**My Air
Conditioner? You're Standing on It!**** **

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

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:

- water
(other coolants may pose an environmental hazard if the pipe leaks).

Design
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:
- At what depth D could
you then use this system
to
heat the home during the winter months?

- What pipe layouts (e.g. parallel pipes, coils, etc.) prove more
effective at capturing the heat? What flow rates correspond to
this?
- 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!!