2.2.4. Governing Equation of the Fluid

For the working fluid, the energy balance equation states

$$
\varepsilon\_f \rho\_f A \frac{\partial T\_f}{\partial t} = \pi d\_{\rm in} h\_f \left( T\_p - T\_f \right) - \dot{m}\_f \mathbf{c}\_f \frac{\partial T\_f}{\partial z} \tag{7}
$$

Equation (6) can be applied directly to the evaluation of performance of horizontal GHE. While for the vertical GHE, the internal pipe diameter *din* was replaced with an equivalent pipe diameter *Deq* since the single U shaped pipe is represented as a pipe, which has an equivalent diameter (refer to Figure 3). The equation to calculate the equivalent diameter has been developed and validated by Gu and Oneal [25] as:

$$D\_{cq} = \sqrt{2D\_p L\_s} \tag{8}$$

**Figure 3.** Schematic of the vertical GHE with an equivalent diameter.

The equation is valid if *Dp* ≤ *Ls* ≤ *rb*
