real-world modeling of distillation ... - clarkson...

28 www.cepmagazine.org July 2003 CEP

Reactions and Separations

HEMICAL ENGINEERS HAVE BEEN solving their distillation problems using theequilibrium stage model since Sorel first used

the model for the distillation of alcohol over 100 yearsago. Seader (1) has provided an elegant history of thefirst century of equilibrium stage modeling. Real distil-lation and absorption processes, however, normally donot operate at equilibrium.

In recent years, it has be-come more common to simu-late distillation and absorptionas a mass-transfer-rate-basedoperation, using what have be-come known as nonequilibri-um, or rate-based, models. Thisarticle presents a brief outlineof nonequilibrium modelingand provides pointers to thegrowing literature in this field.

Modeling the old-fashioned way

To model a plant like theone shown in Figure 1, we de-compose the entire plant intosmaller units. In this case, theplant contains a distillation col-umn that is shown enlarged inthe center panel of the figure.There are many ways to model

an entire column, but the most common approach is todivide the column into a number of discrete “stages,”as depicted in the third panel. Thus, the question to beaddressed first is: How do we model these stages?

The equations that model equilibrium stages areknown as the MESH equations. MESH is an acronymreferring to the different types of equation that are usedin the model:

Previously, simulations based on nonequilibrium, or rate-based, models were considered impractical due to their complexity.However, with ever-increasing computingpower, these simulations are not only feasible,but in some circumstances they should beregarded as mandatory.

Real-WorldMMooddeelliinngg ooff DDiissttiillllaattiioonn

CC

Ross Taylor, Clarkson University and University of TwenteRajamani Krishna,University of AmsterdamHarry Kooijman,Shell Global Solutions International

The Stage Concept

A

A, B, C

C

B

A,C

A

B

V1 Reflux

V2 L1

Vj Lj-1

Vj+1 Lj

Stage 1

Stage j� Figure 1. Decomposition of a chemical plant into unit operations, and decomposition of the distillation into stages.

Taylor 6/13/03 10:00 AM Page 28

CEP July 2003 www.cepmagazine.org 29

• M stands for material balances• E stands for equilibrium relationships (to express the

assumption that the streams leaving the stage are in equi-librium with each other)

• S stands for summation equations (mole fractions areperverse quantities and won’t sum to unity unless youforce them to)

• H stands for heat or enthalpy balances (processes con-serve energy, as well as mass).

There are few mathematical models in any branch ofengineering that are as well-suited to computer solutionsand that have prompted the development of as many differ-ent algorithms as have the MESH equations. It would notbe too far from the truth to claim that it is equilibrium

stage calculations thatbrought computing intochemical engineering —and chemical engineersto computers (1).

The equilibrium stagemodel is so simple inconcept, so elegant fromthe mathematical view-point, the basis for somany commercial col-umn simulation pro-grams, and been used tosimulate and design somany real columns, thatit seems almost hereticalto mention that themodel is fundamentallyflawed. However, chemi-cal engineers have longbeen aware of the factthat the streams leavinga real tray or section of apacked column are not inequilibrium with eachother. In fact, the separa-tion actually achieveddepends on the rates ofmass transfer from thevapor to the liquid phas-es, and these rates de-pend on the extent towhich the vapor and liq-uid streams are not inequilibrium with eachother. The next questionis: What have we doneabout this fundamentalweakness?

The conventional wayaround this shortcoming

of what is referred to frequently as the rigorous model (withsome disregard for semantic accuracy), is to employ effi-ciencies. Several kinds of efficiency have been used in distil-lation column modeling and design, including the overall,Murphree, Hausen and vaporization efficiencies. The Mur-phree efficiency (2) is arguably the most widely employedby distillation engineers and is defined by:

where the overbars indicate the average mole fraction in theentering (E) and leaving (L) streams, as depicted in Figure 2.

For packed columns, we use something analogous to the

stage efficiency called the HETP (Height Equivalent to aTheoretical Plate). In practice, efficiencies and HETPsoften are estimated simply from past experience with simi-lar processes. However, for new processes, this approach isof no use whatsoever (and often fails even for old ones).Chemical engineers have, therefore, devoted a great deal ofeffort to devising methods for estimating efficiencies andHETPs (3, 4).

These different kinds of efficiencies all attempt to repre-sent the extent to which the real trays in a tray column (orthe entire column itself) depart from equilibrium. TheHETP is a number that is easy to use in column design.However, there are several drawbacks to employing effi-ciencies and HETPs in a computer simulation based on theequilibrium stage model:

• There is no consensus on which definition of efficien-cy is best (although many distillation experts will admit toa preference for Murphree-type efficiencies).

• The Murphree vapor-phase efficiency is not the same

Ey y

y yi MViL iE

iL iE, *=

−

−(1)

Nomenclaturec = number of components,

dimensionlessct = total concentration, mol/m3

d = driving force for mass transfer, m–1

Di,k = Maxwell-Stefan diffusivity, m2/sEi,MV = Murphree tray efficiency,

dimensionlessf = proportionality coefficient k = mass transfer coefficient, m/sK = vapor-liquid equilibrium

constant,Ni = molar flux of species i,

mol/m2-sP = pressure, Pap = partial pressure, PaR = gas constant, J/mol-Kt = time, sT = temperature, Ku = average velocityx = mole fraction, dimensionlessy = mole fraction, dimensionless

Greek lettersκ = mass transfer coefficient of

binary pair in multicomponent mixture, m/s

µ = Chemical potential, J/molη = distance along diffusion path,

dimensionless

Subscriptsi = component indexI = referring to interfacej = stage indexk = alternative component indexm = reaction indext = total

SuperscriptsF = referring to feed streamI = referring to interfaceL = referring to liquid phaseV = referring to vapor phase

L

L = LiquidV = VaporyE = Mole fraction in entering streamyL = Mole fraction in leaving stream

L V,yL

V,yE

� Figure 2. Idealized flow patterns on a distillation column tray.

Taylor 6/13/03 10:00 AM 9

as the liquid-phase efficiency on the same tray (the Hausenefficiency does not share this property).

• The generalized Hausen efficiencies (sometimesknown as Standart efficiencies (5)) are the most fundamen-tally sound, but are impractically complicated to calculateand are never used in practice.

• Vaporization efficiencies, favored by some in the pastbecause they are easy to include in computer programs, arenot often used today.

• Efficiencies vary from component to component, andfrom tray to tray, in a multicomponent mixture. Very rarelyis this fact taken into account in a simulation model thatuses efficiencies.

• Efficiencies vary from stage to stage in a tray column.HETPs are a function of height in a packed column. Thesebehaviors of efficiencies and HETPs are often not account-ed for in conventional column simulation software.

These weaknesses of the standard model have beenknown for a long time (6). Thus, our third question is: Howshould we deal with the shortcomings of the standard model?

Modeling in the real worldIn recent years, a new approach to the modeling of dis-

tillation and absorption processes has become available —the so-called nonequilibrium, or rate-based, models. Thesemodels treat these classical separation processes as themass-transfer-rate-governed processes that they really are.

The building blocks of the nonequilibrium model shownin Figure 3 are sometimes referred to as the MERSHQequations, where:

• M represents material balances• E represents energy balances• R represents mass- and heat-transfer rate equations• S represents summation equations• H represents hydraulic equations for pressure drop• Q represents equilibrium equations.Some of these equations are also used in building equi-

librium stage models; however, there are crucial differencesin the way in which the conservation and equilibrium equa-tions are used in the two types of model. In a nonequilibri-um model, separate balance equations are written for eachdistinct phase. Figure 3 shows that the material balance foreach phase includes terms to represent the mass transferredfrom one phase to the other. For the equation used in theequilibrium stage model, the sum of the phase balancesyields the material balance for the stage as a whole. The en-ergy balance is treated in a similar way — it is split intotwo parts, one for each phase, each part containing a termfor the rate of energy transfer across the phase interface.

Modeling distillation and related operations as the rate-based processes that they really are requires us to face up tothe challenge of modeling interfacial mass and energy trans-fer in tray and packed columns. This is something that we donot do in the conventional equilibrium stage model (al-though we face essentially the same problem if efficiencies

are to be estimated from a mathematical model (3, 4)). Themolar fluxes at a vapor liquid interface may be expressed as:

where ciV and ci

L are the molar densities of the superscript-ed phases, yi

V is the mole fraction in the bulk vapor phase,xi

L is the mole fraction in the bulk liquid phase, and xiI and

yiI are the mole fractions of species i at the phase interface.

kiV and ki

L are the mass-transfer coefficients for the vaporand liquid phases.

The inclusion in the model of the mass transport equa-tions introduces the mole fractions at the interface, some-thing we have not had to deal with so far, at least not explic-itly. It is common to assume that the mole fractions at the in-terface are in equilibrium with each other. We may, there-fore, use the very familiar equations from phase equilibriumthermodynamics to relate the interface mole fractions:

where the superscript I denotes the interface compositionsand Ki is the vapor-liquid equilibrium ratio for componenti. These K-values are evaluated at the interface composi-tions and temperature using the same thermodynamic mod-els used in conventional equilibrium stage simulations. Theinterface composition and temperature must, therefore, becomputed during a nonequilibrium column simulation. Inequilibrium stage calculations, the equilibrium equationsare used to relate the composition of the streams leavingthe stage and the K-values are evaluated at the compositionof the two exiting streams and the stage temperature (usu-ally assumed to be the same for both phases).

y K xiI

i iI= (4)

N c k x xiL

iL

iL

iI

iL= −( ) (3)

N c k y yiV

iV

iV

iV

iI= −( ) (2)



Vapor

VaporFilm

MassTransfer

EnergyTransfer

LiquidFilm

Y

VL,yL

LE,xE

LE = Liquid entering streamLL = Liquid leaving streamT = TemperatureVE = Vapor entering streamVL = Vapor leaving stream

VE,yE

LL,xL

TX

Inte

rfac

e

Liquid

xE = Liquid mole fraction in entering streamxL = Liquid mole fraction in leaving streamyE = Vapor mole fraction in entering streamyL = Vapor mole fraction in leaving stream

� Figure 3. Schematic diagram of a nonequilibrium stage.

Taylor 6/13/03 10:00 AM 0

Physical propertiesFigure 4 identifies the major physical property require-

ments. It is obvious that nonequilibrium models are moredemanding of physical property data than are equilibriumstage models (except when tray-efficiency or HETP andequipment-design calculations are carried out, but thoseare done after a simulation and are not needed to carry outthe column simulation). The only physical properties re-quired for an equilibrium stage simulation are those neededto calculate the K-values and enthalpies. Those same prop-erties are needed for nonequilibrium models as well.

Mass-transfer coefficients and interfacial areas mustbe computed from empirical correlations or theoreticalmodels. There are many correlations for mass-transfercoefficients in the literature (3, 4). These coefficientsdepend on the column design, as well as its method ofoperation.

We do not believe that the need for additional physi-cal properties should be a reason not to use a nonequilib-rium stage model. Estimation methods are available forthese properties, although they are typically much lessaccurate than methods for evaluating thermodynamicproperties (7). However, these properties are needed onlyin so far as they are required to estimate mass-transfercoefficients. In fact, the sensitivity of these coefficientsto any of these properties is not that large, and the factthat we do not always have accurate estimation methodsshould not act as a deterrent to their use. Rather, itshould serve as a spur to more research and to the devel-opment of better methods for transport property predic-tion and estimation in much the same way as the need forreliable phase equilibrium models has served as motiva-tion for the development of methods to predict thermo-dynamic properties.

Equipment designThe estimation of mass-transfer coefficients and inter-

facial areas from empirical correlations nearly always re-quires us to know something about the column design. Atthe very least, we need to know the diameter and type ofinternal (although usually we need to know more thanthat, since most empirical correlations for mass-transfercoefficients have some dependency on equipment designparameters, such as the weir height of trays). This need forcomplete equipment design details suggests that nonequi-librium models cannot be used in preliminary process de-sign (before any actual equipment design has been carriedout). However, this is not true. Column design methodsare available in the literature, as well as in most processsimulation programs. It is straightforward to simultane-ously solve equipment sizing calculations and stage-equi-librium calculations (8). This does not add significantly tothe difficulty of the calculations, and it allows nonequilib-rium models to be used at all stages of process simulation,including preliminary design, detailed plant design andsimulation, troubleshooting and retrofitting. In fact,nonequilibrium models can be particularly valuable introubleshooting and retrofitting, even to the point of help-ing identify what particular equipment design detail mightbe responsible for a column failing to do what it was de-signed to do.

Solving the model equationsThere has been so much work done on developing com-

putational methods for solving the equilibrium stage modelequations that we may essentially use the same approachesto solve the nonequilibrium model equations (8). The equa-tions required by the two kinds of model are summarizedin Figure 5. The fact that the nonequilibrium model in-


Physical Property Requirements

Activity CoefficientsVapor Pressures

Fugacity CoefficientsDensities

Enthalpies

DiffusivitiesViscosities

Surface TensionThermal Conductivities

Mass-Transfer CoefficientsHeat-Transfer Coefficients

Interfacial Areas

Activity CoefficientsVapor Pressures

Fugacity CoefficientsDensities

Enthalpies

Model Requirements: Equations

Phase Mass Balances

Phase Energy Balances

Equilibrium Eqs.

Summation Eqs.

Mass-Transfer inVapor Phase

Mass-Transfer inLiquid Phase

Energy Transfer

Mass Balances

Energy Balances

Equilibrium Eqs.

Summation Eqs.

� Figure 4. Physical propertyneeds of equilibrium (right)and nonequilibrium (left) models.

� Figure 5. Equations used inequilibrium (right) andnonequilibrium (left) models.

Taylor 6/13/03 10:00 AM Page 31

volves more equations is not a concern. In our experience,the equations of both models are about equally simple (ordifficult) to solve.

Numerical solution of the nonequilibrium model equa-tions provides the chemical engineer with all of the quan-tities normally associated with the conventional equilibri-um stage model — temperatures, flowrates, mole frac-tions, etc. Nonequilibrium-model calculations also pro-vide a great deal of additional information, such as physi-cal and transport property profiles, and equipment designand operating data.

Example 1: A simple debutanizer. Consider thesimple debutanizer shown in Figure 6. The flowrate andcomposition profiles do not differ to any significant ex-tent from the results that you would obtain with a con-ventional equilibrium stage model (although the numberof stages and feed stage location would be different).However, a nonequilibrium model can also provide con-siderable additional information, such as mass-transferrates and predicted efficiency profiles (Figure 7). The

nonequilibrium model, it must be re-membered, does not use efficiencies.McCabe-Thiele diagrams (9) can beconstructed from the results of anonequilibrium simulation (Figure 8),and are just as useful for understandingcolumn behavior as they are for binarydistillation. Note how the triangles donot touch the equilibrium line.

Example 2: A not-so-simple ab-sorber. Consider the simple packedcolumn depicted in Figure 9. The richammonia and air mixture enters at thebottom where the ammonia is ab-sorbed. The enthalpy of absorption isreleased, causing the temperature ofthe liquid to rise. As a result, waterevaporates. The mass transfer processin the gas therefore involves threespecies — ammonia, water and (essen-tially stagnant) air. Toward the top ofthe column, the gas encounters coldentering water. Therefore, water vaporcondenses near the top of the column,and we now have co-diffusion of am-monia and water through air. Weshould not ignore water vaporization atthe bottom and condensation at the topin the analysis. The resulting tempera-



� Figure 7. Murphree efficiency profiles (predict-ed) for the debutanizer shown in Figure 6.

28 Sieve Traysp = 5.5 to 6 bar

Properties:Peng Robinson

C3: 1.5i-C4: 56.5n-C4: 4.5i-C5: 1.9C6: 2.9C7: 4.9C8: 34.3C9: 3.9

C3: 1.5i-C4: 56.5n-C4: 4.5i-C5: 0.03

i-C5: 1.86C6: 2.9C7: 4.9C8: 34.3C9: 3.9

mol s-1

Downcomer Area10% 11%

Hole Area % of Active12% 10%

Weir Length1.9m 2.1m

0.6m

� Figure 6. Debutanizer adapted from Example 9.1 in Ref. 9. The simulation program created the tray design.

5 10 15 20 25

1.2

1

0.4

0.8

0.6

Mu

rph

ree

Eff

icie

ncy

Stage Number

PropaneIsobutane

n-ButaneIsopentane

n-Hexanen-Heptane

n-Octanen-Nonane

Taylor 6/13/03 10:00 AM Page 32

ture profiles along the column show a pronounced bulgenear the bottom (Figure 9).

The Maxwell-Stefan approachEquations 2 and 3 are included in all basic mass trans-

fer texts and chemical engineering handbooks, and aretaught to all chemical engineers in undergraduate chemi-cal engineering degree programs. Strictly speaking, theseequations are valid only for binarysystems and under conditions wherethe rates of mass transfer are low.Most industrial distillation and absorp-tion processes, however, involve morethan two different chemical species.

The most fundamentally sound wayto model mass transfer in multicompo-nent systems is to use the Maxwell-Ste-fan (MS) theory (11–13). In our opin-ion, the MS approach to mass transfershould be what is taught to students,but rarely is that done, even at the grad-uate level; most texts give little or noserious attention to the matter of masstransfer in systems with more than twocomponents (exceptions include thetexts by Seader and Henley (9) andBenitez (14)).

The MS equation for diffusion in abinary ideal gas mixture is:

where d1 is the driving force for diffusion and ui is the av-erage velocity of species i.

This expression may be derived using nothing morecomplicated than Newton’s second law — the sum of theforces acting on the molecules of a particular species is di-rectly proportional to the rate of change of momentum(Ref. 11 provides a more complete derivation). The rate ofchange of momentum between different species is propor-tional to the concentrations (mole fractions) of the differentspecies and to their relative velocity. In Eq. 5, f12 is the co-efficient of proportionality and is related to a friction fac-tor. Eq. 5 is more often written in the form:

where D12 is the MS diffusion coefficient.The MS equations are readily extended to multicompo-

nent systems simply by adding similar terms on the right-hand side to account for momentum exchanged betweeneach pair of differing types of molecules. For a ternarymixture, for example, we would have two terms on theright, one of momentum exchange between molecules oftypes 1 and 2, and a second term for momentum transferbetween molecules of types 1 and 3:

with the equations for species 2 and 3 obtained by rotatingthe subscripts.

dx x u u

D

x x u u

D11 2 1 2

12

1 3 1 3

13

=−

−−( ) ( )

(7)

dx x u u

D11 2 1 2

12

=−( )

(6)

d f x x u u1 12 1 2 1 2= −( ) (5)


� Figure 8. McCabe-Thiele diagram for the debutanizer shown in Figure 6.

0

1

0.8

0.6

0.4

0.2

0.2 0.4 0.6 0.8 10

YC4

YC4+ YC5

XC4

XC4+ XC5

� Figure 9. Ammonia absorber adapted from Example 8.8 in (10).

Water

Air Ammonia

Gas Temperature

LiquidTemperature

20 25 30 35

Hei

gh

t

Temperature, ˚C

Condensation of WaterAbsorption of Ammonia

Evaporation of WaterAbsorption of Ammonia

Taylor 6/13/03 10:00 AM Page 33

The generalization of this expression to mixtures withany number of different species is:

which is more familiar to us in the form:

where we have replaced the velocities with the molar flux-es Ni = ciui.

For an ideal gas mixture, the driving force is the partialpressure gradient:

Solving the MS equations might involve the computationof various matrices and functions thereof (11). In practice,we most often employ a simple film model for mass transferwith a simple difference approximation to the MS equations:

where xi is the average mole fraction over the film. The MSmass-transfer coefficients κij can be estimated from exist-ing correlations. For a nonideal fluid, the driving force isrelated to the chemical potential gradient:

The difference approximation of this expression issomewhat more involved, since we have to include thederivative of the activity (or fugacity) coefficient (13).

Example 3: The need for rigorous Maxwell-Ste-fan-based nonequilibrium models. The differencesin column composition profiles predicted by a rigorousnonequilibrium model that incorporates the MS equa-tions may differ significantly from those predicted byan equilibrium stage model. Consider the experimentalwork of Springer et al. (15) on the distillation of water(1), ethanol (2) and acetone (3) carried out in a 10-traycolumn operated at total reflux. The residue curve mapfor this system is shown in Figure 10a. This systemshows a binary minimum boiling azeotrope betweenwater and ethanol; an almost-straight distillationboundary connects the azeotrope with pure acetone.

A measured composition profile, carried out in the re-gion to the left of the distillation boundary, is shown inFigure 10b. Simulations of the column, starting with thevapor composition at the column top, are also shown. It isevident that the nonequilibrium model is able to followthe experimentally observed column trajectories muchbetter than the equilibrium model. The differences in thecolumn composition trajectories are due to differences inthe component Murphree efficiencies (Figure 10c).

Differences in component efficiencies could have asignificant impact on a column design that aims for a spe-cific purity at either ends of the column. For example, forthe water (1), ethanol (2) and acetone (3) system operat-

dx

RT

d

dzii i=

µ(12)

∆xx N x N

cii k k i

t ikk

c= −

−∑= κ1

(11)

dP

dp

dz

dx

dzi i

11

= = (10)

dx N x N

c Di k k i

t ikk

c

11

= −−

∑=

(9)

dx x u u

Di k i k

ikk

c

11

= −−

∑=

( )(8)



� Figure 10. Distillation of water (1), ethanol (2) and acetone (3) in a bubble cap tray column: (a) residue curve map; (b) experimental composition trajecto-ry for Run 6, compared with the nonequilibrium and equilibrium simulations; and (c) component Murphree efficiencies for Run 6 (15).

WaterEthanolAcetone

0.0

0.6

0.8

1.0

0.4

0.0

0.2

0.2 0.4

Residue Curve LinesAzeotropeDistillation Boundary

0.6 0.8 1.0

Eth

ano

l Co

mp

osi

tio

n

Water Composition

2 4 6 8 10

0.5

1.0

1.5

2.0

0.0

Co

mp

on

ent

Mu

rph

ree

Eff

icie

ncy

, Ei

Stage Number

a c

0.02

0.6

0.8

1.0

0.40.060.04

Eth

ano

l Co

mp

osi

tio

n

Water Composition

b

Nonequilibrium ModelEquilibrium ModelExperimental Data

-

Taylor 6/13/03 10:00 AM Page 34

ing in the region to the left of the distillation boundary,let us demand a purity of 96% ethanol at the top of thecolumn. For a specified feed composition and refluxratio, the column composition trajectories for thenonequilibrium model and the equilibrium model (assum-ing 60% efficiencies for all components) are presented inFigure 11. The nonequilibrium model suggests that 39

stages are needed to reach the specified 96% ethanol pu-rity at the top, whereas the equilibrium model indicatesthat only 25 stages are needed. In this case, the nonequi-librium model takes “the scenic route” to reach the de-sired top purity. Ignoring the differences in component ef-ficiencies may lead to severe underdesign.

Columns operating close to the distillation boundarymay experience much more exotic differences in the col-umn composition trajectories predicted by the nonequi-librium and equilibrium models. For operation with thesame water, ethanol and acetone system, Figure 12ashows that the experiments cross the straight-line distil-lation boundary (15), something that is forbidden by theequilibrium model (16). The nonequilibrium model isable to retrace this boundary-crossing trajectory, whereasthe equilibrium model remains on one side of the distil-lation boundary. The nonequilibrium model predicts thatthe column gets progressively richer in water as we pro-ceed down the column to the reboiler, whereas the equi-librium model anticipates that the column gets enrichedin ethanol as the reboiler is approached. The root causeof this behavior lies with the differences in the efficien-cies of the individual species (Figure 12b); the compo-

nent efficiency of ethanol varies sig-nificantly from tray to tray. Compar-ing the component efficiency valuesin Figures 10c and 12b reveals thateven though the mass transfer param-eters used in the nonequilibriummodel are identical for these two runs,the calculated component efficiencyvalues bear no resemblance to one an-other. This underlines the difficulty oftrying to emulate the performance ofthe nonequilibrium model by fudgingcomponent efficiency values. There isno way that this can be achieved.

Other applicationsThe principles outlined above are

applicable to a wide range of relatedprocesses. Below, we very brieflyconsider some of these applications.

Three-phase distillation. Three-phase distillation remains relatively poorly understoodcompared to conventional distillation operations involv-ing just a single liquid phase. Simulation methods cur-rently in use for three-phase systems employ the equi-librium stage model (16). It is important to be able tocorrectly predict the location of the stages where a sec-ond liquid phase can form (to determine the appropriatelocation for a sidestream decanter, for example). Thelimited experimental data available suggest that effi-ciencies can be low and highly variable. Clearly, amodel based on the assumption of equilibrium on every


� Figure 11. Comparison of nonequilibrium and equilibrium models fordistillation of water (1), ethanol (2) and acetone (3) in a bubble cap tray col-umn with the objective of reaching 96% ethanol purity at the top.

0.00

1.0

0.8

0.6

0.4

0.2

0.00.04 0.08

Eth

ano

l Co

mp

osi

tio

n

Water Composition

Nonequilibrium ModelEquilibrium Model +60% Efficiency

NEQ = 39 StagesEQ, 60% Efficiency = 25 Stages

� Figure 12. Distillation of water (1), ethanol (2) and acetone (3) in a bubblecap tray column: (a) experimental composition trajectory for Run 26, com-pared with the nonequilibrium and equilibrium simulations; and (b) Compo-nent Murphree efficiencies for Run 26 (15).

0.0

0.6

0.8

1.0

0.4

0.20.20.1

Eth

ano

l Co

mp

osi

tio

n

Water Composition

2 4 6 8 10

0.6

0.8

1.4

1.2

1.0

0.4

0.2

0.0

Co

mp

on

ent

Mu

rph

ree

Eff

icie

ncy

, Ei

Stage Number

Ethanol-WaterBinary Azeotrope

Nonequilibrium ModelExperimental DataDistillation BoundariesEquilibrium Model

WaterEthanolAcetone

a b

Taylor 6/13/03 10:00 AM Page 35

stage cannot predict column performance. Springer andothers (17) stress the limitations of simulation modelsassuming equal Murphree efficiencies for all compo-nents in the mixture.

It is straightforward in principle to extend the ideasthat underlie nonequilibrium models to systems withmore than two phases, as first shown by Lao and Tay-lor (18). A complete nonequilibrium model for the sys-tem depicted in Figure 13 contains three phase bal-ances, each of which contains terms for mass transferto or from both of the other two phases. In addition,the model contains up to six sets of the MS equations,two for each phase boundary (vapor–liquid I,vapor–liquid II, and liquid I–liquid II). Three sets ofequilibrium equations, one for each possible interface,complete the model. In practice, it is quite likely thatthe vapor phase and a dispersed liquid phase see only acontinuous liquid phase, thereby considerably simpli-fying the model (17).

Example 4. Heterogeneous azeotropic system.Sometimes the curvature of the distillation boundary issuch that its crossing by the equilibrium stage model isallowable (16). This is illustrated in Figure 14 for thewater (1), cyclohexane (2) and ethanol (3) system. For acolumn operating at total reflux with the top composi-tion corresponding to the heterogenous ternaryazeotrope, the equilibrium model has no difficultycrossing the curved distillation boundary from the con-vex side, moving in the direction of high water compo-sitions and proceeding down the column. However, theexperimental data of Springer et al. (17) show that theboundary is not crossed in practice and the column com-position trajectories are anticipated very well by anonequilibrium model.

Reactive distillation. The design and operation issues

for reactive distillation processes are considerably morecomplex than those of either conventional reactors orconventional distillation columns. The introduction ofan in situ separation function within the reaction zoneleads to complex interactions between vapor-liquidequilibrium, vapor-liquid mass transfer, intra-catalystdiffusion (for heterogeneously catalyzed processes) andchemical kinetics. For such systems, the chemical reac-tion influences the efficiencies to such an extent that theconcept loses its meaning (19).

Building a nonequilibrium model of a reactive sepa-ration process is not as straightforward as building anequilibrium stage model, in which we simply add aterm to account for reaction to the liquid-phase materialbalances. It must be recognized that no single nonequi-librium model can deal with all possible situations.Separate models are needed depending on whether thereaction takes place within only the liquid phase or if asolid phase is present to catalyze the reaction. Refer toRefs. 16, 19 and 20 for further discussion.

Gas absorption. Efficiencies in gas absorption tendto be much lower than in distillation, sometimes as lowas 5%. In addition, many important gas absorption pro-cesses involve chemical reactions. It does not seem to



� Figure 13. Schematic representation of a three-phase nonequilibrium stage.

Vapor Liquid l

Liquid llTransfer

TransferTransfer

� Figure 14. Distillation of water (1), cyclohexane (2), ethanol (3) and ace-tone (3) in a bubble cap tray column: experimental composition trajectory,compared with the nonequilibrium and equilibrium simulations (17).

0.04

0.6

0.4

0.2

0.00.12 0.160.08

Cyc

loh

exan

e C

om

po

siti

on

Water Composition

Liquid-LiquidPhase

Splitting

HeterogenousTernary Azeotrope

Ethanol-WaterBinary AzeotropeNonequilibrium Model

Experimental DataDistillation BoundariesEquilibrium Model

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make a great deal of sense to employ an equilibriumstage model for systems so far removed from equilibri-um. In fact, although equilibrium stage models forsuch systems are used, it has long been more commonto use mass-transfer-rate-based models to design gasabsorption processes (21). Nonequilibrium modelsapply more or less unchanged in principle to gas ab-sorption (with or without reaction). The only differ-ences between the models are the inclusion of differentsub-models for the reaction kinetics and thermodynam-ic properties. Many absorption processes involve dilutemixtures, and the rate relationships in Eqs. 2 and 3 suf-fice (the latter modified by the inclusion of an en-hancement factor to account for any chemical reaction

in the liquid phase). Models of this sorthave been used with some success inthe modeling of amine-based gas treat-ing processes (22).

Distillation column dynamics. Oneof the key points of this article is thatnonequilibrium models should be usedwhen efficiencies are unknown, cannotbe reliably predicted, and are lowand/or highly variable. Efficiencies inany process depend strongly on theproperties of the mixture, whether ornot chemical reactions are involved,and (last, but by no means least in im-portance) the type of column employedand the way in which it is operated. If acolumn is not at steady state, then effi-ciencies vary with time as a result ofchanges to flowrates and composition.Thus, equilibrium stage models with ef-ficiencies should not be used to modelthe dynamic behavior of distillation andabsorption columns. Nonequilibriummodels for column dynamics are de-scribed in Refs. 23–25.

Nonequilibrium cell model. An issuethat is not adequately addressed by mostmodels is that of vapor and liquid flowpatterns on distillation trays or maldis-tribution in packed columns. Since reac-tion rates and chemical equilibrium con-stants depend on the local concentra-tions and temperature, they may varyalong the flow path of liquid on a tray,or from side to side in a packed column.For such systems, the residence timedistribution could be very important.

To deal with this shortcoming of ear-lier models, nonequilibrium cell modelshave been developed (26–28). The dis-

tinguishing feature of this model is that the stages aredivided into a number of contacting cells (Figure 15).These cells describe just a small section of the tray orpacking, and by choosing an appropriate set of cell con-nections, one can very easily study the influence of flowpatterns on the distillation process.

Flow patterns on distillation trays are modeled bychoosing an appropriate number of cells in each flowdirection. A column of cells can model plug flow in thevapor phase, and multiple columns of cells can modelplug flow in the liquid phase as depicted in Figure 15.Backmixing may also be taken into account by usingan appropriate number of cells. Flow patterns inpacked columns are evaluated by means of a cell flowmodel (27).


VaporLiquidCell

TraySection

PackedSection

� Figure 15. The nonequilibrium cell model.

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Available softwareAspenTech developed RateFrac, in collaboration with

Koch Engineering, Inc. This implementation is based large-ly on the nonequilibrium model described in the originalpapers by Krishnamurthy and Taylor (29, 30), with the im-portant additional capability of being able to handle sys-tems with chemical reactions. The influence of reaction onmass transfer is modeled by means of enhancement factors.RateFrac has one mass-transfer coefficient model for eachtype of column internal, but it has the facilities to add usermodels for the calculation of transfer coefficients, pressure

drop and interfacial area. RateFrac can use any of the ther-modynamic packages that exist within AspenPlus, and canmodel columns with sidestreams, interstage heaters/coolersand pumparounds. Complex specifications can designatedfor product purity or internal streams. RateFrac is especiallyuseful for modeling columns with chemical reactions thatinfluence the separation. Illustrations of the use of RateFracare described in Seader and Henley (9). For more informa-tion, visit www.aspentech.com/includes/product.cfm?Indus-tryID=0&ProductID=110

CHEMCAD from Chemstations, Inc. (www.chemsta-



Literature Cited1. Seader, J. D., “The B. C. (before computers) and A. D. of Equilibri-

um-Stage Operations,” Chem. Eng. Educ., 19 (2), pp. 88–103(Spring 1985).

2. Murphree, E. V., “Rectifying Column Calculations with ParticularReference to n-component Mixtures,” Ind. Eng. Chem., 17, pp.747–750 (1925).

3. Kister, H. Z., “Distillation Design,” McGraw-Hill, New York (1992).4. Lockett, M. J., “Distillation Tray Fundamentals,” Cambridge Uni-

versity Press, Cambridge, MA (1986).5. Standart, G., “Distillation. V. Generalized Definition of Theoretical

Plate or Stage of Contacting Equipment,” Chem. Eng. Sci., 20, pp.611–622 (1965).

6. Seader, J. D., “The Rate-Based Approach for Modeling Staged Sepa-rations,” Chem. Eng. Prog., 85, pp. 41–49 (1989).

7. Poling, B. E., et al., “The Properties of Gases and Liquids,” 5th Edi-tion, McGraw-Hill, New York (2001).

8. Taylor, R., et al., “A 2nd Generation Nonequilibrium Model forComputer-Simulation of Multicomponent Separation Processes,”Comput. Chem. Eng., 18, pp. 205–217 (1994).

9. Seader, J. D., and E. J. Henley, “Separation Process Principles,”John Wiley, New York, NY (1998).

10. Treybal, R. E., “Mass-Transfer Operations,” 3rd Edition, McGraw-Hill, New York, NY (1980).

11. Taylor, R., and R. Krishna, “Multicomponent Mass Transfer,” JohnWiley, New York, NY (1993).

12. Krishna, R., and J. A. Wesselingh, “The Maxwell-Stefan Approachto Mass Transfer,” Chem. Eng. Sci., 52, pp. 861–911 (1997).

13. Wesselingh, J. A., and R. Krishna, “Mass Transfer in Multicompo-nent Mixtures,” Delft University Press, Delft (2000).

14. Benitez, J., “Principles and Modern Applications of Mass TransferOperations,” John Wiley, New York, NY (2002).

15. Springer, P. A. M., et al., “Crossing of the Distillation Boundary inHomogeneous Azeotropic Distillation: Influence of Interphase MassTransfer,” Ind. Eng. Chem. Res., 41, pp. 1621–1631 (2002).

16. Doherty, M. F., and M. F. Malone, “Conceptual Design of Distilla-tion Systems,” McGraw-Hill, New York, NY (2001).

17. Springer, P. A. M., et al., “Composition Trajectories for Heteroge-neous Azeotropic Distillation in a Bubble-cap Tray Column: Influ-ence of Mass Transfer,” Chem. Eng. Res. Des., 81, pp. 413–426(2003).

18. Lao, M. Z., and R. Taylor, “Modeling Mass-Transfer in 3-PhaseDistillation,” Ind. Eng. Chem. Res., 33, pp. 2637–2650 (1994).

19. Taylor, R., and R. Krishna, “Modeling Reactive Distillation,”Chem. Eng. Sci., 55, pp. 5183–5229 (2000).

20. Sundmacher, K., and A. Kienle, “Reactive Distillation. Status andFuture Directions,” Wiley-VCH Verlag, Weinheim, Germany (2003).

21. Cornelisse, R., et al., “Numerical Calculation of SimultaneousMass Transfer of Two Gases Accompanied by Complex ReversibleReactions,” Chem. Eng. Sci., 35, pp. 1245–1260 (1980).

22. Pacheco, M. A., and G. T. Rochelle, “Rate-Based Modeling of Re-active Absorption of CO2 and H2S into AqueousMethyldiethanolamine,” Ind. Eng. Chem. Res., 37, pp. 4107–4117(1998).

23. Kooijman, H. A., and R. Taylor, “A Nonequilibrium Model forDynamic Simulation of Tray Distillation-Columns,” AIChE Journal,41, pp. 1852–1863 (1995).

24. Baur, R., et al., “Dynamic Behaviour of Reactive DistillationColumns Described by a Nonequilibrium Stage Model,” Chem. Eng.Sci., 56, pp. 2085–2102 (2001).

25. Gunaseelan, P., and P. C. Wankat, “Transient Pressure and FlowPredictions for Concentrated Packed Absorbers Using a DynamicNonequilibrium Model,” Ind. Eng. Chem. Res., 41, pp. 5775–5788(2002).

26. Higler, A., et al., “Nonequilibrium Cell Model for Multicomponent(Reactive) Separation Processes,” AIChE Journal, 45, pp.2357–2370 (1999).

27. Higler, A , et al., “Nonequilibrium Cell Model for Packed Distilla-tion Columns — The Influence of Maldistribution,” Ind. Eng. Chem.Res., 38, pp. 3988–3999 (1999).

28. Baur, R., et al., “Dynamic Behaviour of Reactive Distillation TrayColumns Described with a Nonequilibrium Cell Model,” Chem.Eng. Sci., 56, pp. 1721–1729 (2001).

29. Krishnamurthy, R., and R. Taylor, “A Nonequilibrium StageModel of Multicomponent Separation Processes. Part I: Model De-scription and Method of Solution,” AIChE Journal, 31, pp. 449–456(1985).

30. Krishnamurthy, R., and R. Taylor, “A Nonequilibrium StageModel of Multicomponent Separation Processes. Part III: The Influ-ence of Unequal Component Efficiencies in Process Design Prob-lems,” AIChE Journal, 31, pp. 1973–1985 (1985).

31. Kooijman, H. A., and R. Taylor, “The ChemSep Book,” Books onDemand, Norderstedt, Germany (2001).

32. Lewis, W. K., and K. C. Chang, “Distillation. III. The Mechanismof Rectification,” Trans. Am. Inst. Chem. Eng., 21, pp. 127–138(1928).

33. Krishna, R., “A Unified Theory of Separation Processes Based on Irre-versible Thermodynamics,” Chem. Eng. Commun., 59, pp.33-64 (1987).

Further ReadingFor further reading, visit www.chemsep.org/publications

Taylor 6/13/03 10:00 AM Page 38

tions.net) contains a nonequilibrium model for both steady-state and dynamic simulation.

ChemSep (31) incorporates some of the most recent de-velopments in nonequilibrium modeling. Many correla-tions for the mass-transfer coefficients, interfacial area andflow models are built into ChemSep. It also contains a vari-ety of thermodynamic and physical property models.ChemSep can also provide a detailed design of the equip-ment selected for the simulation. This allows the programto simulate columns for preliminary design purposes. It hasa limited component library but allows the user to addcomponents with a databank manager. ChemSep is avail-able through CACHE (www.cache.org) for educational useonly. Applications of ChemSep are discussed in Refs. 9,14, 31. For more information, visit www.chemsep.org.

Many other models have been implemented primarilyfor research purposes and are not available to others.

ConclusionWithin the last two decades, a new way of simulating

multicomponent distillation operations has come of age.These nonequilibrium, or rate-based, models abandon theidea that the vapor and liquid streams in a distillation col-umn ever are in equilibrium with each other. The idea ofmodeling distillation as a mass-transfer-rate-based opera-tion is hardly new. Equations 2 and 3 (albeit in differentunits) actually appear in the classic paper by E.V. Mur-phree (2) that introduced us to efficiencies. Murphree wentso far as to say: “the use of the general [mass-transfer]equation in rectifying column problems would cause thecalculations to become very much involved, and it is there-fore not considered feasible for practical purposes.” Nowa-days, such calculations not only are feasible, there are cir-cumstances where they should be regarded as mandatory.

Of course, models based on equilibrium stage conceptswill not be abandoned, nor is there any need for us to doso. For design of new columns in which the column con-figuration is not fixed, it is best to start with the equilibri-um model to determine the configuration, optimum reflux,etc. (16). The final design should be checked against thenonequilibrium model because, as we have seen, it is pos-sible for the predictions of the nonequilibrium model todiffer considerably from those of the equilibrium model.

Nonequilibrium models are of great value in simulatingexisting columns. No longer is it necessary to guess thenumber of equilibrium stages, the location of the feed andany intermediate product streams, and the individual com-ponent efficiencies in order to try and model a column thatno longer is performing as intended.

Reactive distillation is an emerging application that hasintroduced additional complications. Here it is not uncom-mon to assume equilibrium with regard to mass transfer,but allow for finite reaction rates. This is fine for conceptu-al design. But for equipment sizing, the problem of deter-mining column heights remains. For reactive distillation,

HETPs and efficiencies have no physical meaning, as theseare also influenced by reaction.

Rigorous nonequilibrium models require the use ofthe MS equations to properly describe mass transfer inmulticomponent systems. These equations have, in fact,been with us for much longer than has the equilibriumstage model (see Ref. 11 for original citations). The ap-plication of the MS equations to modeling mass transferin distillation is also not all that recent. Lewis and Chang(32), in a remarkably prescient paper that appears tohave been largely ignored, used the MS equations to in-vestigate the mechanism of rectification. They wrote:“engineers generally are unfamiliar with them” — a situ-ation that has persisted until relatively recent times. Notonly do the MS equations allow us to model mass trans-fer in conventional operations like distillation, absorp-tion and extraction, they also describe transport in manyless common separation processes, such as membraneprocesses. Indeed, the MS formulation of mass transferprovides a rational basis for unifying the treatment ofseparation processes (33). CEP


ROSS TAYLOR is the Kodak Distinguished Professor of Chemical Engineeringat Clarkson Univ. in Potsdam, New York ([email protected]), where hehas been since 1980. He currently serves as chair of the Dept. of ChemicalEngineering. He received his PhD degree from the Univ. of ManchesterInstitute of Science and Technology in England. His research interests arein the areas of separation process modeling, multicomponent masstransfer, thermodynamics, and developing applications of computeralgebra to process engineering (and cartography). He is a coauthor (withKrishna) of the textbook “Multicomponent Mass Transfer” (Wiley, 1993).He also holds a joint appointment as Professor of Reactive Separations inthe Dept. of Chemical Technology at the Univ. of Twente in TheNetherlands, and is a trustee of The CACHE Corp.

R. KRISHNA is a professor at the Univ. of Amsterdam ([email protected]).He graduated in chemical engineering from the Univ. of Bombay and wasawarded a PhD in 1975 from the Univ. of Manchester. He then joined theRoyal Dutch Shell Laboratory in Amsterdam, where he was engaged inresearch, development and design of separation and reaction equipment.After nine years of industrial experience, he returned to India to take overthe Directorship of the Indian Institute of Petroleum. Since 1990, heoccupies the position of Professor of Chemical Reactor Engineering at theUniv. of Amsterdam. His current research interests range from molecularmodeling, bubble and particle dynamics, and reactor scale-up to processsynthesis. Krishna has co-authored three textbooks. His researchcontributions have won him the Conrad Premie of the Royal DutchInstitution of Engineers in 1981, and the Akzo-Nobel prize in 1997.

HARRY KOOIJMAN is a research distillation specialist at the Amsterdamlaboratory of Shell Global Solutions International BV, The Netherlands([email protected]). He graduated from Delft Univ. of Technologyand received his PhD in 1995 from Clarkson Univ. He joined the BOC Groupin 1996 as a senior research engineer, where he was involved with thedevelopment of structured packing for cryogenic distillation. In 1999, hemoved to Germany where he worked at science+computing as a consultantin high-performance computing. He joined Shell Global Solutions in 2002,where he focuses on the development of distillation tray technology andseparation equipment.

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