Heat Transfer and Flow Characteristics of Coal Slurries under the Temperature Difference between Inside and Outside of the Channel  
[PERSON]  
1 Frontier Scientific Research Centre for Fluidized Mining of Deep Underground Resources, China University of Mining and Technology, Xuzhou 221116, China 12 State Key Laboratory for Geomechanics and Deep Underground Engineering, China University of Mining and Technology, Xuzhou 221116, China 23 School of Mechanics and Civil Engineering, China University of Mining and Technology, Xuzhou 221116, China 3  
[PERSON]  
3,*,  
of hydrodynamic instability and the effects of the flow rates, rectangular aspect ratio and thermal fluxes on the flow characteristics and the heat transfer. In another work of [PERSON], they numerically simulated the flow in a rectangular bent duct with the wall heated, incorporating the buoyancy force due to the heating in the formulas [30]. They found that under the wall heated, the Dean vortex in the curved channel is asymmetric compared to the symmetric vortex structure without wall heating, and the secondary vortex flow enhances the convective heat transfer. [PERSON] et al. used a spectral method and numerically calculated the flow in a curved rectangular duct, with the vertical outer wall heated but the inner wall cooled [31]. They found that at a smaller Grashof number, the flow turns from steady to periodic and then to chaotic pattern. Whereas, at a larger Grashof number, before the flow reaches the chaotic state, it experiences three steady-to-periodic cycles. At the periodic and chaotic states, the convective heat transfer is enhanced significantly by the secondary vortex flow. [PERSON] et al. numerically studied the flow and heat transfer in the curved wavy ducts [32]. They concluded that there exits an optimal combination of the wave amplitude and the wavy length to maximize the heat transfer. The flow resistance is more sensitive to the wave length compared to the amplitude because the wave length determines the frequency of the flow disturbance. Apart from these, the surface roughness has been proved to affect the heat transfer and flow behaviors. [PERSON] et al. found that the efficiency of the heat transfer increases as the surface roughness or hydrophilicity increases, particularly at high Reynolds numbers [33]. However, as a cost, at these surface conditions, the pressure drop is larger. For this research topic for non-Newtonian fluids, almost all studies are discussing the flow with heat transfer in geometries without bend. [PERSON] et al. used a density-based topology optimization algorithm to calculate both Newtonian and non-Newtonian thermal fluids in rectangular channels, with the purpose of minimizing the pressure drop but maximizing the heat transfer [34]. The optimal design structures of the channels are different from Newtonian to non-Newtonian fluids, and this difference is maximized below Re = 0.01. Non-Newtonian fluids always perform better in heat transfer, but at the expense of a higher pressure drop. [PERSON] et al. studied the flow of non-Newtonian fluid in circular-cross-section channels under a thermal isolfuv imposed on the channel wall, with the consideration that the viscosity and the thermal conductivity are functions of the temperature [35]. They gave an analytical solution of the fully developed flow and found its deviation from the one using the temperature-independent fluid properties that could reach 32%. [PERSON] et al. used a finite element method to study the flow of power-law fluids with different index numbers \(n\) in an annular gap between two coaxial rotating cylinders [36]. The boundary condition of the simulation is an adiabatic and stationary outer wall and an inner wall with a constant temperature and angular velocity. They found that the increase of \(n\) or the thermal conductivity causes the decrease of the average Nusselt number. For the shear thinning fluids (index smaller than one), the average Nusselt number is obviously affected by the viscosity parameter.