Thermodynamic Calculations in the Modeling of Multiphase Processes and Reactors

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Thermodynamic Calculations in the Modeling of Multiphase

Processes and Reactors

Anna Yermakova* and Vladimir I . AnikeevBoreskov I nstitute of Catalysis, Pr. Lavrentieva, 5, Novosibir sk 630090, Russia

A two-parameter cubic Soave-Redlich-K wong (SRK ) equation of state with modified binaryinteraction coefficients has been used for phase equilibrium calculations of multicomponent

systemsunder supercritical and near-supercritical conditions. Example calculations of the phasebehavior of multicomponent reaction mixtures are presented and used to illustrate the increasedaccuracy of theSRK equation of state with thenew binary interaction coefficients. The homotopycontinuation method was adapted for the solution of the model equations by using temperatureand pressure as continuation parameters. The effective models and algorithms thus createdareused for phase state calculations, localization of thecritical point of multicomponent mixtures,studies of phaseproperties near the critical point, and determination of parameter regions thatare of practical importance for process performance under supercritical conditions. The problemsof existence and uniqueness of a solution are studied. A new method for solving equations forthe critical phaseis suggested. Many examples of the calculation of phasediagrams of mixtureswith a supercritical solvent are presented and the results compared with the reference data. Thermodynamic and macrokinetic models of the Fischer- Tropsch (FT) reaction are developedand used for the calculation of reactor performance.

Contents1. Introduction 1453

2. Base Model for the Description of P-V- T Properties of Multicomponent Mixtures

1454

2.1. Modified Soave-Redlich-KwongEquation of State

1455

2.2. Determination of the BinaryInteraction Coefficients UsingExperimental Data

1455

2.3. Examples and ComparativeAnalysis

1456

3. Method for Phase Diagrams

Calculations

1456

3.1. Mathematical Model for aTwo-Phase Equilibrium

1456

3.2. Equation for Binodal Line as aLimiting Case of the Two-PhaseEquilibrium

1457

3.3. Homotopy ContinuationAlgorithm for Solving the PhaseBoundary Calculation Problem

1458

3.4. Problems of Existence andUniqueness of a Solution

1459

4. Critical State Criterion of aMulticomponent Mixture

1460

4.1. Calculation of a Spinodal L ine 1461

4.2. Localization of a Critical Point 1461

5. Phase Behavior of MulticomponentMixtures

1461

5.1. Critical Point, Calculated andExperimental Data

1461

5.2. Phase Diagrams and thePhase-Transition Features

1462

5.2.1. Model Mixture with aSupercritical Solvent

1462

5.2.2. “Gas-Gas” Equilibrium 1464

6. Fischer-Tropsch Process in aSlurry Reactor

1465

6.1. Kinetic Model of theFischer-Tropsch Reaction

1465

6.1.1. Primary Analysis of Experimental Data

1466

6.1.2. Determination of r as aFunction of the ReactionMixture Composition

1466

6.1.3. Chemical R eactions Used as aBasis for the Kinetic Model

1466

6.1.4. Mathematical Model of theKinetic Experiment

1467

6.1.5. Selection of the RateEquations

1467

6.1.6. Results of the Kinetic ModelIdentification

1468

6.2. Mathematical Model of theReactor Unit

1468

6.2.1. Main Prerequisites of theModel

1468

6.2.2. Algorithm for ProcessScheme Calculations

1470

6.2.3. Calculation Results 1470

7. Conclusion 1471

Acknowledgment 1472Literature Cited 1472

1. Introduction

Three-phasecatalytic reactors havebeen used exten-sively in many chemical, petrochemical, and biochemicalprocess engineeri ng applications. Liquid-phase oxida-

* To whom correspondence should beaddressed. Fax: 7 38323974 47. E-mail: [email protected].

1453I nd. E ng. Chem. Res. 2000, 39, 1453-1472

10.1021/ie9905761 CCC: $19.00 ©2000 American Chemical SocietyPublished on Web 04/13/2000



tion, hydrogenation, alkylation, and so forth are a fewamong theseprocesses. Such multiphasereactors (slurrybubble column) have a significant application in theFischer- Tropsch synthesis of syn gases on a suspendedcatalyst.

Many monographs describing the various aspects of multiphase reactors (heat- and mass-transfer charac-teristics, hydrodynamic behavior, and slurry mixing)appeared in recent years. The development and opti-mization of multiphase processes requires the knowl-edge of the phasebehavior, theamount and compositionof phases, thesolubility of thecomponents, andenthalpyof phase conversions at the reaction conditions subjectto nonideality of reaction mixtures. I n this connectionthe thermodynamic calculations are necessary for math-ematical modeling. Al ong with the experimental meth-ods, mathematical modeling is of great help in obtainingknowledge about processes. Moreover, mathematicalmodeling is preferred in some cases because experimen-tal studies of multiphase multicomponent chemicalreactions are very compli cated.

To model thephase states of multicomponent reactionmixtures, thermodynamic models providing realisticdescriptions of P-V- T properties over a wide range of

temperatures and pressures have been actively used inrecent years.

A large family of thermodynamic models, suggestedfor calculations of phase equilibrium, uses cubic equa-tions of state which include the binary interactioncoefficients. These equations make a nice instrumentfor calculations of thermodynamic properties and phaseequilibrium of multicomponent nonideal mixtures.

Thermodynamic models are especially important forthe study and modeling of multiphase processes occur-ring at or near supercritical states of the reactionmixture.

Such processes usually proceed in the high-pressureand -temperature region, either in thepresence of some

inert supercritical solvent or during the time when oneof the reaction products acts as a supercritical solvent.Peculiar properti es of the “supercritical medium” (lowviscosity, high mutual solubility of components, highdiffusion, etc.) are usually exhibited in the regionranging from the critical temperature and pressure of thesupercritical solvent tothoseof thereaction mixture.Because of its extraordinary properties, this region(hereafter named as the “supercritical” region) is verypromising for the performance of homogeneous andheterogeneous catalysis. One can control the reactionsby choosing a solvent and i ts concentration.

In thesupercritical region (except themixturecriti calpoint), the two-phase state still holds. However, thisstate is not stable and is extremely sensitiveto changesin the mixture composition, temperature, and pressurethat may occur in the course of the chemical reaction.Densities of fluid phases in this region are comparableand range from thedensity of gastothat of liquid. More,the solvent concentration in both phases is approxi-mately similar.

The high sensitivity of the medium properties tochanges in parameters, and the probable appearanceor disappearance of phases in the course of the chemicalreaction, indicate that thermodynamic calculationsshould play a more important role in themathematicalmodeling of multiphase processes, especially undernear-to-critical conditions. Therefore, software and al-gori thms for calculations of phase behavior of complex

mixtures, performed on the basis of rigorous thermo-dynamic models, must be included in programs simulat-ing the operation of reactors and technologies.

In the present paper, we discuss the typical phasebehavior of multicomponent reaction mixtures andproblems associated with calculations of multicompo-nent reactors operating under phase-transition condi-tions. The goal is to develop simple methods and robustalgorithms for calculations of phase states (yield andcomposition and thermodynamic stability of equilibriumphases) in complex mixtures as a part of themultiphasereactor model software. Of special interest are theproblems related to the localization of the critical pointof multicomponent mixtures, studies of phasepropertiesnear the critical point, and determination of parameter(temperature and pressure) regions that are of practicalimportance for process performance under supercriticalconditions. The use of thermodynamic models for cal-culations of reactor performance are exemplified. Themodels discussed in this article are based on thetheoretical fundamentals of multiphasethermodynam-ics.

For a steady-state or quasi-steady-state equilibriumprocess, the thermodynamic model reduces to a system

of nonlinear algebraic equations, whoseparameters areusually temperature and pressure in applied problems. These parameters can be measured and controlledduring experimental tests of themodel. The steady-statemodel of the perfectly mixed reactor also reduces to asystem of nonlinear equations with parameters. T hewell-known homotopy continuation method is adaptedfor the soluti on of model equations by using the tem-perature and pressure as continuation parameters. Thecreated effective algorithms are part of the reactor andthe flow sheet modeling program package.

Because the ki netic model of a chemical reaction isan obligatory element of the reactor mathematicalmodel, we have considered the problems related toconstruction of the“global”kinetic model. On one hand,

this model should show the detailed mechanism of thereaction. On the other hand, one needs “global” experi-mental kinetic data to determine the model parameters,which permits oneto express reaction rates in thesameterms of extensive and intensive values used in thereactor mathematical model. I n a complex multicom-ponent reaction system, the concentration data mea-sured in the kinetic experiments are not always suffi-ciently sensitive to develop a detailed reaction mech-anism. Therefore, the global ki netic data allow the useof somephenomenological elements, provided the modelis consistent with the experimental data. Upon develop-ment of a macrokinetic model of somecomplex reaction,one uses statistical methods for data reconciliation,

providing the information on the experimental dataquality and algorithms for the calculation of modelparameters and analysis of their reliability. The meth-odology used is briefly described in this paper. Thesoftware for statistic processing of the experimental datais also a part of the general software for reactorcalculations.

2. Base Model for the Description of P-V- TProperties of Multicomponent Mixtures

Phase equili brium is usually calculated by three-parameter cubic equations of state (EOS) such asSoave-Redlich-Kwong, Peng-Robinson, Patel- Teja,and soforth.1 In most cases, theseEOSs nicely describe

1454 I nd. E ng. Chem. Res., Vol. 39, No. 5, 2000



theP-V- T properties of all fluid phases. Therefore, theundoubted advantage will be the modeling of a mul-tiphase system using a unified model EOS for all fluidphases.

2.1. Modified Soave-Redlich-Kwong Equationof State. The Soave-Redlich-K wong (SRK ) equationof state for multicomponent mixtures is2

where Tci,Pci, and ωi are the critical temperature,pressure, and the acentric factor of the ith component,xi and x j are the molar fractions of mixture components,Vm) Vm(am,bm, T,P) is themolar volume of themixture,and kij and cij are the empirical mixing parameters.

The fugacity is expressed from equation of state (1):1

where

The fugacity coefficientΦi and chemical potential µi aredetermined as

Molar volumes of phases in eq 10 are calculated aspolynomial roots:

where

One chooses an arbitrary equation of state for calcu-lations of system phase equilibrium. But the chosenequation should beuniversal and applicable for descrip-tion of all fluid phases, when operational conditions andmixture composition vary over a wide range.

From the above reasoning, we decided to tune eq 1using the empirical coefficients kij and cij which allowfor binary interaction between the ith and jth compo-nents (i ) 1, 2, ..., Nc, i * j).

2.2. Determination of the Binary InteractionCoefficients Using Experimental Data.Coefficientskij and cij are determined from the experimental dataon the equili brium i n two-phase systems. I n our ex-amples, we referred to ref 3 which systematizes theexperimental data on the vapor-liquid and gas-liquidequilibrium over a wide range of temperatures andpressures, including the region of critical conditions.

The constants were calculated by minimization of thefunctional as

Subject to the constraints was the equality of thecomponent fugacity in the vapor-gas and liquid phases:

The following conditions were also taken into accountas restrictions:

In eqs 15-17, indices k and i denote the number of theexperi mental point and component, respectively. Theexperimental molar fractions of components in the liquidand vapor-gas phases are denoted as xik and yik, respec-tively, whereas xjik and yjik are their adjusted values,which minimizefunction (15) and satisfy eqs 16 and 17.

The relative errors σ xi and σ yi are usually 0.05xik and0.05yik, respectively. f ik

L and f ikG are the fugacity of the

ith component in theliquid and gas phases in a kth run. The effective numerical algorithms for the solution

of the minimization problems (15)-(17) are describedelsewhere.4,5 The estimated parameters kij and cij fit thepolynomial dependences Tr2 ) T/ Tc2 and Pr2 ) P/Pc2,where Tc2 and Pc2 are the critical temperature andpressure of the heavier component in the binary mix-

ture. Polynomial dependencies providing good approxi-mations of kij and cij for all pairs studied are

Equations 18 and 19 arefor Tr2 ranging from 0.3 to0.97. Tables 1 and 2 illustrate the derived values of thecoefficients in eqs 18 and 19 for a number of binarypairs.

P )RT

vm - bm

-am( T)

vm(vm + bm)

(1)

am )∑ j)1

N

∑i)1

N

xix jaij (2)

bm )∑ j)1

N

∑i)1

N

xix jbij (3)

aii ) Ri( T)(0.42748R2 Tci2/Pci) (4)

bii ) 0.08664RT ci/Pci (5)

aij ) (1- kij) aiia jj (6)

bij ) (1- cij)(bii + b jj)/2 (7)

Ri( T) ) [1+ mi(1- T/ Tci)]2 (8)

mi ) 0.480+ 1.574ωi - 0.176ωi2 (9)

ln f i ) ln[xiRT /(vm - bm)]+ βi/(vm - bm) +γi βi

RT bm2[ln(

vm + bm

vm) -

bm

(vm + bm)] -γi

RT bm

ln(vm + bm

vm) (10)

γi ) 2∑ j)1

Nc

x jaij, βi ) 2∑ j)1

Nc

x jbij - bm (11)

Φi ) f i/Pxi; µi ) RT ln f i (12)

Z3 - Z2 - Z(B2 - B + A) - AB ) 0 (13)

Z ) PVm/RT , A ) amP2/(RT )2, B ) bmP/RT

(14)

S )∑k)1

K

[∑i)1

2

(xik - xjik

σ xi)

2

+∑i)1

2

(yik - yjik

σ yi)

2

] f min (15)

F ik t

f ikL (xj1k,x2k, T,P,k12,c12)- f ik

G(yj1k,yj2k, T,P,k12,c12) ) 0,

i ) 1, 2; k ) 1, 2, ..., K (16)

∑i)1

2

xjik ) 1, ∑i)1

2

yjik ) 1 k ) 1, 2, ... K (17)

k12) k21)∑q)0

1

∑p)0

2

Apq Tr2p Pr2

q (18)

c12) c21)∑q)0

1

∑p)0

2

Bpq Tr2p Pr2

q (19)

I nd. E ng. C hem. Res., Vol. 39, N o. 5, 2000 1455



2.3. Examplesand ComparativeAnalysis.Figures1 and 2 exemplify an increase in the accuracy of theSRK EOS when one uses the above corrections. Thephase diagram “pressure-composition” is built for thesystem H2-C3H8 at Tr2 ) 0.39 (310.8 K). Solid linesdenote calculated curves with allowance for coefficientsk12 and c12, and broken lines indicate calculations atk12) 0and c12) 0. Dots correspond to the experimentaldata.3

As shown i n F igure 1, the saturated vapor line i spractically insensitive to correction terms until a certainpressure valueis reached. Note that this valueis ratherdistant from thecritical point. The saturated liquid linecalculated without corrections differs significantly fromthe experimental data. This data deviation is observed

near the critical pressure. In addition, the coordinatesof the critical point with allowance for coefficients k12

and c12 are reproduced with high accuracy.In F igure 2a,b, theSRK model is used to describethe

solubili ty of liquid components in the compressed gases. This situation is usually observed when chemical reac-tions are performed at high pressures. Anomalousbehaviors of some gas soluti ons, such as benzene incompressed nitrogen, and extreme points on the curveof solubili ty versus pressure have engaged the attentionof researchers.6 I t is evident that theseanomalies cannotbe described by the Raoult-Henry equations. Calcula-tions were performed with and without allowances fork12 and c12. Note that correction terms for the binarysystemN2-C6H6 at 348 and 373 K were obtained using

experimental data.3 A comparison was performed usingthedata of ref 3 and“independent”data of K richevskii.6

The latter data go far beyond thepressure limits wherek12 and c12 were determined. According to Figure 2a,bthe model extrapolated to the region of high pressuresdescri bes with high accuracy the experi mental data of Krichevskii.6 But the possibility of the extrapolationmust not begeneralized and should bechecked experi-mentally in each particular case.

One may conclude that the empirical parameters kij

and cij and their dependence on temperature andpressure permit one to describe accurately thermody-namic properties of mixtures in the region of highpressure and temperature and near the critical points.

3. Method for Phase Diagrams CalculationsOne of the steps of thermodynamic calculations of a

multiphase reactor is the determination of the regionswhere the mixture with a given composition is thermo-dynamically stable and homogeneous and the regionswhere the mixture is unstable and splits into two orseveral phases.7 To solve this problem, it necessary toconstruct the two- or multiphase boundaries on varioustypes of diagrams (pressure-temperature, pressure-composition, and temperature-composition). Criti calpoints on themultiphaseboundaries are alsoestimatedduring diagram construction.

3.1. Mathematical Model for a Two-Phase Equi-librium. L et us consider first the well-known phase-

Table 1. Coefficients in E q 18

binary pair A00 A01 A02 A10 A11 A12 intervals of Pr2

H2-C3H8 -2.932 6.448 -3.266 -0.5055 1.456 -1.044 1-14H2-C6H6 -0.9124 3.457 -2.935 -0.03036 0.1022 -0.09266 5-60H2-NH 3 1.057 0.5694 -1.750 -1.685 3.750 -2.077 0.5-4N2-NH 3 -7.794 21.610 -14.24 1.880 -4.790 2.957 0.5-4N2-C6H6 0.4329 -0.7749 0.0 0.0 0.0 0.0 1-6N2-C10H22 -3.075 6.727 -1.516 0.3184 -0.8094 0.3796 2.5-16CO-C3H8 -0.2061 0.4183 0.0 0.1104 -0.2078 0.0 2-4.5CH 4-C3H8 0.09241 -2.150 4.484 -0.1205 1.676 -3.068 0.5-2.5

CH 4-C10H22 -1.522 4.841 -3.554 0.1429 -0.4334 0.3055 0.2-16C4H10-C10H22 -0.1427 0.5172 -0.3941 -1.819 4.565 -2.845 0.5-4

Table 2. Coefficients in E q 19

binary pair B00 B01 B02 B10 B11 B12 intervals of Pr2

H2-C3H8 -0.8063 2.069 -1.462 -0.1681 0.4666 -0.3162 1-14H2-C6H6 -0.1014 0.3966 -0.4669 -0.01238 -0.03398 0.02428 5-60H2-NH 3 -0.05320 1.262 -1.550 -0.5495 1.2170 -0.6313 0.5-4N2-NH 3 -5.936 15.76 -10.35 2.002 -5.108 3.224 0.5-4N2-C6H6 0.07557 -0.2257 0.0 0.0 0.0 0.0 1-6N2-C10H22 -1.520 4.527 -3.128 0.1202 -0.3619 0.2484 2.5-16CO-C3H8 0.1190 -0.1008 0.0 0.04752 -0.1182 0.0 2-4.5CH 4-C3H8 -0.1586 -0.7920 2.540 -0.0037 0.7994 -1.776 0.5-2.5CH 4-C10H22 -1.082 3.480 -2.626 0.0703 -0.2128 0.1375 0.2-16C4H10-C10H22 -0.7484 2.150 -1.502 -2.647 6.859 -4.399 0.5-4

Figure 1. Phase diagram“pressure-composition” for the binarymixtureH 2-C3H8 at T ) 310.8 K . Solid lines stand for calculationvia the SRK model allowing for kij and cij; dotted lines stand forkij ) 0, cij ) 0. Squares denote experimental data.3




split problem.8 Startingwith a single phase of total molenumber, L (0), component mole numbers, ni

(0), and molefractions, zi (i ) 1, 2, ..., N). It is suggested that thismixture divides into twophases. L et ni

(1) and ni(2) (i ) 1,

2, ..., N) stand for mole numbers of these phases. T heyield of phases, phase concentrations, and balanceconstraints between variables under conditions of mate-rial isolation are determined as

where the equilibrium ratios K i are defined by K i t xi/yi.

The equilibrium condition i s described by a systemof equations expressing the equality of chemical poten-

tials in both phases:

System (26), allowing for eqs 24 and 25, is a closedsystem for the determination of the total mole fractionsof phases Wx and Wy and the phase concentrations xi

and yi in the equilibrium state. Temperature andpressure are theindependent variables of this problem.

3.2. Equation for Binodal Line as a LimitingCase of the Two-Phase Equilibrium. To determinethe phase boundaries, we consider the limiting casewhere the infinitesimal new phase with composition xis derived from the source phase of composition z

(hereafter z and x phases). The x phase differs from thephase z by its thermodynamic properties.

For the phase transition described by eqs 24-26, thechange in the Gibbs free energy is

Let

where is an infinitesimal positive value. As followsfrom (24),

Consequently,

where σ iy and σ ix are the errors of the order . It isevident that the problem is meaningless for ) 0because the phase x is not determined. Therefore, thesystem properties must change abruptly on the binodalline.

Let us determine the function F(x, T,P) t ∆G/RT ,describing the change of the molar Gibbs phase-transi-tion energy when composition z is fixed:

Equation 30 is theknown formulation of the“tangentialplane criterion” which describes the distance betweenthe molar Gibbs energy surface and the tangent planeat a composition z. Equation 30 is widely used to solvevarious thermodynamic problems.9-11

In this work, eq 30 is used as a mathematical modelfor a two-phase boundary l ine (binodal) calculation.

Figure 2. (a,b) Solubility of benzene in compressed nitrogen as a function of pressure. Solid lines denote calculations using the SRK model with allowance for kij and cij; dotted lines stand for kij ) 0, cij ) 0. Points are the experimental data.3,6

L (1) )∑i)1

N

ni(1), L (2) )∑

i)1

N

ni(2), L (0) ) L (1) + L (2)

(20)

xi )ni

(1)

L (1), yi )

ni(2)

L (2), zi )

ni(0)

L (0)(21)

Wx ) L (1)/L (0), Wy ) L (2)/L (0), Wx + Wy ) 1 (22)

Wxxi + Wyyi ) zi, i ) 1, 2, ..., N (23)

yi ) zi/(1+ Wx(K i - 1)), xi ) K izi/(1+ Wx(K i - 1))

(24)

φ t∑i)1

N

(xi - yi) )∑i)1

N zi(K i - 1)

1+ Wx(K i - 1)) 0 (25)

hi t µi(x) - µi(y)) ln K i + ln Φi(x)- ln Φi(y) ) 0,

i ) 1, 2, ..., N (26)

∆G ) G1 + G2 - G0 )

∑i)1

N

ni(1) µi(x) +∑

i)1

N

(ni(0) - ni

(1)) µi(y) -∑i)1

N

ni(0) µi(z) (27)

ni(1) ) i, ∑

i)1

N

i )

limL (1)f

(yi) ) zi + σ iy() limL (1)f

(xi) ) K izi + σ ix() (28)

limL (1)f

(∆G) ) ∑i)1

N

xi{ µi(x) - µi(z)} t ∆G

(29)

F(x, T,P)t1

RT∑i)1

N

xi[ µi(x) - µi(z)])

∑i)1

N

xi[ln xi + ln Φi(x) - ln zi + ln Φi(z)] (30)




Function F(x, T,P) shows thedistance from theenergysurface to the tangent plane for every arbitrary x. Thetangent plane slope is determined by the chemicalpotentials of the components of phase z. According tothe phase-equilibrium conditions, described by an equal-ity of chemical potentials of components in both phases,the equilibrium composition x ) x* should also belongto this plane. It is evident that the function F(x, T,P)must vanish at x ) x* and x ) z, whereas points x*and z must be coordinates of two local minimumsF(x, T,P), according to the condition of the Gibbs freeenergy minimum at an equilibriumof phases. Accordingto the equili brium stability conditions, F(x, T,P)> 0 forx * x* and x * z.

Therefore, the problem of phase-boundary calcula-tions will be formulated as a solution of the system

and its subsequent analysis with respect to the perfor-mance of the stability conditions:

Vector-function h and matrix hx are determined as

According to the l aw of Gibbs’ phase rule, phaseboundaries have only 1 degreeof freedom, and either Por T is independent. Therefore, the phase diagram(binodal line) on the T,P plane can be determined as aset of stable equilibrium states, x1

/( Tb1(Pb1)), x2/( Tb2-

(Pb2)), ..., or x1/(Pb1( Tb1)), x2

/(Pb2/ ( Tb2

/ )), ..., which satisfyeqs 31-33, when conditions (34) and (35) are fulfi lled.

3.3. Homotopy Continuation Algorithm for Solv-ing the Phase Boundary Calculation Problem.Homotopy continuation methodshavebecomeimportantin solving various chemical engineering problems whenlocally convergent methods fail.12,13

As mentioned early, the equations for the mathemati-

cal model of the binodal line are determined by system(31)-(33). Introducing new variables ui, weobtain xi )ui/Su, where

Equations 31-33 can be rewritten i n terms of the newvariables:

Let u be the vector of the solution of subsystem (37),that is, hi(u) ) 0, where i ) 1, 2, ..., N. Solutions existat any arbitrary T and P values which do not belong tothe binodal line. Substitution of u in eq 36 gives theresidual

which can beeither higher or lower than 0 at arbitraryvalues of T and P. The required coordinates of thebinodal line are such solutions -u, at which -lnSu(u) ) 0; therefore,

To trace the homotopy path by integrating along thearc length, theEuler linear step predictor and Newton’scorrection method is applied.12,13

Let us consider the function in (38) and determinethe sequence of temperature and pressure values thatsatisfy eq 39.

The solution is as follows. We set first the startingpoint T0

k,P0k (k ) 0) near the binodal with an “s”-type

trajectory and check the condition |F( T0k,P0

k)| e , wherek is the iteration number on the predictor step. Whenthis condition does not hold, the starting values of T0

k,P0

k are corrected by several of Newton’s iterations. As aresult, one obtains the starting point Tb

0,Pb0 on curve s.

One then singles out the direction parameter σ k to beeither +1 or -1, referring to the direction of furtheriterations.

The next point near trajectory “s” is determined fromthe linear approximation:

where pk controls the length of an all owed step. The derivatives d T/ds and dP/ds are the components

of tangent vector σ kwj s at point Tb0,Pb

0. The vector isnormalized to unit length, and the direction is set byparameter σ k. F rom the solution of equations

one finds vector w, which can be normalized as

Therefore,

If the absolute value of dF/d T in system (41) is lowerthan sing≈ 10-3 to 10-2, onedoes not solve system (41),but rather a similar system obtained by the substitutionof vector |1, 0| for the bottom matrix l ine.

Using subscript “0” for T and P in formula (40), wenote that these values are i n close proximity to thebinodal line (but not on the binodal line) and serve asinitial approximations for theNewton procedureat eachk + 1 step of the predictor.

F(x, T,P) ) 0 (31)

h(x, T,P) ) 0 (32)

Sxt∑i)1

N

xi - 1) 0 (33)

det[hx]x)x* > 0 (34)

det[hx]x)z > 0 (35)

hi t ∂F/∂xi ) ln xi + ln Φi(x) - ln zi - ln Φi(z),

i ) 1, 2, ..., N

hx ) [∂hi/∂x j], i ) 1, 2, ..., N; j ) 1, 2, ..., N

Su )∑i)1

N

ui

F(u, T,P) )∑i)1

N

ui(ln ui + ln Φi(x) - ln zi - ln Φi(z))

(36)

hi(u, T,P) ) ln ui + ln Φi(x) - ln zi -ln Φi(z) ) 0(37)

F(u, T,P) ) -ln Su(u) (38)

∑) 1, u t x* (39)

T0k+1 ) Tb

k + pk d T

ds

(sk), P0k+1 ) Pb

k + pk dP

ds

(sk) (40)

[(∂F/∂ T)(sk) (∂F/∂P)(sk)

0 1 ] × [w1

w2] ) [0

1] (41)

wj 1 )w1

||w||2, wj 2 )

w2

||w||2(42)

d T

ds(sk) ) σ kwj 1

k,dP

ds(sk) ) σ kwj 2

k (43)




The adjustment of T0k+1,P0

k+1 is as follows. Let i bethe number of iterations by the Newton method at thek + 1 step of the predictor. The sequence of theadjustment steps is determined from the solution of thesystem as

Adjustment i s performed until the condition |F| e isfulfilled ( is the preset accuracy). The Tb

k+1 and Pbk+1

values determine a new point on the binodal li ne. To avoid many unnecessary steps and to take care

that the solution is not drifting away from the curve,thealgorithm provides automatic control of step sizebychoosing parameter pk. The sign of parameter σ k is alsochecked upon the predictor iterations.12

3.4. Problems of Existence and Uniqueness of aSolution. To calculate F (eq 38), it is necessary to solvethe nonlinear system (37) at each iteration. We usedboth the N ewton method and the direct substituti onmethod. In some cases where the matrix [∂h

i/∂u

j] i s i ll -

conditioned, the latter method is more reliable androbust. As a result, the direct substitution method ispreferred, despite the larger number of iterations re-quired.

Consider now the features of the soluti on of system(37). As is known,9 the Gibbs’s free energy surface inthe composition space has R local minima, correspond-ingto R equilibrium phases. I n practice, heterogeneoussystems usually contain no more than three phases ata time, that is, two liquids and one vapor-gas phase.

The number and nature of phases depend on thetemperature, pressure, and chemical composition of theinitial mixture. Close to the binodal line, function (36)should have no more than two local minima. Moreover,

according to the formulated problem, coordinates of oneminimum must coincide with composition z. Such“trivial”u ) z solution of system (37) is one of the truesolutions of problem.

The other local minimum of function (36) must beconsistent with thesolution u* z, corresponding eitherto a “strained”equilibrium composition x, if Su(u) * 1,or a true equilibrium composition x*, if Su(u) ) 1.

We havenumerically studied the problem (37) usingmodel mixtures of different compositions. The mainresults will be ill ustrated usingparticular examples. Asan illustrative example, a binary system containing N2

(z1 ) 25%) and NH3 (z2 ) 75%) has been chosen.Figure 3a-d presents the h1-ln(K 1) plots, whereK 1 ) u1/z1 and h1 ) ∂F(u)/∂u1. According to F igure 3a-c, h1 ) 0 vanishes three times, which corresponds tothree extrema of function (36). One of the intersectionpoints (h1 ) 0) corresponds to maximum F, and theother two points correspond to local minima. One of theminima (Figure 3a,b) fits the composition z (“trivial”solution, ln K 1 ) 0), and the coordinates of the otherminimum is a desired solution, u * z. I n these cases(Figure 3a,b), the chosen parameters T,P are near thebinodal line.

In Fi gure3c, thetrivial solution u) z is a maximumof F. This means that the initial mixture with composi-tion z is thermodynamically unstable and divides into

two phases whosecompositions are given by coordinatesln K 1

(1) ) ln(u1(1)/z1) and ln K 1

(2) ) ln(u1(2)/z1) of the other

twoi ntersection points of thecurveh1 and x axis whichdenote two local minima of function F. Consequently,the chosen parameters T,P (Figure3c) are not near thebinodal l ine. These belong to the two-phase region.

Figure 3d illustrates the situation when a mixture zis thermodynamically stableat a given T,P. F unction Fhas only oneextremumwhich is a stable minimumwiththe u ) z coordinates.

Notethe position of thedesired solution with respectto the trivial one (Figure 3a,b). In the first case, thesolution is to the left and in the second to the right of the trivial solution. I t is obvious that the desired

Figure 3. (a-d) F unction h1 ) ∂F(u)/∂u1 vs ln K 1 ) ln(u1/z1), for a binary mixture N2-NH 3. I ndex “1”denotes nitrogen. The points h1 )0 correspond to three points of function F (36) extrema. Heavy points, minima points; light points, maxima points.

[(∂F/∂ T)( T ik+1,P i

k+1) (∂F/∂P)( T ik+1,P i

k+1)(d T/ds)(sk) (dP/ds)(sk) ] × [∆ T i

k+1

∆P ik+1] )

[-F( T ik+1,P i

k+1)

0 ] (44)

T i+1k+1

) T i

k+1

+ ∆ T i

k+1

, P i+1k+1

) P i

k+1

+ ∆P i

k+1

(45)




stationary points will be found by the iterative proce-dure only if a correct choice is made for the initial u0

approximations.We have shown that the iterative convergence is

always guaranteed to find the nontrivial solution u *z, if the initial approximations u0 meet the followingconditions:

As initi al approximations ui0 at the k + 1 iteration of the predictor, one chooses xi values found at the kthstep, if these approximations satisfy condition (46). Agood estimation of ui0 is otherwise obtained by14

or by the reverse value K i0 ) 1/K i

0.I n Figure 4a,b, function F(x) versus x1 is shown for

T,P occurri ng directly on the binodal line. Buil dingsimilar construction of an F surface in the space of x

compositions is a problem of calculating a binodal l inefor a multicomponent mixture. For theparameters givenin Figure 4a, z is the denser phase and x the lighterphase. In Figure 4b, the situation is reversed. The Tvalue is increased and the P value is decreased, point“2”, moving along the binodal line in the direction of themixture critical point. This point is a singular pointand cannot be attained by the algorithm describedabove.

4. Critical State Criterion of a MulticomponentMixture

Gibbs classical theory of critical points lead to twononlinear equations in the intensive variables:6,15

Elements of matrix M1 are thesecond derivatives of theGibbs free energy:

Matrix M2 is derived from M1 by substituting the rowvector for any matrix row as

If only eq 48 is met, the set of temperature-pressurepairs forms a spinodal line, or a stability limit l ine oneither the T,P or on theP,V planes.9,16 The spinodal l ineseparates the region of possible metastable equilibriumfrom the region where the mixture is absolutely un-stable and metastable states are impossible.

The simultaneous fulfillment of eqs 48 and 49 deter-mines the mixture critical point coordinates Tc,Pc.

Calculations of the critical point, based on the directsolution of eqs 48 and 49, are “rigorous”, and theobtained data agree well with experimental results.15

However, these equations are of such complexity as todiscourage application to large systems. In thestandardapproach, as employed by Peng and Robinson,15 it isnecessary to find 3(N2 - 2N + 3) determinants of theorder (N - 1) at each test point merely to evaluate thefunctions.

In this paper we suggest a simpler system of equa-tions for the critical phase, which preserves all basicproperti es of the rigorous model (48)-(49).

Consider theN components of a homogeneous mixturewith composition z0. Let µi(z0) stand for the chemical

potential of the ith component, calculated at somefixed T and P values. The function µi(z) may beapproximatedby expansion into a Taylor series around the test pointz0.

Derivatives in the right part of eq 52 are calculated atz ) z0, and ∆z j,∆zk denote the vector elements as

Summing eq 52 by index i with weights ∆zi, one has

where∆ µi ) µi(z) - µi(z0). The right-hand part of eq 54 is an approximation of

the fundamental Gibbs relation of the second dif-ferential of the internal energy.9 When T and P are

constant, eq 54 approximates the second differential of the Gibbs free energy function.At predetermined pressure and temperature the

homogeneous mixture of z0 is thermodynamically stableif function S is positive at any arbitrary variations of composition near z0. The limits of mixture stability are

T and P values, at which thequadratic form in theright-hand part of eq 54 is positive semidefinite. A set of values ( Tsp,1,Psp,1), ( Tsp2,Psp,2), ..., satisfying thecondition

form a spinodal line on the T,P plane.

Figure 4. (a, b) Function F(x) (eq 31) for a binary mixture of N2-NH 3. “1” denotes the given compositi on z1 ) 0.25, z2 ) 0.75. “2”

denotes the desired equilibrium compositi on: (a) x1 ) 0.4335,x2 ) 0.5665, Fx ) 401.6 g/L , Fz ) 427.6 g/L , T ) 360.4 K , and P )541.3 atm. (b) x1 ) 0.0415, x2 ) 0.9585, Fx ) 314.8 g/L , Fz ) 142.9g/L, T ) 383.2 K, and P ) 148.9 atm.

S0t∑

i)1

N

zi/K 10 . 1 where K 1

0 ) ui0/zi (46)

K i0t

ui0

zi

)Pci

Pexp[5.3727(1+ ωi)(1-

Tci

T )] (47)

F 1( T,P) ) Det(M1) ) 0 (48)

F 2( T,P) ) Det(M2) ) 0 (49)

M1 ) { ∂2G

∂zi∂z j} ) {∂ µi(z)

∂z j},

i ) 1, 2, ..., N, j ) 1, 2, ..., N (50)

[∂F 1/∂z1, ∂F 1/∂z2, ..., ∂F 1/∂zN] (51)

µi(z) )

µi(z0) +∑ j)1

N ∂ µi

∂z j

∆z j +1

2∑k)1

N

∑ j)1

N ∂2 µi

∂zk∂z j

∆z j∆zk + 0(∆z3),

i ) 1, 2, ..., N (52)

∆z ) [z1-z1,0, z2-z2,0, ..., zN-zN,0] (53)

S t∑i)1

N

∆ µi∆zi )∑i)1

N

∑ j)1

N ∂ µi

∂z j

∆z j∆zi +

1

2∑

i

N

∑k

N

∑ j

N ∂2 µi

∂zk∂z j

∆zi∆zk∆z j + 0(∆z4) (54)

S1t∑ j)1

N

∑i)1

N ∂ µi(z0)

∂z j

∆zi∆z j ) 0 (55)




The critical point Tc,Pc of a mixture is a marginallystable point in the spinodal line.9 At this point, if condition (55) is met, thefollowing conditions must alsobe satisfied:

Equations 55 and 56 are taken as the bases of the“critical phase” model.

4.1. Calculation of a Spinodal Line. Consider firstthe trajectory of the successive soluti ons ( Tsp,1,Psp,1),( Tsp,2,Psp,2), ..., satisfying eq 55 at the preset mixturecomposition z0.

F unction (55) may be rewritten in the matrix form

where H is the symmetric matrix as

whose elements are calculated at z ) z0.Using the well-known methods of linear algebra,

matrix H may be transformed to the diagonal form

where L ) diag( λi), λi are the eigenvalues of H, and Uis the orthogonal matri x of eigenvectors u of matrix H.By orthogonality of U, thesevectors satisfy thefollowingconditions:

Let us determinetheprincipal components∆ψi as a newsystem of variables:

and rewrite function (57) in a new coordinate system:

Let um, the eigenvector of H, correspond to the smallesteigenvalue λm. The principal component ∆ψm will be

If the variations of ∆zi are chosen so that ∆zi ) ξum,i,where ξ is an arbitrary scalar multiplier (for example,ξ ) 1), then according to (60), ∆ψm ) 1,∆ψ j ) 0 ( j ) 1,2, ..., N; j * m), and eq 62 becomes

It i s evident that S1 vanishes at λmf 0. Therefore, theproblem reduces to an evaluation of the T,P values atwhich λm( T,P)) 0. Because at a fixed mixture composi-tion the degree of freedom along the spinodal line is 1,the desired values of T and P form dependent pairs of parameters.

Taking into account theabovediscussion, thesecondequation for the critical phase may be presented as

Equations 64 and 65 are equivalent to eqs 55 and 56.A simultaneous fulfillment of equalities (64) and (65)determines Tc,Pc coordinates of the mixture criticalpoint.

Using the above homotopy continuation algorithm,

and replacing function F with function S1 in eqs 41-44, one solves eq 64. So algorithm (41)-(44) is thestandard unit in the reactor modeling software. Ele-ments of matrix H are normalized by the correspondingsubstitution of variables. Then, matrixes L and U arefound by the J acobi method, and partial derivatives ∂ λm/∂zi in eq 65are calculated by numerical differentiation.

4.2. Localization of a Critical Point.In thecourseof iteration, function S2 is determined only at pointswhere eq 64 is fulfilled. Moving along the spinodalcurve, one determines Tsp,L ,Psp,L , Tsp,R,Psp,R values,between which function S2 changes its sign. Thereafter,the Tc,Pc values are found with high accuracy byNewton’s method.

5. Phase Behavior of Multicomponent Mixtures

5.1. Critical Point,Calculated andExperimentalData. The critical point of a multicomponent mixtureof a given composition is one of the most important andnecessary characteristics of a technological processperformed in the reactor under near- or supercriticalconditions. T herefore, it is important to compare ourresults with the reference data.8,15,16

Figure 5 illustrates the “cubic” forms calculated byeq 49 (curve 2) and eq 65 (curve 1), respectively. Themodel mixture has the following composition (molefactions): CH 4, 0.4590; C2H9, 0.1315; C3H8, 0.0830;n-C5H12, 0.1314; n-C7H16, 0.1925.

S2t∑i

N

∑k

N

∑ j

N ∂2 µi

∂zk∂z j

∆zi∆zk∆z j ) 0 (56)

S1t ∆z TH∆z (57)

hij )∂ ln f i(z)

∂z j

, i ) 1, 2, ..., N; j ) 1, 2, ..., N

(58)

U THU ) L, H ) ULU T (59)

ui Tu j ) 0 at i * j; ui

Tu j ) 1 at i ) j (60)

∆ψi ) ui T∆z )∑

j)1

N

uij∆z j, i ) 1, 2, ..., N (61)

S1(z0, T,P) )∑i)1

N

λi∆ψi2 ) 0 (62)

∆ψm ) um,1

∆

z1 + um,2∆

z2 + ... + um,N∆

zN (63)

S1(z0, T,P)t λm( T,P) ) 0 (64)

Figure 5. Graphics of “cubic form”, calculated using differentmodels. Curve 1 calculations by eqs (64) and (65); curve 2

calculations by eqs (48) and (49). “C” denotes the critical point.

S2(z0, T,P)t∑i)1

N ∂ λm

∂zi

um,i ) 0 (65)




I t is obvious that the curves obtained by differentmodels are different. However, the coordinates of thecritical mixture point coincide completely. In this case,thecalculated critical point coordinates are Tc) 436.933K and Pc ) 119.389 atm.

The next example concerns the following model

mixture: CH 4, 0.9430; C2H6, 0.0270; C3H8, 0.0074;n-C4H10, 0.0049; n-C5H12, 0.0010; n-C6H14, 0.0027; N2,0.0140. Thi s model mixture is used for comparativeanalysis in a number of works.8,15,16 The experimentalcritical parameters of this mixture are given in ref 15.

Table 3 presents calculated and experimental param-eters of the critical point.

Heidemann et al.16 and Michelsen8 use differentmodels of the critical phase. The mixture P-V- Tproperties were calculated by the SRK equation withoutallowance for binary coefficients kij and cij. Therefore,we do not use these corrections in this example. P engand Robinson15 used model (48)-(49) and the P-Requation of state. They did not report on the useof anycorrections.

As shown in Table3, theresults obtained for differentmodels agree well. It is well-known that, without kij andcij corrections, the SRK equation somewhat overesti-mates values of the critical pressure compared to theexperimental data. This is also the case here.

The next example is used for comparison between thecalculations and experimental data as well as compari-son with thedata calculated in ref 15 using a “rigorous”model of (48)-(49) and the P-R equation of state.Binary mixtures of n-butane and carbon dioxide areconsidered (Table 4). In this exampleweused kij and cij

corrections.

According to Table 4, the data obtained by differentmodels deviatewithin the limits of experimental error.As was noted above, our model is more efficient regard-ing the calculation procedure.

Composition dependencies of the critical temperatures(Figure 6a) and critical pressures (Figure 6b) werecalculated for mixtures prepared by the mixing of fractions A and B15 (see Table 5). The molar fraction of A is plotted on the x axis, and the calculated criticalparameters obtained by eqs 64 and 65 and by a“ri gorous” model15 are plotted on the y axis.

The points show a good agreement between theresults. Note that the critical temperature changesmonotonically with the changing A fraction. The criticalpressure passes a maximum, which is greater than thecritical pressure for either pure component. This phe-nomenon has been observed and systematically studiedfor many binary systems.

5.2. Phase Diagrams and the Phase-Transition

Features. 5.2.1. Model Mixture with a Supercriti-cal Solvent. Table 6 presents the composition (molefractions) and physical properties of the individualcomponents considered.

Now we shall consider the areas of mixture stabilityand their mutual arrangement on the P-V plane. Thephasediagram of themodel mixture in P-V coordinatesis shown in F igure 7.

Contour I corresponds to a bimodali line, contour I Icorresponds to the spinodal line, and “C” is the criticalmixture point. The P-V isotherms are represented bydotted l ines, which are calculated by the SRK EOS,

Table 3. Coordinates of the Critical Point of a MethaneMixture

this work Michelsen8Pengand

Robinson15Heidemannand Khalil16 expt.15

Tc, K 202.77 203.13 202.44 202.20 201.09Pc, atm 59.20 58.11 59.04 58.89 55.78

Table 4. Coordinates of the Critical Point of theCO2-n-C4H10 Mixtures

this work,

calc

Pengand Robinson,

calc15 exp15molefractionof C4H10 T C, K PC, atm TC, K PC, atm TC, K PC, atm

0.1694 328.77 76.85 326.8 77.43 325.90 76.280.3334 352.65 82.78 356.30 81.50 351.7 81.700.4984 380.70 76.58 381.70 74.28 377.20 75.360.6740 401.21 62.98 401.50 61.31 398.8 62.810.8273 411.27 53.31 414.30 49.73 412.3 51.09

Figure 6. (a, b) Cri tical pressure and critical temperature vs fraction A in the mixture.

Table 5. Composition of the Mixed Fractions

species fraction A fraction B

methane 0.156 0.762ethane 0.111 0.152propane 0.108 0.058n-butane 0.246 0.022n-heptane 0.379 0.006

Table 6. Model Mixture: Mole Fractions and PhysicalProperties of Components

species formula molefraction Tc, K Pc, atm

nitrogen N2 0.03 126 33.5carbon dioxide CO2 0.05 304 72.8ammonia NH3 0.73 406 111water H2O 0.02 647.3 218toluene C7H8 0.10 562 48.34-methylpyridine C6H7N 0.05 699 52.4benzal dehyde C7H6O 0.01 699 41.61-methylnaphthalene C11H10 0.01 789 38




and line I I I connects theextreme’s points, in which dP/dV ) 0. The area between contours I and II representsthe area where metastable states of the mixture arepossible. The supercooled vapor states can be realizedto the right of a critical point between I and I I , and totheleft, thecondition of the preheated liquid. The area

inside contour I I is the area of diffusion instability of the mixture, where metastable states are impossible,and the mixture of overall composition z cannot exist.

The area, determined by line II I , is thearea of mechan-ical instability of the mixture. From F igure 7 it followsthat when we move on any isotherm, the diffusioninstability of a mixture comes before a mechanicalinstability. This phenomenon takes place just for themulticomponent mixtures, whil e for pure componentsareas II and II I coincide. As shown on the diagram inFigure 7, along the mixture critical isotherm the condi-tion of mechanical stability is realized.

The peculiarities of phase behavior for the abovemixture are in many respects common for other mix-tures in which a component plays therole of solvent (in

this case ammonia). This “near-criti cal area”, which liesbetween the critical parameters of pure ammonia ( Tc)406 K and Pc ) 111 atm) and the critical point of themixture( Tc) 472.2 K andPc) 164.2atm), has a uniqueset of properties (low viscosity, high diffusivity, and highsolubility of gases in liquid phase), which provide theadvantages of the supercri tical technologies.

Simil ar phase diagrams are typical for hydrocarbonmixtures containing, for example, a large amount of methane7,8 and for working mixtures obtained at sepa-ratestages of theFischer- Tropsch synthesis. When theprocess is carried out under “supercritical” conditions,one of the reaction products, propane19 or hexane,20 isused usually as a solvent.

The yield of the equilibrium phases and their com-position, determined from eqs 24-26 at preset T and Pin the region enclosed by binodal l ines, are very impor-tant for the characterization of mixture phase composi-tion in the reactor. L et us discuss these features. Theyield of the more dense phase (condensate) versuspressure at different isotherms is shown in F igure 8.

I t is of interest to note how the condensate yieldchanges depending on the isotherm position relative tothe critical mixture point. At T g 493 K , thei sothermssituated to the right of the critical point exhibit someretrograde phenomena.9 Under isothermal compression,the condensate of a mixture begins to form when thepressure reaches a value of Pb,1. A s the pressure rises,the condensate fraction passes a maximum and van-

ishes at Pb,2. At this point the mixture becomes ahomogeneous compressed gas. K richevskii6 was thefirstto establish the retrograde evaporation caused by anonlinear dependence between pressure and the solubil-ity of the high-boiling components in the compressedgases. The retrogradeevaporation is alsocharacteristicof hydrocarbon mixtures containing individual compo-nents with a wide spectrum of critical parameters,among them supercritical, near-critical, and under-critical components.

In F igure 8, for the isotherm at T ) 473 K , which isvery close to the critical mixture temperature, the yieldof a more dense phase increases sharply with theincreasing pressure and the two-phasestatedisappears

abruptly. In i sotherms at lower temperatures ( T < 433K ), a two-phase mixture consists predominantly of theliquid phase, which increases monotonically with theincreasing pressure.

As mentioned above, the near-critical P-V regionwhich begins at critical parameters of ammonia andends at themixture critical point is very interesting forthe chemical processes. Table 7 presents the phasesplitcharacteristics at the critical parameters of ammonia.Phase compositions are distinguished by approximatelyequal parts of the solvent between both phases. Alongwith the solvent, one phase contains predominantlylight gases, while the other contains high molecularweight organics. Such a phase distribution (Table 7) is

Figure 7. P-V diagrams for a model mixture isotherms. T, K :(1) 400, (2) 425, (3) 450, (4) 472, critical, and (5) 500.

Figure 8. Mole fractions of a denser phase (condensate) in anequili brium state vs pressure at different temperatures. Numbers

in the curves denote the temperature, K .

Table 7. Composition (mol %) and Properties of Equilibrium Phases: T ) 406 K, P ) 111 atm

species phase 1 phase 2

N2 2.403 14.259CO2 4.791 8.944NH 3 73.017 72.673H2O 2.071 0.662C6H7N 5.253 0.228C7H8 10.363 3.164C7H6O 1.050 0.051C11H10 1.052 0.019W 94.961 5.039F, g/L 456.24 115.59




typical for the whole near-critical region, but phaseyields change with respect to temperature and pressure.

5.2.2. “Gas-Gas”Equilibrium. The phenomenon of limited solubility of fluids in the super-high-pressure

region was first observed for binary nitrogen-ammoniamixtures of various compositions. van der Waal firstproposed the possibility of a “gas-gas” equilibrium.K richevskii supported this proposal with the experi-mental data6 and explained qualitatively this phenom-enon.

In this work wemade an attempt todescribethegas-gas equilibrium using theSRK EOS. Figure9a-c showsphase diagrams of “pressure-composition” for binaryN2-NH 3 mixtures at 310.8, 344.1, and 377.4 K. Thesolid lines stand for calculations with a model, andindividual symbols stand for the experimental data.6

I n the low-temperature region (Fi gure 9a), the twobranches of the phase equilibrium diagram run almost

parallel and do not merge together at the critical pointwith thei ncreasing pressure up to 2200 atm. The rightbranch shows the maximum of ammonia solubility incompressed nitrogen at P ≈ 1400 atm, and the leftbranch shows the maximum of nitrogen solubility inliquid ammonia at P ≈ 1000 atm.

The phase diagram changes significantly as thetemperature increases. At T ) 344.1 K and T ) 777.4K (Figure 9 b,c) the right and left branches come closeand mergetogether at the critical points (P ≈ 1000atmat T ) 344.1 K and P ≈ 350 atm at T ) 377.4 K).

As shown in Figure 9b, themonophase condition overcritical pressure holds at a pressure interval 1000-2500-2600 atm. At higher pressures the homogeneoussupercritical fluid splits again into two phases; that is,

the gas-gas equilibrium takes place. The gas-gasequilibrium lines with a critical point at P = 2500 atmwere calculated by RKS EOS. They describe well theregularities observed in experiments.6 Yet, at T ) 377.4K (Figure 9c), wedid not succeed in simulating a gas-gas equilibrium because theexperimental data were notsufficient for calculations of the binary interactioncoefficients.

Thegas-gas equilibrium is alsoobserved in nitrogen-n-heptane binary mixtures (Fi gure 10).

A S-shaped binodal linein theP- T diagram obtainedfor a mixture of 70% N2/30% C7H16 divides the phaseplane into homogeneous (I) and heterogeneous (II)regions; “C” is the critical point. The calculated coordi-

nates of C (P ) 300.5 atm and T ) 453 K ) agree wellwith the experimental data.21 The line of thegas-liquidequilibrium goes along the bottombranch and smoothlytransforms into thegas-gas equilibrium at C, which isobserved at high pressure and temperature, exceedingthe critical temperature of n-heptane.

In Figure 10, the “liquid-gas” phase diagrams areshown in thepressure-composition coordinates (bottompart) at T ) 453 K (curve 1) and 497 K (curve 2).Symbols denotethe experimental data,21 and solid linesdenote model calculations. The letter “C” indicatescritical points.

The curves for the gas-gas equilibrium were calcu-lated at 453 and 553 K (Fi gure 11, curves 1′ and 2′).Circles show critical points. Notethat the curves of thegas-gas equilibrium are nearly symmetric with respectto the critical composition (45% N 2, 55% C7H16). In

Figure 11 (top), the left branches exhibit nitrogensolubility in supercritical heptaneand theright branchesshow heptanesolubility in supercritical nitrogen. As thepressure increases to 900-1000 atm, the mutual solu-bility of the components drops and the composition of each phase tends to a pure component. Such behaviorof the gas-gas equilibrium curves has been observedexperi mentally in some other systems.6

The above examples illustrate the potentials of theSRK EOS for modeling almost all of the experimentallyobserved high-density equilibrium phenomena. Suchhigh-density phaseseparations are important in variousapplications, one of which wewill discuss in the follow-ing section.

Figure 9. (a-c) “Pressure-composition” diagrams of the N2-NH 3 mixtures.

Figure 10. P- T phase diagram of the N2-n-C7H16 mixtures.




6. Fischer-Tropsch Process in a Slurry Reactor

The chemistry of theF ischer- Tropsch (FT) synthesisinvolves thehydrogenation of carbon monoxide, in whicha broad spectrum of products including paraffins (al-kanes), olefins (alkenes), alcohols, aldehydes, ketones,and so forth are produced. The FT slurry reactor is amultiphase system of a solid catalyst, liquid products,and gas-phasereactants. Under reaction conditions thechemical reactions are accompanied by phase conver-sions. Using this process as an example, wemay discusssome general problems encountered i n modeli ng reac-tors with modeling involving phase transitions. Thecatalytic process occurring in thereactor determines the

choice of the main technological stages that directlyinteract with the reactor. Therefore, in choosing theoptimal process parameters, the whole reactor unit asa part of the general process schememust becalculated.

We would like to discuss briefly the literature datarelated to theF ischer- Tropsch (FT) process. Saxenna22

has reviewed the publications dedicated to mathemati-cal modeling of three-phase reactors, including theslurry reactors for FT synthesis. The list of citedpublications includes 258 works published since 1950until 1993. The review concerns a large number of problems, including hydrodynamics, heat and masstransfer, and mathematical models for F T synthesis.Different concepts on mathematical modeling of F Tprocesses i n a slurry reactor are discussed and experi-mental results obtained by different authors are com-pared.

Regarding the F T process, the general assumptionsare as follows:

(1) The suspension phaseis perfectly back-mixed, thecatalyst is uniformly distributed in the liquid, and thegradients of concentration and temperature for theliquid phase are not found.

(2) The slurry reactor is at steady state, implying thatthermodynamic equilibrium exists with respect to vari-ous states variables in each of the individual phases aswell as at the interfaces of two different phases. Thesteady-state operation also implies that there is noproduct accumulation in the reactor.

At the same time the following problems were notproperly reflected in theknown models of theFT slurryreactor:

(1) Complex compositi on of the reaction mixture.(2) Complex reaction kinetic.(3) P hase transiti ons in multi component mixtures.Wewant todiscuss theseproblems briefly and tostate

our approach to the modeling of F T slurry reactors.As is known, the list of components participating in

the FT synthesis comprises 50-100 individual chemicalcomponents, which exist in various thermodynamicstates in thesynthesis reactor and units, providing theirsubsequent treatment. Thus, CO, H2, CO2, CH 4, andC2-C3 are in the supercritical state, C4-C10 hydrocar-bons arein thestate that is to a lesser or greater extentclose to the critical state, and heavier fractions are inthe liquid-only phase. I n the general case, all compo-nents of thereaction mixturearepresent in both phases.

Despite this fact, the known mathematical models23-25

consider only light gases (CO, H2, CO2, and CH4) in thegas phase. The liquid phase is represented as a hydro-carbon “lump”which is determined component-wise bythe Anderson-Shultz-Flory (ASF) distribution.26 The

reaction kinetic models consider thetotal rate of CO andH2 conversion and the rate of the water-gas shiftreaction.

It is obvious that these model conceptions do notadequately descri be the actual process. A sufficientmodel must consider the complex composition of reac-tion mixtures and thecomplex reaction kinetics. Phaseconversions must be calculated using rigorous thermo-dynamic models. In commercial slurry reactors,27 chemi-cal processes are accompanied by product separation bydistillation. To obtain a better primary separation of heavy and light fractions, a condensate, obtained uponvapor gas cooling, is returned to the reactor. To designsuch a reactor andthebest technology for theseparationof products, the mathematical model must contain anextended kinetic model. The latter in combination withthe thermodynamic models of phase split calculationswill permit one to solve the problem set.

6.1. Kinetic Model of the Fischer-Tropsch Reac-tion. At present many research groups try to developdetailed kinetic models of F T reactions in the presenceof various catalysts.28-30 This is now possible becauseof extended potentialities for the experimental study of kinetics and the improvement of algorithms and calcu-lation techniques required for mathematical processingof the experimental data.

Let us consider somemethodical questions related tothe development of the F T kinetic model using theexperimental data of ref 31.

The experiment was performed in a slurry reactorover a finely dispersed alumina-supported cobalt cata-lyst promoted with zirconium. The catalyst properti es,the procedures of preparation and activation, and theexperi mental data are reported elsewhere.31

To illustratethe method of thekinetic model develop-ment, we have chosen 16 isothermal (533 K ) andisobaric (20 atm) runs from a series of extended tests,31

in which the catalyst was treated with syngas flow fora long time with a view to obtain its steady-stateactivity.

The space velocity of the syngas flow at the reactorinlet (under normal conditions) was 1, 1.5, and 2 nL /gcat/h, the inlet ratioof H2/CO being 0.49, 0.75, and 1.0.

Figure 11. P-x phase diagrams of the N2-n-C7H16.




The experimental data provide information about therate of formation of 120 individual hydrocarbons (C1-C40, linear and branched paraffin, and R-olefins). Forthe ki netic model, data on l inear and branched satu-rated hydrocarbons were summarized.

6.1.1. Primary Analysis of Experimental Data. Toaccurately balance the chemical elements at the reactorinlet and outlet, it is necessary to adjust the experi-mental data. Data were adjusted by the known recon-ciliation procedure32a,b which permits one to evaluatethe experiment quality. I f balance errors exceed theaccuracy of measurements, one cannot develop anextended kinetic model. In the experiments, the ad-

justed molar fluxes of the components differ from theexperimental data by 3-8%, which is within the limitsof experimental error. The balance of chemical elementsbefore and after adjustment i s illustrated in Table 8.

The experimental data can be modeled using ASF -determined distributions. A typical distribution curvefor C1-C40 yield (mass %) versus the number of carbonatoms in a molecule is shown in F igure 12.

The hydrocarbon yield is maximum at n ) 6 (n is thenumber of carbon atoms in a molecule). Thereafter, asn grows, the yield of hydrocarbons decreases exponen-tially. Note that theASF distribution predicted with theR coefficient, characterizing the hydrocarbon chaingrowth probability26 is valid only at n g 6 (see F igure

12). The chain growth probability, R, is a parameter of the kinetic model and i s determined by the knownstatistical methods for processing the experimentaldata. T hese are discussed below.

6.1.2. Determination of r as a Function of theReaction Mixture Composition. L et the sumof ratesof n-hexaneand n-hexeneformation bethefirst elementin a sequence approximated by the ASF distribution.Denote this rate as Rn

0, where n0 ) 6. Hereafter, therateis expressed in gmol/(kgcat h). The rateof formationof every subsequent hydrocarbon with a number of carbon atoms, n0 + i, is determined as

where Q ) N - n0, N is the greatest number of carbonatoms i n a molecule, determined i n the run (N ) 40).

Summarizing series (66), we have

where

By normalizing the rates from eq 66 against sum SR j,one obtains dimensionless rates rn0+i:

Estimates for R were determined from each run byminimization of the objective function as

where rn0+i represents the experimental values of rn0+i.Equation 70 was solved using theGauss-Markwardt

method.32 The average relative residuals were 18-20%.Usually, R is considered (except works in refs 28 and

29) as a constant value, which does not depend on thereaction mixture composition. But in the experimentsdiscussed here,R depends on the current concentrationof CO and H 2. Values of R increase with the carbondioxide concentration and decrease with increasinghydrogen concentration.

To develop an explicit dependence of R on the con-centrations of CO and H2, we have tested severaldifferent equations, including those obtained by thereaction mechanism analysis.28 On the basis of thestatistics, the most reliable result was obtained for thefollowing empirical equations:

The estimated constants A and B are 0.2332 ( 0.0740and 0.6330 ( 0.0420, respectively.

6.1.3. Chemical Reactions Used asa Basis for theKinetic Model. The kinetic model of the F T processuses the following stoichiometric equations for theformation of individual hydrocarbons (methane, olefins,and paraffins):

where n ) 2-40 is the number of carbon atoms in amolecule.

Figure 12. Hydrocarbons (alkanes + alkenes) mass percent vsthe carbon number in the molecule. T he l ine shows the ASFpredicted distribution.

Table 8. Atomic Balance before and after Adjustment

atom

experimentinlet,

g atom/h

experimentoutlet,

g atom/h

adjustedoutlet,

g atom/hrelativeerror, %

C 3.3868 3.6739 3.3868 7.81O 3.3868 3.2816 3.3868 -3.21H 6.7736 6.3646 6.7736 -6.43

SR )∑i)0

Q

Rn0Ri ) Rn0(1+ SR) (67)

SR )∑i)1

Q

Ri ) R(1- RQ)

1- R(68)

rn0+i tRn0+i

SR

)Rn0R

i

SR

) Ri 1- R1- RQ+1

i ) 0, 1, 2, ..., Q (69)

ψ(R) )∑i)0

Q{ln rn0+1 -

[i ln R + ln(1- R) - ln(1- RQ+1)]}2f min (70)

R ) AyCO

yCO + yH2

+ B (71)

CO + 3H2a CH 4 + H2O (72)

CO + 2H2f 1n

CnH2n + H2O (73)

CO +2n + 1

nH2f

1n

CnH2n+2 + H2O (74)

Rn0+i ) RnoRi i ) 1, 2, ..., Q (66)




The reaction mixture also contains carbon dioxideformed by the following reaction:

The set of reactions (72)-(75) was used to describe therates of individual component formation at n e 6. Todescribe the rate of saturated hydrocarbon formationfor n > 6, we used eq 66 with allowance for thedependence between R and the reaction mixture com-

position from eq 71. N ote that the amount of unsatur-ated hydrocarbons with n > 6 was rather small i n theruns.

The reaction set (72)-(75) is simplified because paraf-fin hydrocarbons may result fromboth consecutive andparallel reactions. F or instance, olefins may beprimaryproducts which arethen hydrogenated toform saturatedhydrocarbons.33 A hydrocarbon chain may also growbecauseof theinsertion of new molecules of CO and H2

into the molecule. Nevertheless, on the basis of theexperimental data fromLox et al.,28,29 weconcluded thateach individual hydrocarbon could be considered as aprimary reaction product.

6.1.4. Mathematical Model of theKinetic Experi-

ment. The general mathematical model of the isother-mal isobar steady-statekinetic experiment,30 performedin a perfectly mixed flow reactor, can be presented as asystem of nonlinear algebraic equations written invector terms:

where M0 ) [m10, m2

0, ..., mNk

0 ] T is the Nk-dimensionalvector of component molar fluxes (mol/h) at the reactorinlet, M ) [m1, m2, ..., mNk] T is the same vector at thereactor outlet, Z is a stoichiometric matrix of Nr × Nk

dimension, R ) [R1, R2, ..., RNr] T is the Nr -dimensional

vector of reaction rates, y ) [y1, y2, ..., yNg] T is the Nk-

dimensional vector of the molar fractions of componentsin the vapor-gas phase, K ) [K 1, K 2, ..., K Np] T is the Np-dimensional vector of kinetic constants, Nk is thenumber of components, Nr is the number of reactions,Np is the number of kinetic constants (here, Np ) Nr),and gcat is the catalyst sample (kg). T he composition of the liquid phase, x ) [x1, x2, ..., xN] T, is determined onthe basis of the assumption of a thermodynamic equi-librium between phases:

where K eq isthe Nk × Nk diagonal matrix of distributionconstants determined by thermodynamic calculations.

Thekineticmodel identification problem is formulated

by minimizing the error norm function,

subject to the constraints

Here, Nexp is the number of experiments and Vi (i ) 1,2, ..., Nexp) is the Nexp × Nexp diagonal matrix of dispersions of the ith run. Thesigns“∼”and“∧” denotethe experi mental and adjusted variables, respectively.For this problem of the model identification, both thekinetic constants involved in eq 80 and adjusted vari-ables Mi(y) minimizing functional (78) are sought. Interms of themathematical statistics, theobtained resultmeets the principle of maximum likelihood.34 Theproblem is solved by the extended Marquardt-typeprocedure descri bed elsewhere.4,5

6.1.5. Selection of theRate Equations. To describereaction rates, one sets an explicit form of the R vectorcomponents, that is, functions R j ) R j(y,K ( T,P)), j ) 1,2, ..., Nr. N ote that R j may be expressed by concentra-tions x in the l iquid phase using the distributionconstants (77).

The major part of the kinetic studies did not gobeyond rate expressions of the empirical power law typerate for the rate of carbon monoxide. Some authorsderived Langmuir-Hinshelwood rate equations. T hereview of the kinetic equations is given in refs 22 and26.

In refs 28 and 29 a detailed kinetic model was derived.

The rateexpressions for thehydrocarbon-forming reac-tions are based on elementary reactions correspondingto the carbidemechanism. Usually, kinetic experimentsare not sensitive enough to all parameters responsiblefor the detailed reaction mechanism. Therefore, thechosen kinetic functions contain phenomenological ele-ments. T he best agreement between the experi mentaldata and the model as well as a nice “conditionality”of rateconstants were taken as a criterion of choosingthesuitable approximations for rate equations.

Analyses of different versions of explicit R j functionshave shown that statistically reliable estimations maybe obtained only for expressions that are l inear inconstants. Studying the power functions as

and comparing the experimental data, we have estab-lished somegeneral regulariti es. Thus, for componentsC2-C5, the rate of unsaturated hydrocarbon formationis best described by

The rate of saturated hydrocarbon formation is de-scribed by

Unsaturated hydrocarbons are almost unobserved in aC2 fraction and arealso insignificant in theC6 fraction.For thelatter case, a rather good approximation of datawas obtainedfor a number of saturated and unsaturatedhydrocarbons as eq 82.

Wesuggest describingthe rates of methaneformationand vapor shift by the mass action law with allowancefor their reversibility:

CO + H2O a CO2 + H2 (75)

M0- M + (gcat)Z TR(y,R(y),K ) ) 0 (76)

x ) K eqy (77)

Q ) ∑i)1

Nexp

(Mi - Mi) TV

-1i(Mi - Mi)f min (78)

M10- M1 + (gcat)Z TR(y1,R(y1),K ) ) 0

M20- M2 + (gcat)Z TR(y2,R(y2),K ) ) 0

l

MNexp

0 - MNexp

+ (gcat)Z TR(yNexp,R(yNexp

),K ) ) 0 (79)

R j ) K jyCOq yH2

p (80)

R j ) K jyCOyH2

2 (81)

R j ) K jyH2

2

(82)

RCH4) K CH4(yCOyH2

3 -yCH4

yH2O

K CH4

eq ) (83)




The equilibrium constants are respectively K CH 4

eq ) 2×1010 and K CO2

eq ) 74.1474 at 533 K and 2.0 MP a. To efficiently solvea minimization problem, (78)-(79),

it is necessary to choose good initial approximations of

the constants and variance error matrix. These valueswere found for every individual hydrocarbon, C1-C6, bythe least-squares technique when the functions set byeqs 81 and 82were studied at once.30 Relativeresiduals,obtained upon such primary processing of data, wereused later for the estimation of V matrix elements.

To reduce theproblem dimensionality, weused eq 66with an allowance for eq 71, which nicely described thecarbon series starting from C7. Becauseof this arrange-ment, in each run one can group all individual hydro-carbons starting from C7, as an averaged componentCxHy, where x and y are thenumbers of atoms of carbonand hydrogen and index i indicates the experimentalnumber. Values of x and y were found by

R was found for each run by eq 71. Thus, the dimen-sionality of problem (81)-(84) was reduced to 14, andvector M was formed frommolar fluxes of the followingcomponents: CO, H2, CO2, H2O,CH4, C2H6, C3H6, C3H8,C4H8, C4H10, C5H10, C5H12, C6H14, and CxHy. The forma-tion of the group component may be written as

The rate of this reaction in the kinetic model is set bythe equation as

where RC6 is the total rate of hexane and hexeneformation.

6.1.6. Results of the Kinetic Model Identifica-tion. The statistics obtained by identification of thekinetic model is as follows:

(1) Number of degrees of freedom, 228

(2) Value of t criterion at the 95% confidence level,1.966

(3)Root-mean-square absolute deviation of experimen-tal and calculated values of molar fluxes (gmol/h),0.052 11

(4) Root-mean-square relative deviation of experimen-tal and calculated molar fluxes (%), 17.29

(5) Root-mean-square disbalance of eq 80 (gmol/h),0.000 028 85

The parameter estimates aregiven in Table 9. Figure13a-d shows a comparison between the experi mentaland the calculated outlet mole flows of the components.

As shown by statistical summary, providing an aver-age relative error of 17%, the model fits the experimen-tal data well and may be used for mathematicalmodeling of the reactor.

6.2. Mathematical Model of the Reactor Unit. Ascheme of the reactor unit as an element of the wholeprocess scheme is shown in F igure 14.

Syngas (FL O_G) with a set ratio of CO/H2 is directedto the reactor (unit B1). Condensate (FL0_L) is alsodirected to B1 after separation of water in B3 (separator).

T1 and P1 are the working parameters of the reactor. The liquid (FL 1_L ) and vapor-gas (FL 1_G) reactionproducts leave the reactor. The latter goes to a cooler(unit B2). Here, at T2 and P2 (temperature and pressureof the cooler), a part of the high-boiling synthesisproducts is condensed. Vapor-gas F L 2_G, containing apart of unreacted CO and H2 and light products of synthesis, is directed for further processing. The liquidcondensate (FL 2_L) is fed into the separator (unit B3),where it is divided into two liquid phases, such as awater phase (FL 3_L 3) and hydrocarbon phase (FL 3-L1)at T3 and P3 (temperature and pressure in the separa-tor). I n this case, a new vapor-gas phase (FL3_G) may

form. The hydrocarbon fraction (FL 3_L1) is divided intotwo fluxes: FL O_L is returned to the reactor, andFL 4_L is directed for further processing. The ratiobetween the fluxes F L0_L and FL 3_L 1 is set by coef-ficient β. Coefficient β, T2, P2, T3, and P3 are thecontrolling process parameters. The fluxes dimension-ally are kmol/h.

6.2.1. Main Prerequisites of the Model. Reflux ismeant to provide the steady-state operation of thereactor. At 530-550 K and under reaction conditions,the dry syngas fed into the reactor is permanentlysaturated with hydrocarbon vapors, which are releasedtogether with thegas reaction products. As a result, theliquid volume may gradually decrease in the reactor.

Table 9. Results of Kinetic Model I dentification

reactions rate equations model parameters, mol/(gcat h)

1 CH + 3H2S CH 4 + H2O R1 ) K 1(yCOyH2

3 - (1/K CH4

eq )yCH 4yH2O) 76.246( 1.312

2 CO + H2O S CO2 + H2 R2 ) K 2 (yCOyH2O) - (1/K CO2

eq )yCO2yH2) 5.653( 1.477

3 CO + (5/2)H 2f (1/2)C2H6 + H2O R3 ) K 3yH2

2 4.514( 0.746

4 CO + (7/3)H 2f (1/3)C3H8 + H2O R4 ) K 4yH2

2 6.344( 1.489

5 CO + (9/4)H 2f (1/4C4H10+ H2O R5 ) K 5yH2

2 6.949( 1.168

6 CO + (11/5)H2 f (1/5)C5H12+ H2O R6 ) K 6yH2

2 7.520( 0.826

7 CO + (13/6)H2 f (1/6)C6H14+ H2O R7 )

K 7y

COy

H2

2 49.203( 0.612

8 CO + 2H2 f (1/3)C3H6 + H2O R8 ) K 8yCOyH2

2 5.978( 1.728

9 CO + 2H2 f (1/4)C4H8 + H2O R9 ) K 9yCOyH2

2 9.153( 0.823

10 CO + 2H2 f (1/5)C5H10+ H2O R10) K 10yCOyH2

2 7.85( 1.249

11 CO + (1+ y/2x)H2 f (1/x)CxHy + H2O R11) R7R(1- RN)/(1- R)

RCO2) K CO2(yCOyH2O -

yCO2yH2

K eqCO2

) (84)

xi )

∑ j)n0+1

N jR j-n0

∑ j)n0+1

N

R j-n0

; yi ) 2xi + 2, i ) 1, 2, ..., Nexp

(85)

CO +yi + 2xi

2xi

H2f 1xi

CxiHyi

+ H2O (86)

RCxHy) RC6 ∑R j-6 (87)




So the steady-state process regime is not provided.Reflux compensates removal of the liquid phase fromthe reactor and provides the steady-state flow of theliquid phase in the reactor. In the limiting case, thisflow may be zero, but negative values arenot permitted.

It is expected that phases arecompletely mixed in thereactor (isothermal and isobar regimes). Changes in theoverall molar flow of components are caused by chemicalreactions proceeding in the reactor. Phase distributionof components and yields of each phaseare determinedby equilibrium phase transitions.

The list of components comprises 35 individual sub-stances (Nk ) 35) as CO, H2, CO2, H2O, CH4, C3-C5

olefins, and C2-C28 n-paraffins.

The mathematical description of the above processscheme with allowance for the prerequisites discussedis stated below.

(a) Reactor. I nlet fluxes are set mi,G(0) and mi,L

(0) (miL(0) )

βmiL(3)):

The design of reactors is reduced to solution of the

following algebraic equations regarding the unknownsmi(1), miL

(1), and miG(1):

and with allowance for balance relations and otherobvious restrictions:

Figure 13. Comparison between experimental (B) and calculated (A) molar fluxes (mol/h) of CO and H2 (a), C5H12 and C6H14 (b), C2H6,C3H8, and C4H10 (c), and CxHy (d) at the reactor outlet.

Figure 14. Scheme of reactor uni t.

mi(0) ) mi,L

(0) + mi,G(0) (88)

m1G(0) ) mCO

(0) ; m2G(0) ) mH2

(0); miG(0) ) 0;

i ) 3, 4, ..., Nk (89)

FL0_G ) mCO0 + mH2

0 ; FL 0_L )∑i)1

Nk

miL0 (90)

mi0 - mi

(1) + qi(y(1), T1,P1) ) 0 (91)

qi ) gcat∑ j)1

Nr

Z jiR j(y(1), T1,P1) (92)

µiG( T1,P1,y(1)) - µi

L( T1,P1,x(1)) ) 0 (93)




equal tothat of thecooler, and thepressurein thewholesystem was 20atm. Spacevelocity (0.68 nm3/kgcat/h) anda ratio between CO and H2 (1:1) in the initial gas werekept constant.

Figure 16a,b shows graphs of the hydrocarbon dis-tribution in fluxes of FL1_L (liquid hydrocarbons afterthe reactor), FL 3_L1 (liquid hydrocarbons after theseparator), and F L2_G (hydrocarbons in the vapor gasafter the cooler) at β ) 1 (Figure 16a) and β ) 0.5(Figure 16b), respectively. The pressure in the systemis 20 atm, and the temperature of the cooler is 373 K .

At β ) 1, FL 1_L and FL 2_G fluxes of semiproductsare released from the reactor. At 1 > β > 0, (1 - β)-FL 3_L1 is also observed among the product fluxes.

At β ) 1, thegraphs of thelight and middle fractionsare easily distinguished. T he light fraction consists of hydrocarbons of FL2_G and themiddle fraction consistsof hydrocarbons of flux F L 1_L. At β ) 0.5, threefractions of semiproducts that aredistinctly divided intolight, middle, and heavy hydrocarbons are chosen. Here,the light fraction consists of hydrocarbons of flux (1 - β)FL 3_1, and the heavy fraction consists of hydrocar-bons of flux FL1_L.

I n F igure 16a,b, similar distributions are given for β ) 0.5, thetemperature of the cooler being 423K (17a)and 473 K (17b). The figure shows that, at the given β,the composition and the fraction yield can becontrolled

by cooler temperature. As thetemperature in thecoolerrises, separation is more pronounced.

In conclusion, we want to emphasize that reflux isrequired for steady-stateoperation of theslurry reactorupon F T synthesis. Note that laboratory systems in-tended to study the kinetics of the F T synthesis areusually supplied with a “reverse cooler” ( β ) 1).35,36 Intheprototypes of theFT liquid-phase reactors,37,38 someportions of liquid feedstock were also returned in theprocess. The commercial slurry reactor (SSPD, Sasolslurry phase distillate reactor) works on the sameprinciple at the SASOL-II plant.27

Corresponding equations for the enthalpy balance aresupplied to the model for the technological calculations.Enthalpies are calculated at stationary values of fluxesand concentrations. As a result, heat fluxes, providingisothermal regimes in each device, are determined.

7. Conclusion

A linefrom thepoem by Aris R., “Do all that you knowand try all that you don’t,” 39 stimulated the perfor-mance of this work. We emphasize the leading role of thermodynamic calculations in the modeling of mul-tiphase processes and reactors. Because the division of fluxes into equilibrium phases in a reactor unit requiresknowledge of the composition of every reaction mixture

Figure 15. (a, b) Distribution of hydrocarbons in the product fluxes released from the reactor unit. The temperature of the cooler is 373K. (a) β ) 1, (b) β ) 0.5. (1) vapor-gas flow after the cooler, (2) liquid flow after the reactor, and (3) liquid flow after the separator.

Figure 16. (a, b) Distribution of hydrocarbons in the product fluxes released from the reactor unit. β ) 0.5. The temperature of the

cooler is (a) 423 K , (b) 473 K . (1) vapor-gas flow after the cooler, (2) liquid flow after the reactor, and (3) li quid flow after the separator.




component, it is necessary to have an expanded kineticmodel of thereaction, which helps to predict thedetailedcomposition of reaction products. Though mathematicalmodels of reactors incorporating the kinetics, thermo-dynamics, and multicomponent structure of thereactionmixture are rather cumbersome, they help to under-stand the process essence, and thus to develop theoptimal technology.

We donot claim that our models are complete or thatour solution algorithms are necessarily optimal. I n ourminds, the doors are open for further investigations.

Acknowledgment

Theauthors would like tothank Dr. J effrey A. Manionfrom the National Institute of Standards and Technol-ogy, Gaithersburg, MD for useful comments regardingthis manuscript.

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Received for review August 2, 1999Revised manuscript received J anuary 6, 2000

Accepted J anuary 11, 2000

I E9905761


Thermodynamic Calculations in the Modeling of Multiphase Processes and Reactors

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