# An Enhanced Vertical Ground Heat Exchanger Model for Whole-Building Energy Simulation

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## Abstract

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## 1. Introduction

^{®}[1] and TRNSYS [2], use equipment-loop simulation algorithms that pass component (e.g., pumps, boilers, chillers, heat exchangers, etc.) entering and exiting conditions (e.g., flow rates, temperatures, humidity ratios, etc.) flow-wise from up-stream components to down-stream components. These components are generally connected in a loop so that every component on the loop is in some way connected to every other component on the loop. Equipment that is active at any given timestep may be controlled by an individual controller or by a larger control algorithm that specifies control for the entire system. These components may also make requests to other components for specific performance. For example, a chiller model that has a minimum required flow rate may require that all other components on the loop run at its minimum flow rate, even though they could be operated at a lower flow rate.

## 2. Literature Review

#### 2.1. Analytical Models

#### 2.2. Response Factor Models

#### 2.3. Thermal Resistance-Capacitance Models

#### 2.4. Numerical Models

## 3. Methodology

#### 3.1. Enhanced Response Factor Model

- Significant effort has been expended to understand and improve the original response factor model. Since its publication, researchers have performed many studies that enhance understanding of the methods and improve on the original work. This work similarly builds on the original model, about which much is already known, and which has already been widely adopted.
- Because the formulation builds on the historical response factor model, software or other programs that already have this model implemented can more easily make modifications to incorporate the enhancements. This simplifies adoption of the model in WBES environments.
- Again because the formulation builds on the historical response factor model, methods for generating standard borehole wall temperature g-functions are still applicable (as are load aggregation procedures, which are critical to maintaining low simulation times).
- By clearly defining the heat transfer rate applied in this model as the fluid’s heat transfer rate, the domain inside the borehole and the domain between the borehole wall and the far-field soil temperature can be coupled together easily through two separate response factor computations. This more easily allows the transient effects to be handled, even down to timesteps below the transit time of the GHE circulation fluid.

#### 3.2. Model Reformulation for WBES Usage

#### 3.3. Dynamic Borehole Model

#### 3.3.1. Simple Borehole TRC Model

#### 3.3.2. Dynamic Pipe Model

#### 3.3.3. Dynamic Borehole Model Validation

#### 3.4. Exiting Fluid Temperature Response Factor Generation

#### 3.5. Borehole Wall Temperature Response Factor Generation

#### 3.6. Methodology Summary and Discussion

- Long-timestep borehole wall temperature g-functions could be computed with any number of approaches outlined previously. In this work, an FLS approach was used by utilizing the pygfunction library by Cimmino [42]. No modifications are required to the methodology used to compute these g-functions. Cimmino has produced significant quantification regarding the speed of these methods [39] and has shown that for borefields with up to 64 boreholes—which is realistically expected to capture most of the WBES use cases—g-functions can be computed in times from a few seconds to a few minutes. Note that the code by Cimmino is written in Python; the computation time could likely be reduced to some extent after being implemented in a compiled programming language.
- Short-timestep borehole wall temperature g-functions could also be computed using any accurate borehole model that captures the dynamic effects of the borehole thermal capacity on the borehole wall temperature. In this work, the model by Xu & Spitler [58] was used because it uses a simplified geometry that accounts for the borehole thermal capacity. In addition, it is a 1D, radial finite-volume model that can be solved rapidly using a tri-diagonal matrix formulation. Note that the original model has to be slightly modified to compute the temperature at the borehole wall, and not at the fluid. This model is highly efficient and can compute the short-timestep g-functions in a matter of seconds.
- Exiting fluid temperatures g-functions are computed with a simplified dynamic borehole model that accounts for the transit delay of the circulation fluid via utilization of a transit delay pipe model. As before, the method presented in this paper does not rely on this specific dynamic GHE model; rather, any dynamic GHE model that accurately captures the transit delay effects of the borehole could be used, assuming that it is sufficiently fast and accurate. The borehole wall boundary temperature was updated by computing the heat flux at the borehole wall and then using that information to inform the original heat-input-formulated response factor model (Equation (4)). The borehole wall temperature resulting from that was then set as the borehole wall temperature boundary condition and updated at each timestep. Note that the short-timestep g-functions computed in the previous step are used here for determining the updated borehole wall boundary temperature. The dynamic model used here was able to compute the exiting fluid temperature g-functions for a single flow rate in $7.8$ $\mathrm{s}$, which is an acceptable time for WBES. Note that this should be repeated for different flow rates and that additional work should be done to determine how many different ${g}_{b}$ values are needed.

## 4. Validation

#### 4.1. High-Flow MFRTRT

#### 4.2. Low-Flow MFRTRT

## 5. Conclusions

- The pipe model currently used does not account for laminar flow; therefore, the pipe model should be enhanced to accurately account for these conditions.
- Any dynamic borehole model could be used to generate ExFT g-functions; therefore, a study should be performed to assess this model along with other existing models for accuracy and performance.
- The methods described here could also be used to model double U-tube or coaxial ground heat exchangers. Dynamic borehole models for these configurations would be needed in order to calculate the ExFT g-functions.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ExFT | exiting fluid temperature |

FLS | finite line source |

GHE | ground heat exchanger |

GSHP | ground-source heat pump |

ICS | infinite cylinder source |

ILS | infinite line source |

MFRTRT | multi-flowrate thermal response test |

TRC | thermal resistance-capacitance |

TRT | thermal response test |

WBES | whole-building energy simulation |

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**Figure 6.**Pareto values for variations in dynamic borehole model input parameters (note: RMSE = root mean squared error).

${\mathit{N}}_{\mathit{it}}$ | ${\mathit{N}}_{\mathit{seg}}$ | $\mathsf{\Delta}\mathit{t}$ | ${\mathit{f}}_{\mathit{g}}$ | RMSE [C] | MBE [C] | Time [s] |
---|---|---|---|---|---|---|

3 | 1 | 60 | 0.75 | 0.229 | −0.073 | 8.5 |

2 | 1 | 60 | 0.5 | 0.322 | −0.169 | 7.8 |

1 | 1 | 60 | 0.5 | 0.634 | −0.448 | 6.9 |

${\dot{\mathit{m}}}_{\mathit{f}}$ | Re | $ln(\mathit{t}/{\mathit{t}}_{\mathit{s}})$ | ${\mathit{g}}_{\mathit{b}}$ | ${\mathit{T}}_{\mathit{out}}$ | ${\mathit{T}}_{\mathit{b}}$ | ${\mathit{q}}_{\mathit{f}}$ | ${\mathit{R}}_{\mathit{b}}$ | ${\mathit{T}}_{\mathit{out}}-{\mathit{T}}_{\mathit{b}}$ | ${\mathit{q}}_{\mathit{f}}\phantom{\rule{0.166667em}{0ex}}\mathit{\xb7}\phantom{\rule{0.166667em}{0ex}}{\mathit{R}}_{\mathit{b}}$ |
---|---|---|---|---|---|---|---|---|---|

kg/s | - | - | - | °C | °C | (W/m) | °C/(W/m) | °C | °C |

0.02 | 1400 | −8.8 | 0.67 | 18.80 | 17.34 | 9.989 | 0.216 | 1.45 | 2.16 |

0.03 | 1967 | −8.8 | 0.71 | 18.89 | 17.35 | 9.991 | 0.217 | 1.55 | 2.16 |

0.04 | 2534 | −8.8 | 0.74 | 18.92 | 17.35 | 9.993 | 0.211 | 1.57 | 2.11 |

0.05 | 3079 | −8.8 | 0.75 | 18.62 | 17.35 | 9.994 | 0.169 | 1.27 | 1.69 |

0.1 | 5894 | −8.8 | 0.81 | 18.62 | 17.36 | 9.996 | 0.154 | 1.26 | 1.54 |

0.3 | 17,209 | −8.8 | 0.91 | 18.73 | 17.36 | 9.996 | 0.151 | 1.37 | 1.51 |

0.5 | 28,535 | −8.8 | 0.94 | 18.76 | 17.36 | 9.994 | 0.150 | 1.40 | 1.50 |

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## Share and Cite

**MDPI and ACS Style**

Mitchell, M.S.; Spitler, J.D.
An Enhanced Vertical Ground Heat Exchanger Model for Whole-Building Energy Simulation. *Energies* **2020**, *13*, 4058.
https://doi.org/10.3390/en13164058

**AMA Style**

Mitchell MS, Spitler JD.
An Enhanced Vertical Ground Heat Exchanger Model for Whole-Building Energy Simulation. *Energies*. 2020; 13(16):4058.
https://doi.org/10.3390/en13164058

**Chicago/Turabian Style**

Mitchell, Matt S., and Jeffrey D. Spitler.
2020. "An Enhanced Vertical Ground Heat Exchanger Model for Whole-Building Energy Simulation" *Energies* 13, no. 16: 4058.
https://doi.org/10.3390/en13164058