Modelling reactive distillation

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5208 R. Taylor, R. Krishna / Chemical Engineering Science 55 (2000) 5183}5229uncertainty in the parameters that appear in theequations. Berg and Harris (1993) have developed a detailedmodel of di!usion and reaction in MTBE synthesis.However, their model focuses only on thetransport and reaction issues and they have not developeda complete column/process model. Sundmacher,Ho!mann and their co-workers (Sundmacher &Ho!mann, 1992, 1993, 1994a,b, 1995; Sundmacher et al.,1994; Sundmacher, Zhang & Ho!mann, 1995) have carriedout an extensive study of mass transfer and activityin and around RD catalysts. Sundmacher and Ho!mann(1992) recommend the MS equations over Fick's law formodelling mass transfer in the liquid-phase surroundingan ion-exchange catalyst particle. The reaction ismodelled by an activity-based Langmuir}Hinshelwoodexpression derived from an extensive experimental studyby Reh"nger and Ho!mann (1990a,b). The model is usedto determine the catalyst e!ectiveness factor, and themodel is in good agreement with extensive experimentaldata. Sundmacher and Ho!mann (1994a) developeda detailed model to investigate the interaction betweenmass and energy transport and reaction in an ion-exchangeresin. Catalyst e!ectiveness factors and selectivitiescomputed from the model are in good agreementwith experimental data for MTBE formation obtained ina CSTR. In other papers Sundmacher & Ho!mann(1994, 1995) use the Maxwell}Stefan equations as a basisfor a "lms-in-series mass transfer model for a vapor}liquid-porouscatalyst system. However, di!usion inthe catalyst is described using an e!ective di!usivitymodel that is equivalent to a rearrangement of the dusty#uid model while ignoring the Knudsen di!usion terms.The e!ect of the reaction is lumped into a componentconsumption term, corrected by the catalyst e$ciency.The reaction term is evaluated at bulk liquid conditions.Non-idealities in the liquid-phase were described withthe UNIQUAC equation. The SRK equation of state wasused for the gas phase. Sundmacher and Ho!mann introduce`arti"cial inertia termsa into the interfacemass balances in order to create a system of di!erentialand algebraic equations (DAEs). The DAE systemwas solved with the LIMEX solver (Deu#hard, Hairer& Zugck, 1987). Important conclusions from theirtheoretical work are that the mass transfer resistanceinside the catalyst can vary signi"cantly along the packingand that the main resistance to mass transfer is on theliquid side.5.1. The conventional NEQ modelIt will be helpful to begin with a brief review of theNEQ model for conventional distillation operations.A schematic representation of the NEQ stage is shown inFig. 23. This NEQ stage may represent a tray or a crosssectionof a packed column. The component molar balancesfor the vapor and liquid-phases are< y !< y !f # "0, (23) x !f ! "0, (24)¸x !¸ where is the interfacial mass transfer rate and is theproduct of the molar #ux and the net interfacial area. Theoverall molar balances are obtained by summing Eqs.(23) and (24) over the total number (c) of components inthe mixture. The are obtained from the Maxwell}Stefan(16) modi"ed as follows:x μ ¹ η " x !x c (κ a) (25)with a similar relation for the vapor phase. The κ representsthe mass transfer coe$cient of the i!k pair inthe liquid-phase; this coe$cient is estimated from informationon the corresponding Maxwell}Stefan di!usivityn using the standard procedures discussed inTaylor and Krishna (1993). a is the interfacial area. Onlyc!1 of Eq. (25) are independent. The mole fraction ofthe last component is obtained by the summation equationsfor both phases. The enthalpy balances for bothvapor and liquid-phases are< H!< H !FH# #Q"0, (26)¸H H !FH!#Q"0. !¸ (27)5. Non-equilibrium (NEQ) stage modellingThe NEQ stage model for RD follows the philosophyof rate-based models for conventional distillation(Krishnamurthy & Taylor, 1985; Taylor & Krishna,1993; Taylor, Kooijman & Hung, 1994; Seader & Henley,1998).Fig. 23. The non-equilibrium stage (cell) for homogeneous liquid-phasereaction.
R. Taylor, R. Krishna / Chemical Engineering Science 55 (2000) 5183}5229 5209The interphase energy transfer rates (equal in bothphases) have conductive and convective contributions "!ha ¹ η # H (28) with a similar relation for the vapor phase. h is the heattransfer coe$cient in the liquid-phase. The conductivecontributions are ignored in some modelling studies.This omission results in liquid-phases being (predicted tobe) slightly superheated and vapor phases that are subcooled(Taylor et al., 1994).At the vapor}liquid interface we assume phase equilibriumy "K x , (29)where the subscript I denotes the equilibrium compositionsand K is the vapor}liquid equilibrium ratio forcomponent i on stage j. The K-values are evaluated at thetemperature, pressure and composition of the interfacefrom appropriate thermodynamic models (the samemodels used in conventional equilibrium stage models).In addition to Eqs. (23)}(29), we have the summationequations for the mole fractions in the vapor and liquidphaseand equations expressing the continuity of #uxes ofmass and energy across the interface. Furthermore, in theNEQ model we take account of the pressure drop acrossa stagep !p !(῀p )"0, (30)where p and p are the stage pressures and ῀p is the pressure drop per tray from stage (j!1) to stage j.The pressure drop over the stage is considered to bea function of the stage #ows, the physical properties andthe hardware design. In the NEQ model, hardware designinformation must be speci"ed so that mass transfercoe$cients, interfacial areas, liquid hold-ups, etc. can becalculated. The NEQ model requires thermodynamicproperties, not only for calculation of phase equilibriumbut also for calculation of driving forces for mass transferand, in RD, for taking into account the e!ect of non-idealcomponent behaviour in the calculation of reaction ratesand chemical equilibrium constants. In addition, physicalproperties such as surface tension, di!usion coe$cients,viscosities, etc. for calculation of mass (and heat) transfercoe$cients and interfacial areas are required. Thesteady-state model equations most often are solved usingNewton's method or by homotopy-continuation. A reviewof early applications of NEQ models is available inTaylor & Krishna (1993, Chapter 14).5.2. NEQ modelling of RDBuilding an NEQ model of a reactive separation processis not as straightforward as it is for the EQ stagemodel in which we simply (or not so simply) add a termto account for reaction to the liquid-phase material balances.For a reactive separation process, we "rst need toknow whether the reaction is heterogeneous or homogeneous.For homogeneous systems the component molar balancefor the liquid-phase becomesx !f ! ! ¸x !¸ ν R ε "0, (31)where R is the rate of reaction m on stage j. ν representsthe stoichiometric coe$cient of component i in reaction m and ε represents the reaction volume on stage j.For homogeneous reactions this is given by the totalliquid hold-up on stage j and, in an NEQ model, isobtained directly from the column internals speci"cationsand appropriate hydrodynamic correlations.If it is su$ciently rapid, the reaction will also takeplace in the liquid "lm adjacent to the phase interface,and very fast reactions may occur only in the "lm. Ineither case the continuity equations for the "lm are requiredfor taking into account the e!ect of the reactionon the interphase mass transfer rates. The combined setof MS and continuity equations usually must be solvednumerically.The phase equilibrium equations for the interface mayneed to be modi"ed for the in#uence of additional specieson the thermodynamic properties at the interface.Amine-based gas treating again provides a case in pointwhere reactions in the liquid-phase create additionalspecies (including ions) that a!ect the interfacial equilibrium(Glasscock & Rochelle, 1989).For a heterogeneous reaction, there are two optionsfor the description of the reaction term. The simplestapproach is to treat the reaction pseudo-homogeneously,whereby catalyst di!usion and reaction is lumped into anoverall reaction term. For heterogeneous reactions thatare modelled in this way the liquid-phase material balanceis as given above and ε is given by the total amountof catalyst present on the stage under consideration. Inthis case, one only needs to specify catalyst mass andactivity. A more rigorous approach would involve the useof the dusty #uid model discussed above if the catalyst isporous, or reaction at the surface if not. In this case onealso needs information about the catalyst geometry (surfacearea, mean pore diameter, etc). In either case it isunnecessary to allow for reaction in the vapor}liquid "lmand the vapor}liquid transport equations are exactly asgiven above.5.3. NEQ modelsSawistowski and Pilavakis (1979, 1988) modelleda packed RD column for the esteri"cation of methanoland acetic acid to methyl acetate. They used an e!ective
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