My Air Conditioner?  You're Standing on It! 

Burt Tilley, Franklin W. Olin College of Engineering, Needham, MA


Roman spas - sorry, not this week!Geothermal energy harnessing has been a technique used since Roman times.  Although, most people do not live near naturally heated springs, the idea of harnessing heat from the Earth (or depositing heat in the summer) could be a cost-effective option.  Your job this week is to determine how this option could work in a rural American setting.  The materials to be used in the system are:
Some typical configurationsDesign an array of these pipes running below the ground that optimizes the amount of heat exchange from the humid confines of a 2000 sq. ft. home in the summer to the relatively cool 50-55 F subterranean environment.   The figure below (from Reference 2) shows some common configurations for these systems.  However, it is not clear which of these systems would be beneficial, and under what situations.  Devise a mathematical model that determines the optimal flow rate of the water through these pipes for these parameters so that the maximal amount of heat is transferred from the water to the ground.  Note that the least efficient way of doing this would be to toss the equations into a CFD package and hope for the best!  You should from your model be able to back out the different mechanisms contributing to heat transport and determine some qualitative principles that can be used in designing new systems.  Some questions you may want to consider are:
  1. At what depth D could you then use this system to heat the home during the winter months? 
  2. What pipe layouts (e.g. parallel pipes, coils, etc.) prove more effective at capturing the heat?  What flow rates correspond to this?
  3. At what acreage does such a system become economically competitive to a central air conditioning system for a 2000 sq. ft. home? 

References

1. C.C. Lin and L.A. Segel, Mathematics Applied to Deterministic Problems in the Natural Sciences, (1988, SIAM).
2.
W.A. Duffield and J.H. Sass, Geothermal Energy -- Clean Power from the Earth's Heat, U.S. Geological Survey Circular 1249, Reston, VA (2003).
3. Bring your favorite numerical methods book along with your favorite fluid mechanics/heat transfer/partial differential equations text!!