An Analytical Method for Determining the Convection Heat Transfer Coefficient Between Flowing Fluid and Rock Fracture Walls

• Published in: Rock mechanics and rock engineering 2017-07, Vol.50 (7), p.1787-1799
• Main Authors: Bai, Bing, He, Yuanyuan, Hu, Shaobin, Li, Xiaochun
• Format: Article
• Language: English
• Subjects: Analytical methods; Civil Engineering; Convection; Convective heat transfer; Derivation; Dirichlet problem; Distribution; Earth and Environmental Science; Earth Sciences; Flow rates; Fluids; Formulas (mathematics); Fracture toughness; Geophysics/Geodesy; Green's function; Heat transfer; Heat transfer coefficients; High flow; Inlets (waterways); Laplace equation; Low flow; Mathematical analysis; Mathematical models; Original Paper; Polynomials; Program verification (computers); Rocks; Temperature; Temperature distribution; Temperature effects; Walls; Water temperature
• ISSN: 0723-2632; 1434-453X
• DOI: 10.1007/s00603-017-1202-6

The convective heat transfer coefficient (HTC) is a useful indicator that characterizes the convective heat transfer properties between flowing fluid and hot dry rock. An analytical method is developed to explore a more realistic formula for the HTC. First, a heat transfer model is described that can be used to determine the general expression of the HTC. As one of the novel elements, the new model can consider an arbitrary function of temperature distribution on the fracture wall along the direction of the rock radius. The resulting Dirichlet problem of the Laplace equation on a semi-disk is successfully solved with the Green’s function method. Four specific formulas for the HTC are derived and compared by assuming the temperature distributions along the radius of the fracture wall to be zeroth-, first-, second-, and third-order polynomials. Comparative verification of the four specific formulas based on the test data shows that the formula A corresponding to the zeroth-order polynomial always predicts stable HTC values. At low flow rates, the four formulas predict similar values of HTC, but at higher flow rates, formulas B and D, respectively, corresponding to the first- and third-order polynomials, predict either too large or too small values of the HTC, while formula C, corresponding to the second-order polynomial, predicts relatively acceptable HTC values. However, we cannot tell which one is the more rational formula between formulas A and C due to the limited information measured. One of the clear advantages of formula C is that it can avoid the drawbacks of the discontinuity of temperature and the singular integral of HTC at the points (± R , 0). Further experimental work to measure the actual temperature distribution of water in the fracture will be of great value. It is also found that the absorbed heat of the fluid, Q , has a significant impact on the prediction results of the HTC. The temperatures at the inlet and the outlet used for Q should be consistent with the assumptions adopted in the derivation of its corresponding HTC formula. A mismatched value of Q might be the reason that some existing HTC formulas predict negative or extremely large HTCs at high flow rates.