Non-equilibrium factors

In the derivation of equation (4.42), it has been tacitly assumed that the cluster partition function Zjy remains unaffected by the flux of clusters to higher classes, i.e., Z] f = Z. A correction for the depletion of the cluster population due to this flux can be introduced by the factor r, known as the Zeldovich or non-equilibrium factor [4.16]. 

The multiphase method provides a practical screening tool for industrial process research and development, even though under many circumstances the nonequilibrium effects such as supersaturation of solutions, retarded mass transfer or reaction kinetics and inhomogeneity of suspensions limit the applicability of the thermodynamic calculations. When the thermodynamic multiphase models are developed towards process simulation tools, one should incorporate such methods that include the effects of these non-equilibrium factors. They must be based on... 

Zone Spreading. The net forward progress of each solute is an average value with a normal dispersion about the mean value. The increased band or zone width which results from a series of molecular difrusion and non-equilibrium factors is known as zone spreading. 

The Fine Porous Model as presented by Xu and Spencer (1997), describes equilibrium and non-equilibrium factors of rejection. Only coupling between solvent and solute is taken into account, and no solute-solute coupling is permitted. Equilibrium parameters dominated separation, and these are described by the reflection coefficient (J in equation (3.28), where kii is the solute mass transfer coefficient in the membrane. 

Figure 9-6. Energy cost of acetylene production from methane in plasma , Babaritsky s microwave plasma experiments, p = 40 Torr , Babaritsky s microwave plasma experiments, p = 80 Torr o, a, experiments with thermal arc discharges. Curves represent calculations (1) quasi-equilibrium plasma modeling (2-5) non-equilibrium kinetic modeling at different values of the non-equilibrium factor y = (T, — To)/To = 0.1,0.2, 0.3, 0.5.

Figure 9-7. Temperature dependence of the reaction rate coefficients (1) methane conversion into ethane (CH4 —>-1/2 C2H6 + 1 /2H2) at different values of the non-equilibrium factor y = (Tv — To)/To, (2) ethane conversion into ethylene (C2H,5 — C2H4-f H2) (3) ethylene conversion into acetylene (C2H4 —> C2H2 -f H2) (4) acetylene conversion into soot (C2H2 — 2Ccond + H2 ).

In the equation for the nucleation rate J the classical expression was supplemented by the non-equilibrium factor Z, introduced by Zeldovich, and by the so-called non-isothermal factor 0 (cf. [21 ) ... 

Note that the rate expression can be written as the product of three terms. The first one [/(n )] is the product of p times the surface area of the critical nucleus, and represents the frequency of arrival of single molecules to the critical nucleus. The third term, N(n ), is the equilibrium concentration of critical nuclei. Therefore, the second term, Z, can be interpreted as a factor that corrects for the fact that the concentration of critical nuclei differs from the equilibrium value. This term is frequently referred to as the Zeldovich non-equilibrium factor [9]. For incompressible embryos (e.g., droplets in a supercooled vapor) use of (11)-(15) in (21) yields... 

For the harmonic oscillator model, the non-equilibrium factor is specified by the vibrational temperature T, and can be calculated using the expression ... 

Figure 2 presents the temperature dependence of the non-equilibrium factor Z(T,Ti,U) in nitrogen for fixed vibrational temperature values. The non-equilibrium factor is calculated for both anharmonic (104) and harmonic (105) oscillator models. We can see that for minor deviations from the equilibrium (Tj/T 1), both models yield similar results, whereas for the ratio Tj/T essentially different from unity, the values of Z for harmonic and anharmonic oscillators differ considerably. In particular, for the selected dissociation model, the non-equilibrium factor and hence the dissociation rate coefficient of harmonic oscillators at Tj/T > 1 significantly exceed Z and respectively, when calculated for anharmonic oscillators. For Tj/T < 1, the use of the harmonic oscillator model yields lower Z and k - than those obtained taking into account anharmonic effects. 

Fig. 2. The non-equilibrium factor Z in N2 as a function of temperature T for fixed temperatures Ti and U = D/(6fc). The solid lines represent anharmonic oscillators, dashed — harmonic osdUators. The curves 1,1 — T = 3000 2,2 — Tj = 5000 3, 3 — T = 7000 K.

Note in addition that the ratio of the dissociation and recombination rate coefficients Kj-ec-diss under the non-equilibrium conditions can also be expressed in terms of the averaged non-equilibrium factor ... 

The previous sections demonstrated that reaction rates may be limited by intermolecular energy transfer at low pressure. As a consequence, transition-state theory overestimates the rate constant, and the appropriate expression for the rate constant should contain energy-transfer parameters and non-equilibrium factors. 

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