The fast method and convergence analysis of the fractional magnetohydrodynamic coupled flow and heat transfer model for the generalized second-grade fluid

In this paper,we first establish a new fractional magnetohydrodynamic(MHD)coupled flow and heat transfer model for a generalized second-grade fluid.This coupled model consists of a fractional momentum equation and a heat conduction equation with a generalized form of Fourier law.The second-order fractional backward difference formula is applied to the temporal discretization and the Legendre spectral method is used for the spatial discretization.The fully discrete scheme is proved to be stable and convergent with an accuracy of O(τ^(2)+N-r),whereτis the time step-size and N is the polynomial degree.To reduce the memory requirements and computational cost,a fast method is developed,which is based on a globally uniform approximation of the trapezoidal rule for integrals on the real line.The strict convergence of the numerical scheme with this fast method is proved.We present the results of several numerical experiments to verify the effectiveness of the proposed method.Finally,we simulate the unsteady fractional MHD flow and heat transfer of the generalized second-grade fluid through a porous medium.The effects of the relevant parameters on the velocity and temperature are presented and analyzed in detail.

Analysis of coupled flow-reaction with heat transfer in heap bioleaching processes

A mathematical model for heap bioleaching is developed to analyze heat transfer, oxygen flow, target ion distribution and oxidation leaching rate in the heap. The model equations are solved with Comsol Multiphysics software. Numerical simulation results show the following facts： Concentration of oxygen is relatively high along the boundary of the slope, and low in the center part where leaching rate is slow. Temper- ature is relatively low along the slope and reaches the highest along the bottom region near the slope, with difference being more than 6℃. Concentration of target mental ions is the highest in the bottom region near the slope. Oxidation leaching rate is relatively large in the bottom and slope part with a fast reaction rate, and small in the other part with low oxygen concentration.

Preliminary design method for absorber pipe length of tunnel lining ground heat exchangers based on energy efficiency of heat pump

Many ongoing tunnel projects provide a favorable opportunity for the investigation and application of tunnel lining ground heat exchangers(GHEs).Tunnel lining GHEs can be connected to a heat pump to extract geothermal energy for heating and cooling buildings.Numerous studies have focused on the thermal performance of tunnel lining GHEs;however,the studies on the interaction between heat pumps and tunnel lining GHEs are relatively rare.In this study,a coupled heat transfer model of heat pumps and tunnel lining GHEs was proposed and then calibrated based on in situ test results.The model was used to evaluate the energy efficiency of a heat pump with tunnel lining GHEs under different conditions.The results show that the energy efficiency ratio(EER)increases exponentially with the absorber pipe length and thermal conductivity of the surrounding rock.The EER is governed by the convection heat transfer coefficient,which varies exponentially;meanwhile,the EER decreases approximately linearly with the annual average air temperature in the tunnel.Different types of heat pumps affect the EER significantly,and the EER of a Type-3 heat pump is higher than that of a Type-1 heat pump by 27.1%.Based on the aforementioned results,an empirical formula for the EER and absorber pipe length was established.Moreover,a preliminary design method for the absorber pipe length based on this empirical formula was developed.The method was employed to determine the appropriate absorber pipe length for the tunnel lining GHEs in the Shapu tunnel in Shenzhen,China.Finally,groups of absorber pipe layouts with a pipe spacing of 0.5 m,area of 135 m2,and length of 293.5 m were preliminarily determined.