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Energies 2022, 15, 6385 The cubic law, shown in Equation (7) [19] shows that the flow through the fracture is proportional to the cubic of fracture width (e), and Equation (8) [20] shows convective heat transfer between fracture water and the tube, so it can be concluded that the energy car- ried by fracture water flow was significant compared with the non-fracture case. The equations are as follows: 9 of 13 qfr = e3g J (7) the porous media and Darcy’s velocity field was coupled by the energy balance presented 12ν where qfr is the fracture water flow of single-width, m2/s; e is the width of the fracture,m; in Equation (5). The cubic law, shown in Equation (7) [19] shows that the flow through the fracture is v is viscosity of the fracture water,m2/s; g is the acceleration of gravity, m/s2; J is the pres- proportional to the cubic of fracture width (e), and Equation (8) [20] shows convective heat sure drop along the fracture flow direction. transfer between fracture water and the tube, so it can be concluded that the energy carried by fracture water flow was significant compared with the non-fracture case. The equations are as follows: Qw =hΔtw−fπde e3g (8) where Qw is the heat transfer power betwqeen=the UJpipe and the fracture water flow, W;(h7) fr where q is2 the fracture water flow of single-width, m2/s; e is the width of the fracture, water, w/f(rm ∙k); Δtw-f is the temperature difference between the wall of the U pipe and the m; v is viscosity of the fracture water, m2/s; g is the acceleration of gravity, m/s2; J is the fracture water, K; d is the depth of the fracture, m; ē is the mean value of the width of the 12ν is the surface heat transfer coefficient between the outer wall of the U pipe and the fracture pressure drop along the fracture flow direction. fracture, m. Qw =h∆tw−fπde (8) The top surface of the model was set as no mass and energy flux, and the other sides 3.3. Boundary and Initial Conditions where Qw is the heat transfer power between the U pipe and the fracture water flow, W; h were set as open boundaries. The porous pressure gradient was established by assigning is the surface heat transfer coefficient between the outer wall of the U pipe and the fracture initial pressur2es on each boundary. Both the initial temperatures of the rock mass and the water, w/(m ·k); ∆tw-f is the temperature difference between the wall of the U pipe and the U pipes were set as 289.15 K [18]. The hydraulic head different was set as 10 m. fracture water, K; d is the depth of the fracture, m; e ̄ is the mean value of the width of the fracture, m. 3.4. Meshing 3.3. Boundary and Initial Conditions The area with a gentler temperature gradient far away from the U pipes was meshed byareTghuelatorpgrsiudr,fancedothfethaeremaowdhelewreaasgseretatsernotemapsesratnudrengerargdyieflnutxe,xaisnteddthweaostmheersshieddes bwyearfeinsetragsriodp(eFnigbuoruen8dAa)r.ies.Theporouspressuregradientwasestablishedbyassigning initial pressures on each boundary. Both the initial temperatures of the rock mass and the The out temperatures in different mesh division formed near the U pipes were sim- U pipes were set as 289.15 K [18]. The hydraulic head different was set as 10 m. ulated to select the most suitable mesh method. It was found that similar results were acquired from the finer and ultra-fine meshes (Figure 8B), while the running time under 3.4. Meshing the ultra-fine mesh was three times that of the finer mesh. Coupled failure or the non- The area with a gentler temperature gradient far away from the U pipes was meshed convergence of the calculation also appeared, so it can be considered that the finer mesh- by a regular grid, and the area where a greater temperature gradient existed was meshed ing was sufficient. The mesh unit was from 0.6 m to 6.6 m. by a finer grid (Figure 8A). 290 289 eruterep 288 meteltu 287 O 286 285 Figure 8. Illustration of the model meshing. (A) chart of meshing way; (B) comparison chart of the Figure 8. Illustration of the model meshing. (A) chart of meshing way; (B) comparison chart of the fifinneeggrirdidanandduultlrtara-f-ifinneeggrirdid.. The out temperatures in different mesh division formed near the U pipes were sim- ulated to select the most suitable mesh method. It was found that similar results were acquired from the finer and ultra-fine meshes (Figure 8B), while the running time under the ultra-fine mesh was three times that of the finer mesh. Coupled failure or the non- convergence of the calculation also appeared, so it can be considered that the finer meshing was sufficient. The mesh unit was from 0.6 m to 6.6 m. 0 5 10 15 20 25 30 35 40 (Ms) (K) B A dir eni gr f-a enif rtlu dirg e miTPDF Image | Research on the Application of Fracture Water
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