Local equilibrium assumption

The standard wall function is of limited applicability, being restricted to cases of near-wall turbulence in local equilibrium. Especially the constant shear stress and the local equilibrium assumptions restrict the universality of the standard wall functions. The local equilibrium assumption states that the turbulence kinetic energy production and dissipation are equal in the wall-bounded control volumes. In cases where there is a strong pressure gradient near the wall (increased shear stress) or the flow does not satisfy the local equilibrium condition an alternate model, the nonequilibrium model, is recommended (Kim and Choudhury, 1995). In the nonequilibrium wall function the heat transfer procedure remains exactly the same, but the mean velocity is made more sensitive to pressure gradient effects. 

Valocchi, A.J., 1985, Validity of the local equilibrium assumption for modeling sorbing solute transport through homogeneous soils. Water Resources Research 21, 808-820. 

For fast equilibrium chemistry (Section 5.4), an equilibrium assumption allowed us to write the concentration of all chemical species in terms of the mixture-fraction vector c(x, t) = ceq( (x, 0). For a turbulent flow, it is important to note that the local micromixing rate (i.e., the instantaneous scalar dissipation rate) is a random variable. Thus, while the chemistry may be fast relative to the mean micromixing rate, at some points in a turbulent flow the instantaneous micromixing rate may be fast compared with the chemistry. This is made all the more important by the fact that fast reactions often take place in thin reaction-diffusion zones whose size may be smaller than the Kolmogorov scale. Hence, the local strain rate (micromixing rate) seen by the reaction surface may be as high as the local Kolmogorov-scale strain rate. 

In groundwater and soil pollution problems, there is sometimes discussion of fast sorption and slow sorption, where the local equilibrium assumption would not be valid. How would you formulate a diffusion equation to deal with both the fast and slow forms of adsorption and desorption ... 

In this summary, the local thermal equilibrium model has been used to derive the energy equation. This model is much simpler than the two-phase model however, the local thermal equilibrium model is most likely not adequate to describe the transport of energy when the temperature of the fluid and solid are undergoing extremely rapid changes. Although such extremely rapid temperature changes are not expected, in most RTM, IP, and AP processes the correctness of the local thermal equilibrium assumption can be verified by following the procedure discussed by Whitaker [28]. 

Thus summarizing, we note that at the leading order the asymptotic solution constructed is merely a combination of the locally electro-neutral solution for the bulk of the domain and of the equilibrium solution for the boundary layer, the latter being identical with that given by the equilibrium electric double layer theory (recall (1.32b)). We stress here the equilibrium structure of the boundary layer. The equilibrium within the boundary layer implies constancy of the electrochemical potential pp = lnp + ip across the boundary layer. We shall see in a moment that this feature is preserved at least up to order 0(e2) of present asymptotics as well. This clarifies the contents of the assumption of local equilibrium as applied in the locally electro-neutral descriptions. Recall that by this assumption the electrochemical potential is continuous at the surfaces of discontinuity of the electric potential and ionic concentrations, present in the locally electro-neutral formulations (see the Introduction and Chapters 3, 4). An implication of the relation between the LEN and the local equilibrium assumptions is that the breakdown of the former parallel to that of the corresponding asymptotic procedure, to be described in the following paragraphs, implies the breakdown of the local equilibrium. 

In many non-equilibrium situations, this local equilibrium assumption holds for the crystal bulk. However, its verification at the phase boundaries and interfaces (internal and external surfaces) is often difficult. This urges us to pay particular attention to the appropriate kinetic modeling of interfaces, an endeavour which is still in its infancy. 

The applicability of the preceding pseudocontinuum approach to convective heat transfer of gas-solid systems without heat sources depends not only on the validity of the phase continuum approximation but also on the appropriateness of the local thermal equilibrium assumption. The local thermal equilibrium may be assumed only if the particle-heating... 

Most field-scale modeling studies of contaminant plumes make the local equilibrium assumption (LEA) [18,19]. The LEA is based on the premise that the interactions between the contaminant and the aquifer material are so rapid compared to advective residence times that it can be assumed that the interactions are instantaneous [3]. Linear equilibrium sorption assumes that the binding of contaminants to aquifer solids is instantaneous and that the concentration of sorbed contaminant is directly proportional to the concentration of the dissolved contaminant. This can be modeled by a linear sorption isotherm [2] ... 

The theory treating near-equilibrium phenomena is called the linear nonequilibrium thermodynamics. It is based on the local equilibrium assumption in the system and phenomenological equations that linearly relate forces and flows of the processes of interest. Application of classical thermodynamics to nonequilibrium systems is valid for systems not too far from equilibrium. This condition does not prove excessively restrictive as many systems and phenomena can be found within the vicinity of equilibrium. Therefore equations for property changes between equilibrium states, such as the Gibbs relationship, can be utilized to express the entropy generation in nonequilibrium systems in terms of variables that are used in the transport and rate processes. The second law analysis determines the thermodynamic optimality of a physical process by determining the rate of entropy generation due to the irreversible process in the system for a required task. 

Local Equilibrium Assumption. There exists a quasi-equilibrium between the reactant and a system crossing the TS from the reactant to the product. This precise representation was taken from an insightful article on the observability of the invariant of motion in the transition state by Marcus [16] The motions along the reaction coordinate at the transition state was... 

Recently, however, experimental studies of reaction processes have cast doubt on the local equilibrium assumption. When that assumption is not valid, understanding of reaction processes requires the smdy of global aspects of the phase space in multidimensional chaotic dynamics [1]. 

Applicability of the Local Equilibrium Assumption to Transport through Soils of Solutes Affected by Ion Exchange... 

Subsurface solute transport is affected by hydrodynamic dispersion and by chemical reactions with soil and rocks. The effects of hydrodynamic dispersion have been extensively studied 2y 3, ). Chemical reactions involving the solid phase affect subsurface solute transport in a way that depends on the reaction rates relative to the water flux. If the reaction rate is fast and the flow rate slow, then the local equilibrium assumption may be applicable. If the reaction rate is slow and the flux relatively high, then reaction kinetics controls the chemistry and one cannot assume local equilibrium. Theoretical treatments for transport of many kinds of reactive solutes are available for both situations (5-10). 

It is often desirable, where applicable, to use the local equilibrium assumption when predicting the fate of subsurface solutes. Advantages of this approach may include 1) data such as equilibrium constants are readily available, as opposed to the lack of kinetic data, and 2) for transport involving ion exchange and adsorption, the mathematics for equilibrium systems are generally simpler than for those controlled by kinetics. To utilize fully these advantages, it is helpful to know the flow rate below which the local equilibrium assumption is applicable for a given chemical system. Few indicators are available which allow determination of that critical water flux. 

As q decreases, the terms Fp and F become small compared to D, and Equation 7 becomes more like Equation 6. This indicates that the local equilibrium assumption is applicable when q is sufficiently small. 

Inequality 11 was substituted into Equation 8, together with reasonable values of other parameters and 0.1 cm < a < 0.2 cm as a was found to be in this study. This leads to the conclusion that, for systems in which r is less than 0.1 cm, the local equilibrium assumption is applicable (i.e., q is sufficiently small) when D is nearly equal to as observed in the experiments. In soils in which the exchanging particles are not spherical, r would represent approximately the mean diffusion path within clay aggregates or within clay coatings on coarse particles. 

For soils without appreciable clay aggregation, the experimental results and theoretical analysis described here indicate that when diffusion is rate-limiting, the ratio of the hydrodynamic dispersion coefficient to the estimated soil self-diffusion coefficient may be a useful indicator of the applicability of the local equilibrium assumption. For reacting solutes in laboratory columns such as those used in this study, systems with ratios near unity can be modeled using equilibrium chemistry. 

Lichtner P. C. (1993) Scaling properties of time-space kinetic mass transport equations and the local equilibrium assumption. Am. J. Sci. 293, 257-296. 

Several key questions must be answered initially in a study of reaction chemistry. First, is the reaction sufficiently fast and reversible so that it can be regarded as chemical-equilibrium controlled Second, is the reaction homogeneous (occurring wholly within a gas or liquid phase) or heterogeneous (involving reactants or products in a gas and a liquid, or liquid and a solid phase) Slow reversible, irreversible, and heterogeneous (often slow) reactions are those most likely to require interpretation using kinetic models. Third, is there a useful volume of the water-rock system in which chemical equilibrium can be assumed to have been attained for many possible reactions This may be called the local equilibrium assumption. 

Contrast the applicability of equilibrium and kinetic models to some environmental problems. Explain local and partial equilibrium assumptions. 

Both the absolute- and local Maxwellians are termed equilibrium distributions. This result relates to the local and instantaneous equilibrium assumption in continuum mechanics as discussed in chap. 1, showing that the assumption has a probabilistic fundament. It also follows directly from the local equilibrium assumption that the pressure tensor is related to the thermodynamic pressure, as mentioned in sect. 2.3.3. 

James, R.V., and J. Rubin. 1979. Applicability of local equilibrium assumptions to transport through soil of solutes affeeted by ion exchange, p. 225-235. In E.A. Jenne (ed.) Chemical modeling in aqueous systems. Symp. Ser. Vol. 93. ACS, Washington, DC. 

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