The k Equation

Note that the right-hand side of this expression is an E x I null matrix, and thus element conservation must hold for any choice of e e l,E and i e 1Moreover, since the element matrix is constant, (5.10) can be applied to the scalar transport equation ((1.28), p. 16) in order to eliminate the chemical source term in at least E of the K equations.9 The chemically reacting flow problem can thus be described by only K - E transport equations for the chemically reacting scalars, and E transport equations for non-reacting (conserved) scalars.10... 

It is important to note that each of the K equations represented by Eqs. 3.105 are not all independent. The mole fractions must sum to unity, which means that VX = 0. It is also the case that... 

The solution of equilibrium problems is based on algebraic manipulation of the equilibrium constant expression. In typical equilibrium problems the unknown is one part of the K equation shown above and the other parts are given. 

Analogous to the transport equation of turbulence kinetic energy, k, the first term on the right-hand side represents production or generation , the second represents diffusion and the final term represents dissipation . The third term of the right-hand side, which has no counterpart in the k equation, represents redistribution . It is the correlation between fluctuations in pressure and velocity gradients, which results in enhancement of velocity fluctuations in one direction at the expense of those in the other directions. It is necessary to model these terms in order to close the set of equations. 

However, further simplifications can be achieved focusing on the log-law region. Considering the -equation (1.414) experimental observations indicate that the production and dissipation terms balance (i.e., thus the dissipation and production terms are in local equilibrium), hence in the k-equation the diffusion term vanishes. This implies that k is approximately uniform. 

Normally, as discussed earlier, in reactor modeling we are interested in the energy associated with the velocity fluctuations v only. The appropriate turbulent kinetic energy balance equation, the k equation, has therefore been derived via an equation for the Reynolds stress tensor. 

If we compare the k equation (1.401) with the mean kinetic energy equation (1.453) we see that they both contain a term describing the interaction between the mean flow and turbulence. We are of course referring to the velocity... 

At this point it is important to notice that the interfacial source and dissipation terms occurring in the k- equation above are consistent with the source and sink terms in the two-phase k - equation developed by [74]. Lopez de Bertodano et al [93] recognized that the terms pitsi + are a linear superposition of shear and bubble induced dissipation. Similarly, the terms Pk,si + were found to denote the linear superposition of the shear and bubble induced sources of turbulence. Note that the latter term, i.e., by introducing the suggested parameterizations, is identical to the bubble induced source of turbulence (Pf,) which was taken to be proportional to the work done on the liquid by the bubbles as shown in equation (5.3). The k - equation used by [93] is thus identical to equation (5.3). The only modification... 

FIG. 2-13 Pressure-enthalpy diagram for nitrogen. Properties computed with the NIST REFPROP Database, Version 7.0 (Lemmon, E. W., McLinden, M. O., and Huber, M. L., 2002, NIST Standard Reference Database 23, NIST Reference Fluid Thermodynamic and Transport Properties—REFPROP, Version 7.0, Standard Reference Data Program, National Institute of Standards and Technology), based on the K> equation of state of Span, R., Lemmon, E. W., Jacobsen, R. T., Wagner, W., and Yokozeki, A,. A Reference Equation of State for the Thermodynamic Properties of Nitrogen for Temperatures from 63.151 to 1000... 

These ten equations can. with a reasonable computer, be solved for the ten unknowns in question. Alternatively, by carefully organizing the calculations and making some simplifying assumptions, for a simple system such as CI2-H2O trial and error using a calculator is also feasible. What has been understated thus far is that, embedded in equations 1-5, the K equations, is the essential complexity of the electrolyte calculations. The variables. K(T,P) and Y(T,P,m) are often highly nonlinear functions of the state variables shown. The purpose of this book is thus to describe ... 

The k-equations of the last relation type form together with the energy minimization condition a closed system leaving with determination of the -atomic partition functions in molecule, afr), and the Lagrange multiplier n(x). The phenomenological relationship between these two last AIM measures can be deduced, if the last relation (3.19) is multiplied with a.(x) to get the form ... 

A stable starting point for the kinds of flows enconntered in a stirred tank is the k-8 model. This model assumes that the normal stresses are roughly equal and are adequately represented by k. Two differential eqnations are used to model the production, distribution, and dissipation of tmbulent kinetic energy the k-equation, and the s-equation. These equations were developed for free shear flows, and experimentally determined constants are established for the model parameters. One of these constants is nsed to relate local values of k and e to an estimate of (uv) using a modified tmbulent viscosity approach ... 

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