Constitutive equations, diffusion

The next step is the formulation of an equation of motion. We assume for this moment that h x) can only vary by surface diffusion, i.e., by peripheral diffusion of h along x. The classical conservation law holds that (5/5t)A + divy /, = 0. For the current the constitutive equation is, according to classical thermodynamics, j = n = 6F/6h = -V A,... 

It can be noted that other approaches, based on irreversible continuum mechanics, have also been used to study diffusion in polymers [61,224]. This work involves development of the species momentum and continuity equations for the polymer matrix as well as for the solvent and solute of interest. The major difficulty with this approach lies in the determination of the proper constitutive equations for the mixture. Electric-field-induced transport has not been considered within this context. 

The mass conservation equation only relates concentration variation with flux, and hence cannot be used to solve for the concentration. To describe how the concentrations evolve with time in a nonuniform system, in addition to the mass balance equations, another equation describing how the flux is related to concentration is necessary. This equation is called the constitutive equation. In a binary system, if the phase (diffusion medium) is stable and isotropic, the diffusion equation is based on the constitutive equation of Pick s law ... 

The constitutive equations use a thermodynamic framework, that in fact embodies not only purely mechanical aspects, but also transfers of masses between the phases and diffusion of matter through the extrafibrillar phase. Since focus is on the chemo-mechanical couplings, we use experimental data that display different salinities. The structure of the constitutive functions and the state variables on which they depend are briefly motivated. Calibration of material parameters is defined and simulations of confined compression tests and of tree swelling tests with a varying chemistry are described and compared with available data in [3], The evolution of internal entities entering the model, e.g. the masses and molar fractions of water and ions, during some of these tests is also documented to highlight the main microstructural features of the model. 

Abstract We formulate the balance principles for an immiscible mixture of continua with micro structure in the broadest sense for include, e.g., phenomena of diffusion, adsorption and chemical reactions. After we consider the flow of a fluid/adsorbate mixture through big pores of an elastic solid skeleton and propose suitable constitutive equations to study the coupling of adsorption and diffusion under isothermal conditions. 

The constitutive equation of the A-V model, when lattice and grain boundary diffusion are taken into account, is written ... 

In the case of YTZP, on which a large number of studies have been performed, the data could be fitted to a constitutive equation, which is identical to that found in metals when lattice diffusion is the rate-controlling mechanism 29... 

The constitutive equations of transport in porous media comprise both physical properties of components and pairs of components and simplifying assumptions about the geometrical characteristics of the porous medium. Two advanced effective-scale (i.e., space-averaged) models are commonly applied for description of combined bulk diffusion, Knudsen diffusion and permeation transport of multicomponent gas mixtures—Mean Transport-Pore Model (MTPM)—and Dusty Gas Model (DGM) cf. Mason and Malinauskas (1983), Schneider and Gelbin (1984), and Krishna and Wesseling (1997). The molar flux intensity of the z th component A) is the sum of the diffusion Nc- and permeation N contributions,... 

In the DGM, the solid phase is modeled as giant dust molecules held motionless in space with which the diffusing gas molecules collide. The constitutive equations governing the diffusion molar flux intensities Nf for both MTPM and DGM are the generalized Maxwell-Stefan equations... 

In this treatment only the ordinary and Knudsen diffusion mechanisms will be considered. Then, mass transport in isothermal, multicomponent gas phase systems is described by the following constitutive equation ... 

The following summary is from Jou and Casas-Vazquez (2001). In the extended nonequilibrium thermodynamics for a binary liquid mixture, the viscous pressure tensor Pv and the diffusion flux J are considered as additional independent variables. The viscous pressure tensor, Pv, by the simplest Maxwell model, is defined by the following constitutive equation ... 

At present two models are available for description of pore-transport of multicomponent gas mixtures the Mean Transport-Pore Model (MTPM)[4,5] and the Dusty Gas Model (DGM)[6,7]. Both models permit combination of multicomponent transport steps with other rate processes, which proceed simultaneously (catalytic reaction, gas-solid reaction, adsorption, etc). These models are based on the modified Maxwell-Stefan constitutive equation for multicomponent diffusion in pores. One of the experimentally performed transport processes, which can be used for evaluation of transport parameters, is diffusion of simple gases through porous particles packed in a chromatographic column. 

Olir discussion on diffusion will be restricted primarily to binary systems containing only species A and B. We now wish to determine how the molar diffusive flux of a species (i.e., Ja) is related to its concentration gradient. As an aid in the discussion of the transport law that is ordinarily used to describe diffusion, lesll similar laws ftom other trans K)it processes. For example, in conductive heat transfer the constitutive equation relating the heat flux q and the temperature gradient is Fourier s law ... 

Step 2 For eveiy mole of O2 that diffuses into the spherical pellet, 1 mol of CO2 diffuses out = — Wq ), that is, EMCD. The constitutive equation for constant total concentration becomes... 

It is valid for each continuum independent of the individual material properties and is therefore one of the fundamental equations in fluid mechanics and subsequently also in heat and mass transfer. The movement of a particular substance can only be described by introducing a so-called constitutive equation which links the stress tensor with the movement of a substance. Generally speaking, constitutive equations relate stresses, heat fluxes and diffusion velocities to macroscopic variables such as density, velocity and temperature. These equations also depend on the properties of the substances under consideration. For example, Fourier s law of heat conduction is invoked to relate the heat flux to the temperature gradient using the thermal conductivity. An understanding of the strain tensor is useful for the derivation of the consitutive law for the shear stress. This strain tensor is introduced in the next section. 

In order to analyze multicomponent diffusion processes we must be able to solve the continuity equations (Eq. 1.3.9) together with constitutive equations for the diffusion process and the appropriate boundary conditions. A great many problems involving diffusion in binary mixtures have been solved. These solutions may be found in standard textbooks, as well as in specialized books, such as those by Crank (1975) and Carslaw and Jaeger (1959). 

Both formulations of the constitutive equations for multicomponent diffusion, the Maxwell-Stefan equations and the generalized Fick s law, are most compactly written in matrix form. It might, therefore, be as well to begin by writing the continuity equations (Eq. 1.3.9) in n - 1 dimensional matrix form as well... 

Constitutive equations, which quantitatively describe the physical properties of the fluids. The most important constitutive equations used in this book are the Newton s viscosity law, the Fourier s law of heat conduction, and the Pick s law of mass diffusion. The equation of state and more empirical relations for the physical properties of the fluid mixture also belong to this group of equations. 

Figure 2 shows the comparison of the fractal-layer (solid line a) and two-timescale (solid line b) models with the simulations in terms of effective diffusivity, eq. (13). Both the models furnish a satisfactory level of agreement with simulation data. We may therefore conclude that approximate models based on a Riemann-Liouville constitutive equation are able to furnish an accurate description of adsorption kinetics on fractal interfaces. These models can also be extended to nonlinear problems (e.g. in the presence of nonlinear isotherms, such as Langmuir, Freundlich, etc.). In order to extend the analysis to nonlinear cases, efficient numerical sJgorithms should be developed to solve partied differential schemes in the presence of Riemann-Liouville convolutional terms. 

Here we have used the approximation that can be replaced by Dj y and that variations of D y can be ignored within the averaging volume. The fact that only a single tortuosity needs to be determined by equations 1.152 and 1.153 represents the key contribution of this study. It is important to remember that this development is constrained by the linear chemical kinetic constitutive equation given by equation 1.113. The process of diffusion in porous catalysts is normally associated with slow reactions and equation 1.93 is satisfactory however, the first-order, irreversible reaction represented by equation 1.113 is the exception rather than the rule, and this aspect of the analysis requires further investigation. The influence of a non-zero mass average velocity needs to be considered in future studies so that the constraint given by equation 1.97 can be removed. An analysis of that case is reserved for a future study which will also include a careful examination of the simplification indicated by equation 1.117. 

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