Ground-source Heat Engine Prototype Design and Test

Figure 1. Generalized Schematic

A company wished to explore feasibility and performance of ground-source heat engines designed to generate a tiny amount of electrical power from the daily fluctuating temperature difference between the air and the ground using solid-state thermoelectric generators. 

 Figure 1 shows the general structure of the device. Multiple thermoelectric modules were sandwiched between air-side and ground-side heat exchangers. The thermoelectric modules provided the electrical power generation from heat that flowed as a result of the air-ground temperature difference. The design consisted of appropriate geometric design for both heat exchangers, selection and interface configuration of the thermoelectric modules, and thermal resistance matching between the heat exchangers and the thermoelectric modules. 

Figure 2. Instrumented Prototype

 Prototype devices were designed and built. Figure 2 shows one of the prototypes situated in the test area and instrumented for performance evaluation. This prototype included fins on the air-side heat exchanger visible in the figure. Extended test results indicated peak power of 5 milliwatts and average power of 1 milliwatt could be obtained from the prototype configurations. It was determined that approximately 50% of the power generated in extended tests could be attributed to direct solar insolation. 

 Figure 3 shows a sample of measured power generation rates over a short part of the test period, and Figure 4 shows the total energy generated by hour of the day for the entire test period.

Figure 3. Measured Power

Figure 4. Total Energy

Parameter Simplification

It is often desirable to simplify heat transfer problems by working in dimensionless quantities. In some situations it is desirable to apply formal non-dimensionalization approaches such as the Buckingham Pi theorem. In other cases, there are obvious non-dimensional groups. Sometimes it is most convenient to only partly non-dimensionalize a problem. Normally, this process yields three significant benefits:

(1) it often simplifies the governing differential equation or the boundary conditions
(2) it results in a general solution (creates a fixed scale for the problem)
(3) it identifies significant groups

As an example, we can examine a simple textbook case.

      Consider 1-D, transient conduction in a plane wall with constant properties and a convective boundary condition at the front surface and a symmetry boundary condition at the rear surface.  The governing differential equation is:

and the boundary conditions are:

Note that there are 5 parameters in this problem (including boundary conditions): k, rho, c, L, and h.

We can non-dimensionalize using the following definitions:

in preparation for substituting into the original problem, these definitions can be re-arranged as follows:

substituting into the original differential equation:

cancelling terms leaves:

for the boundary conditions:

The group hL/k is commonly referred to as the Biot Number, Bi. From the definition of T*, it can be seen that the rest of the right hand side is simply T*(x*=1).

The final problem with boundary and initial conditions is now:

–Note that the parameter count has shrunk from five to one (Bi)

–The group that was termed t* is often called the Fourier Number, Fo.