McCormack Kinetic Model Research Articles

Oscillatory pressure-driven rarefied binary gas mixture flow between parallel plates.

The rarefied, oscillatory, pressure-driven binary gas mixture flow between parallel plates is computationally investigated in terms of the mixture molar fraction and molecular mass ratio of the species, in a wide range of gas rarefaction and oscillation frequency. Modeling is based on the McCormack kinetic model. The output quantities are in dimensionless form and include the flow rate, wall shear stress and pumping power of the mixture, as well as the velocity and shear stress distributions and flow rates of the species. The presented results are for He-Xe and Ne-Ar. The heavier species are affected more drastically than the lighter ones from the inertial forces, resulting to large differences between the flow rate amplitudes of the species, which are increased as the flow becomes less rarefied, provided that the oscillation frequency is adequately high. At very high frequencies the ratio of the flow rate amplitudes of the light over the heavy species tends to the inverse of their molecular mass ratio in the whole range of gas rarefaction. The velocity overshooting effect becomes more pronounced as the molecular mass is increased. The mixture flow rate amplitude is larger, while its phase angle is smaller, than the corresponding ones of single gas, and they both vary nonmonotonically with the molar fraction. The effect of the mixture composition on the wall shear stress and pumping power is small. The present work may be useful in the design of gas separation devices, operating at moderate and high frequencies in rarefied and dense atmospheres.

Transient heat transfer in a rarefied binary gas mixture confined between parallel plates due to a sudden small change of wall temperatures

Transient behavior of the heat transfer through binary gas mixture, confined between two infinite parallel plates, caused by the sudden change of the plates’ temperatures, is studied for two monoatomic gas mixtures: Ne–Ar and He–Ar. The walls’ temperature changes are considered small compared to the equilibrium temperature of the system, so the McCormack kinetic model is used for the numerical simulations. The time evolution of the main macroscopic parameter is investigated for various species concentrations and for different gas rarefactions ranging from near the free molecular to slip flow regime. It is found that the mixture heat flux takes several characteristic times, which is defined by the distance between the plates over the most probable molecular speed, to achieve its new equilibrium state. This time of the steady state flow establishment depends strongly on the gas rarefaction, mixture nature and composition.

Isothermal flows of rarefied ternary gas mixtures in long tubes

A method is presented to determine isothermal flows of three-component rarefied gaseous mixtures in long tubes. The flow is subject to arbitrary but small local density gradients, when linearization can be applied. The mixture is described by the McCormack kinetic model, which is solved by discretizing the spatial and velocity spaces. Results are presented for the realistic and hard-sphere molecular interactions. The dimensionless kinetic coefficients are presented and commented for He–Ar–Xe mixture in a wide range of the rarefaction and numerous values of the mole fractions. The velocity profiles of the components are also shown. An additional method is also developed to solve the flow problem of the mixture driven by a global pressure difference between the two ends of the tube. The flow rates and the distributions of the mole fractions and the pressure along the axis of the tube are presented for pressure-driven flows of He–Ar–Xe mixture. The flow of the ternary mixture exhibits the separation phenomenon. The profiles of the mole fractions are non-uniform.

Accelerated discrete velocity method for axial-symmetric flows of gaseous mixtures as defined by the McCormack kinetic model

An accelerated discrete velocity method is presented to calculate the steady axial-symmetric flows of gaseous mixtures defined by the McCormack kinetic model. The scheme is formulated in cylindrical coordinates. Diffusion equations for the macroscopic velocity and the heat-flow are derived on the basis of the projected McCormack equations. The solutions of the kinetic equations are carried out iteratively by using the discrete velocity method. The diffusion equations are also solved in each stage of the iteration in order to accelerate the scheme. Pressure driven flows of He/Xe and Ne/Ar mixtures through a cylindrical tube are simulated in order to study the computational performance of the approach. It is shown that the required number of iterations and the computational times are significantly reduced at intermediate and large values of the rarefaction parameter by using the accelerated method. In the hydrodynamic limit, the flow rates of the components converge to the corresponding slip flow results. Flows driven by mole fraction gradient are also successfully simulated. Typical velocity and heat-flow profiles for pressure driven flow of He/Xe mixture are shown and commented on.

Rarefied gas mixture flow between plates of arbitrary length due to small pressure difference

The linearized flow of a binary gas mixture between two parallel plates of any length driven by a small pressure difference is numerically solved based on the McCormack kinetic model. In this case the flow is not fully developed and the simulated flow field includes, in addition to the channel, efficiently large upstream and downstream regions significantly increasing the required computational effort. In the present work results are provided for one gas mixture, namely He-Ne with a specific concentration of 0.5 in a wide range of gas rarefaction. The ratio of the length over the height of the channel is taken to be equal to 0, 1 and 5 corresponding to flow through channels of very short (slit), short and moderate lengths respectively. Results are provided for the two kinetic coefficients related to Poiseuille and barodiffusion flow rates and the velocity profiles of each species. The effect of the length to height ratio and of the gas rarefaction on the flow quantities is investigated deducing that as the ratio in increased the separation phenomenon becomes more dominant, while a Knudsen minimum is observed at the ratio equal to 5.

Flow rate measurements of binary gas mixtures through long trapezoidal microchannels

The flow rate of two noble gas mixtures, namely He/Ar and He/Kr, is measured through a microsystem containing 400 long trapezoidal microchannels placed in parallel configuration. Each microchannel has a trapezoidal cross section with long base 5.38 micrometers and height 1.90 micrometers, while its length is 5000 micrometers. The experiment is based on the constant volume method. The flow is driven by pressure gradient. The flow rate measurements refer to downstream pressures of 15.1 kPa and 8.05 kPa. The pressure ratio is in the range of 3-7 and 4-7 for the larger and smaller downstream pressures, respectively. The investigated rarefaction range is in the slip and early transition regions. The concentration of He varies from zero to one. The measured flow rates are compared to the corresponding computational ones obtained by the numerical solution of the McCormack kinetic model. Very good agreement between the experimental and computational results is reached. The difference between the corresponding results is less than the experimental uncertainty. Typical pressure and concentration profiles along the axis and the velocity profiles in the center of the channel obtained from the numerical solution are also presented.

Variance-reduced DSMC for binary gas flows as defined by the McCormack kinetic model

A variance-reduced direct simulation Monte Carlo method is presented for binary gas flows as defined by the McCormack kinetic model. Two types of flow configuration, namely pressure driven flow between two parallel plates and through rectangular channels, are considered. The kinetic model is transformed into one- and two-dimensional projected formalisms. Both cases are modeled by the variance-reduced DSMC. Simulations are performed to compare the method to the analytical discrete ordinate and discrete velocity methods. For both flow configurations, very good agreement is obtained between the results of different approaches. The results of the approach are also compared to the prediction of the moment method and the results of hard-sphere gases. The model can be beneficial for computing slow rarefied gaseous mixture flows, especially in nano- and microscale devices.

Comparative study between computational and experimental results for binary rarefied gas flows through long microchannels

A comparative study between computational and experimental results for pressure-driven binary gas flows through long microchannels is performed. The theoretical formulation is based on the McCormack kinetic model and the computational results are valid in the whole range of the Knudsen number. Diffusion effects are taken into consideration. The experimental work is based on the Constant Volume Method, and the results are in the slip and transition regime. Using both approaches, the molar flow rates of the He–Ar gas mixture flowing through a rectangular microchannel are estimated for a wide range of pressure drops between the upstream and downstream reservoirs and several mixture concentrations varying from pure He to pure Ar. In all cases, a very good agreement is found, within the margins of the introduced modeling and measurement uncertainties. In addition, computational results for the pressure and concentration distributions along the channel are provided. As far as the authors are aware of, this is the first detailed and complete comparative study between theory and experiment for gaseous flows through long microchannels in the case of binary mixtures.

Rarefied gas flow of binary mixtures through long channels with triangular and trapezoidal cross sections

The flow of binary gas mixtures through long micro-channels with triangular and trapezoidal cross sections is investigated in the whole range of the Knudsen number. The flow is driven by pressure and concentration gradients. The McCormack kinetic model is utilized to simulate the rarefied flow of the gas mixture, and the kinetic equations are solved by an upgraded discrete velocity algorithm. The kinetic dimensionless flow rates are tabulated for selected noble gas mixtures flowing through micro-channels etched by KOH in silicon (triangular and trapezoidal channels with acute angle of 54.74°). Furthermore, the complete procedure to obtain the mass flow rate for a gas mixture flowing through a channel, based on the dimensionless kinetic results, which are valid in each cross section of the channel, is presented. The study includes the effect of the separation phenomenon. It is shown that gas separation may change significantly the estimated mass flow rate. The presented methodology can be used for engineering purposes and for the accurate comparison with experimental results.

Heat flux between parallel plates through a binary gaseous mixture over the whole range of the Knudsen number

The heat flux problem for a binary gaseous mixture confined between two parallel plates with different temperatures is studied on the basis of the McCormack kinetic model equation, which was solved by the discrete velocity method. The calculations were carried out for three mixtures of noble gases: neon–argon, helium–argon and helium–xenon. The heat flux and distributions of temperature, density and concentration were calculated for several values of rarefaction in the range from 0.01 to 40 and for three values of the concentration: 0.1 , 0.5 and 0.9 . The numerical data together with an analytical solution based on the temperature jump boundary condition cover the whole range of the gas rarefaction beginning from the free-molecular regime to the hydrodynamic one. It was shown that the heat flux significantly depends on the intermolecular interaction law.

A Legendre expansion and some exact solutions basic to the McCormack model for binary gas mixtures

A Legendre expansion of the scattering kernel, a conservation condition and some exact solutions are reported for the McCormack kinetic model that is used to describe a binary mixture of rarefied gases.

The McCormack model for gas mixtures: The temperature-jump problem

An analytical version of the discrete-ordinates method (the ADO method) is used to establish a concise and particularly accurate solution to the temperature-jump problem for a binary gas mixture described by the McCormack kinetic model. The solution yields, in addition to the temperature-jump coefficient for the general (specular–diffuse) case of Maxwell boundary conditions for each of the two species, the density and temperature profiles for both types of particles. Numerical results are reported for two binary mixtures (Ne-Ar and He-Xe) with various molar concentrations. The algorithm is considered especially easy to use, and the developed (FORTRAN) code requires typically less than a second on a 2.2 GHz Pentium 4 machine to compute all quantities of interest.

The McCormack model for gas mixtures: Plane Couette flow

An analytical version of the discrete-ordinates method is used to establish a concise and particularly accurate solution to the problem of plane Couette flow for a binary gas mixture described by the McCormack kinetic model. The solution yields, for the general (specular-diffuse) case of Maxwell boundary conditions for each of the two species, the velocity, heat-flow, and shear-stress profiles for both types of particles, as well as the particle-flow and heat-flow rates associated with each of the two species of gas particles. Highly accurate numerical results are reported for the case of a helium–argon mixture confined between molybdenum and tantalum plates. The algorithm is considered especially easy to use, and the developed (FORTRAN) code requires typically less than a second on a 2.2GHz Pentium 4 machine to compute all quantities of interest with at least five figures of accuracy.

Velocity slip and temperature jump coefficients for gaseous mixtures. IV. Temperature jump coefficient

The temperature jump coefficient was calculated for a binary gaseous mixture on the basis of the McCormack kinetic model of the Boltzmann equation, which was solved by the discrete velocity method. The calculations were carried out for the three mixtures of noble gases: neon–argon, helium–argon and helium–xenon. A strong influence of the intermolecular interaction on the temperature jump coefficient was observed by comparing the results based on the model of rigid spheres with those obtained for a realistic potential.

Velocity slip and temperature jump coefficients for gaseous mixtures. III. Diffusion slip coefficient

The diffusion slip coefficient is calculated for binary gaseous mixtures on the basis of the McCormack kinetic model of the Boltzmann equation, which is solved by the discrete velocity method. The calculations are carried out for the three mixtures of noble gases: neon–argon, helium–argon, and helium–xenon. Two models of the intermolecular interaction potential were considered. It was shown that this coefficient strongly depends on the potential. It was concluded that the use of the rigid spheres model does not provide physical results. An example of application of the diffusion slip coefficient is given.

The McCormack model for gas mixtures: Heat transfer in a plane channel

An analytical version of the discrete-ordinates method (the ADO method) is used to establish a concise and particularly accurate solution to the heat-transfer problem in a plane channel for a binary gas mixture described by the McCormack kinetic model. The solution yields for the general (specular-diffuse) case of Maxwell boundary conditions for each of the two species, the density and temperature profiles for both types of particles, as well as the overall heat flow associated with each of the two species of gas particles. Numerical results are reported for two binary mixtures (Ne–Ar and He–Xe). The algorithm is considered especially easy to use, and the developed (FORTRAN) code requires typically less than a second on a 2.2 GHz Pentium 4 machine to compute all quantities of interest.

Velocity slip and temperature jump coefficients for gaseous mixtures. II. Thermal slip coefficient

The thermal slip coefficient was calculated for a binary gaseous mixture on the basis of the McCormack kinetic model of the Boltzmann equation, which was solved by the discrete velocity method. The calculations were carried out for the three mixtures of noble gases: neon–argon, helium–argon, and helium–xenon. A strong influence of the potential of intermolecular interaction upon the thermal slip coefficient was observed by comparing the results based on the model of rigid spheres with those obtained for a realistic potential. An example of application of the thermal slip coefficient is given.

Velocity slip and temperature jump coefficients for gaseous mixtures. I. Viscous slip coefficient

The viscous slip coefficient was calculated for binary gaseous mixtures on the basis of the McCormack kinetic model of the Boltzmann equation, which was solved by the discrete velocity method. The calculations were carried out for the three mixtures of noble gases: neon–argon, helium–argon, and helium–xenon. It was showed that for the mixture of helium and xenon, which has a large ration of the molecular masses, the slip coefficient significantly differs from that for a single gas. A comparison of the present results with those obtained by the moment method applied to the Boltzmann equation showed that the McCormack model equation provides reliable results with modest computational efforts. An example of application of the viscous slip coefficient was given.