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Abstract: The possibility of modeling the Navier–Stokes equations and together with the conventional second order slip boundary condition at high Knudsen numbers is explored in this paper by incorporating the Knudsen diffusion phenomenon in rarefied gases. An effective mean free path (MFP) model is augmented to the governing equation and the slip boundary condition, as gas transport properties can be related to the MFP. This simple modification is shown to implicitly take care of the complexities associated in the transitional flow regime, without necessitating dependency of the slip coefficients on the Knudsen number. Unique analytical model with fixed values of slip coefficients is proposed and rigorous comparisons with the experimental and simulation data for pressure driven and thermally driven rarefied gas flows support this conjecture. First and second order slip coefficients have been proposed as 1.1466 and 0.9756 for rectangular channels and 1.1466 and 0.14 for the capillaries, from the continuum to the transition flow regime. The current work is significant from the numerical simulation point of view because simulation tools are better developed for Navier–Stokes equations. [Copyright &y& Elsevier]

Abstract: Earth heat exchangers are essential parts of the ground-source heat pumps and the accurate prediction of their performance is of fundamental importance. This paper presents the development and validation of a numerical model for the simulation of energy flows and temperature changes in and around a borehole heat exchanger when a fluid circulates through a U-tube. Based on the time-dependent convection–diffusion equation, the FlexPDE software package is employed to solve the resulting boundary value problem that model a heat exchanger. First, the mathematical model is validated through a comparison with data obtained from experiments with real borehole set-ups in Cyprus. Then the validated model is used to study the heat flow and the temperature variation in the heat-exchanger. Finally conclusions are extracted on how various parameters like the U-tube diameter, the variation of the ground thermal conductivity and specific heat and the borehole filling material affect the temperature of the inlet and outlet fluid. [Copyright &y& Elsevier]