Equilibrium-stage Model

Different model assumptions reflect the relation between the mass transfer and reaction rates. The definition of the Hatta number, representing the maximum reaction rate with reference to that of the mass transfer, helps to discriminate between very fast, fast, average and slow chemical reactions [56, 57]. 

If the reaction system considered is fast, the process can be satisfactorily described assuming reaction equilibrium. Here, a proper modeling approach is based on the nonreactive equilibrium-stage model, extended by the chemical equilibrium relationship. An alternative approach proposed by Davies and Jeffreys [13] includes two separate steps. First, the concentrations and flow rates of the leaving streams are calculated with the simple nonreactive equilibrium-stage model. Afterwards, the leaving concentrations are adapted by using an additional equilibrium reactor concept. However, the latter approach does not consider direct interactions between the chemical and thermodynamic equilibrium. 

Such descriptions can be appropriate for instantaneous and very fast reactions. In contrast, if the chemical reaction is slow, the reaction rate dominates the whole process, and therefore, the reaction kinetics expression has to be integrated into the mass and energy balances. This concept has been used in a number of RD studies (e.g., [58, 59]). 

In practice, thermodynamic equilibrium can seldom be reached within a single stage. Therefore, some correlation parameters, like tray efficiencies or HETS values, have been introduced to adjust the equilibrium-based theoretical description to the reality. For multicomponent mixtures, however, the application of this concept is often difficult due to diffusional interactions of several components [60, 61], These effects cause an unpredictable behavior of the efficiency factors, which are different for each component, vary along the column height and show a strong dependency on the component concentration [42, 61, 62]. 

The acceleration of mass transfer due to chemical reactions in the interfacial region is often accounted for via the so-called enhancement factors [57, 63, 64]. 

M equations—material balance for each component (C equations for each stage)  

E equations—phase equilibrium relation for each component, here modified to include the Murphree efficiency defined by equation (4-55) (C equations for each stage)  

The MESH equations can be applied to all of the equilibrium stages in the column, including the reboiler and condenser. The result is a set of nonlinear equations that must be solved by iterative techniques. 

If a very fast reaction is considered, the reactive separation process can be satisfactorily described assuming reaction equilibrium. Here, a proper modeling approach is based on the non-reactive equilibrium stage model, extended by simultaneously 

The acceleration of mass transfer due to chemical reactions in the interfacial region is often accounted for via the so-called enhancement factors [19, 26, 27]. These parameters are defined as a relationship between the mass transfer rate with reaction and mass transfer rate without reaction, assuming the same mass transfer driving force. 

The enhancement factors are either obtained by fitting experimental results or are derived theoretically on the grounds of simplified model assumptions. They depend on reaction character (reversible or irreversible) and order, as well as on the assumptions of the particular mass transfer model chosen [19, 26]. For very simple cases, analytical solutions are obtained, for example, for a reaction of the first or pseudo-first order or for an instantaneous reaction of the first and second order. Frequently, the enhancement factors are expressed via Hatta-numbers [26, 28]. They can be used in combination with the HTU/NTU-method or with a more advanced mass transfer description method. However, it is generally not possible to derive the enhancement factors properly from binary experiments, and a theoretical description of reversible, parallel or consecutive reactions is based on rough simplifications. Thus, for many reactive absorption processes, this approach appears questionable. 

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Availability of large digital computers has made possible rigorous solutions of equilibrium-stage models for multicomponent, multistage distillation-type columns to an exactness limited only by the accuracy of the phase equilibrium and enthalpy data utilized. Time and cost requirements for obtaining such solutions are very low compared with the cost of manual solutions. Methods are available that can accurately solve almost any type of distillation-type problem quickly and efficiently. The material presented here covers, in some... 

Naplitali-Sandholm SC Method This method employs the equilibrium-stage model of Figs. 13-48 and 13-49 but reduces the number of vari les by 2N so that only N(2C + 1) equations in a hke number of unknowns must be solved. In place of Vj, Lj, Xij, and iji j, component flow rates are used according to their definitions ... 

The iC values (vapor-liquid equihbrium ratios) in Equation (13-123) are estimated from the same equation-of-state or aclivity-coefficient models that are used with equilibrium-stage models. Tray or packed-section pressure drops are estimated from suitable correlations of the type discussed by Kister (op. cit.). 

FIVE STAGE COUNTERCURRENT EXTRACTION CASCADE EQUILIBRIUM STAGE MODEL... 

Five stage countercurrent extraction cascade with backmixing Equilibrium stage model... 

The variation of efficiencies is due to interaction phenomena caused by the simultaneous diffusional transport of several components. From a fundamental point of view one should therefore take these interaction phenomena explicitly into account in the description of the elementary processes (i.e. mass and heat transfer with chemical reaction). In literature this approach has been used within the non-equilibrium stage model (Sivasubramanian and Boston, 1990). Sawistowski (1983) and Sawistowski and Pilavakis (1979) have developed a model describing reactive distillation in a packed column. Their model incorporates a simple representation of the prevailing mass and heat transfer processes supplemented with a rate equation for chemical reaction, allowing chemical enhancement of mass transfer. They assumed elementary reaction kinetics, equal binary diffusion coefficients and equal molar latent heat of evaporation for each component. 

Before leaving this section we consider a slightly different optimization problem that may also be expensive to solve. In flowsheet optimization, the process simulator is based almost entirely on equilibrium concepts. Separation units are described by equilibrium stage models, and reactors are frequently represented by fixed conversion or equilibrium models. More complex reactor models usually need to be developed and added to the simulator by the engineer. Here the modular nature of the simulator requires the reactor model to be solved for every flowsheet pass, a potentially expensive calculation. For simulation, if the reactor is relatively insensitive to the flowsheet, a simpler model can often be substituted. For process optimization, a simpler, insensitive model will necessarily lead to suboptimal (or even infeasible) results. The reactor and flowsheet models must therefore be considered simultaneously in the optimization. 

The equilibrium-stage model seems to be suitable for esterification reaction in CD processes (see Refs. 35 and 74). However, it cannot be recommended for all reaction types, especially those with higher reaction rates. 

All results demonstrate a good agreement between the simulations and experiments. The simulation studies warn of equilibrium stage model application, which appears to be completely inappropriate for the case of finite rate reactions. The film reaction consideration is found to be crucial. 

Krishnamurthy and Taylor [51] developed a so-called non-equilibrium stage model , the characteristics of which were the balance of mass and energy for each component in the two phases. These are coupled over the energy and mass flows in the boundary layers and are at equilibrium at the phase boundary. 

Schultes presented an absorption model for packed columns including simulations [81], while Eden [18] developed a non-equilibrium stage model describing the absorption of electrolytes in co-current and countercurrent scrubbers. The simultaneous view of phase and reaction equilibrium and the existence of solids in the liquid phase were emphasized in these reports. 

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... 

To do so, we must use a model of the TMB that enables us to focus the influence of the system efficiency on purities. Many different models have been applied to the modeling of chromatographic processes.11 The equilibrium stage model has been proven to be suitable under the usual conditions of high-performance preparative chromatography31 and can also be applied to TMB.32... 

Chemical engineers have been solving distillation problems by using the equilibrium-stage model since 1893 when Sorel outlined the concept to describe the distillation of alcohol. Since that time, it has been used to model a wide variety of distillation-like processes, including simple distillation (single-feed, two-product columns), complex distillation (multiple-feed, multiple-product columns), extractive and azeotropic distillation, petroleum distillation, absorption, liquid-liquid extraction, stripping, and supercritical extraction. 

With 11 stages and 5 components the equilibrium-stage model has 143 equations to be solved for 143 variables (the unknown flow rates, temperatures, and mole fractions). Convergence of the computer algorithm was obtained in just four iterations. Computed product flows are shown in Fig. 13-37. 

Although the widely used equilibrium-stage models for distillation, described above, have proved to be quite adequate for binary and closeboiling, ideal and near-ideal multicomponent vapor-liquid mixtures. 

The sum of the phase and interface balances yields the component material balance for the stage as a whole, the equation used in the equilibrium-stage model. 

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. 

Eer stage. As with the equilibrium-stage model discussed above, we ave not included the feed mole fraction summation equation, or those for the vapor and hquid streams coming from adjacent stages. 

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. 

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... 

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. 

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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