Local Equilibrium Flows

In order to solve the conservation or transport equations (mass, momentum, energy, and entropy) in terms of the dependent variables n, Vo,U, and , we must further resolve the expressions for the flux vectors— P, q, and s and entropy generation Sg. This resolution is the subject of closure, which will be treated in some detail in the next chapter. However, as a matter of illustration and for future reference, we can resolve the flux vector expression for what is called the local equilibrium approximation, i.e., we assume that the iV-molecule distribution function locally follows the equilibrium form developed in Chap. 4, i.e., we write [cf Eq. (4.34)] 

let s look at each flux term utilizing the distribution function, Eq. (5.92). We require both the first-order and second-order distribution functions, which under local equilibrium conditions follow Eq. (5.92) as [see Eqs. (4.47), (4.66), and (4.72)] 

by inspection it can be seen that under local equilibrium conditions the integrals defining q, s, and Sg [Eqs. (5.66), (5.67), (5.86), and (5.87), respectively] are antisymmetric with respect to the sign of p and therefore must vanish. Thus 

Example 5.1 Isothermal Compressible Flow For local equilibrium, isothermal flows in the absence of external forces, the conservation of mass and momentum are, respectively. 

Substituting into Eqs. (5.106) and (5.107) and retaining terms through first-order in disturbance quantities gives 

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In Table 6.7, C is the Martinelli-Chisholm constant, / is the friction factor, /f is the friction factor based on local liquid flow rate, / is the friction factor based on total flow rate as a liquid, G is the mass velocity in the micro-channel, L is the length of micro-channel, P is the pressure, AP is the pressure drop, Ptp,a is the acceleration component of two-phase pressure drop, APtp f is the frictional component of two-phase pressure drop, v is the specific volume, JCe is the thermodynamic equilibrium quality, Xvt is the Martinelli parameter based on laminar liquid-turbulent vapor flow, Xvv is the Martinelli parameter based on laminar liquid-laminar vapor flow, a is the void fraction, ji is the viscosity, p is the density, is the two-phase frictional... 

Potentiometric transducers now belong to the most mature transducers with numerous commercial products. For potentiometric transducers, a local equilibrium is established at the transducer interface at near-zero current flow, where the change... 

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. 

Beginning in the late 1980s, a number of groups have worked to develop reactive transport models of geochemical reaction in systems open to groundwater flow. As models of this class have become more sophisticated, reliable, and accessible, they have assumed increased importance in the geosciences (e.g., Steefel et al., 2005). The models are a natural marriage (Rubin, 1983 Bahr and Rubin, 1987) of the local equilibrium and kinetic models already discussed with the mass transport... 

Only for the intermediate cases - those with velocities in the range of about 100 m yr-1 to 1000 m yr-1 - does silica concentration and reaction rate vary greatly across the main part of the domain. Significantly, only these cases benefit from the extra effort of calculating a reactive transport model. For more rapid flows, the same result is given by a lumped parameter simulation, or box model, as we could construct in REACT. And for slower flow, a local equilibrium model suffices. 

To remove momentum fluctuations from the problem, BCAH assume that in the creeping flow limit, in which a system of small mass interacts strongly with a thermally equilibrated solvent, the distribution of values for the momenta for fixed coordinate values stays very near a state of local equilibrium, in which... 

To quantify this treatment of migration as influenced by kinetics, a model has been developed in which instantaneous or local equilibrium is not assumed. The model is called the Argonne Dispersion Code (ARDISC) ( ). In the model, adsorption and desorption are treated independently and the rates for adsorption and desorption are taken into account. The model treats one dimensional flow and assumes a constant velocity of solution through a uniform homogeneous media. 

The work that was performed in this set of experiments was an extension of work performed by Inoue and Kaufman (7). In the previous work, the migration of strontium in glauconite was modeled using conditions of local equilibrium for flows up to 6.3 kilometers per year (72 cm/hr). The differences between the predicted and experimental results in the experiments performed by Inoue and Kaufman may be due to the existence of non-equilibrium behavior. 

The smoothing of a rough isotropic surface such as illustrated in Fig. 3.7 due to vacancy flow follows from Eq. 3.69 and the boundary conditions imposed on the vacancy concentration at the surface.12 In general, the surface acts as an efficient source or sink for vacancies and the equilibrium vacancy concentration will be maintained in its vicinity. The boundary condition on cy at the surface will therefore correspond to the local equilibrium concentration. Alternatively, if cy, and therefore Xy, do not vary significantly throughout the crystal, smoothing can be modeled using the diffusion potential and Eq. 3.72 subject to the boundary conditions on a at the surface and in the bulk.13... 

The equations for a local equilibrium cell model of pressure swing adsorption processes with linear isotherms have been derived. These equations may be used to describe any PSA cycle composed of pressurization and blowdown steps and steps with flow at constant pressure. The use of the equations was illustrated by obtaining solutions for a single-column recovery process and a two-column recovery and purification process. The single-column process was superior in enrichment and recovery of the light component at large product cuts. The two-column process was superior at small cuts ... 

The plug-flow model indicates that the fluid velocity profile is plug shaped, that is, is uniform at all radial positions, fact which normally involves turbulent flow conditions, such that the fluid constituents are well-mixed [99], Additionally, it is considered that the fixed-bed adsorption reactor is packed randomly with adsorbent particles that are fresh or have just been regenerated [103], Moreover, in this adsorption separation process, a rate process and a thermodynamic equilibrium take place, where individual parts of the system react so fast that for practical purposes local equilibrium can be assumed [99], Clearly, the adsorption process is supposed to be very fast relative to the convection and diffusion effects consequently, local equilibrium will exist close to the adsorbent beads [2,103], Further assumptions are that no chemical reactions takes place in the column and that only mass transfer by convection is important. 

It can be shown from the equation of continuity that the ratio of tangential to normal fluid velocities within the double layer is 0(kL). This implies that the flow in the diffuse layer is primarily tangential to the particle surface and the 0( /kL) normal velocity can be neglected. Based on this fact, it is found from Eq. (38) that to first-order approximation, fij is constant across the double layer, indicating that the double layer is in local equilibrium. Integrating Eq. (39) over the double layer yields a slip velocity at the outer edge of the ion cloud... 

The kinetic theory leads to the definitions of the temperature, pressure, internal energy, heat flow density, diffusion flows, entropy flow, and entropy source in terms of definite integrals of the distribution function with respect to the molecular velocities. The classical phenomenological expressions for the entropy flow and entropy source (the product of flows and forces) follow from the approximate solution of the Boltzmann kinetic equation. This corresponds to the linear nonequilibrium thermodynamics approach of irreversible processes, and to Onsager s symmetry relations with the assumption of local equilibrium. 

Chemical process rate equations involve the quantity related to concentration fluctuations as a kinetic parameter called chemical relaxation. The stochastic theory of chemical kinetics investigates concentration fluctuations (Malyshev, 2005). For diffusion of polymers, flows through porous media, and the description liquid helium, Fick s and Fourier s laws are generally not applicable, since these laws are based on linear flow-force relations. A general formalism with the aim to go beyond the linear flow-force relations is the extended nonequilibrium thermodynamics. Polymer solutions are highly relevant systems for analyses beyond the local equilibrium theory. 

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. 

In this theorem, the inflection point refers to the existence of a point within a shear layer (at y = ys where the local velocity is given by Ug) and where the second derivative of the equilibrium flow profile vanishes i.e. 

C. Marchioli, M. Picciotto, and A. Soldati. Particle dispersion and wall-dependent fluid scales in turbulent bounded flow Implications for local equilibrium models. J. Turh., 7(60) 1-12, 2006. 

We can now employ our usual methods to calculate, at any point in the tube, the net flows of species 1 and 2 across a unit area. If we take a plane perpendicular to the direction of the concentration gradients (i.e., the X axis) then there will be a net flow of species 1 across this plane owing to the fact that there is a difference in the number of molecules of species 1 that strike it from opposite sides (Fig. VIII.5). By making the usual assumption of local equilibrium distributions we can use our formula [Eq. (VII.6.6)] for the number of collisions per unit area per second. We then have for the number of collisions of species 1 with a unit area per second, made on the high-concentration side,... 

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. 

For both soils studied, comparison of calcium-effluent histories predicted by the solution to Equation 6 with those obtained from experimental columns gave good agreement only for the lowest flow rates. For the three higher water fluxes, more apparent dispersion was observed than could be explained by predictions that assume local equilibrium. Examples of these comparisons are shown in Figures 1 and 2. 

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