Number Kn Research Articles

On the properties of the amorphous state of a single-component substance

It is shown that on the nonlinear dependence of the first coordination number (kn) versus the packing coefficient (kp) of the structure of a single-component substance, three special points corresponding to the amorphous structures can be distinguished. Based on the pairwise interatomic potential of Mie–Lennard-Jones, the state equation and properties of iron for both these three amorphous structures and the crystal state are calculated. It is shown that at kp = 0.45556 and kn = 6.2793 the minimum chemical potential is reached, i.e. this point is corresponding a thermodynamically stable amorphous structure into the liquid phase. An energetically equivalent point with the same kn value, but with kp = 0.6237, is a thermodynamically unstable amorphous structure corresponding to the solid phase. The following properties are calculated for amorphous iron: specific volume, chemical potential, entropy, Debye temperature, elastic modulus, thermal expansion coefficient, isobaric heat capacity, surface energy, and Poisson's ratio. It is shown that the specific surface energy of an amorphous solid metal is greater than that of an amorphous liquid phase, but less than that of a metal in the crystalline state. This should lead to the fact that the surface of the crystal metal should tend to amorphization.

Simplified unified wave-particle method with quantified model-competition mechanism for numerical calculation of multiscale flows.

A quantified model-competition (QMC) mechanism for multiscale flows is extracted from the integral (analytical) solution of the Boltzmann-BGK model equation. In the QMC mechanism, the weight of the rarefied model and the weight of the continuum (aerodynamic and hydrodynamic) model are quantified. Then, a simplified unified wave-particle method (SUWP) is constructed based on the QMC mechanism. In the SUWP, the stochastic particle method and the continuum Navier-Stokes method are combined together. Their weights are determined by the QMC mechanism quantitatively in every discrete cell of the computational domain. The validity and accuracy of the present numerical method are examined using a series of test cases including the high nonequilibrium shock wave structure case, the unsteady Sod shock-tube case with a wide range of Kn number, the hypersonic flow around the circular cylinder from the free-molecular regime to the near continuum regime, and the viscous boundary layer case. In the construction process of the present method, an antidissipation effect in the continuum mechanism is also discussed.

Interface Width Effect on the Weakly Nonlinear Rayleigh–Taylor Instability in Spherical GeometrySupported by the National Natural Science Foundation of China (Grant Nos. 11575033, 11675026, and 11975053).

Interface width effect on the spherical Rayleigh–Taylor instability in the weakly nonlinear regime is studied by numerical simulations. For Legendre perturbation mode Pn with wave number kn and interface half-width L, the commonly adopted empirical linear growth rate formula is found to be sufficient in spherical geometry. At the weakly nonlinear stage, the interface width affects the mode coupling processes. The development of the mode P2n is substantially influenced by the interface width. Moreover, the nonlinear saturation amplitude increases with the interface width.

Molecular dynamics simulation on evaporation of a suspending difluoromethane nanodroplet

Nanodroplet evaporation is a basic process widely existing in nature and industrial applications. Difluoromethane (CH2F2, also called “R32”) has attracted more and more attention in refrigeration, heat pump, Organic Rankine Cycle and other fields because of its excellent thermophysical characteristics. In this work, the evaporation process of R32 nanodroplet in large space is studied by means of molecular dynamics simulation. This work focuses on the dynamic evaporation characteristics of droplets under different conditions (droplet size, initial temperature, and ambient temperature) in the evaporation process, and quantitatively analyzes the effects of different parameters on the evaporation rate. The simulation results were compared with the results calculated from the diffusion-based model and the kinetic model. Based on the dynamic law of evaporation process, such as density, temperature field and velocity distribution, the calculation accuracy of each model is evaluated, the reasons for the deviation between model prediction and simulation results are also discussed. The results show that the increase of ambient temperature and initial saturation temperature can lead to a significant increase of droplet evaporation rate, and shorten the pre-heating time needed for droplet to reach quasi-steady evaporation stage. The increase of droplet size also increases pre-heating time, but has no significant effect on evaporation rate. For the evaporation scenario studied in this paper (Knudsen number (Kn) ~ 1), the existing modification methods of the kinetic model can enhance the prediction accuracy of the evaporation rate of R32 droplets in the quasi-steady evaporation stage. The prediction deviation of diffusion-based model applied directly is large, the accuracy of prediction can be significantly improved by correcting the thermal conductivity considering the scale effect.

Laboratory Research on Gas Transport in Shale Nanopores Considering the Stress Effect and Slippage Effect

AbstractThe low permeability of shale, in accretionary complexes and passive continental margins, influences pore pressure generation, and thus induces deformation and fault slip behavior. It is also key in hindering the ability to evaluate shale gas production. Gas is commonly used in measuring the permeability of shale and generally yields high values than that of liquids, due to slippage effect. Separation of the slippage effect from gas permeability is a critical task in the study of low‐permeability media. This study measured the gas permeability (Kg) of the Longmaxi shale parallel to bedding (L1P) and perpendicular to bedding (L1V) under different confining pressures (Cp) and pore pressures (Pp), while simultaneously measuring porosity for different values of Cp. Additionally, a new approach is proposed to define the effective stress coefficient and boundary condition for the Klinkenberg correction. The results show that the Kg of L1P is almost unaffected by the slippage effect, while, for L1V, the Klinkenberg correction is required. The Klinkenberg plot of L1V follows the quadratic model from the study by Ashrafi Moghadam and Chalaturnyk (2014, https://doi.org.10/1016/j.coal.2013.10.008), rather than the classic linear Klinkenberg equation. The results also show that, as the Cp increases and/or the Pp decreases, the contribution of the slippage effect to the L1V Kg increased from 20% to 86.5%. Finally, based on the Knudsen number (Kn), the flow regime of L1P changes from Darcy flow (Kn < 0.01) to slip flow (0.01 < Kn < 0.1), and that of L1V is slip flow.

A model of micro lubrication between two walls with an arbitrary temperature difference based on kinetic theory

Micro lubrication of a gas between two walls with an arbitrary temperature difference is studied on the basis of the Bhatnagar–Gross–Krook–Welander model of the Boltzmann equation. Applying the slowly varying approximation, the kinetic equation is studied analytically when the Knudsen number based on the gap size is large. The leading order approximation, which ought to be the solution of the nonlinear heat transfer problem, is replaced by its free molecular solution. Due to this crude approximation, a macroscopic lubrication model of Reynolds-type equation is derived in a closed form. For an assessment of the model, a direct numerical analysis of the kinetic equation is also conducted. The lift calculated using our model approximates that of the direct numerical analysis within the error of 4% uniformly in the range of the temperature ratio between 0.75 and 2 and the Knudsen number Kn between 0.1 and 10. A heating of the moving wall reduces the lift acting on the other wall when Kn is sufficiently large, whereas it is enhanced when Kn is sufficiently small.

Analysis of radiation pressure and aerodynamic forces acting on powder grains in powder-based additive manufacturing

Selection of process parameters is an important step in Powder-Based Additive Manufacturing (PBAM) of metals. In order to achieve an optimal parameter set, current literature is mainly focused on the understanding of powder dynamics by analysing the aerodynamic forces. In this letter, however, we show the importance of the laser induced force (radiation pressure) on the powder dynamics. Generalised Lorenz-Mie theory has been employed to accurately estimate the radiation pressure and it is shown that its magnitude is significant in comparison to various aerodynamic forces and the grains weight, hence, can significantly contribute to denudation and spatter observed in the manufacturing process. Furthermore, the importance of compressibility and rarefaction effects on the magnitude of drag and lift forces that a particle experiences is demonstrated by estimating the Ma and Kn numbers under process conditions, which directly impact the powder dynamics.

Effect of pore throat structure on micro-scale seepage characteristics of tight gas reservoirs

At present, the effects of pore throat structure on micro-scale seepage characteristics of tight gas reservoirs are less researched, and traditional numerical simulation methods are faced with a great number of challenges in the study of micro-scale flow. In this paper, the flow pattern of tight gas was studied based on the actual temperature and pressure of tight gas reservoir and the characteristic size of reservoir pore throat, and the rationality of tight gas flow was simulated by means of lattice Boltzmann method. Then, considering the influences of micro-scale effect, slippage effect and other factors, a tight gas flow model was established on the basis of LBGK-D2Q9 model, and its calculation results were compared with the analytical solutions and the numerical solutions listed in the literature. Finally, the influential laws of pore throat structure on the micro-scale seepage characteristics of tight gas were discussed. And the following research results were obtained. First, when the pressure is in the range of 3–70 MPa and the temperature is in the range of 293.15–373.15 K, the Knudsen number (Kn) is less than 0.1 and the gas flow is in the pattern of slippage flow and weak continuous flow. And in this case, it is reasonable to adopt the LBGK-D2Q9 model to simulate tight gas flow. Second, the effect of the characteristic size of the flow channel on the Kn is much greater than that of the pressure change. When the pore–throat ratio is constant, the Kn increases slowly along the throat. And its increasing trend gets more obvious with the increase of pore–throat ratio. Third, the presence of the throat makes the non-linear distribution characteristics of the pressure in the pore throat significant, and the pressure drop mainly lies in the throat. And the higher the pore–throat ratio is, the larger the pressure drop range in the throat is. Fourth, the non-linear distribution of pressure decreases the gas flow speed significantly, thus reducing the mass flow rate in the flow channel. In conclusion, the simulation result of the model established in this paper is highly coincident with the analytical solutions and the numerical solutions calculated by DSMC and IP methods in the literature, which verifies that this proposed model is reliable. The research results reveal the importance of “connecting fracture and expanding throat” in the practical development engineering of tight gas reservoirs.

Viscous heating effects on heat transfer characteristics of laminar compressible channel flow

The present paper investigates viscous heating effects on heat transfer characteristics of laminar Newtonian compressible channel flow. At first, incompressible Poiseuille flow case is addressed analytically by including viscous dissipation term in energy conservation equation. Temperature distributions in the channel cross-section and local Nusselt numbers are derived as functions of the Brinkman number, highlighting the role of the viscous term. The analysis is numerically supported by a set of CFD simulations. Additional CFD results are employed to extend the analysis to compressible flow case resulting in dedicated local Nusselt number correlations, function of Brinkman and Mach numbers, and expansion-altered temperature profiles. The study assumes no-slip flow and no temperature-jump boundary conditions at the channel wall (Knudsen number Kn<10−3), and addresses both fluid heating and cooling, both uniform heat flux (UHF) and uniform wall temperature (UWT) boundary condition (BC), and both circular (CR) and parallel-plate (PP) channel cross-section.

Flow and heat transfer characteristics in slip regime for an annulus of evacuated collector tube subjected to non-uniform solar flux boundary condition

Due to the accumulation of residual gas, vacuum failure may occur in the annulus of evacuated collector tube in solar trough thermal power system, resulting in the changes of flow and heat transfer mechanism and characteristics. In this paper, based on the actual situation of solar radiation in summer of Chongqing, China, the flow and heat transfer characteristics of the annulus in slip regime were numerically studied. The effects of gas pressure, gas types (N2, H2 and He), heat transfer fluid temperature were analyzed and discussed. The results show that the temperature jump under lower gas pressure directly affects the heat loss and temperature distribution of the annulus. The temperature jump is most obvious for the residual gas of H2 at gas pressure of 130Pa, while it basically disappears for N2 and He (the corresponding Knudsen numbers Kn are 0.0019 and 0.0040; 0.0022 and 0.0046; 0.0026 and 0.0052 respectively under heat transfer fluid temperatures of 473.15 K, 573.15 K and 673.15 K). The heat loss and the outer wall temperature of glass envelope rise with the increasing of heat transfer fluid temperature and gas pressure, and are related to the thermo-physical properties and thermal adaptation coefficient of gas. For gas pressure being less than 70Pa (Kn > 0.0048) or greater than 20000Pa (continuous gas flow regime), the heat loss of N2 keeps increasing with gas pressure, and the heat losses of H2 and He are even more than 1200 W·m−1 under heat transfer fluid temperature of 673.15 K. When the temperature of heat transfer fluid is 573.15 K and the pressure increases from 1.83Pa to 130Pa, the declining relative values of first law efficiency (thermal efficiency) and second law efficiency (exergic efficiency) for N2, H2 and He are 1.59%, 10.07% and 8.48% respectively.

Fractional parabolic two-step model and its accurate numerical scheme for nanoscale heat conduction

Simulation of nanoscale heat transfer phenomena has attracted great attention, particularly the nanoscale heat conduction induced by ultrashort-pulsed laser heating. In this article, we present a nanoscale fractional parabolic two-step model and obtain an accurate numerical scheme for nanoscale heat conduction in metals. The model is obtained by introducing the Knudsen number (Kn) and the fractional-order derivative in time to the original parabolic two-step energy transport, and then coupling them with the Kn dependent and temperature-jump boundary condition. The numerical scheme is developed based on the fourth-order compact finite difference method and the L1 approximation for fractional derivatives. The well-posedness of the model, and the stability and convergence of the scheme are analyzed theoretically. We finally test the accuracy and applicability of the new model and the obtained numerical method by two examples. By changing values of the Knudsen number and fractional-order derivative as well as the parameter in the boundary condition, the simulation could be a tool for analyzing the nanoscale heat conduction in porous media such as porous thin metal film exposed to ultrashort-pulsed lasers.

Уравнение состояния и поверхностные свойства аморфного железа

It is shown that on the nonlinear dependence of the first coordination number (kn) versus the packing coefficient (kp) of the structure of a mono-component substance, three special points corresponding to the amorphous structure can be distinguished. Based on the pairwise interatomic potential of Mie–Lennard-Jones, the state equation and properties of iron for both these three amorphous structures and the crystal state are calculated. It is shown that at kp = 0.45556 and kn = 6.2793 the minimum chemical potential is reached, i.e. this point is corresponding a thermodynamically stable amorphous structure into the liquid phase. An energetically equivalent point with the same kn value, but with kp = 0.6237, is a thermodynamically unstable amorphous structure corresponding to the solid phase. It is shown that the specific surface energy of an amorphous solid metal is greater than that of an amorphous liquid phase, but less than that of a metal in the crystalline state. This should lead to the fact that the surface of the crystal metal should tend to amorphize.

An immersed boundary-lattice Boltzmann method for gaseous slip flow

An immersed boundary (IB)-lattice Boltzmann (LB) method is proposed for microchannel slip flow encountered in microfluidics applications such as microelectromechanical Systems, filtration applications with nanofibers, polymer processing, and unconventional shale gas and coal seam gas applications. The LB method is theoretically analyzed to have an intrinsic ability to model velocity discontinuities at finite Knudsen numbers (Kn) when a sufficiently fine grid spacing and an external continuous perturbation, e.g., the body force of an IB method, are applied. Based on this analysis, an IB method coupled with a LB framework without ghost grids in nonfluid areas is proposed to study gaseous slip flow at finite Kn. In addition, an improved treatment to the suspending grids in nonfluid areas is proposed to assist the IB-LB method. In the simulations of two-dimensional Poiseuille and Couette flows for 0.01 ≤ Kn ≤ 1, the slip flow predicted by the proposed nonghost-grid IB-LB method achieves good agreement with that predicted by the linearized Boltzmann and/or Direct Simulation Monte Carlo methods up to Kn = 0.2. Since the proposed IB-LB method is free of adjustable parameters and slip velocity models, it provides a simple and promising pathway for modeling microscale flow applications for the validated regime, i.e., Kn &amp;lt; = 0.2.

Parametric Design and Static Performance Comparison of Six Typical Spherical Semi-open Reticulated Shell

By using ANSYS Parametric Design Language (APDL), this paper compiled the parametric design macro program for six types of single-layer semi-open spherical reticulated shell structure. Parametric design of the six types of reticulated shell structure was realized with given parameters of span S, the vector high F, radial node cycles number Nx, ring to symmetric regional copies number Kn, and the upper removed laps Ns. The stress performance of the six types of reticulated shell structure was compared and analyzed. Modelling examples demonstrated that the parametric design macro program is simple, practical and improves the efficiency of the selection of the reticulated shell design and the stress analysis under different geometric parameters. Through the comparative analysis of stress performance, some conclusions with engineering significance were obtained.

Modeling the effects of gas slippage, cleat network topology and scale dependence of gas transport in coal seam gas reservoirs

Gas flow in coal cleats plays a significant role for the permeability of coal seams. Investigations of gas transport in coal are carried out in this work using a numerical scheme with a coal sample scanned by the micro-computed tomography. We analyse a coal sample from the Bowen Basin (Australia) to characterize the natural discrete fracture network defined by connected cleats and matrix blocks. Due to the heterogeneity of coal micro-structures, the numerical simulator is combined with a grid-adaptive scheme. The accuracy of the simulator is validated for gas slip regime (0.01<Kn<1.0) in the micro-straight channel. Predictions of permeability and porosity are conducted through the method. The results show that the porosity and permeability of the digitalized coal model change along with the length and stabilizes at >400 voxels and >700 voxels, respectively. The Knudsen number (Kn) has an important effect on the gas slippage which can enhance the coal seam gas transport for narrow cleat networks. The increase of Kn ranging from 0.0005 to 0.01 increases the mass flux as well as the permeability of the coal for a given cleat aperture. The investigations of the gas transport in cleats prove the capability of the numerical simulator for modeling gas slip flow. It also provides a potential tool for solving gas transport in the complex geometry of a natural coal seam gas (CSG) reservoir.

Stability analysis of cantilever functionally graded material nanotube under thermo-magnetic coupling effect

In this paper, the Galerkin method is used to study the stability of cantilever functionally graded material (FGM) nanotube under thermo-magnetic coupling effect. The influences of external magnetic field and temperature on the stability of the FGM nanotube are investigated, and the influences of non-local parameters (en), Knudsen number (Kn), elastic coefficient of elastic matrix (ks) and volume fraction exponent (n) on the dynamic instability critical velocity (ucr) of FGM nanotube are discussed. It can be concluded that additional axial magnetic field, increase of temperature and increase of elastic coefficient will lead to the increase of critical instability velocity, while the increase of Kn, n and en will lead to the decrease of the critical instability velocity. Based on the conditions assumed in this paper, under the coupling effect of temperature, magnetic field and ks and the coupling effect of temperature, magnetic field and n, the ucr increases by 29.63% and 26.28% respectively. However, under the coupling effect of temperature, magnetic field and en and the coupling effect of temperature, magnetic field and Kn, the ucr decreases by 16.95% and 10.25% respectively.

Hydrodynamic instability of nanofluids in round jet for small Stokes number

The flow instability of particle-laden jet has been widely studied for large Stokes numbers. However, there is little attention on the case with small Stoke number, which often occurs in practical applications with nanoparticle-laden fluid. In this paper, the instability of nanofluids in round jet is studied numerically for [Formula: see text]. The results show that the law of nanofluids instability is quite similar to regular particle instability for axisymmetric azimuthal mode [Formula: see text]. However, for asymmetric azimuthal mode [Formula: see text], the regular pattern of instability is quite complex and different compared to common particle instability. The variations of wave amplification with wave number for different jet parameter [Formula: see text], Reynolds number Re, particle mass loading [Formula: see text], Knudsen number Kn, Stokes number St and the azimuthal modes [Formula: see text] are given. The flow usually gets more unstable as Knudsen number Kn increases, but the varying law gets inverse at high Reynolds number and at [Formula: see text]. The flow gets more unstable as Stokes number St increases at [Formula: see text] but gets more stable at [Formula: see text]. The decreases in wave number stimulate the flow instability at [Formula: see text] which shows distinct difference for the case at [Formula: see text]. Some unusual results of the effect of B, Re, Z on the flow instability are also discussed.

Origin of spurious oscillations in lattice Boltzmann simulations of oscillatory noncontinuum gas flows.

Oscillatory noncontinuum gas flows at the micro and nanoscales are characterized by two dimensionless groups: a dimensionless molecular length scale, the Knudsen number Kn, and a dimensionless frequency θ, relating the oscillatory frequency to the molecular collision frequency. In a recent study [Shi et al., Phys. Rev. E 89, 033305 (2014)10.1103/PhysRevE.89.033305], the accuracy of the lattice Boltzmann (LB) method for simulating these flows at moderate-to-large Kn and θ was examined. In these cases, the LB method exhibits spurious numerical oscillations that cannot be removed through the use of discrete particle velocities drawn from higher-order Gauss-Hermite quadrature. Here, we identify the origin of these spurious effects and formulate a method to minimize their presence. This proposed method splits the linearized Boltzmann Bhatnagar-Gross-Krook (BGK) equation into two equations: (1) a homogeneous "gain-free equation" that can be solved directly, containing terms responsible for the spurious oscillations; and (2) an inhomogeneous "remainder equation" with homogeneous boundary conditions (i.e., stationary boundaries) that is solved using the conventional LB algorithm. This proposed "splitting method" is validated using published high-accuracy numerical solutions to the linearized Boltzmann BGK equation where excellent agreement is observed.

A generalized model for gas flow prediction in shale matrix with deduced coupling coefficients and its macroscopic form based on real shale pore size distribution experiments

As an unconventional natural gas resource, shale gas is attracting increasing attention worldwide because of its potential to strengthen global energy security and reduce carbon emissions. The shale gas flow in nanopores is one of the main concerns affecting its development process. However, due to the strong nonlinearity and complexity of gas flow at the nanoscale, there is no satisfactory consensus on a physically sound flow mechanism scheme and its corresponding coupling method. Furthermore, although the integration method using specific functions has been proposed to facilitate the consideration of various pore sizes in shale matrix, the real shale experiments are rarely involved to realize this integration method with definitely determined parameters. In this study, the concept of “wall-associated diffusion” is firstly utilized to clarify the relationship among several flow mechanisms, and thereby a comprehensive flow mechanism scheme differing from common research is proposed, which physically considers both the division of mechanical mechanisms in nanopores and partition of flow space. Besides, the deficiencies in the mathematical models of viscous flow and several diffusion processes for flow description are illustrated. Using the molecular collision frequency expressions, a new coupling coefficient-based flow model in nanopores is derived to eliminate the aforementioned deficiencies by strengthening the correspondence between the flow models and Knudsen number (Kn), which is applicable in the full flow regime scope and avoids segment processing. Furthermore, based on the pore size distribution experiments of the real shale samples from a gas field, the fitting parameters needed for the macroscopic form of the above coupled model are obtained, so as to establish an integration method-based unified model for gas flow prediction in shale matrix. Results show that the prediction accuracy of the new model for the two tested shale samples is at least 2.2 times higher than several typical models and is also 1.9 and 7.5 times more accurate than the model without coupling coefficients, respectively. And the application of the proposed model to the flow experiment on another shale sample at higher pressure levels is in accordance with these observations. Interestingly, as temperature increases, the flow rate decreases with minor amplitudes. Under the conditions studied, the pressure change can cause over 10 times variation of the flow rate, revealing measures should be taken to maintain relatively high reservoir pressures to guarantee appreciable gas production from matrix if the other factors remain unchanged. In addition, the difference between the average flow shape in the whole rock and the gas motion in the pore of mean radius is revealed. Sounder in theoretical bases and better in application effects, the proposed model is expected to be of practical significance for guiding shale gas reservoir development.

Parabolic two-step model and accurate numerical scheme for nanoscale heat conduction induced by ultrashort-pulsed laser heating

Simulation of nanoscale heat transfer phenomena has attracted great attention, particularly the nano-scale heat conduction induced by ultrashort-pulsed laser heating. In this article, we propose a nanoscale parabolic two-step model and an accurate numerical scheme for thermal analysis of the nanoscale heat conduction induced by ultrashort-pulsed laser heating. To this end, we first introduce the Knudsen number (Kn) into the original parabolic two-step heat conduction equations and couple them with the Kn-dependent and temperature-jump boundary condition. We then develop a fourth-order accurate compact finite difference method for solving the nanoscale model. Stability and convergence of the obtained numerical scheme are analyzed theoretically. We finally test the accuracy and applicability of the nanoscale model and the obtained numerical scheme in three examples. By choosing various values of the Kn and the parameter α in the boundary condition, the simulation could be a tool for analyzing the nanoscale heat conduction induced by ultrashort-pulsed laser heating.