## Ratebased Models

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 [AIChEJ., 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 Qj". 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 N,j.

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 + iff)Vjyij-Vj+! y,j+!-tyj- Vj -1 yt,j -1 -fj - I Gvtjv + Nt]

= 0 i = 1, 2,..., c Component for the liquid phase:

ML = (1 + rj + )LjXj - Lj - 1Xi,j - 1 - + 1Lj + 1Xi,j + 1 - f - Gi]L - Nij v= 1

Total for the vapor phase:

Mj = (1 + rj + tyjj - Vj+1 - j 1J -1 - Fj - I I Gjv + Nj i = 1 V = 1

Total for the liquid phase:

ML = (1 + r!f + tyL )Lj - Lj -1 - j 1Lj+1 - FL - I IGj- Nj i = 1 V = 1

Energy Balances (3 Equations) For the vapor phase:

Ej = (1 + rj + )VjHjV - Vj+Hj+1 -tyj- Jj - Hj-1 - FjVHjVF

EjL = (1 + rj + $f)LjHjL - Lj - 1 - 1Lj+HL+1- FjLHjLF

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:

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