The Local Equilibrium Assumption

Correcting the titration acidity for the presence of competing acids in order to obtain a better estimate of mH2CO presents the same problems as discussed previously ( 3.9.3), and the same remedy is proposed, i.e., the solution analysis should be titrated using a program such as phreeqc, as discussed in Chapter 8. 

2 we pointed out that thermodynamically based geochemical models can in principle only be used successfully for natural systems which exhibit areas of local equilibrium. We must now examine this idea more closely, and develop criteria for deciding whether or not the local equilibrium assumption (LEA) is a valid approximation in given systems. The conditions under which the LEA is applicable to groundwater and geological systems have been discussed extensively (Bahr, 1990 Bahr and Rubin, 

0121 liters acid x O.lOmoles H+ per liter x 1000ml/250ml 

0058liters base x O.lOmoles OH- per liter x 1000ml/250ml = 0.00232 moles OH- per liter of sample 

This number (0.00716) would be entered as the carbonate basis species in programs which do not accept an alkalinity. 

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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. 

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 ... 

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. 

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. 

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. 

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. 

The fact that the local equilibrium assumption is highly questionable highlights the importance of kinetics (Chapter 11) and may appear to invalidate much of geochemical modeling. This is not the case. In our view, modeling is always valuable, whatever the state of our knowledge. Refer to 1.4.2, page 16, for a more detailed discussion of this point. 

Unfortunately, lack of accuracy and internal consistency in our thermodynamic data may well be the least of our problems. It is easy to imagine that even if we had absolutely accurate data for every aqueous species and mineral, and accurate models for activity coefficients in all phases, predictions based on present models would still be inaccurate. This is the point made by Kraynov (1997). The deeper problem is the inadequacy of the conceptual framework of our geochemical models themselves, including their reliance for the most part on the local equilibrium assumption ( 3.11). 

Most applications of coupled models use the local equilibrium assumption. It is well known that most heterogeneous and redox reactions in the low temperature, near-surface systems are kinetically controlled (Hunter et al., 1998). However, we lack both theory and data to model kinetic reactions satisfactorily. In addition, inclusion of kinetic reactions makes the results even more difficult to comprehend completely and hence more costly. 

Reactive transport simulation is more realistic because of the consideration of SrSOj and BaS04 precipitation in reservoir (Bertero et al., 1988), predicting a lower scaling in the production well than the conservative transport simulation (Figure 6). This can be justified by the fact that in a reactive transport simulation part of the mineral that could cause scale in the production well has already precipitated in the reservoir. For kinetic reasons, they are called potentially precipitated minerals because the local equilibrium assumption at the production well is not necessarily valid. Produced water can be SrSO and BaS04... 

We have completed the analysis of system with linear isotherm, and local equilibrium between the two phases. The local equilibrium assumption is generally valid in many systems the linearity between the two phases is, however, restricted to systems with very low concentration of adsorbate, usually not possible with many... 

Most of the models considered in this chapter rely on the local equilibrium assumption. This assumption requires that the rates of chemical reaction and local mass transfer within the model s spatial domain are fast relative to the residence time of a slug of solution within that domain. Knapp (1989) and Bahr and Rubin (1987) have evaluated conditions where the local equilibrium assumption is valid and the Knapp treatment is summarized by Zhu and Anderson (2002). 

For the local equilibrium assumption to be valid, both the mineral and solution reaction rates and the transport rate to and from the minerals surfaces must be fast. These constraints are best tested using the first Damkohler number, Dai, and the Peclet number, Pe. Dai compares the rate of consumption (or production) of a species by chemical reaction to the rate of delivery (or removal) of that species by advection. 

When Pe < 50, diffusion and dispersion dominate and aqueous species are delivered to or removed from the surface faster than they are added to or removed from the domain by advection. When Pe > 100, the rate of advection is so fast that the reacting species are transported into or out of the domain before they can migrate to or from minerals surfaces. Figure 8.1 shows a map of Da and Pe. The local equilibrium assumption is valid in the upper right corner of the map where Da > 10 and Pe > 100 but reactions will be unlikely to reach equilibrium for the other conditions shown because they are either transport or reaction limited, or both. 

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