Asymptotic form of the admixture transport equation for a channel flow with an anisotropic diffusion tensor

The problem of the asymptotically correct reduction of a 3-D mass (heat) transfer equation to a 1-D equation in a flow with anisotropic diffusion properties is considered. The convective mass (heat) transfer domain is a cylindrical channel of arbitrary cross section. The diffusion coefficient matrix is assumed to be independent of the spatial coordinates. In the equivalent diffusion equation constructed, a certain effective diffusion (dispersion [1]) coefficient is introduced. Formulas for this coefficient are obtained. A relation between the effective diffusion coefficient calculations and the problem of minimization of a certain functional is established, i. e. the possibility of calculations based on variational methods is noted. An example of an exact calculation of the effective diffusion coefficient is considered. The possibility of a generalization of the problem, in which the effective diffusion (heat conduction) equation is essentially a nonlinear equation of general form for the one-dimensional case, is indicated. Sankt-Peterburg. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 110–123, March–April, 2000.     

Determination of heat fluxes in the neighborhood of a double curvature stagnation point in a flow of dissociating gas of arbitrary chemical composition

Numerical solutions are obtained for the equations of a uniform compressible boundary layer with variable physical properties in the vicinity of a stagnation point with different principal curvatures in the presence of an injected gas with the same properties as the incident flow. The results of the numerical solutions are approximated for the heat flux in the form of a relation that depends on the variation of the product of viscosity and density across the boundary layer and on the ratio of the principal radii of curvature.Using the concepts of effective diffusion coefficients in a multicomponent boundary layer, previously introduced by the author in [1], and the generalized analogy between heat and mass transfer in the presence of injection, together with the numerical solutions obtained, it is always possible, even without additional solutions of the boundary-layer equations, to derive final formulas for the heat fluxes in a flow of dissociating gas of arbitrary chemical composition, provided that we make the fundamental assumption that all recombination reactions take place at the surface.By way of example, formulas are given for the heat transfer to the surface of a body from dissociating air, regarded as a five-component mixture of the gases O, N, NO, O₂, N₂, and from a dissociating mixture of carbon dioxide and molecular nitrogen of arbitrary composition, regarded as an eleven-component mixture of the gases O, N, C, NO, C₂, O₂, N₂, CO, CN, C₃, CO₂.In the process of obtaining and analyzing these solutions it was found that, in computing the heat flux, a multicomponent mixture can be replaced with an effective binary mixture with a single diffusion coefficient only when the former can be divided into two groups of components with different (but similar) diffusion properties. In this case the concentrations of one group at the surface must be zero, while the diffusion flows of the second group at the surface are expressible, using the laws of mass conservation of the chemical elements, in terms of the diffusion flows of the first. Then the single effective diffusion coefficient is the binary diffusion coefficient D(A,M), where A relates to one group of components and M to the other.In view of the small amount of NO(c(NO) < 0.05), the diffusion transport of energy in dissociated air maybe described with the aid of a single binary diffusion coefficient D(A, M)(A=O, N, M=O₂, N₂, NO). However even in the case of complete dissociation into O and C atoms at the outer edge of the boundary layer, the diffusion transport of energy in dissociated carbon dioxide can not be described accurately enough by means of a model of a binary mixture with a single diffusion coefficient, since the diffusion properties of the O and C atoms are distinctly different.     

Exact solution of the capillary diffusion equation for a three‐component mixture

The paper gives an exact solution of the steady system of equations for stable threecomponent diffusion in the entire range of concentrations for a long capillary under a controlled capillary pressure differential. The solution allows one to calculate the distributions of component concentrations and mixture density along the capillary. It is shown that if the diffusion coefficients are markedly different, an extremum of mixture density can arise inside the capillary. In particular, if the density of the mixture in the upper flask is higher than that in the lower flask and the stratification of the system is generally stable, a region with a reverse density gradient that is unstable against gravity convection can appear inside the capillary. A comparison with experimental results shows that the resistance to gravity convection is disturbed when an extremum of mixture density arises in the channel during steady diffusion.     

对流扩散方程的迎风变换及相应有限差分方法

本文提出所谓迎风变换,将对流扩散方程分解为对流迎风函数和扩散方程,并构造相应的有限差分格式。对流迎风函数以简明的指数解析形式反映对流扩散现象的迎风效应,原则上消除了源于不对称对流算子的困难,能够便利对流扩散方程的数值求解。有限差分格式具有二阶精度和无条件稳定性,算例表明其准确性、收敛速度及对边界层效应的适应能力均明显优于中心差分格式和迎风差分格式。     

Simulation and transport phenomena of a ternary two-phase flow

A chemical flood model for a three-component (petroleum, water, injected chemical) two-phase (aqueous, oleic) system is presented. It is ruled by a system of nonlinear partial differential equations: the continuity equation for the transport of each of its components and Darcy's equation for the two-phase flow. The transport mechanisms considered are ultralow interfacial tension, capillary pressure, dispersion, adsorption, and partition of the components between the fluid phases (including solubilization and swelling).The mathematical model is numerically solved in the one-dimensional case by finite differences using an explicit and direct iterative procedure for the discretization of the conservation equations. Numerical results are compared with Yortsos and Fokas' exact solution for the linear waterflood case including capillary pressure effects and with Larson's model for surfactant flooding. The effects of the above-mentioned transport mechanisms on concentration profiles and on oil recovery are also analyzed.     

Thermocapillary instability of the equilibrium of a flat layer containing internal heat sources

In the absence of body forces, a factor which has a strong influence on the equilibrium stability of a nonuniformly heated liquid is the dependence of the coefficient of surface tension on the temperature and the thermocapillary effect generated by it. If the equilibrium temperature gradient is sufficiently great, then the presence of the thermocapillary forces on the free surface can lead to the occurrence of convective motion. The monotonie instability of the equilibrium of a flat layer was investigated in [1–3]. Analysis of nonmonotonic disturbances [4] showed that in the case of an undeformable free surface there is no oscillatory instability. In [5] it was found that oscillatory instability is possible if there is a nonlinear dependence of the coefficient of surface tension on the temperature. The present paper is devoted to numerical investigation of the equilibrium stability of a flat layer with respect to arbitrary disturbances. It is shown that for a deformable free boundary there appears an additional neutral curve, which corresponds to monotonie capillary instability. In addition, when the capillary convection mechanism is taken into account, there appears an oscillatory instability, which becomes the most dangerous in the region of small Prandtl and wave numbers.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 27–31, March–April, 1991.I thank V. K. Andreev for a helpful discussion of the work.     

Effect of the capillary forces on the moisture saturation distribution during the thawing of a frozen soil

Heterogeneous water-air mixture flows in the presence of capillary forces are investigated. It is shown that for moderate pressure and temperature gradients the distribution of the water saturation function can be determined from a nonlinear differential equation with a coefficient dependent on the porous medium parameters, the water viscosity, and the capillary pressure. The water-air mixture flow behind the ice melting front is considered.     

Condition of the onset and nonlinear regimes of convection of a three-component isothermal mixture in a porous rectangle under modulation of the concentration gradient

Convection of an isothermal three-component mixture saturating a porous block with a rectangular cross-section is investigated numerically when the concentration gradient of one of the components is modulated about a certain mean value. The shape of the neutral curves separating the growing and decaying disturbances in the amplitude-modulation frequency plane is investigated for various diffusion Rayleigh numbers. The results of calculations based on the linearized and complete systems of equations which describe convection of a mixture in a porous medium are analyzed and compared. The supercritical regimes of concentration convection are studied for steady-state boundary conditions and in the case of a periodic modulation of the concentration gradient of one of the components.     

Convection in a low prandtl number fluid with internal heating

This paper studies the problem of non-linear thermal convection in a horizontal layer of a low Prandtl number fluid with nearly insulating boundaries and in the presence of horizontally uniform internal heat sources. Two-dimensional rolls and hexagonal cells are found to be the only possible stable convection cells. Subcritical instability associated with the hexagons can occur for a range of the amplitude of convection. It is found that non-uniform internal heating can affect various flow features and the stability of the convective motion. A new subcritical instability which exists even in a symmetric layer with arbitrary Prandtl number is also found for the case where the variations of the internal heating with respect to the vertical variable is sufficiently high.     

Stationary Fingering Instability in a Non-equilibrium Porous Medium with Coupled Molecular Diffusion

Finger type double diffusive convective instability in a fluid-saturated porous medium is studied in the presence of coupled heat-solute diffusion. A local thermal non-equilibrium (LTNE) condition is invoked to model the Darcian porous medium which takes into account the energy transfer between the fluid and solid phases. Linear stability theory is implemented to compute the critical thermal Rayleigh number and the corresponding wavenumber exactly for the onset of stationary convection. The effects of Soret and Dufour cross-diffusion parameters, inter-phase heat transfer coefficient and porosity modified conductivity ratio on the instability of the system are investigated. The analysis shows that positive Soret mass flux triggers instability and positive Dufour energy flux enhances stability whereas their combined influence depends on the product of solutal Rayleigh number and Lewis number. It also reveals that cell width at the convection threshold gets affected only in the presence of both the cross-diffusion fluxes. Besides, asymptotic solutions for both small and large values of the inter-phase heat transfer coefficient and porosity modified conductivity ratio are found. An excellent agreement is found between the exact and asymptotic solutions.     

Multi-scale Analysis of Gas Transport Mechanisms in Kerogen

Quantification of natural gas transport in organic-rich shale is important in predicting natural gas production. However, laboratory measurements are challenging due to tight nature of the rock and include large uncertainties. The emphasis of this work is to understand mass transport mechanisms inside the organic nanoporous material known as kerogen under subsurface conditions and describe its permeability. This requires a multi-scale theoretical approach that includes flow measurements in model nanocapillaries and within their network. Molecular dynamics simulation results of steady-state supercritical methane flow in single-wall carbon nanotube are presented in this article. A transition from convection to molecular diffusion is observed. The simulation results show that the adsorbed methane molecules are mobile and contribute a significant portion to the total mass flux in nanocapillaries with diameter \({<}\)10 nm. They experience cluster diffusion that is dependent on the applied pressure drop across the capillary. A modified Hagen–Poiseuille equation is proposed considering the convective–diffusive nature of the overall transport in nanocapillary. The molecular-level study of steady-state transport is extended to a simple network of interconnected nanocapillaries representing kerogen. The modified Hagen–Poiseuille equation leads to a representative elementary volume of the model kerogen. The estimated permeability of the volume is sensitive to compressed and adsorbed fluids density ratio and to surface properties of the nanocapillary walls, indicating that fluid–wall interactions driven by molecular forces could be significant during the large-scale transport within shale. A modified Kozeny–Carman correlation is proposed, relating kerogen porosity and tortuosity to the permeability.     

Quantification of Density-Driven Natural Convection for Dissolution Mechanism in CO<Subscript>2</Subscript> Sequestration

Dissolution of CO₂ into brine causes the density of the mixture to increase. The density gradient induces natural convection in the liquid phase, which is a favorable process of practical interest for CO₂ storage. Correct estimation of the dissolution rate is important because the time scale for dissolution corresponds to the time scale over which free phase CO₂ has a chance to leak out. However, for this estimation, the challenging simulation on the basis of convection–diffusion equation must be done. In this study, pseudo-diffusion coefficient is introduced which accounts for the rate of mass transferring by both convection and diffusion mechanisms. Experimental tests in fluid continuum and porous media were performed to measure the real rate of dissolution of CO₂ into water during the time. The pseudo diffusion coefficient of CO₂ into water was evaluated by the theory of pressure decay and this coefficient is used as a key parameter to quantify the natural convection and its effect on mass transfer of CO₂. For each experiment, fraction of ultimate dissolution is calculated from measured pressure data and the results are compared with predicted values from analytical solution. Measured CO₂ mass transfer rate from experiments are in reasonable agreement with values calculated from diffusion equation performed on the basis of pseudo-diffusion coefficient. It is suggested that solving diffusion equation with pseudo diffusion coefficient herein could be used as a simple and rapid tool to calculate the rate of mass transfer of CO₂ in CCS projects.     

Equivalent Diffusion in a Thin Film Flow with a Non-One-Dimensional Velocity Field and Anisotropic Diffusion Tensor

For describing the mass transfer processes in channels, Taylor's dispersion theory is widely used. This theory makes it possible, with asymptotic rigor, to replace the complete diffusion (heat conduction) equation with a convective term that depends on the coordinate transverse to the flow by an effective diffusion (dispersion) equation with constant coefficients, averaged over the channel cross-section. In numerous subsequent studies, Taylor's theory was generalized to include more complex situations, and novel algorithms for constructing the dispersion equations were proposed. For thin film flows a theory similar to Taylor's leads to a matrix of dispersion coefficients.In this study, Taylor's theory is extended to film flows with a non-one-dimensional velocity field and anisotropic diffusion tensor. These characteristics also depend to a considerable extent on the spatial coordinates and time. The dispersion equations obtained can be simplified in regions in which the effective diffusion coefficient tensor changes sharply.     

Longitudinal diffusion of solid particle admixture in a flow through a tube

The propagation of solid particle admixture in a flow through a flat channel is studied.The processes of diffusion and convective transfer as well as solid particle deposition due to gravity result in varying admixture concentration both in depth and longtitudinally.The study of admixture longitudinal distribution is of great interest in a lot of applications, therefore this paper gives the derivation of longitudinal diffusion equation for a mean cross-section admixture concentration.The equation contains three effective parameters; i.e. convective tranfer velocity, longitudinal diffusion coefficient and particle deposition time. These parameters integrally reflect local processes of matter transfer as well as momentum.The proposed model is specific and differs from Taylor equation for longitudinal diffusion, since the fact of particle deposition and adhesion is taken into account. As a result of particle deposition a sediment layer is formed on the channel bottom which increases in thickness with time. To describe this process balance conditions for the whole flow mass and admixture mass on sediment sediment surface are formulated and a condition for matter movement towards the channel bottom is derived that is different from zero due to particle adhesion.     

Numerical and experimental investigation of convective drying in unsaturated porous media with bound water

The heat and mass transfer in an unsaturated wet cylindrical porous bed packed with quartz particles was investigated theoretically for relatively low convective drying rates. Local thermodynamic equilibrium was assumed in the mathematical model describing the multi-phase flow in the unsaturated porous media using the energy and mass conservation equations to describe the heat and mass transfer during the drying. The drying model included convection and capillary transport of the free water, diffusion of bound water, and convection and diffusion of the gas. The numerical results indicated that the drying process could be divided into three periods, the temperature rise period, the constant drying rate period and the decreasing drying rate period. The numerical results agreed well with the experimental data verifying that the mathematical model can evaluate the drying performance of porous media for low drying rates. The effects of drying conditions such as the ambient temperature, the relative humidity, and the velocity of the drying air, on the drying process were evaluated by numerical solution.     

Anomalous gravitational instability of mechanical equilibrium during diffusion mixing in isothermal three-component gas mixtures

A theoretical model determining the condition of onset of free gravitational convection in an isothermal ternary gas mixture with inhomogeneous concentrations and stable vertical density stratification (the density is lower at the top than at the bottom) is developed. The model is in satisfactory agreement with the experimental data on the determination of the boundaries between diffusion zones under conditions of mechanical equilibrium and convective mixing in a gravity force field in a vertical capillary connecting two vessels with ternary gas mixtures. Almaty. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 185–192, May–June, 2000. The work was carried out with financial support from the Science Foundation of the Kazakhstan Republic and the Russian Foundation for Basic Research (project No. 98-01-00879).     

Solute dispersion in a pulsating flow

The one-dimensional model proposed by Taylor [1] of the dispersion of soluble matter describes approximately the distribution of the solute concentration averaged over the tube section in Poiseuille flow. Aris [2] obtained more accurately the effective diffusion coefficient in Taylor's model and solved the problem for the general case of steady flow in a channel of arbitrary section. Many papers have been published in the meanwhile devoted to particular applications of this theory (for example, [3–5]). Various dispersion models have been constructed [6–8] that make the Taylor—Aris model more accurate at small times and agree with it at large times. The acceleration of the mixing of the solute considered in these models in the presence of the simultaneous influence of molecular diffusion and convective transport also operates in unsteady flows. In particular, the presence of velocity pulsations influences the growth of the dispersion even if the mean flow velocity is equal to zero at every point of the flow. In the present paper, the Taylor—Aris theory is extended to the case of laminar flows with periodically varying flow velocity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 24–30, September–October, 1982.     

Calculation of convective heat flux in the vicinity of the stagnation point for partially ionized gas flow past a body

From numerical solutions of the boundary layer equations for a four-component gas mixture (E, N⁺, N₂, and N) with gas injection, approximate formulas for the heat flux as a function of the variation of λρ/c_p and h^* across the boundary layer and the magnitude of the objection are obtained (λ is the thermal conductivity of the mixture,ρ is density, c_p is the specific heat, and h^* is the enthalpy of the ideal gas state of the mixture). An effective ambipolar diffusion coefficient D^(a)(i) is introduced, making possible finite formulas for the convective heat fluxes in the “frozen” boundary layer. We study the behavior of these coefficients within the boundary layer. A formula is obtained for convective heat flux to the wall from partially ionized air for a nine-component mixture (E, O⁺, N⁺, NO⁺, O, N, NO, O₂ N₂). Even for simpler four-component gas model three effective ambipolar diffusion coefficients are necessary: $$\begin{gathered} D^{(a)} (A) = D (A, M) D^{(a)} (I) = 2D (A, M), \hfill \\ D^{(a)} (M) = [ 1 + c_e (I)] D(A, M). \hfill \\ \end{gathered} $$ Here D(A, M) is the binary diffusion coefficient of the atoms into molecules, and c_e(I) is the ion concentration at the outer edge of the boundary layer. The assumption of an infinitely large charge-exchange cross section and the other simplifying assumptions used in [1] lead to overestimation of the magnitude of the dimensionless heat flux by 7–15% for the “frozen” boundary layer case.