The Equilibrium Stage Model

The next task is to set up the equations that model a complete distillation column. As noted in the introduction to this chapter, simulation of multicomponent distillation operations usually is carried out using the equilibrium stage model introduced below. 

The equations are the Equilibrium relations, here modified to include the Murphree efficiencies defined by Eq. 13.1.5 

If we count the equations listed, we will find that there are 2n + 4 equations per stage. However, only 2 n + 3 of these equations are independent. These independent equations are generally taken to be the n component mass balance equations, the n equilibrium relations, the enthalpy balance, and two more equations. These two equations can be the two summation equations or the total mass balance and one of the summation equations (or an equivalent form). The 2n + 3 unknown variables determined by the equations are the n vapor mole fractions the n liquid mole fractions, the stage temperature 7 and the vapor and liquid flow rates LJ and Ly. Thus, for a column of 5 stages, we must solve s 2n + 3) equations. 

For these special stages it is common to use some specification equation instead of the enthalpy balance. Common specifications include 

The flow rate of the distillate-bottoms product stream. 

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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 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. 

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. 

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. 

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. 

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. 

From the above list of rate-based model equations, it is seen that they total 5C -H 6 for each tray, compared to 2C + 1 or 2C -H 3 (depending on whether mole fractions or component flow rates are used for composition variables) for each stage in the equilibrium-stage model. Therefore, more computer time is required to solve the rate-based model, which is generally converged by an SC approach of the Newton type. 

The equilibrium stage model reported by Zhang et al. [7], which accounts for the influence of fluid flow rate on fee column efficiency, has been used for the optimization simulations. 

Fig. 5 shows good agreement between the experimental and simulation results of dynamic liquid bulk concentrations. Because of its complexity the rate-based model is not suitable for controller design and optimization of the RD process. Therefore, an extended equilibrium stage model, which includes a reaction kinetic, is used for these tasks. Fig. 6 shows comparisons of simulation results of the rate-based model (RBA) and the equilibrium stage model for a typical trajectory of input variables. The dynamic behavior is covered well by the simplified model and the deviations between the absolute values are acceptable for control purposes. The advantage of substantially reduced computing time motivates the use of the simplified model for control and optimization purposes. 

Another feature of the two-tier method is the use of special iteration variables for improved convergence stability. The equilibrium stage model is formulated as follows, beginning by writing the component material balances and equilibrium relations in terms of component molar flow rates instead of mole fractions ... 

It is obvious from the conditions defined above that the rate-based model equations and variables are more numerous and complex than those in the equilibrium stage model described in Chapter 13. Other features of the rate-based model are that the exiting liquid and vapor from a stage can be at different temperatures since separate balance equations are written for each phase. Each phase on a stage can have a different externally transferred heat duty. The exiting phases in general are not at equilibrium the liquid may be subcooled and the vapor may be superheated. In a rate-based model the phase interface must be defined. The variables defining the interface include the liquid and vapor compositions and the temperature at the interface, and the molar flux across the interface. 

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