Nonequilibrium local

The first and second integrals have their coordinate systems centered on the catalytic C and noncatalytic N spheres, respectively. The local nonequilibrium average microscopic density field for species a is pa(r) = [Y = 5(r - ( )) The solution of the diffusion equation can be used to estimate this nonequilibrium density, and thus the velocity of the nanodimer can be computed. The simple model yields results in qualitative accord with the MPC dynamics simulations and shows how the nonequilibrium density field produced by reaction, in combination with the different interactions of the B particles with the noncatalytic sphere, leads to directed motion [117],... 

We will look at the three variables that may cause zone spreading, that is, ordinary diffusion, eddy diffusion, and local nonequilibrium. Our approach to this discussion will be from the random walk theory, since the progress of solute molecules through a column may be viewed as a random process. 

Early explanations about the effect of mechanical energy on the reactivity of solids are the hot-spot-model [23] and the magma-plasma-model [8]. The generation of hot-spof may be used to explain the initiation of a self-sustained reaction such as explosion, deflagration, or decomposition. Temperatures of over 1000 K on surfaces of about 1 pm2 for KM to 10-3 s can be created. These temperatures can also be found near the tip of a propagating crack [24]. Typically nonequilibrium thermodynamics are used to describe these phenomena. The magma-plasma-model allows for local nonequilibrium states on the solid surface during impact however, due to the very short time scale of 1(H s of these states only statistical thermodynamics can describe the behavior. 

We now briefly consider another important aspect of nonequilibrium thermodynamics, namely phase transformations and how they are modelled. Galenko and Jou198 develop a thermodynamic formalism for rapid phase transformations within a diffuse interface of a binary system in which the system is in a state of local nonequilibrium. The phase-field method, in which the phase- field variable O varies smoothly and continuously between one pure phase (in which O = +1) and another (in which -1), is used to derive... 

Figure 21.5. Illustration of the influence of local nonequilibrium on band dispersion. Dashed lines equilibrium concentration profile solid lines actual concentration profile.

The Knudsen layer plays a fundamental role in the slip flow regime. This thin layer, between one and two molecular mean free paths in thickness, is a region of local nonequilibrium which is observed in any gas flow near a surface. In nonrarefied flows, the Knudsen layer is too thin for having any significant influence, but in the slip flow regime, it should be taken into account. 

Sorption and desorption (due to a local nonequilibrium condition created by the process of sorption and desorption) ... 

The third approach is called the thermodynamic theory of passive systems. It is based on the following postulates (1) The introduction of the notion of entropy is avoided for nonequilibrium states and the principle of local state is not assumed, (2) The inequality is replaced by an inequality expressing the fundamental property of passivity. This inequality follows from the second law of thermodynamics and the condition of thermodynamic stability. Further the inequality is known to have sense only for states of equilibrium, (3) The temperature is assumed to exist for non-equilibrium states, (4) As a consequence of the fundamental inequality the class of processes under consideration is limited to processes in which deviations from the equilibrium conditions are small. This enables full linearization of the constitutive equations. An important feature of this approach is the clear physical interpretation of all the quantities introduced. 

The breakdown and repair of a passive film prior to pitting dissolution creates a kind of nonequilibrium fluctuation all over the electrode surface, which results from the localized inequality of film dissolution and formation. Since this type of film is too thin for direct observation of the... 

A passive film is stable in the region between the passivation and breakdown potentials if any part of the film is broken, it is rapidly repaired. Therefore it is necessary to derive a model that depicts the processes by which such local destruction and restoration are continuously repeated. This process can be regarded as a kind of nonequilibrium fluctuation concerning passivity. Using energetics, Sato7 analyzed such fluctuation processes as follows. 

In all these treatments, nonequilibrium fluctuation plays the most important role. This is defined as the fluctuation of a physical quantity that deviates from the standard state determined by the nonzero flux in a nonequilibrium state. Such fluctuation has a kind of symmetry in that the area average is equal to zero although the flux changes locally. Therefore, macroscopically, such fluctuation does not affect the flux itself. This means that the flux must be determined a priori and is indifferent to the fluctuations. 

Computer codes usually calculate only the thermodynamically most stable configuration of a system. Modifications can simulate nonequilibrium, but there are limitations on the extent to which codes can be manipulated to simulate processes that are kinetically (rate) controlled the slow reaction rates in the deep-well environment compared with groundwater movement (i.e., failure to attain local homogeneous or heterogeneous reversibility within a meter or so of the injection site) create particular problems. 

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. 

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