A novel solution for vertical borehole heat exchangers in multilayered ground with the Robin boundary condition
Bibliographic record
Abstract
The ground surface condition is crucial in the thermal analysis of borehole heat exchange (BHE) systems, as it directly influences the thermal regime within the subsurface. However, most existing analytical solutions assume a constant temperature at the ground surface equal to the undisturbed ground temperature, overlooking the impact of various anthropogenic and environmental factors that alter ground surface conditions. This study addresses this gap by proposing a new semi-analytical solution that incorporates the Robin boundary condition for BHEs in multilayered ground. The solution, developed using the unsteady surface element method, balances computational efficiency with analytical precision. The proposed solution was verified with an exact solution in a homogeneous body, showing a maximum error of approximately 0.2 %. Compared with prescribing a constant temperature at the ground surface, using a Robin boundary condition better captures the heat exchange between the ground and the surrounding environment. The results show that variations in the heat transfer coefficient at the ground surface can cause differences in the mean borehole wall temperature change of up to 7.45 % over about 31.7 years (109 s) for a 70-metre borehole. Meanwhile, a unit heat flux at the ground surface can raise the mean borehole wall temperature by 0.56 °C for a heat transfer coefficient of 0.25 Wm−2K−1 over 31.7 years. Additionally, the circulating fluid can transmit the impact of the ground surface condition to greater depths. The results indicate that even at the bottom of a 70-metre borehole, using a fixed temperature can underestimate the temperature change by up to 6.11 % over 31.7 years compared with using the Robin boundary condition. These findings emphasize the indispensability of the proposed solution for accurately predicting the thermal performance of BHEs.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.002 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".