Nonequilibrium Modeling

The nonequilibrium-model equations for the stage in Fig. 13-56 are as follows in residual form, where i = component (i = 1 to C), j = stage number (j = 1 to N), and V = a stage in another column that supplies an interlink. 

A second shortcoming that arises at this stage of evaluation is that in order to conduct the evaluation much more information is required, i.e. soil and sediment degradation rates and hydrolysis and photolysis rates. At this point, more complex nonequilibrium models may be more useful. If and when methods of estimating degradation process become available, this level of evaluation will become more useful. 

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. 

Artola-Garicano et al. [27] compared their measured removals of AHTN and HHCB [24] to the predicted removal of these compounds by the wastewater treatment plant model Simple Treat 3.0. Simple Treat is a fugacity-based, nine-box model that breaks the treatment plant process into influent, primary settler, primary sludge, aeration tank, solid/liquid separator, effluent, and waste sludge and is a steady-state, nonequilibrium model [27]. The model inputs include information on the emission scenario of the FM, FM physical-chemical properties, and FM biodegradation rate in activated sludge. 

Hem J. D. and Lind C. J. (1983). Nonequilibrium models for predicting forms of precipitated manganese oxides. Geochim. Cosmochim. Acta, 47 2037-2046. 

Goltz, M. N., and Roberts, P. V. (1986). "Interpreting organic solute transport data from a field experiment using physical nonequilibrium models." J. Contam. Hydrol., 1(1), 77-93. 

Higler AP, Krishna R, Taylor R. Nonequilibrium modeling of reactive distillation a dusty fluid model for heterogeneously catalyzed processes. Ind Eng Chem Res 2000 39 1596-1607. 

Sundmacher K, Hoffmann U. Development of a new catalytic distillation process for fuel ethers via a detailed nonequilibrium model. Chem Eng Sci 1996 51 2359-2368. 

A number of investigators have modeled solute transport in soils assuming an equilibrium occurs between solution and solid phases. This assumption is often not valid in heterogeneous soil systems, and has been the impetus for the development of a number of nonequilibrium models. Some researchers have assumed that the nonequilibrium is caused by stagnant zones, which result in tortuous diffusional processes between solution and sorbed phases (Rao et al., 1979). Other researchers have attributed the nonequilibrium to kinetic effects. 

Many of the nonequilibrium models that have been used to describe soil chemical reactions take into account the heterogeneous nature of soils. Some of these models have described sorption using a two-site model (Selim et al., 1976b Cameron and Klute, 1977) characterized by fast and slow binding sites. 

The relaxation, inside-out, and bomotopy-continuation methods are extensions of whole or part of the first four methods in order to solve difficult systems or columns. The nonequilibrium models are rate- or transport phenomena-based methods that altogether do away with the ideal-stage concept and eliminate any use of efficiencies. They are best suited for columns where a theoretical stage is difficult to define and efficiencies are difficult to predict or apply. 

In nonequilibrium models, as in the other models, the subscript j is for the stage. In a trayed column, it is the actual tray. In a packed column, j is a section of packing. By convention, transfer is to be from the liquid to the vapor with the mass transfer rate to the vapor, Nf, taken as positive. 

The Taylor method. Krishnamurthy and Taylor (88, 89) present and test a nonequilibrium model which includes rate equations for mass transfer, and sometimes reaction, among the traditional MESH equations. These include individual mass and energy balances in the vapor and the liquid and across the interface. An equilibrium equation exists for the interface only. The solution method for these equations is the same as that of the block-banded matrices of the global Newton methods and the style of the method is similar to the Naphtali-Sandholm (Sec. 4.2.9). 

The number of equations, M5C + 1), for a large number of trays and components, can be excessive. The global Newton method will suffer from the same problem of requiring initial values near the answer. This problem is aggravated with nonequilibrium models because of difficulties due to nonideal if-values and enthalpies then compounded by the addition of mass transfer coefficients to the thermodynamic properties and by the large number of equations. Taylor et al. (80) found that the number of sections of packing does not have to be great to properly model the column, and so the number of equations can be reduced. Also, since a system is seldom mass-transfer-limited in the vapor phase, the rate equations for the vapor can be eliminated. To force a solution, a combination of this technique with a homotopy method may be required. 

The methods based on the equilibrium stage model have existed for over 30 years and refinements continue, but serious development of nonequilibrium models has begun only recently. These methods are an alternative means to the stage model for predicting column performance. They are expected to make inroads, especially for systems for which stage efficiency prediction is very difficult, such as reactive distillation, chemical absorption, and three-phase distillation. However, their progress into systems where efficiency prediction is well-established is likely to be slower. Their complexity due to the restriction to... 

Recently, kinetic models have been combined with the equilibrium data of the interfacial processes, taking into account that soils and rocks are heterogeneous and consequently have different sites. These models are called nonequilibrium models (Wu and Gschwend 1986 Miller and Pedit 1992 Pedit and Miller 1993 Fuller et al. 1993 Sparks 2003 Table 7.2). These models describe processes when a fast reaction (physical or chemical) is followed by one or more slower reactions. In these cases, Fick s second law is expressed—that the diffusion coefficient is corrected by an equilibrium thermodynamic parameter of the fast reaction (e.g., by a distribution coefficient), that is, the fast reaction is always assumed to be in equilibrium. In this way, the net processes are characterized by apparent diffusion coefficients. However, such reactions can be equally well described using Equation 1.126. 

The prediction and use of stage efficiencies are described in detail in Sec. 14. Alternative approaches based on mass-transfer rates are preferred, as described in the subsection below. Nonequilibrium Modeling. 

The computer simulation of distillation processes, whether done using equilibrium or nonequilibrium models, requires us to address the following topics ... 

In a nonequilibrium model, separate balance equations are written for each distinct phase. The material balances for each species in the vapor and liquid phases on an arbitrary stagej are... 

In equilibrium-stage models, the compositions of the leaving streams are related through the assumption that they are in equilibrium (or by use of an efficiency equation). It is important to recognize that efficiencies are not used in a nonequilibrium model they may, however, be calculated from the results obtained by solving the model equations. 

Physical Properties The on equilibrium-stage simulation are i and enthalpies these same properties are needed for nonequilibrium models as well. Enthalpies are required for the energy balance equations vapor-liquid equilibrium ratios are needed for the calculation of driving forces for mass and heat transfer. The need for mass- (and heat-) transfer coefficients means that nonequilibrium models are rather more demanding of physical property data than are equilibrium-stage models. These coefficients may depend on a number of other physical properties, as summarized in Table 13-12. 

TABLE 13-12 Physical Property Needs of Equilibrium and Nonequilibrium Models... 

Solving the NEQ Model Equations In general, a nonequilibrium model of a column has many more equations than does an equivalent equilibrium-stage model. Nevertheless, we use may essentially the same computational approaches to solve the nonequilibrium model equations simultaneous convergence (Krishnamurthy and Taylor, op. cit.) and continuation methods [Powers et al., Comput Chem. Engng., 12, 1229 (1988)]. Convergence of a nonequilibrium model is likely to be slower than that of the equilibrium model because of the greater... 

Example 13 A Nonequilibrium Model of a C4 Splitter Consider, again the C4 splitter that formed the basis of Examples 7,8, and 11. 

In this particular case the converged composition and temperature profiles have the same shape as those obtained with the equilibrium-stage model (with specified efficiency) and, therefore, are not shown. The reason for the similarity is that, as noted above, this is basically a binary separation of very similar compounds. The important point here is that, unlike the equilibrium-stage model simulations, the nonequilibrium model predicted how the column would perform no parameters were adjusted to provide a better jit to the plant data. That is not to say, of course, that NEQ models cannot be used to fit plant data. In principle, the mass-transfer coefficients and interfacial area (or parameters in the equations used to estimate them) can be tuned to help the model better fit plant data. 

To converge the nonequilibrium model at the specified reflux ratio, it was necessary first to solve the problem at a much lower reflux ratio R = 2 and then increase R in steps until the desired value of 7.5 was reached... 

Individual component efficiencies can vary as much as they do in this example only when the diffusion coefficients of the three binary pairs that exist in this system differ significantly For ideal or nearly ideal systems, all models lead to essentially the same results. This example demonstrates the importance of mass-transfer models for nonideal systems, especially when trace components are a concern. For further discussion of this example, see Doherty and Malone (op. cit.) and Baur et al. [AIChE J. 51,854 (2005)]. It is worth noting that there exists extensive experimental evidence for mass-transfer effects for this system, and it is known that nonequilibrium models accurately describe the behavior of this system, whereas equilibrium models (and equal-efficiency models) sometime... 

Nonequilibrium models should be preferred to equilibrium models when efficiencies are unknown, cannot be reliably predicted, and are low and/or highly variable in nonideal systems and in processes where trace components are a concern. 

There is a rapidly growing body of literature on nonequilibrium modeling of distillation and absorption processes. An extended bibliography is available at www.chemsep.org/publications. A brief review of other applications follows. 

There is now an extensive literature on using nonequilibrium models for reactive distillation see, e.g., Taylor and Krishna [Chem. Eng. 

Even at steady state, efficiencies vary from component to component and with position in a column. Thus, if the column is not at steady state, then efficiencies also must vary with time as a result of changes to flow rates and composition inside the column. Thus, equilibrium-stage models with efficiencies should not be used to model the dynamic behavior of distillation and absorption columns. Nonequilibrium models for studying column dynamics are described hy, e.g., Kooijman and Taylor [AlChE 41, 1852 (1995)], Baur et al. [Chem. 

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