Objective Identification of the Stabilized Regime in Thermal Response Tests: A Log-Derivative Alternative to Semilog Slope Fitting
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Fabrice Toussaint,MSEng., Voraggo Limited — Sofia, Bulgaria
Abstract
Standard thermal response test (TRT) interpretation reads ground thermal conductivity (ks) from the slope of a straight line fitted to mean fluid temperature versus the logarithm of elapsed time, and the borehole thermal resistance (Rb) from that same fit's intercept. The method is simple, but the choice of where the straight-line segment begins is left to the analyst's judgment, and different reasonable choices can produce materially different conductivity and resistance values from the same dataset — the two parameters are coupled, so a biased slope carries a biased intercept with it. This paper proposes the logarithmic (Bourdet-style) derivative of the mean fluid temperature as a diagnostic that removes this ambiguity: the derivative becomes flat precisely when, and only when, the response has reached the regime the straight-line method assumes. We show, using a synthetic case with a known ground truth and two competing fit windows that are both individually plausible on the semilog plot, that the two windows disagree on ks by roughly 20% and on Rb by a similar margin, while a fit anchored to the derivative-identified plateau recovers both parameters within about 1–2% of their true values. We propose that a derivative plot become a standard, reportable piece of supporting evidence alongside the conventional semilog fit.
1. Introduction
Ground-source heat pump design depends on an accurate estimate of the undisturbed ground thermal conductivity, ks, and the borehole thermal resistance, Rb, both obtained from a thermal response test. In the conventional interpretation (the Kelvin infinite line-source method, as codified in ASHRAE and IGSHPA guidance), the mean fluid temperature is plotted against the natural logarithm of elapsed time; ks is recovered from the slope of the straight-line portion of that plot, and Rb is recovered from its intercept.
The method's validity rests on a theoretical result: at sufficiently large time, the line-source (or cylindrical-source) solution becomes asymptotically linear in ln(t). At early time it is not linear — the response is shaped by the thermal capacitance of the circulating fluid, the pipe, and.................