Local mass equilibrium

When a two- or higher-phase system is used with two or more phases permeable to the solute of interest and when interactions between the phases is possible, it would be necessary to apply the principle of local mass equilibrium [427] in order to derive a single effective diffusion coefficient that will be used in a one-equation model for the transport. Extensive justification of the principle of local thermdl equilibrium has been presented by Whitaker [425,432]. If the transport is in series rather than in parallel, assuming local equilibrium with equilibrium partition coefficients equal to unity, the effective diffusion coefficient is... 

For the case of a three-phase problem, where the solute is accessible to the a, (3, and y phases, Whitaker [427] finds the overall average phase concentration for the case of local mass equilibrium given by... 

A complete description of the mass transfer process requires a connection between the surface concentration, and the bulk concentration, c. One classic connection is based on local mass equilibrium, and for a linear equilibrium relation this concept takes the form... 

The condition of local mass equilibrium can exist even when adsorption and chemical reaction are taking place (Whitaker, 1999, Problem 1-3). When local mass equilibrium is not valid, one must propose an interfacial flux constitutive equation. The classic linear form is given by (Langmuir, 1916, 1917)... 

Secondary Ion Yields. The most successful calculations of secondary in yields are based on the local thermal equilibrium (LTE) model of Andersen and Hinthorne (1973), which assumes that a plasma in thermodynamic equilibrium is generated locally in the solid by ion bombardment. Assuming equilibrium, the law of mass action can be applied to find the ratio of ions, neutrals and electrons, and the Saha-Eggert equation is derived ... 

Alternate mass-core hard potential channel In the two billiard gas models just discussed there is no local thermal equilibrium. Even though the internal temperature can be clearly defined at any position(Alonso et al, 2005), the above property may be considered unsatisfactory(Dhars, 1999). In order to overcome this problem, we have recently introduced a similar model which however exhibits local thermal equilibrium, normal diffusion, and zero Lyapunov exponent(Li et al, 2004). 

Up to now we have presented this example without any regard for consistency, i.e. satisfying thermodynamic and conservation principles. This fuel mass flux must exactly equal the mass flux evaporated, which must depend on q and h(g. Furthermore, the concentration at the surface where fuel vapor and liquid coexist must satisfy thermodynamic equilibrium of the saturated state. This latter fact is consistent with the overall approximation that local thermodynamic equilibrium applies during this evaporation process. 

A fluid composed of a single species is described by five fields the three components of the velocity, the mass density, and the temperature. This is a drastic reduction of the full description in terms of all the degrees of freedom of the particles. This reduction is possible by assuming the local thermodynamic equilibrium according to which the particles of each fluid element have a Maxwell-Boltzmann velocity distribution with local temperature, velocity, and density. This local equilibrium is reached on time scales longer than the intercollisional time. On shorter time scales, the degrees of freedom other than the five fields manifest themselves and the reduction is no longer possible. 

Latent heat associated with phase change in two-phase transport has a large impact on the temperature distribution and hence must be included in a nonisothermal model in the two-phase regime. The temperature nonuniformity will in turn affect the saturation pressure, condensation/evaporation rate, and hence the liquid water distribution. Under the local interfacial equilibrium between the two phases, which is an excellent approximation in a PEFG, the mass rate of phase change, ihfg, is readily calculated from the liquid continuity equation, namely... 

The mass fraction Gi of the mass velocity M differs from the mass fraction of the local mixture when diffusion is taking place. Ki is the ordinary reaction rate of chemical kinetics, expressed as mole/unit volume/unit time, which is measured in a static experiment as a function of temperature and composition if the flame is in local thermal equilibrium. If not, this equation serves only as a measure of the nonequilibrium local reaction rate. 

Engineering systems mainly involve a single-phase fluid mixture with n components, subject to fluid friction, heat transfer, mass transfer, and a number of / chemical reactions. A local thermodynamic state of the fluid is specified by two intensive parameters, for example, velocity of the fluid and the chemical composition in terms of component mass fractions wr For a unique description of the system, balance equations must be derived for the mass, momentum, energy, and entropy. The balance equations, considered on a per unit volume basis, can be written in terms of the partial time derivative with an observer at rest, and in terms of the substantial derivative with an observer moving along with the fluid. Later, the balance equations are used in the Gibbs relation to determine the rate of entropy production. The balance equations allow us to clearly identify the importance of the local thermodynamic equilibrium postulate in deriving the relationships for entropy production. 

With a simplifying assumption that local adsorption equilibrium is instantaneously attained between gas and particles, the mass-transfer process is expressed by Fig. 58 for the cloud-overlap region. Here the influence of bubble wall curvature is neglected, since the region is very thin. When no catalyst is suspended in the bubble void, the equations of continuity for the reactant gas are as follows (M30) ... 

Prediction of Elution Profiles (Linear Equilibrium). For the case of local linear equilibrium (infinite rate of mass transfer), Lapidus and Amundson (25) derived equations for computing concentration distributions in a packed column. With concentrations at the inlet of the column, and initial conditions throughout the column known, concentration profiles at a specific distance from the column inlet can be computed. The derivation was based on a semi-infinite column, which differs mathematically from a finite column, in that effects of the mobile phase leaving the stationary phase are not modeled. Nonetheless, the solution obtained is useful for giving a qualitative picture of important parameters in column performance. The equation is ... 

Local thermodynamic equilibrium in space and time is inherently assumed in the kinetic theory formulation. The length scale that is characteristic of this volume is i whereas the timescale is xr. When either L i, ir or t x, xr or both, the kinetic theory breaks down because local thermodynamic equilibrium cannot be defined within the system. A more fundamental theory is required. The Boltzmann transport equation is a result of such a theory. Its generality is impressive since macroscopic transport behavior such as the Fourier law, Ohm s law, Fick s law, and the hyperbolic heat equation can be derived from this in the macroscale limit. In addition, transport equations such as equation of radiative transfer as well as the set of conservation equations of mass, momentum, and energy can all be derived from the Boltzmann transport equation (BTE). Some of the derivations are shown here. 

Using the precondition of local geochemical equilibrium, the mass balance equation for concentration TDS C can be written as ... 

Now, calculate the normal component of the total local molar flux of species A at the nondeformable zero-shear interface. Since the radial component of the flnid velocity vector vanishes at r = R, species A is transported across the interface exclusively via concentration diffnsion (i.e.. Pick s law). Then, the diffusional flux of species A in the radial direction, evalnated at the interface, is equated to the product of a local mass transfer coefficient and the overall concentration driving force for mass transfer (i.e., Ca. equilibrium — CA.buik)- The... 

Basic assumptions in both models include (1) membrane/bulk and membrane/lnternal phases are immiscible, (2) local phase equilibrium between membrane and internal phases, (3) no internal circulation in the globule, (4) uniform globule size, (5) mass transfer is controlled by globule diffusion, (6) internal droplets are solute sinks with finite capacity, (7) reaction of solute in the internal phase is instantaneous, (8) no coalescence and redistribution of globules, and (9) a well-mixed tank with an exponential residence time distribution of emulsion globules. 

CTE is not realized in laboratory plasmas wich are optically thin (Planck s law is not valid). But the mass action law, the Boltzmann distribution and Maxwell distribution may be obeyed by a unique local temperature such that T, = T e = T = T, then one introduces the (complete) local thermodynamic equilibrium (LTE). Boltzmann and Saha s law are often obeyed only for highly excited levels, the plasma is then said to be in Local Partial Thermodynamic Equilibrium (LPTE). 

Besides, small subsystems of the solution relax, i.e., reach equilibrium much sooner than the entire solution. As a result, chemical equilibrium in separate parts of the solution is reached at different times. Equilibrium, reached in a separate part of the solution, is called local chemical equilibrium. The local equilibrium principle maintains that each small (but macroscopic) element of volume in a nonequilibrium overall system at any moment in time is in the state of equilibrium. Special significance is attributed to local equilibrium at the boxmdary of different media, which determines the nature and rate of the mass exchange between them. 

Open models of local chemical equilibrium are object balanced only chemically. That is why they accept groimd water flow, may be nonuniform but ignore processes of mass transfer between water and rock. Their main assumption is that the relaxation time At = 0 and flow time At > 0. These models consider hydrochemical processes relative to time, distance and events scales, which characterize change of rock properties and composition, head gradient, etc. However, they do not consider kinetics of chemical processes. 

First, we shall use a quasi-stationary approach already mentioned earlier, based on the assumption that characteristic times of heat and mass transfer in the gaseous phase are much shorter than in the liquid phase, since the coefficients of diffusion and thermal conductivity are much greater in the gas than in the liquid. Therefore the distribution of parameters in the gas may be considered as stationary, while they are non-stationary in the liquid. On the other hand, small volume of the drop allows us to assume that the temperature and concentration distributions are constant within the drop, while in the gas they depend on coordinates. Another assumption is that the drop s center does not move relative to the gas. Actually, this assumption is too strong, because in real processes, for example, when a liquid is sprayed in a combustion chamber, drops move relative to the gas due to inertia and the gravity force. However, if the size of drops is small (less than 1 pm) and the processes of heat and mass exchange are fast enough, then this assumption is permissible. As usual, we assume the existence of local thermodynamic equilibrium at the drop s surface, as well as equal pressures in both phases. The last condition was formulated at the end of Section 6.7. 

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