2.2.3. Governing Equation of the Pipe

The energy balance equation for the pipe is

$$c\_p \rho\_p V\_p \frac{\partial T\_p}{\partial t} = A h\_f \left( T\_f - T\_p \right) + \frac{k\_\mathcal{\mathcal{S}} A}{0.5 \Delta r} \left( T\_s - T\_p \right) \tag{6}$$

The area of the pipe for the vertical GHE was determined based on an equivalent pipe diameter, which is given later by Equation (8).
