur at different temperatures to keep the story simple, but the results are the same: the entire cyclic process if it occurred, cannot be more efficient than an ideal air conditioner or a Carnot engine.

From the previous post on air conditioners and the efficiency of an ideal air conditioner, it should be clear that the full cycle is no different.  Over the full cycle more heat must be released to the surroundings than is absorbed from the evaporation of the water.

But wait!  Is it not true that for any irreversible process the entropy of the system plus the surrounding must increase?  If a swamp cooler cools without heating, regardless of whether the process is cyclic, is it not violating the second law of thermodynamics?

Entropy and Swamp Coolers

For any process, the entropy of the system and its surroundings must stay the same or increase, and for any irreversible process, the entropy must increase.  Even non-cyclic processes must obey this constraint.

Consider the reservoir of water itself.  At the same temperature, water vapor contains more latent heat than liquid water.  There is a change in the entropy of the water from the liquid state to the vapor state.

Consider water that is exactly at its boiling point under constant pressure.  The system is at equilibrium. The constant pressure heat of vaporization is the same as  a term called the enthalpy of vaporization.

     qvap = ΔHvap

At the boiling point the liquid is in equilibrium with the vapor; therefore the entropy change must be:

     ΔSvap = ΔHvap/Tboil

For water, the result works out to be 109.1 Joules per Kelvin per mole (J/K*mol). As the water evaporates, its entropy increases; so the process is in compliance with the second law of thermodynamics.

To understand how this entropy change in water occurs it is important to understand entropy from a different standpoint than I have presented it thus far.  It is time to discuss entropy and statistical thermodynamics.

Sources

• Atkins, P. W. Physical Chemistry, W. H. Freeman and Company, New York, 3rd edition, 1986
• McQuarrie, Donal d A., Statistical Thermodynamics,  University Science Books, Mill Valley, CA, 1973 
• Bromberg, J. Philip, Physical Chemistry, Allan and Bacon, Inc., Boston, 2nd Edition, 1984
• Anderson, H.C., Stanford University, Lectures on Statistical Thermodynamics, ca. 1990.
• Camaro, José Ru, Evaporative Cooling: Water for Thermal Comfort, Ambiente y Agua - An Interdisciplinary Journal of Applied Science, vol. 3, no. 2, pp 51-61 (2008)

Contents

1. Introduction
2. What the Second Law Does Not Say
3. What the Second Law Does Say
4. Entropy is Not a Measure of Disorder
5. Reversible Processes
6. The Carnot Cycle
7. The Definition of Entropy
8. Perpetual Motion
9. The Hydrogen Economy
10. Heat Can Be Transferred From a Cold Body to a Hot Body: The Air Conditioner
11. The Second Law and Swamp Coolers
12. Entropy and Statistical Thermodynamics
13. Fluctuations
14. Partition Functions
15. Entropy and Information Theory
16. The Second Law and Creationism
17. Entropy as Religious, Spiritual, or Self-Help Metaphor
18. Free Energy
19. Spontaneous Change and Equilibrium
20. The Second Law, Radiative Transfer, and Global Warming
21. The Second Law, Microscopic Reversibility, and Small Systems
22. The Arrow of Time
23. The Heat Death of the Universe
24. Gravity and Entropy
25. The Second Law and Nietzsche's Eternal Recurrence
26. Conclusion