Borehole Heat Exchanger (BHE)¶

The Borehole Heat Exchanger (BHE) models the ground thermal response when heat is extracted or injected by the ground-source heat pump.

Modeling Assumptions¶

BHE Circulation Pump: The circulation pump operates at a constant power (\(E_\text{pmp}\)) and flow rate during active heating cycles, adding heat to the evaporator inlet fluid.
Ground Thermal Response: Long-term thermal drift and short-term transient responses are calculated using precomputed step-response g-functions.

Mathematical Modeling¶

Circulation Pump Heat Addition¶

The fluid circulating through the BHE absorbs the heat generated by the circulation pump (\(E_\text{pmp}\)) before entering the heat pump evaporator:

\[T_\text{evap,in} = T_\text{bhe,f,out} + \frac{E_\text{pmp}}{\dot{m}_\text{bhe} \cdot c_{p,\text{w}}}\]

The actual heat extracted from the ground (\(Q_\text{bhe}\)) accounts for the heat added by the circulation pump, given the total evaporator heat absorption rate (\(Q_\text{ref,evap}\)):

\[Q_\text{bhe} = Q_\text{ref,evap} - E_\text{pmp}\]

Ground Thermal Response (g-functions)¶

The transient temperature change at the borehole wall (\(T_\text{bhe}\)) is evaluated using the temporal superposition of step heat extraction pulses and the spatial-response g-function (\(g(t)\)):

\[T_\text{bhe}(t) = T_s - \sum_{i=1}^{n} \frac{q_i - q_{i-1}}{2 \pi k_s} g(t_n - t_{i-1})\]

Where:

\(T_s\) is the undisturbed ground temperature.
\(q_i\) is the specific heat extraction rate per unit borehole length at time step \(i\).
\(k_s\) is the soil thermal conductivity.

The average circulating fluid temperature (\(T_\text{bhe,f}\)) is subsequently derived using the effective borehole thermal resistance (\(R_b\)):

\[T_\text{bhe,f} = T_\text{bhe} - q \cdot R_b\]