Macroscopic Conservation Equations Research Articles

Kinetic equation for stochastic vector bundles

The kinetic equation is crucial for understanding the statistical properties of stochastic processes, yet current equations, such as the classical Fokker–Planck, are limited to local analysis. This paper derives a new kinetic equation for stochastic systems on vector bundles, addressing global scale randomness. The kinetic equation was derived by cumulant expansion of the ensemble-averaged local probability density function, which is a functional of state transition trajectories. The kinetic equation is the geodesic equation for the probability space. It captures global and historical influences, accounts for non-Markovianity, and can be reduced to the classical Fokker–Planck equation for Markovian processes. This paper also discusses relative issues concerning the kinetic equation, including non-Markovianity, Markov approximation, macroscopic conservation equations, gauge transformation, and truncation of the infinite-order kinetic equation, as well as limitations that require further attention.

The lattice Boltzmann model for the heat and mass transfer of latent functional thermal fluid using enthalpy-transforming method

An enthalpy-transforming model based on lattice Boltzmann method is established in this paper to solve incompressible thermal flow problem of latent functional thermal fluid. Using the Chapman-Enskog expansion, the macroscopic conservation equation of enthalpy is obtained by introducing a new type of equilibrium distribution. It is validated by its application to simulate the natural convection of latent functional thermal fluid in a square cavity, which is heated differentially at two vertical side walls. Good agreement was obtained between the present results and those from other works. Furthermore, natural convection of latent functional thermal fluid in two-dimension and three-dimension cases are considered respectively. The influence of Stefan numbers, Rayleigh numbers and phase change temperature on heat transfer was studied. The relative errors of average Nusselt number between the present model and references are less than 2 %. In the case with periodic boundary condition, the present model can reveal heat transfer process of LFTF with different frequencies, amplitudes and volume fractions. The simulation results show that the present model can be used in the flow and heat transfer simulation of latent functional thermal fluid.

Derivation of a second-order acoustic wave equation in an air-saturated porous medium from a volume-averaged transport model

For propagation of high-amplitude acoustic waves in a rigid porous frame saturated by air, significant departures from a constant flow resistivity model have been observed. This departure can be modeled as a quadratic modification to the linear Darcy’s drag law, known as Forchheimer’s correction. To study the physical cause and relative significance of this nonlinear correction with respect to convective nonlinearity, a second-order acoustic wave equation is derived from volume-averaged equations of mass, momentum, and entropy conservation. Due to the scale separation between the wavelength and the porous structure, the particle velocity is approximated as the sum of irrotational macroscopic and incompressible microscopic fields. The porosity, tortuosity, and permeability are defined and incorporated into the macroscopic conservation equations. Porous drag terms are also derived in the macroscopic momentum equation for Darcy’s law and Forchheimer’s correction. Conservation equations for fluid parameter perturbations are obtained, and a second-order wave equation is derived. Dimensional analysis, in terms of the acoustic Reynolds number and Darcy number, describes the criteria for significant Forchheimer and convective nonlinearity. A case is obtained for a Westervelt-like equation for approximately progressive plane waves. These results inform modeling of nonlinearities for sound propagation in porous media under practical conditions.

A comparative study on the accuracy and conservation properties of the SPH method for fluid flow interaction with porous media

Smoothed Particle Hydrodynamics (SPH) has successfully been employed in the last decade for macroscopic simulation of fluid flow in porous media. However, there are still certain inconsistencies in the existing models with regard to both the definition of the governing equations and the adopted numerical treatments. In this study, thus, in order for a comparative investigation of the existing practices, the method of spatial averaging is employed to derive two types of macroscopic conservation equations of mass and momentum through phase averaging and intrinsic phase averaging of fluid density. Consistent discretisation methods are adopted for the determination of particle's volume, in the Weakly Compressible SPH form, and velocity by which particles move; and they are used to construct three numerical models. The models are then employed to solve two different types of problems: i) a fully saturated soil medium under the closed channel flow condition, and ii) a porous medium under a free surface flow condition with varying saturation. The results of the models are compared with analytical and experimental data; and particle distribution, water surface elevation, conservation of fluid volume, and pressure distribution are investigated. It is shown that only the latter type of modelling (i.e., intrinsic phase averaging of fluid density) leads to the conservation of fluid volume for both types of the problems, while the former type of formulation (i.e., based on phased averaged density, also known as apparent density) is suitable for closed porous media flow conditions only. Comparisons with other numerical models and the author's previous study indicate that the adopted numerical treatments, especially the δ-SPH formulation, have led to an improvement in the smoothness of particles distribution and pressure field. A few recommendations are finally made about potential further improvements of the developed models and extensions of their applicability.

How to build coarse-grain transport models consistent from the kinetic to fluid regimes

In this paper, we examine how to build coarse-grain transport models consistently from the kinetic to fluid regimes. The internal energy of the gas particles is described through a state-to-state approach. A kinetic equation allows us to study transport phenomena in phase space for a non-homogeneous gas mixture. Internal energy excitation is modeled using a binary collision operator, whereas gas chemical processes rely on a reactive collision operator. We obtain an asymptotic fluid model by means of a Chapman–Enskog perturbative solution to the Boltzmann equation in the Maxwellian reaction regime. The macroscopic conservation equations of species mass, mixture momentum, and energy as well as expressions of the transport properties are given. Reversibility relations for elementary processes are formulated in the coarse-grain model at the kinetic level and are enforced in the collision algorithm of the direct simulation Monte Carlo method used to solve the kinetic equation. Furthermore, respecting these reversibility relations is key to deriving a fluid model that is well-posed and compatible with the second law of thermodynamics. The consistency between the kinetic and fluid simulations is assessed for the simulation of a shock wave in a nitrogen gas using the uniform rovibrational collisional coarse-grain model. The kinetic and fluid simulations show consistency for the macroscopic properties and transport fluxes between both regimes.

Force approach for the pseudopotential lattice Boltzmann method.

One attractive feature of the original pseudopotential method consists on its simplicity of adding a force dependent on a nearest-neighbor potential function. In order to improve the method, regarding thermodynamic consistency and control of surface tension, different approaches were developed in the literature, such as multirange interactions potential and modified forcing schemes. In this work, a strategy to combine these enhancements with an appropriate interaction force field using only nearest-neighbor interactions is devised, starting from the desired pressure tensor. The final step of our procedure is implementing this external force by using the classical Guo forcing scheme. Numerical tests regarding static and dynamic flow conditions were performed. Static tests showed that current procedure is suitable to control the surface tension and phase densities. Based on thermodynamic principles, it is devised a solution for phase densities in a droplet, which states explicitly dependence on the surface tension and interface curvature. A comparison with numerical results suggest a physical inconsistency in the pseudopotential method. This fact is not commonly discussed in the literature, since most of studies are limited to the Maxwell equal area rule. However, this inconsistency is shown to be dependent on the equation of state (EOS), and its effects can be mitigated by an appropriate choice of Carnahan-Starling EOS parameters. Also, a droplet oscillation test was performed, and the most divergent solution under certain flow conditions deviated 7.5% from the expected analytical result. At the end, a droplet impact test against a solid wall was performed to verify the method stability, and it was possible to reach stable simulation results with density ratio of almost 2400 and Reynolds number of Re=373. The observed results corroborate that the proposed method is able to replicate the desired macroscopic multiphase behavior.

On fine particles synthesis using three-zone reactor

This paper reports a numerical study on thermal and mass dispersion in carbon combustion synthesis of oxides (CCSO) in porous media. A volume averaging of the microscopic conservation equations over an elementary volume is applied to derive the macroscopic conservation equations for convective and conductive heat and mass transfer in a porous media consisting of gas components mixture and solid particles of reactant species. The governing system of macroequations is the conservation of momentum, mass and energy for multicomponent gas-solid media using the heat and mass exchange between gas and solid phases. The model developed in dimensionless variables and similarity parameters is applied to numerical simulation of barium titanate micron particles synthesis in three-zone reactor using the thermal and mass dispersion. The results are in satisfactory agreement with experimental measurements. The dispersion strongly influences on the thermal front propagation as well as on the barium titanate synthesis rate. The three-zone reactor is more efficient compared to the flow-reactor.

Natural convection of hybrid nanofluids inside a partitioned porous cavity for application in solar power plants

The present article deals with the CFD simulation of natural convection heat transfer of a hybrid nanofluid in an inverted T-shaped cavity partitioned and saturated by two different types of porous media. Suspensions of organic and inorganic nanoparticles, i.e., MWCNTs and Fe3O4, in water were selected as the working fluid. The macroscopic conservation equations for the flow field and heat transfer were modeled via volume averaging the microscopic equations inside porous media over a representative elementary volume. The effects of many parameters were investigated. The parameters included the Rayleigh number (Ra = 103–106), porosity coefficient ratio of two porous media (er = 0.5–1.8), volume fraction of the dispersed nanoparticles ( $$\varphi$$ = 0–0.003), Richardson number (Ri = 0.1–20), Darcy number ratio of two porous media (Dar = 0.01, 1, 100) and thermal conductivity ratio of two porous media (kr = 0.2, 0.4, 1, 5). The results showed that, with an increase in the Rayleigh number, porosity ratio and Darcy number ratio and decrease in the thermal conductivity ratio, the averaged Nusselt number increased.

Modeling the Effect of Relative Humidity on Adsorption Dynamics of Volatile Organic Compound onto Activated Carbon.

A two-dimensional heterogeneous mathematical model was developed and validated to study the effect of relative humidity on volatile organic compound (VOC) adsorption onto activated carbon. The dynamic adsorption model consists of the macroscopic mass, momentum, and energy conservation equations and includes a multicomponent adsorption isotherm to predict the competitive adsorption equilibria between VOC and water vapor, which is described by an extended Manes method. Experimental verifications show that the model predicted the breakthrough profiles during competitive adsorption of the studied VOCs (2-propanol, acetone, n-butanol, toluene, 1,2,4-trimethylbenzene) at relative humidity range 0-95% with an overall mean relative absolute error (MRAE) of 11.8% for dry (0% RH) conditions and 17.2% for humid (55 and 95% RH) conditions, and normalized root-mean-square error (NRMSE) of 5.5 and 8.4% for dry and humid conditions, respectively. Sensitivity analysis was also conducted to test the robustness of the model in accounting for the impact of relative humidity on VOC adsorption by varying the adsorption temperature. Good agreement was observed between the experimental and simulated results with an overall MRAE of 12.4 and 7.1% for the breakthrough profiles and adsorption capacity, respectively. The model can be used to quantify the impact of carrier gas relative humidity during adsorption of contaminants from gas streams, which is useful when optimizing adsorber design and operating conditions.

Transport coefficients in ultrarelativistic kinetic theory

A spatially periodic longitudinal wave is considered in relativistic dissipative hydrodynamics. At sufficiently small wave amplitudes, an analytic solution is obtained in the linearized limit of the macroscopic conservation equations within the first- and second-order relativistic hydrodynamics formulations. A kinetic solver is used to obtain the numerical solution of the relativistic Boltzmann equation for massless particles in the Anderson-Witting approximation for the collision term. It is found that, at small values of the Anderson-Witting relaxation time $\ensuremath{\tau}$, the transport coefficients emerging from the relativistic Boltzmann equation agree with those predicted through the Chapman-Enskog procedure, while the relaxation times of the heat flux and shear pressure are equal to $\ensuremath{\tau}$. These claims are further strengthened by considering a moment-type approximation based on orthogonal polynomials under which the Chapman-Enskog results for the transport coefficients are exactly recovered.

Unified Lattice Boltzmann Method for Electric Field–Space Charge Coupled Problems in Complex Geometries and Its Applications to Annular Electroconvection

A lattice Boltzmann method (LBM) is developed to solve the electric field–space charge coupled problems. Instead of solving the macroscopic charge conservation equation and the Poisson's equation for electric potential, two discrete lattice Boltzmann equations are formulated and solved. A nonequilibrium extrapolation scheme is used to treat the boundary conditions with complex geometry. Our method is validated with several test cases for which analytical solutions and/or reference numerical results exit. An attractive feature of this methodology lies in its natural coupling with the LBM for the fluid flow. As a demonstration and also an application, the unipolar injection induced electroconvection of dielectric liquids in annular geometries is considered. The different flow patterns of the concentric and eccentric configurations are highlighted.

A novel lattice Boltzmann model for the solid–liquid phase change with the convection heat transfer in the porous media

A novel lattice Boltzmann (LB) model with double distribution functions is proposed to simulate the solid–liquid phase change problems with the convection heat transfer in the porous media at the representative elementary volume scale. A generalized LB model is adopted to simulate the velocity field of the liquid region in the presence of the porous media. Based on the framework of a LB model for nonlinear convection–diffusion equation, an LB evolution equation without a phase change source term, as well as a newly modified equilibrium distribution function including the effective total enthalpy and thermal capacity ratio, are constructed to solve the effective total enthalpy energy conservation equation for the porous media. The macroscopic effective total enthalpy energy conservation equation can be exactly recovered from the present LB model by the Chapman–Enskog procedure. As compared with the most previous models, the present model can provide a higher computational efficiency, because the iterative step or linear equations solving procedure for the latent heat source term can be avoided. Furthermore, the temperature and heat flux continuity conditions at the interface between phases of different thermophysical properties are inherently satisfied for the present model. In such a case, the differences in both thermal conductivity and specific heat between phases, as well as the variation of porosity can be tackled by independently adjusting the relaxation time and the thermal capacity ratio included in the equilibrium distribution function, respectively. Additionally, the numerical diffusion across the phase interface for temperature, induced by solid–liquid phase change, can be dramatically reduced in the present model through the MRT collision scheme with a special configuration for the relaxation matrix. Numerical tests for several benchmark problems are carried out to validate the wide practicability and the accuracy of the present model. Very good agreements with analytical solutions or previous experimental and numerical results can be achieved, demonstrating the present model can be served as a promising numerical technique for simulating the transient solid–liquid phase change problems in the porous media.

A multiscale SPH method for simulating transient viscoelastic flows using bead-spring chain model

In this article, a multiscale smoothed particle hydrodynamics (SPH) method is proposed to simulate transient viscoelastic flows by using a bead-spring chain description of polymer molecule. This methodology couples macroscopic conservation equations for mass and momentum with a stochastic differential equation for bead-spring chain dynamics, which can be solved by a three-step semi-implicit algorithm to obtain the polymeric stress. Accordingly, a closed-form constitutive equation is not required. The multiscale SPH method is firstly verified by the plane Couette flow with Hookean and finitely extendable non-linear elastic (FENE) bead-spring chain models with the number of beads M=2, which correspond to Hookean and FENE dumbbell models, respectively. Results for the velocity and shear stress are in excellent agreement with the available numerical and analytical solutions in literature. Then, the numerical method is extended to simulate the plane Couette flow and Poiseuille flow with FENE bead-spring chain model with M >2. In particular, some molecular information of bead-spring chain such as the molecular stretch, the orientation angle and the mean configuration thickness are displayed. The influence of the number of beads M on the evolutions of the molecular information is also analyzed in detail. It is demonstrated that the multiscale SPH method proposed here is a promising computational tool for the multiscale simulations of transient viscoelastic flows.

Numerical Simulation of Titanium Alloy Ingot Solidification Structure during VAR Process Based on Three-Dimensional CAFE Method

Abstract A multiscale three-dimensional (3D) mathematical model has been established for simulating the temperature field, the fluid flow and the solidification structure of Ti-6Al-4V alloy ingot during vacuum arc remelting (VAR) process, which consists of the macroscopic mass, the momentum and energy conservation equations and the macroscale model of the nucleation and growth of grains. On the basis of heat transfer and fluid flow calculation, the 3D solidification structure formation of the ingot during whole VAR process has been obtained. Comparing the simulation result with the experimental observation, a reasonably qualitative agreement is achieved on grain structure and grain growth pattern. In particular, when taking the heat radiation into consideration during the calculation, the columnar grains on the ingot top are predicted well. Furthermore, a sensitivity study of the effect of natural convection on the grain structure has been carried out. The results show that the natural convection has a great influence on columnar-to-equiaxed transition (CET) and grain size, expressing as promoting the CET and refining the grain.

Modeling Competitive Adsorption of Mixtures of Volatile Organic Compounds in a Fixed-Bed of Beaded Activated Carbon

A two-dimensional mathematical model was developed to study competitive adsorption of n-component mixtures in a fixed-bed adsorber. The model consists of an isotherm equation to predict adsorption equilibria of n-component volatile organic compounds (VOCs) mixture from single component isotherm data, and a dynamic adsorption model, the macroscopic mass, energy and momentum conservation equations, to simulate the competitive adsorption of the n-components onto a fixed-bed of adsorbent. The model was validated with experimentally measured data of competitive adsorption of binary and eight-component VOCs mixtures onto beaded activated carbon (BAC). The mean relative absolute error (MRAE) was used to compare the modeled and measured breakthrough profiles as well as the amounts of adsorbates adsorbed. For the binary and eight-component mixtures, the MRAE of the breakthrough profiles was 13 and 12%, respectively, whereas, the MRAE of the adsorbed amounts was 1 and 2%, respectively. These data show that the model provides accurate prediction of competitive adsorption of multicomponent VOCs mixtures and the competitive adsorption isotherm equation is able to accurately predict equilibrium adsorption of VOCs mixtures.

Flow rate and time mean speed predictions for the urban freeway network using state space models

Short-term predictions of traffic parameters such as flow rate and time mean speed is a crucial element of current ITS structures, yet complicated to formulate mathematically. Classifying states of traffic condition as congestion and non-congestion, the present paper is focused on developing flexible and explicitly multivariate state space models for network flow rate and time mean speed predictions. Based on the spatial–temporal patterns of the congested and non-congested traffic, the NSS model and CSS model are developed by solving the macroscopic traffic flow models, conservation equation and Payne–Whitham model for flow rate and time mean speed prediction, respectively. The feeding data of the proposed models are from historical time series and neighboring detector measurements to improve the prediction accuracy and robustness. Using 2-min measurements from urban freeway network in Beijing, we provide some practical guidance on selecting the most appropriate models for congested and non-congested conditions. The result demonstrates that the proposed models are superior to ARIMA models, which ignores the spatial component of the spatial–temporal patterns. Compared to the ARIMA models, the benefit from spatial contribution is much more evident in the proposed models for all cases, and the accuracy can be improved by 5.62% on average. Apart from accuracy improvement, the proposed models are more robust and the predictions can retain a smoother pattern. Our findings suggest that the NSS model is a better alternative for flow rate prediction under non-congestion conditions, and the CSS model is a better alternative for time mean speed prediction under congestion conditions.

Lattice-Boltzmann-based two-phase thermal model for simulating phase change

A lattice Boltzmann (LB) method is presented for solving the energy conservation equation in two phases when the phase change effects are included in the model. This approach employs multiple distribution functions, one for a pseudotemperature scalar variable and the rest for the various species. A nonideal equation of state (EOS) is introduced by using a pseudopotential LB model. The evolution equation for the pseudotemperature variable is constructed in such a manner that in the continuum limit one recovers the well known macroscopic energy conservation equation for the mixtures. Heats of reaction, the enthalpy change associated with the phase change, and the diffusive transport of enthalpy are all taken into account; but the dependence of enthalpy on pressure, which is usually a small effect in most nonisothermal flows encountered in chemical reaction systems, is ignored. The energy equation is coupled to the LB equations for species transport and pseudopotential interaction forces through the EOS by using the filtered local pseudotemperature field. The proposed scheme is validated against simple test problems for which analytical solutions can readily be obtained.

Direct numerical simulation of complex viscoelastic flows via fast lattice-Boltzmann solution of the Fokker–Planck equation

Micro–macro simulations of polymeric solutions rely on the coupling between macroscopic conservation equations for the fluid flow and stochastic differential equations for kinetic viscoelastic models at the microscopic scale. In the present work we introduce a novel micro–macro numerical approach, where the macroscopic equations are solved by a finite-volume method and the microscopic equation by a lattice-Boltzmann one. The kinetic model is given by molecular analogy with a finitely extensible non-linear elastic (FENE) dumbbell and is deterministically solved through an equivalent Fokker–Planck equation. The key features of the proposed approach are: (i) a proper scaling and coupling between the micro lattice-Boltzmann solution and the macro finite-volume one; (ii) a fast microscopic solver thanks to an implementation for Graphic Processing Unit (GPU) and the local adaptivity of the lattice-Boltzmann mesh; (iii) an operator-splitting algorithm for the convection of the macroscopic viscoelastic stresses instead of the whole probability density of the dumbbell configuration. This latter feature allows the application of the proposed method to non-homogeneous flow conditions with low memory-storage requirements. The model optimization is achieved through an extensive analysis of the lattice-Boltzmann solution, which finally provides control on the numerical error and on the computational time. The resulting micro–macro model is validated against the benchmark problem of a viscoelastic flow past a confined cylinder and the results obtained confirm the validity of the approach.

Acoustic multipole sources for the lattice Boltzmann method

By including an oscillating particle source term, acoustic multipole sources can be implemented in the lattice Boltzmann method. The effect of this source term on the macroscopic conservation equations is found using a Chapman-Enskog expansion. In a lattice with q particle velocities, the source term can be decomposed into q orthogonal multipoles. More complex sources may be formed by superposing these basic multipoles. Analytical solutions found from the macroscopic equations and an analytical lattice Boltzmann wavenumber are compared with inviscid multipole simulations, finding very good agreement except close to singularities in the analytical solutions. Unlike the BGK operator, the regularized collision operator is proven capable of accurately simulating two-dimensional acoustic generation and propagation at zero viscosity.

From vehicle–driver behaviors to first order traffic flow macroscopic models

This paper proposes a two scale modeling approach to vehicular traffic, where macroscopic conservation equations are closed by models at the microscopic scale obtained by a mathematical interpretation of driver behaviors to local flow conditions. The paper focuses on the closure of the mass conservation equations by phenomenological models derived by a detailed analysis at the scale of individual vehicles.