Although the widely used equilibrium-stage models for distillation, described above, have proved to be quite adequate for binary and close-boiling, ideal and near-ideal multicomponent vapor-liquid mixtures, their deficiencies for general multicomponent mixtures have long been recognized. Even Murphree [Ind. Eng. Chem., 17, 747-750 and 960-964 (1925)], who formulated the widely used plate efficiencies that carry his name, pointed out clearly their deficiencies for multicomponent mixtures and when efficiencies are small. Later, Walter and Sherwood [Ind. Eng. Chem., 33, 493 (1941)] showed that experimentally measured efficiencies could cover an enormous range, with some values less than 10 percent, and Krishna et al. [Trans. Inst. Chem. Engr., 55, 178 (1977)] showed theoretically that the component mass-transfer coupling effects discovered by Toor [AIChE J., 3, 198 (1957)] could cause the rate of mass transfer for components having small concentration driving forces to be controlled by the other species, with the result that Murphree vapor efficiencies could cover the entire range of values from minus infinity to plus infinity.
The first major step toward the development of a more realistic rate-based (nonequilibrium) model for distillation was taken by Krish-namurthy and Taylor [AIChE J., 31, 449-465 (1985)]. More recently, Taylor, Kooijman, and Hung [Comp. Chem. Engng., 18, 205-217 (1994)] extended the initial development so as to add the effects of tray-pressure drop, entrainment, occlusion, and interlinks with other columns. In the augmented MESH equations, which they refer to as the MERSHQ equations, they replace the conventional mass and energy balances around each stage by two balances each, one for the vapor phase and one for the liquid phase. Each of the component-material balances contains a term for the rate of mass transfer between the two phases; the energy balances contain a term for the rate of heat transfer between phases. Thus, each pair of phase balances is coupled by mass or heat-transfer rates, which are estimated from constitutive equations that account, in as rigorous a manner as possible, for bulk transport, species interactions, and coupling effects. The heat and mass-transfer coefficients in these equations are obtained from empirical correlations of experimental data and the Chilton-Colburn analogy. Equilibrium between the two phases is assumed at the phase interface. Thus, the rate-based model deals with both transport and thermodynamics. Although tray efficiencies are not part of the modeling equations, efficiencies can be back-calculated from the results of the simulation. Various options for vapor and liquid flow configurations are employed in the model, including plug flow and perfectly mixed flow on each tray.
A schematic diagram of the nonequilibrium stage for the Taylor et al. model is shown in Fig. 13-56. Entering the stage are the following material streams: FV = vapor feed; FL = liquid feed; Vj +1 = vapor from stage below together with fractional-liquid entrainment, +1; Lj -1 = liquid from stage above with fractional-vapor occlusion, j 1; GV = vapor interlink; and GL = liquid interlink. Leaving the stage are the following material streams: Vj = vapor with fractional withdrawal as sidestream, rj, and fractional-liquid entrainment, j and Lj = liquid with fractional withdrawal as sidestream, rjL, and fractional-vapor occlusion, 4>J. Also leaving the stage are heat-transfer streams, QjV and QL. The rate of heat transfer from the vapor phase to the liquid phase is Ej and the rate of component mass transfer from the vapor phase to the liquid phase is Nij.
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.
Material Balances (2C + 2 Equations) Component for the vapor phase:
Mj = (1 + rj +§])Vjytj-Vj+1 y„+1 Vj -1 yt,j -1 -fj - X Glv + N
= 0 i = 1, 2,..., c Component for the liquid phase:
ML = (1 + rj + % j - Lj - 1Xi'j - 1 + 1Lj + 1Xi'j + 1 - fi] - ^X GijL - Nij v = 1
Total for the vapor phase:
Mj = (1 + rj + ^Vj - Vj +1 -%L -1 Vj -1 - FV - X XGVv + N
Total for the liquid phase:
ML = (1 + rf + %L)Lj - Lj -1 - Lj+1 - FL - XX X,QLjv- N
Energy Balances (3 Equations) For the vapor phase:
Ej = (1 + rj + fVjHj - Vj + 1HV+1 - j 1Vj - Hj-1 - FjHjF
E]L = (1 + rj + %]')LjH]L - Lj - 1HF1- 1Lj+HL+1 - F]LH]LF
Continuity across the phase interface:
Mass-Transfer Rates (2C - 2 Equations) Component in the vapor phase:
Component in the liquid phase:
Summation of Mole Fractions (2 Equations) Vapor-phase interface:
Hydraulic Equation for Stage Pressure Drop (1 Equation)
Vapor-phase pressure drop:
Interface Equilibrium (C Equations) Component vapor-liquid equilibrium:
Equations (13-111) to (13-114), (13-118) and (13-119), contain terms, Nij, for rates of mass transfer of components from the vapor phase to the liquid phase (rates are negative if transfer is from the liquid phase to the vapor phase). These rates are estimated from diffusive and bulk-flow contributions, where the former are based on interfacial area, average mole-fraction driving forces, and mass-
transfer coefficients that account for coupling effects through binary-pair coefficients. Although the stage shown in Fig. 13-56 appears to apply to a trayed column, the model also applies for a section of a packed column. Accordingly, empirical correlations for the interfacial area and binary-pair mass-transfer coefficients cover bubble-cap trays, sieve trays, valve trays, dumped packings, and structured packings. The average mole-fraction driving forces for diffusion depend upon the assumed vapor and liquid-flow patterns. In the mixed-flow model, both phases are completely mixed. This is the simplest model and is usually suitable for small-diameter trayed columns. In the plug-flow model, both phases move in plug flow. This model is applicable to packed columns and certain trayed columns.
Equations (13-115) to (13-117) contain terms, e,, for rates of heat transfer from the vapor phase to the liquid phase. These rates are estimated from convective and bulk-flow contributions, where the former are based on interfacial area, average-temperature driving forces, and convective heat-transfer coefficients, which are determined from the Chilton-Colburn analogy for the vapor phase and from the penetration theory for the liquid phase.
The K values (vapor-liquid equilibrium ratios) in Equation (13-123) are estimated from the same equation-of-state or activity-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.).
From the above list of rate-based model equations, it is seen that they total 5C + 6 for each tray, compared to 2C + 1 or 2C + 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.
A potential limitation of the application to design of a rate-based model compared to the equilibrium-stage model is that the latter can be computed independently of the geometry of the column because no transport equations are included in the model. Thus, the sizing of the column is decoupled from the determination of column operating conditions. However, this limitation of the early rate-based models has now been eliminated by incorporating a design mode that simultaneously designs trays and packed sections.
A study of industrial applications by Taylor, Kooijman, and Woodman [IChemE. Symp. Ser. Distillation and Absorption 1992, A415-A427 (1992)] concluded that rate-based models are particularly desirable when simulating or designing: (1) packed columns, (2) systems with strongly nonideal liquid solutions, (3) systems with trace compo nents that need to be tracked closely, (4) columns with rapidly changing profiles, (5) systems where tray-efficiency data are lacking. Besides the extended model just described, a number of other investigators, as summarized by Taylor, Kooijman, and Hung (op. cit.), have developed rate-based models for specific applications and other purposes, including cryogenic distillation, crude distillation, vacuum distillation, catalytic distillation, three-phase distillation, dynamic distillation, and liquid-liquid extraction. Commercial computerized rate-based models are available in two simulation programs: RATEFRAC in ASPEN PLUS from Aspen Technology, Inc., Cambridge, Massachusetts and NEQ2 in ChemSep from R. Taylor and H. A. Kooijman of Clarkson University. Rate-based models could usher in a new era in trayed and packed-column design and simulation.
Example 8: Calculation of Rate-Based Distillation The separation of 655 lb'mol/h of a bubble-point mixture of 16 mol % toluene, 9.5 mol % methanol, 53.3 mol % styrene, and 21.2 mol % ethylbenzene is to be carried out in a 9.84-ft diameter sieve-tray column having 40 sieve trays with 2-inch high weirs and on 24-inch tray spacing. The column is equipped with a total condenser and a partial reboiler. The feed will enter the column on the 21st tray from the top, where the column pressure will be 93 kPa, The bottom-tray pressure is 101 kPa and the top-tray pressure is 86 kPa. The distillate rate will be set at 167 lb'mol/h in an attempt to obtain a sharp separation between toluene-methanol, which will tend to accumulate in the distillate, and styrene and ethyl-benzene. A reflux ratio of 4.8 will be used. Plug flow of vapor and complete mixing of liquid will be assumed on each tray. K values will be computed from the UNIFAC activity-coefficient method and the Chan-Fair correlation will be used to estimate mass-transfer coefficients. Predict, with a rate-based model, the separation that will be achieved and back-calculate from the computed tray compositions, the component vapor-phase Murphree-tray efficiencies.
The calculations were made with the RATEFRAC program and comparisons were made with the companion RADFRAC program, which utilizes the inside-out method for an equilibrium-based model.
The rate-based model gave a distillate with 0.023 mol % ethylbenzene and 0.0003 mol % styrene, and a bottoms product with essentially no methanol and 0.008 mol % toluene. Murphree tray efficiencies for toluene, styrene, and ethyl-benzene varied somewhat from tray to tray, but were confined mainly between 86 and 93 percent. Methanol tray efficiencies varied widely, mainly from 19 to 105 percent, with high values in the rectifying section and low values in the stripping section. Temperature differences between vapor and liquid phases leaving a tray were not larger than 5°F.
Based on an average tray efficiency of 90 percent for the hydrocarbons, the equilibrium-based model calculations were made with 36 equilibrium stages. The results for the distillate and bottoms compositions, which were very close to those computed by the rate-based method, were a distillate with 0.018 mol % ethylbenzene and less than 0.0006 mol % styrene, and a bottoms product with only a trace of methanol and 0.006 mol % toluene.
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