Evaluative Analysis of Formulas of Heat Transfer Coefficient of Rock Fracture
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Evaluative Analysis of Formulas of Heat Transfer Coefficient of Rock Fracture Yinqiang Jiang1,2 · Huayan Yao1 · Yinxiang Cui2,3 · Hongwu Lei2,3 · Yuanyuan He2 · Bing Bai2,3 Received: 17 March 2020 / Accepted: 5 May 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Heat transfer coefficient (HTC) is a useful concept to characterize the heat exchange performance of rock fracture. However, existing study has preliminarily found that some HTC formulas would lead to abnormal values, which claims a systematical evaluation of existing formulas. In this paper, 214 test records are employed to eval‑ uate 8 existing formulas from literature. It is found that, for all these experimental data, Bai’s formula (Formula D) developed in 2017 shows fairly good numerical sta‑ bility, while the rest show numerical oscillations and anomalies in different degrees. Moreover, the numerical oscillations and anomalies would be exacerbated with the increase of flow rates, which leads to poor applicability. Error propagation theory in numerical analysis is used to analyze the mechanism of the numerical oscillations and anomalies which can be effectively explained with the condition number and the relative error of the corresponding formula. Keywords Condition number · Heat transfer coefficient · Numerical stability evaluation · Relative error
1 Introduction Heat transfer coefficient (HTC) is a useful concept to characterize the heat exchange performance of fluid flowing on the surface, which has also been used for rock fractures especially involved of geothermal reservoir engineering. For * Bing Bai [email protected] 1
School of Civil and Hydraulic Engineering, Hefei University of Technology, Hefei 230041, China
2
State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan 430071, China
3
Hubei Key Laboratory of Geo‑Environmental Engineering, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan 430071, China
13
Vol.:(0123456789)
104
Page 2 of 20
International Journal of Thermophysics
(2020) 41:104
a single rock fracture, several formulas with different expressions have been developed based on different simplified methods or assumptions [1–8]. However, Bai et al. [1] preliminarily found that some formulas showed negative value and numerical oscillation especially usually appeared when the flow rate increased. Although preliminary analysis was conducted, the mechanism that how this is produced and how different parameters act in the process is still not well under‑ stood. Moreover, although the test data they used are very limited, this anomalous phenomenon indicates that the rationality or applicability of the formulas might be questionable. Therefore, a systematical evaluation of existing HTC formulas will be valuable for guiding the selection and use of the existing formulas. This paper aims to conduct a systematical review, evaluation and comparison of exist‑ ing HTC formulas. In Sect. 2, t
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