Engineers (Chemical Industries) by Peter Englezos [Cap 14]

8/22/2019 Engineers (Chemical Industries) by Peter Englezos [Cap 14]

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14Param eter Estimation in NonlinearThermodynamic Models:Cubic Equations of State

Thermodynamic models are widely used for the calcula t ion of equ i l i b r iumand thermophysical propert ies of fluid mixtures . Two types of such model s wil lbe examined: cubic equations of state and activity coefficient models. In thischapter cubic equations of state models are used. Volumet r ic equations of state( E o S ) are employed for the calculation of fluid phase equi l ib r ium and thermo-physical propert ies requi red in the design of processes involv ing non-ideal fluidmixtures in the oil and gas and chemical industries. It is wel l known that theintroduction of empirical parameters in equation of state m i x i n g rules enhancesth e ability of a given EoS as a tool fo r process design although th e n u mbe r ofinteraction parameters should be as small as possible. In general , the phaseequi l ib r ium calculations with an EoS are very sensitive to the values of thebinary interaction parameters .

1 4 . 1 E Q U A T I O N S O F STATEA n equation of state is an algebraic relationship that relates th e intensivethermodynamic variables (T, P, v, xb x 2 , . . . , x N c) for systems in thermodynamicequil ibr ium. In this relationship, T is the temperature, P is the pressure, v is themolar volume an d X ; are the mole fractions of component i with i= l , . . . ,N c , Ncbeing the number of species present in the system.

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P a r a m e t e r E s t i ma t i on in C u b i c Eq u a t i o n s of State 227

The EoS may be written in ma ny forms. I t may be impl ic i tF ( T , P , v , x , , x 2 , . . . , x N ) = 0 (14 .1 )

or explicit in any of the variables, such as pressure. In a system of one component(pure fluid) th e above equation becomesF(T,P ,v) = 0 or P = P(T,v) (14 .2 )

The above equations describe a three-dimensional T-P-v surface which is arepresentation of all the experimental or statistical mechanical informationconcerning the EoS of the system. It should also be noted that it is observedexperimentally that.. P vl i m = 1

The quantity Pv /R T is called compressibility factor. The PvT or volumetricbehavior of a system depends on the intermolecular forces. Sizes, shapes andstructures of molecules determine th e forces between them (Tassios, 1993).Volumetr ic data are needed to calculate t he rmodynamic properties(enthalpy, entropy). They are also used for the metering of fluids, the sizing ofvessels and in natural gas and oil reservoir calculations.

14.1.1 Cubic Equations of StateThese equat ions orig ina te from van der Waals cub ic equat ion of state thatwas proposed more than 100 years ago. The EoS has the form (P+a/v 2) (v -b )=RTwhere the constants a and b are specific for each subs tance . This was asignif icant improvement over the ideal gas law because it could describe thevolumetr ic behavior of both gases and l iquids. Subsequently , improvements inthe accuracy of the calculations, especially for l iquids, were made. One of themost widely used cubic equat ions of state in industry is the Peng-Robinson E oS(Peng and Robinson , 1976). The mathematica l form of this equation was given

in Chap te r 1 as Equat ion 1 .7 and it is given again b elow:

(14.3)v - b v(v + b) +b(v - b)where P is the pressure, T the temperature and v the molar vo lume . When theabove equation is applied to a single fluid, for example gaseous or l iquid CO2,th e parameters a and b are specific to CO2. Values for these parameters can be

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readi ly computed us ing Equa t ions 14.5a - 14 .5d . Whenever the EoS (Equa t ion14.3) is appl ied to a m ix tu re , parameters a and b are now mix tu re parameters .The way these mix tu re parameters are defined is important because it affects th eability of the E oS to represent f luid phase behavior . The so-called van der Waalscombining rules g iven below have been widely used. The reader is advised toconsul t the thermod ynam ics l i terature for other c o m b i n i n g rules.( 1 4 - 4 a )

NX j b j (14 .4b)

where a;, and bj are parameters specific for the individual component i andan empir ical interaction parameter characterizing the binary forme ycomponent i and component j . The individual parameters for component i aregiven by

isformed byt i and component j . The individual parameters for comgiven by

(14.5a)

K ; = 0 . 3 7 4 6 4 + 1 . 5 4 2 2 6 0 ) ; -0.26992m2 (14 .5b)cbi = 0 .0 77 80 ^- ( i 4 .5c )

T r f i = - - (14 .5d )c , iI n th e above equations, co is the acentric factor, and T c, P c are the criticaltemperature and pressure respec t ive ly . These quant i t ies are readi ly avai lable formost components .The Trebble-Bishnoi E oS is a cubic equat ion that may util ize up to fourbinary interaction parameters, k=[k a , k b , k c, k d] r . This equation with i ts quadraticcomb ining rules is presented nex t (Trebble and Bishnoi , 1987;1988).



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+ ( b m + c m ) v - ( b m c m + d 1 )where

Nc Nc

(14.6)

( 1 4 - 7 b )

and NC is the n u mbe r of components in the fluid mixture . The quantities a,,b j , c , , d j , i=l , . . ,Nc are the individual component parameters .

14.1.2 Estimation of Interaction ParametersA s w e ment ioned ear l ier , a vo lumet r i c E oS expresses the re la t ionshipa m o n g pressu re , P , m o l a r vo lume , v , temperature , T and composi t ion 2 . for a

fluid mixture . This relationship for a pressure-explicit EoS is of the formP = P(v,T;z;u;k) (14.8)

where th e p-dimensional vector k represents the unknown binary interactionparameters and T . is the composi t ion (mole fractions) vector. The vector u is theset of EoS parameters which are assumed to be precisely known e.g., purecomponent critical propert ies.Given an EoS, th e object ive of the parameter estimation prob lem is tocompute optimal values for the interaction parameter vector , k, in a statisticallycorrect and computat ional ly efficient manner . Those values are expected toenhance th e correlational abi l i ty of the EoS without compromis ing its abil i ty topredict the correct phase behavior .Copyright 2001 by Taylor & Francis Group, LLC


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230 C h a p t e r 14

14.1.3 Fugacity Expressions Using the Peng-Robinson EoSA ny cubic equation of state can give an expression for the fugacity ofspecies i in a gaseous or in l iquid mixture. For example, the expression for the

fugacity of a componen t i in a gas mixture, f v , based on the Peng-Robinson EoSis th e following

R T

2 V 2 b R T In-0 + V 2 ) b PR T

(14.9)

R T ;T o calculate the fugacity of each species in a gaseous m i x t u r e using theabove equation at specified T, P, and mole fractions of all components y\, y 2 , . . . ,th e fo l lowing procedure is used

1 . Obtain th e parameters a, and b ( fo r each pure species (component) of themixture from Equation 14.5.2. Compute a and b parameters for the mixture from Equat ion 14.4.3. Write the Peng-Robinson equation as fol lows

z 3-(l-B)z 2 +(A-3B2 -2B)z-(AB-B2 -B3) = 0where

A = aP( R T ) "

(14.10)

( 1 4 . 1 la )

R TPvR T (14.1 Ic)

4. Solve the above cubic equation for the compressibil i ty factor, zv(corresponding to largest root of the cub ic EoS) .5. U se the value of zv to compute the vapor phase fugacity fo r each speciesfrom Equation 14.9.



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P a r a m e t e r Est imat ion in C u b i c E q u a t i o n s of State 231

The expression for the fugacity of component i in a l iquid mixture is asfollows

In VPV ' J R T

I n R TRT

( 1 4 . 1 2 )

where A and B are as before and Z L is the compressib i l i ty factor of the l iquidphase (corresponding to smallest root of the cubic EoS) . It is also noted that inEquat ions 14.4a and 14.4b the mixture parameters a and b are computed usingthe l iqu id phase mole fractions.

14.1.4 Fugacity Expressions Using the Trebble-Bishnoi EoSThe expression for the fugacity of a component j in a gas or l iquidmixture , f j , based on the Trebble-Bishnoi EoS is avai lable in the literature(Trebble and Bishnoi , 1988) . This exp ression is given in Ap p e n d i x 1 . In addition

the partial der ivat ive , (9/fj/9xj)T P, for a binary mixture is also provided. Thisexpression is very useful in the parameter estimation methods that wil l b epresented in this chapter .

1 4 . 2 PA R A METER ESTIMATIO N USING B I N A R Y V L E D A T ATraditionally, the binary interaction parameters such as the k a, k b , k c, k^ inth e Trebble-Bishnoi E oS have been est imated from th e regression of binary vapor-

l iquid equi l ibrium (V EE ) data . It is assumed that a set of N experiments have beenperformed and that at each of these experiments , four state variables weremeasured. These variables are the temperature (T),pressure (P), l iquid (x) andvapor (y) phase mole fractions of one of the components. The measurements ofthese variables are related to the "true" but unknown values of the state variablesby th e equations given nextT, = (14.13a)

P , = P ; + e Pi (14 .13b)

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232 C h a p t e r 14

X j = X j +e x i i = l , 2 , . . . ,N (14.13c)y , =y,+e y , i = l ,2 , . . . ,N (14 .13d )

where e T j , e P l , e x ; and e y j , are the corresponding errors in the measurements .G i v e n th e m odel described by Equation 14.8 as wel l as the above exper imenta linformation, th e objective is to estimate th e parameters (k a, k b, k- and k d) bymatching the data with the EoS-based calculated v alues.Least squares (LS) or m a x i m u m l ikel ihood ( M L ) estimation methods can beused for the estimation of the parameters . Both methods invo lve th e minimizat ionof an objective function which consists of a weighted sum of squares of deviat ions(residuals). The objective function is a measure of the correlational abili ty of theEoS. Depending on how the residuals are formulated we have expl ic i t or implici testimation methods (Englezos et al. 1990a). In expl ic i t formulations, thedifferences between the measured values and the EoS (model) based predict ionsconstitute th e residuals. A t each iteration of the m inim iza t ion algori thm, expl ic i tformulations involve phase equi l ibr ium calculat ions at each experimental point .Expl ic i t methods often fail to converge at "difficult" points (e.g. at high pressures).A s a consequence, these data points are usually ignored in the regression, withresulting inferior matching ab i l i ty of the model (Michelsen, 1993). On the otherhand, implicit estimation has the advantage that one avoids the i terative phaseequi l ib r ium calculations and thus has a parameter estimation method which isrobust, and computat ional ly efficient (Englezos et al. 1990a; Peneloux et al. 1990).I t should be kep t in m i n d that an objec t ive function w h i c h does not requireany phase equ i l ib r ium calculat ions during each m inim iza t ion step is the basis fora robust and efficient est imation method . The deve lopment o f impl i c i t object ivefunct ions is based on the phase equ i l i b r ium cri teria (Englezos et al. 1990a) .Final ly , i t shou ld be noted that one importan t underly ing assumpt ion in apply ingM L es t imat ion is that the m o d e l is capable of representing th e data without anysystematic devia t ion . Cubic equat ions of state compute equ i l ib r ium properties offluid mix tu res with a v ar iab le degree of success and hence the ML methodshould b e used with caut ion .

14.2 .1 Maximum Likelihood Parameter and State EstimationOver th e years two ML estimation approaches have evolved : (a) parameterestimation based an impl ic i t formulation of the objective function; and (b)parameter and state estimation or "error in variables" method based on an expl ic i tformulation of the objective function. In the first approach only the parameters areestimated whereas in the second th e true values of the state variables as well as the

values of the parameters are estimated. In this section, we are concerned with th elatter approach.



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Parameter E s t i m a t i o n in C u b i c E q u a t i o n s o f State 233

O n l y two of the four state var iab les measured in a b inary V L B e x p e ri m en tare independent . Hence, one can arb i t rar i ly select two as the independentvariables and use the EoS and the phase equi l ib r ium criteria to calculate valuesfo r the other two (dependent variables) . Let Q j ( i = l , 2 , . . . ,N and j=1 ,2 ) be theindependent variables. Then the dependent ones, , can be obtained from thephase equi l ib r ium relationships (Modell and Reid, 1983) using the EoS. Therelationship between the independent and dependent variables is nonlinear and iswritten as fol lows

S y ^ h j t a j j k j u ) ; j = l,2 and i = l ,2 , . . . ,N (14.14)In this case, the ML paramete r estimates are obtained b y m i n i m i z i n g the

fo l lowing quadrat ic opt imal i ty cr i te r ion (Anderson e t al, 1978; Salazar-Sotelo etal. 1986)

N 2 k;u)?' (14 .15)This is the so-called error in variables method. The formulat ion of theabove opt imal i ty criterion was based on the fo l lowing assumptions:

(i) The EoS is capab le of representing the data without any systematicdeviation.(ii) The exper iments are independent,(iii) The errors in the measurements are normally distr ibuted with zero

mean and a known covariance matrix diag(a\ { , O p, C T J ; f o^,).Unless very few experimental data are available, the dimensionali ty of theproblem is extremely large and hence difficult to treat with standard nonlinear

minimization methods. Schwetlick and Tiller (1985), Salazar-Sotelo et al. (1986)and Valko and Vajda (1987) have exploited the structure of the problem andproposed computationally efficient algorithms.

14.2.2 Explicit Least Squares EstimationThe error in variables method can be s impl i f ied to weigh ted least squaresestimation if the independent v ar iables are assumed to be known precisely or if

they have a negl igib le error variance compared to those of the dependentvariables. In practice howev er , the V L B behavior of the binary system dictatesth e choice of the pairs (T,x) or (T,P)as independent variables. In systems with aCopyright 2001 by Taylor & Francis Group, LLC


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234 C h a p t e r 14

spar ingly so luble component in the l i qu id phase the ( T , P ) pa i r should b e chosenas i ndependen t var iab les . On the other hand , in systems with azeotropicbehav ior the ( T , x ) pai r should b e chosen.A s s u m i n g that the variance of the errors in the m e a su r e m e n t of eachdependent variable is kn o wn , th e following explicit LS objective functions maybe formulated:

'P.i Sv(14.16a)

(14.16b)

The calculation of y and P in Equation 14.16a is achieved by bubblepoint pressure-type calculat ions whereas that of x and y in Equat ion 14.16b isb y isothermal-isobaric flash-type calculat ions. These calculat ions h a ve to beperformed dur ing each i terat ion of the m i n i m i z a t i o n procedure us ing the currentestimates of the parameters. G ive n that both the bubb le po in t and the flashcalculat ions are i terat ive in nature the overal l computational requ i remen t s aresignificant. Furthermore, convergence problems in the thermodynamiccalculations could also be encountered when the parameter values are awayfrom their optimal values .

14.2.3 Implicit Maximum Likelihood Parameter EstimationImpl ic i t estimation offers the opportunity to avoid the computationallydemanding state estimation by formulating a suitable optimality criterion. Thepenalty one pays is that additional distributional assumptions must be made.Implici t formulation is based on residuals that are impl ic i t functions of the statevariables as opposed to the explicit estimation where the residuals are the errorsin the state variables. The assumptions that are made are the fol lowing:

(i) The EoS is capable of representing the data without any systematicdeviation.( i i ) The experiments are independent ,( i i i ) The covariance matrix of the errors in the measurements is known and it isoften of the form: diag( oy j , a P , , a x f o y j ) .

( i v ) The residuals employed in the optimality criterion are normally distributed.Copyright 2001 by Taylor & Francis Group, LLC


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P a r a m e t e r Es t i m a t i o n i n C ub i c E q ua t i ons o f State 235

The formulation of the residuals to be used in the object ive function is basedon the phase equi l ibr ium criterion; i = l , 2 , . . . , N a n d j = l , ( 14 .17 )

where f is the fugacity of component 1 or 2 in the vapor or l iquid phase. The aboveequation may be written in the following form=0 ; i = 1,2,. . . N ; j = 1 ,2 (1 4 .1 8 )

The above equations hold at equil ibr ium. However, when the measurementsof the temperature, pressure and mole fractions are introduced into theseexpressions th e resul t ing values are not zero even if the EoS were perfect. Thereason is the random experimental error associated with each measurement of thestate variables. Thus, Equation 14.18 is written as follows

; i = 1,2, . . .N; j = 1,2 (14 .19)The estimation of the parameters is now accomplished by minimiz ing th eimplicit ML objective function

i! 2 i2(14.20)

where cr. ( j = l , 2 ) is computed by a first order variance approximation alsok n o wn as the error propagation la w (Bev ing ton and Rob inson , 1992) as fo l lows

3T 5P ox. a y , (14.21)

3 T A < 5 T 'T.i 5 P ( 3 P a x , A a x , (14.22)



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where al l the derivat ives are evaluated at T J , P J , X J , y j . T h u s , the errors in themeasurement of all four state variables are taken into account.The above form of the residuals was selected based on the fol lowingconsiderations:

The residuals r , j should be approximately norm ally dis tributed .The residuals should be such that no iterative calcu lations such asbubble point or flash-type calculations are needed during each step ofthe min imiza t ion .The residuals are chosen in a away to avoid numerical instabil i t ies, e.g.exponent overf low.Because the ca lcu la t ion of these res iduals does not requi re any i terative

calculations, the overall computational requirements are significantly less thanfo r the explicit estimation method using Equation 14.15 and the explici t LSestimation method using Eq uations 14.16a and b (Englezos et al. 1990a).

14.2 .4 Implic i t Least Squares EstimationIt is wel l known that cubic equations of state have inherent l imitat ions indescribing accurately th e fluid phase behavior. Thus our objective is oftenrestricted to the determination of a set of interaction parameters that will yield an

"acceptable fit" of the binary V LB data. The following implicit least squaresobjective function is suitable for this purposeN 2

/ I 7 1Z (14 .23)

where Q J J are user-defined weighting factors so that equa l attention is paid to alldata points. If the I n f y are within th e same order of magnitude, Q y, can be setequal to one.

14.2 .5 Constrained Least Squares EstimationIt is wel l k n o w n that c u b i c equat ions of state m ay predict erroneous b inaryva p o r l i q u id equ i l i b r i a w h e n using interact ion parameter est imates from anunconstrained regression of binary V LE data (Schwar tzentruber et al., 1987;Englezos et al. 1989). In other words, th e l iquid phase stability criterion is

violated. Modell and Reid (1983) discuss extensively the phase stability criteria.A general method to alleviate the problem is to perform th e least squaresestimation subject to satisfying the l iquid phase stabi l i ty criterion. In otherCopyright 2001 by Taylor & Francis Group, LLC


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P a r a m e t e r E s t i ma t i on i n C u b i c E q u a t i o n s of State 237

words, parameters which yield the optimal fit of the V L B data subject to thel iquid phase s tab i l i ty cri terion are sought. O ne such method w h i c h ensures thatth e stability constraint is satisfied over the entire T-P-x surface has beenproposed by Englezos et al. (1989) .Given a set of N binary VLB (T-P-x-y) data and an EoS, an efficientmethod to estimate the EoS interaction parameters subject to the l iquid phasestability criterion is accompl ished by solving the fol lowing problem

Minimize /f!L-//7f,VIn f2L -/f2V/f ~/ (14 .24)

subject to dlnf = cp(T,P,x;k) > 0, (T,P,x) e QT ,P

(14.25)

where Q , is a user-defined weighting matrix (the identity matrix is often used), kis the u n k n o w n pa ramete r vector, f,L is the fugacity of componen t i in the l iquidphase, f,v is the fugacity of component i in the vapor phase, < p is the stabilityfunction and Q is the EoS computed T-P-xsurface over which th e stabilityconstraint should be satisfied. This is because the EoS-user expects and demandsfrom th e equation of state to calculate the correct phase behavior at anyspecified condition. This is not a typical constrained minimiza t ion problembecause the constraint is a function of not only th e u n kn o wn parameters, k, butalso of the state variables. In other words, although the objective funct ion iscalculated at a finite n u mbe r of data points, the constraint should be satisfiedover the entire feasible range of T, P and x.

14.2.5.1 Simplified Constrained Least Squares EstimationSolut ion of the above const ra ined least squares problem requi res therepeated computation of the equi l ibr ium surface at each iteration of theparameter search. This can be avoided by using the equi l ibr ium surface definedb y the exper imenta l V L B data points rather than the EoS computed ones in thecalculation of the stability function. The above minimiza t ion problem can be

further simplified b y satisfying the constraint only at the given exper imenta ldata points (Englezos e t a l . 1989). In this case, the constraint (Equa t ion 14.25)is replaced by

subject to (14.26)where ^ is a smal l posi t ive n u m b e r given by



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04.27)

I n Equation 14.27, OT, C J P and o x are the standard deviat ions of themeasurements of T, P and x respectively. A ll the derivatives are evaluated at thepoint where the stability function c p has its lowest value. We call them inim iza t ion of Equa t ion 14.24 subject to the above const ra int simplifiedConstrained Least Squares (simplif ied C L S ) est imat ion .

14 .2 .5 . 2 A Potential Problem with Sparse or Not W ell Distributed DataThe problem with th e above simplif ied procedure is that it may yieldparameters that result in erroneous phase separation at conditions other than th egiven experimental ones. This problem arises when the given data are sparse ornot well distributed. Therefore, we need a procedure that extends the region overwhich the stability constraint is satisfied.The object ive here is to construct th e equi l ib r ium surface in the T-P-x spacefrom a set of available experimental V LB data. In general, this can beaccomplished by using a suitable three-dimensional interpolation method.However, if a sufficient number of well distributed data is not available, this

interpolation should be avoided as it may misrepresent the real phase behavior ofthe system.I n practice, VLE data are available as sets of isothermal measurements. Then u m b e r of isotherms is usual ly smal l ( typically 1 to 5). Hence, we are often leftwith limited information to perform interpolation with respect to temperature. O nth e contrary, one can readily interpolate within an isotherm (two-dimensionalinterpolation) . In particular , for systems with a sparingly soluble component , ateach isotherm one interpolates the liquid mole fraction values for a desiredpressure range. For any other binary system (e.g., azeotropic), at each isotherm,

one interpolates the pressure for a given range of liquid phase mole fraction,typically 0 to 1 .For simplicity and in order to avoid potential misrepresentation of theexperimental equil ibr ium surface, we recommend the use of 2-D interpolation.Extrapolation of the experimental data should generally be avoided. It should bekept in mind that, if prediction of complete miscibility is demanded from the EoSat conditions where no data points are available, a strong prior is imposed on theparameter estimation from a B ayesian point of view.



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Parameter E s t i m a t i on i n C u b i c E q u a t i o n s of State 239

Location of (p mmThe set of points over which th e m i n i m u m of ( p is sought has now beene x p a n d e d with the addi t ion of the in terpolated points to the exper imental ones .

In th is expanded set, instead of using any gradient method for the location of them i n i m u m of the stability function, w e advocate the use of direct search. Therationale behind this choice is that first we avoid any local m i n i m a and secondth e computa t ional requ i rements for a direct search over th e interpolated and thegiven experimental data are rather negl igib le . Hence, th e minimiza t ion ofEqua t ion 14.24 should b e pe r fo rmed subject to the fo l lowing constraint

c p o = cp(T0, P 0 , x c; k) > E , > 0. (14 .28)where (T0, P 0 , x 0 ) is the po in t on the expe r imen ta l equ i l i b r ium surface( in terpo la ted or exper imenta l ) where the stabil i ty func t ion is m i n i m u m . W e callth e minimiza t ion of Equation 14.24 subject to the above constraint (Equation14.28) Constrained Least S quares ( C LS) estimation.

I t is noted that , should be a small positive constant with a valueapproximately equal to the difference in the value of q > when evaluated at theexperimental equil ibr ium surface and the EoS predicted one in the vicinity of the(T 0, P O , x 0). N amely, if we have a system with a sparingly soluble component then4 is given by

s - Lexp\X0 ~ EoSX0 (14.29a)

For all other systems, E , can be computed by

SP -p,E oS0 / (14.29b)In th e above two equations, th e superscripts "exp" and "EoS" indicatethat the state var iable has been obtained from the exper imenta l equi l ibr ium

surface or by EoS calculations respectively. The value of which is used is them a x i m u m between the Equa t ions 14.27 and 14.29a or 14.27 and 14.29b.Equa t ion 14.27 accounts for the uncer ta in ty in the measurement of theexpe r imen ta l data, whereas Equat ions 14 .29a and 14.29b account for thedeviation between the model prediction and the experimental equi l ibr iumsurface. The m i n i m u m of c p is computed by using th e current estimates of theparameters during the minimizat ion .



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The problem of m i n i m i z i n g Equat ion 1 4 . 2 4 sub ject to the constraint givenby Equat ion 1 4 . 2 6 or 1 4 . 2 8 is t ransformed into an unconstrained one byin t roducing th e Lagrange mul t ip l i e r , 00, and augmen t ing the LS objec t ivefunction, S L S(k ) , to yieldS L G ( M ) = SLs(k) + e o ( q > o - ) (14 .30)

The minimizat ion of S LG (k,co) can now be accomplished by applying theG a u ss - N e w t o n method with Marquardt ' s modif icat ion and a step-size pol icy asdescribed in earlier chapters.

14.2 .5 .3 Constrained Gauss-Newton Method for Regression of BinaryVLE D ataA s w e m e n t i o n e d earl ier , th is is not a typ ica l const ra ined m inim iza t ionproblem although the development of the solution method is very similar to thematerial presented in Chap te r 9. If we assume that an estimate k is available atthe j th iteration, a better estimate, k^+ 1) , of the parameter vector is obtained asfollows.

around k1^ at the i* data point y ie lds

e,(kV A k (14.31a)

Furthermore , l inearization of the stability function yields-+ l ) 0 ) I O W n 1 (J+1)J) = cp0(kV Up- A k U (14 .31b)

Substi tution of Equations 14. 3 la and b into th e object ive function SLG (k,ci>) andSSioCk^a)) 5 S L G ( k c i + 1 ) , a ) )use of the stat ionary cri teria = 0 and = 05ko+n a c o

yield the fo l lowing system of l inear equat ions:

A A k G + l > = b--c (14.32a)and



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c p 0 ( k ( j ) ) + c T A k ( J + 1 ) = (14.32b)where

NA = Z6^6' ( 1 4 - 3 3 a )i= l

andN e; (14 .33b)

wh e r e Qj is a user-defined weigh t ing matrix (often taken as the identity matr ix) ,-T VG. is the (2xpj sensitivity matr ix evaluated at x ; and k^ and c is1

Equation 14.32a is solved for A k and the result is sub stituted inok I . dk > \

fo r A k and tEquation 14.32b which is then solved fo r co to yield

(14.34a)

Sub sti tut ing the above expression for the Lagrange mult ip l ier into Equation14.32a we arrive at the following linear equation for A k^+1),

(14.34b)

A ssumi ng that an estimate of the param eter vector, k , is available at the j thiteration, one can obtain the estimate for the next iteration by following th e nextsteps:(i ) Compute sensitivity matrix Gj, residual vector e \ and stability function c p ateach data point (exp erimental or interpolated),

( i i ) Using direct search determine the experimental or interpolated point (T 0,P 0, X Q ) where q > i s min imum.( i i i ) Set up matrix A, vector b and compute vector c at (T 0, PO, x 0).( i v ) Decompose matrix A us ing eigenvalue decomposition and compu te c o fromEquation 14.34a.



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242 C h a p t e r 14( j - i - i )(v ) C o m p u t e A k from Equat ion 14.32b and update the parameter vector b yG + i ) 0 ) ( J + i )setting k = k + uA k where a stepping parameter, u ( 0 < u < 1), isused to avoid th e problem of overs tepping . The bisection rule can be usedto arrive at an acceptable value for u ..

14.2.6 A Systematic Ap proach for Regression of Binary VLE DataA s it was explained earlier the use of the objective function described byEquation 14.15 is not advocated due to the heavy computational requirementsexpected and the potential convergence problems. Calculations by using th eexplicit LS objective function given by Equations 14.16a and 14.16b are notadvocated for the same reasons. In fact, Englezos et al . (1990a) have shown thatthe computational time can be as high as two orders of magnitude compared toimplicit LS estimation using Equation 14.23. The computational t ime was alsofound to be at least twice the t ime required for ML est imation using Equat ion14.20. The implici t LS, ML and Constra ined LS (CLS) estimation methodsare now used to synthesize a systematic approach for the parameter estimationprob l em when no pr ior knowledge regarding the adequacy of thethermodynamic model is avai lable . G iven the availabil i ty of methods to estimatethe interaction parameters in equat ions of state there is a need to fol low asystematic and computa t ional ly efficient approach to deal with all possible cases

that could be encountered during th e regression of binary VLE data. Thefol lowing step by step systematic approach is proposed (Eng lezos et al. 1993)Step 1 . Data Compilation and EOS Selection:A set of N VLE experimental data points have been made avai lab le . Thesedata are the measurements of the state variables (T, P, x, y) at each of theN performed exper iments . Prior to the estimation, one should plot the dataand look for potential outliers as discussed in Chapter 8 . In addition, asuitable E oS with the corresponding mix ing rules should be selected.Step 2 . Best Set of Interaction Parameters:The first task of the estimation procedure is to quickly and efficientlyscreen all poss ible sets of interaction parameters that could be used. Forex amp le if the Trebble-Bishnoi E oS were to be employed which can util izeup to four binary interaction parameters , the n u m b e r of possiblecombinat ions that should be examined is 15. The impl ic i t LS estimationprocedure provides the most efficient means to determine the best set ofinteraction parameters. The best set is the one that results in the smallest

value of the LS objective function after convergence of the minimizat ionalgorithm has been achieved. O ne should not readily accept a set thatCopyright 2001 by Taylor & Francis Group, LLC


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Parame te r E s t i ma t i on in C u b i c Eq u a t i o n s of State 243

corresponds to a marginal ly smaller LS function if it utilizes moreinteraction parameters .

Step 3. Computation of VLE Phase Equilibria:O n c e the best set of interaction parameters has been found , theseparameters should b e used with the EoS to perform the VLE calculations.The computed values should be plotted together with th e data. Acomparison of the data with the EoS based calculated phase behaviorreveals whether correct or incorrect phase behavior (erroneous l iquid phasespli t t ing) is obtained.CASE A: C orrect Phase Behavior:If the correct phase behav io r i.e. absence of erroneous l iqu id phase spli ts ispredicted by the EoS then the overall f i t should be examined and it shouldbe judged whether it is "excellent". If the fit is simply acceptable ratherthan "excellent", then th e previously computed LS parameter estimates

should suffice. This w as found to be the case for the n-pentane-acetoneand the methane-ace tone systems presented later in this chapter .

Step 4a. Implicit M L Estimation:When the fi t is j udg e d to be "excellent" the statistically best interactionparameters can be eff iciently obtained by pe r fo rming impl ic i t M Lestimation. This w as found to be the case with th e methane-methanol andth e nitrogen-ethane systems presented later in this chapter.

CASE B: Erroneous P hase BehaviorStep 4b. CLS Estimation:If incorrect phase behavior is predicted by the EOS then constrained leastsquares (CLS) es t imat ion should be performed and new pa ramete restimates b e obtained. Subsequently , the phase behavior should b ecomputed again and if the fit is found to be acceptable for the intendedapplications, then the CLS estimates should suffice. This was found to be

the case for the carbon diox ide-n-hexane system presented later in thischapter .Step 5b. Modify EoS/M/xing Rules:In several occasions the overall f i t obtained by the CLS estimation coulds imply be found unacceptable despite the fact that the predictions oferroneous phase separation have been suppressed. In such cases, oneshould proceed and either modify the employed m i x i n g rules or use adifferent EoS al l together. O f course, th e estimation should start from Step

1 once the new thermodynamic model has been chosen. Calculations usingCopyright 2001 by Taylor & Francis Group, LLC


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244 C h a p te r 14

the data for the propane-methanol system il lustrate this case as discussednext .

14.2.7 Numerical ResultsIn this section we consider typical examples . They cover all possiblecases that could be encountered during th e regression of binary V L B data.Illustration of the methods is done with the Trebble-Bishnoi (Trebble andBishnoi, 1988) EoS with quadratic mix ing rules and temperature-independentinteraction parameters. It is noted, however, that the methods are not restrictedto any particular EoS/mix ing rule.

A Data by Campbell et al. { 1966} Calculations with LS parameters

0.00.0 0.2 0.4 0.6x, y n-Pentane

0.8 1 .0

Figure 14.1 Vapor-liquid equilibrium data and calculated values for the n-pentaneacetone system, x andy are the mole fractions in the liquidand vapor phase respectively [reproduced with permission fromCanadian Journal of Chemical Engineering].

14 .2 .7 .1 The n-Pentane-Acetone SystemExperimental data are available fo r this system at three temperatures by

Campbe l l et al. (1986). Interaction parameters were estimated by Englezos et al.(1993). It was found that the Trebble-Bishnoi EoS is able to represent the correctphase behavior with an accuracy that does not warrant subsequent use of MLCopyright 2001 by Taylor & Francis Group, LLC


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Parameter E s t i m a t i o n i n C u b i c E q u a t i o n s o f State 245

estimation. The best set of interaction parameters was found by implicit LSestimation to be (k a=0.0977, k b=0,1^=0, kd=0). The standard deviation fo r k a wasfound to be equa l to 0.0020. For this system, the use of more than one interactionparameter did not result in any improvement of the overall fit.The deviationsbetween the calculated and the experimental values were found to be larger thanthe experimental errors. This is attributed to systematic deviations due to modelinadequacy. Hence, one should not attempt to perform M L estimation. In this case,the LS parameter estimates suffice. Figure 14.1 shows the calculated phasebehavior using the LS parameter estimate. The plot shows the pressure versus thel iquid and vapor phase mole fractions. This plot is known as partial phasediagram.

14.2.7.2 T h e M e t h a n e - A c e t on e SystemData for the methane-acetone system are avai lable by Y ok oyama et al.(1985) . The impl ic i t LS est imates were computed and found to be sufficient todescribe the phase behavior. These estimates are (k a=0.0447, k b=0 , k c=0 , kd=0).The standard deviation for k a was found to be equal to 0 .0079 .

8

03Q_

Z3to

7165-4-

32

Temperature - 240K

Data by Zeck and Knapp (1986)

Figure

10.0 0.2 0,4 0,6 0.8Nitrogen Mole Fraction

14.2 Vapor-liquid equilibrium data and calculated values forthe nitrogen-ethane system [reprinted from the CanadianJournal of Chemical Engineering with permission].Copyright 2001 by Taylor & Francis Group, LLC


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246 C h a p t e r !4

14.2.7.3 The Nitrogen-Ethane SystemData at two temperatures were obtained from Zeck and Knapp (1986)

for the n i t rogen -e thane system. The impl ic i t LS est imates of the b inaryinteraction parameters are k a=0 , k b=0 , k c=0 and k d=0.0460. The standarddeviation of k d was found to be equal to 0 .0040 . The vapor l iquid phaseequi l ibr ium was computed and the fit was found to be excellent (Englezos et al.1993). Subsequent ly , impl ic i t M L calculations were performed and a parametervalue of k d=0.0493 with a standard deviation equa l to 0 .0070 was computed.Figure 14.2shows the experimental phase diagram as wel l as the calculated oneusing the impl ic i t M L parameter estimate.

14.2.7.4 The Methane-Methanol SystemThe methane-methanol binary is another system where the EoS is alsocapable of matching the exper imenta l data very well and hence, use of MLestimation to obtain the statistically best estimates of the parameters is just if ied.Data for this system are available from Hong et al. (1987) . Using these data, th ebinary interaction parameters were estimated and together with their standarddeviations are shown in Table 14.1. The values of the parameters not shown inth e table (i.e. , k a, k b, k c) are zero.

Table 14.1 Parameter Estimates for the Methane-Methanol SystemParameter Value

k d =-0.1903k d =-0.23 17k d =-0.25 15

Standard Deviation0.02840 .00700 .0025

Objective FunctionImpl ic i t LS(Equat ion 14.23)I mp l i c i t M L(Equat ion 14.20)

Expl ic i t LS (T,P)(Equation 14.16b)Source: Englezos et al . (1990a).

14 .2 .7 .5 Carbon Dioxide-Methanol SystemData for the carbon dioxide-methanol binary are available from Hong and

Kobayashi (1988) . The parameter values and their standard deviations estimatedfrom the regression of these data are shown in Table 14.2.Copyright 2001 by Taylor & Francis Group, LLC


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P a r a m e t e r E s t i ma t i on i n C u b i c E q u a t i o n s o f State 247

Table 14.2 Parameter E stimates for the Carbon Dioxide-Methanol SystemParameter V a lu e

k a=0.0605kd=-0 .1137k a=0 .0504k d=-0.0631k a=0.0566k d=-0.2238

Standard D eviation0 .00580 .01800 .00160 .00740 .00010.0003

Objective FunctionImpl ic i t LS

(Equat ion 14.23)Exp l ic i t LS (T,x)(Equat ion 14.16a)

Implici t M L(Equa t ion 14.20)Note: Exp l ic i t LS (Equa t ion 14.16a) d id not conv erge with a zerovalue for Marquardt 's directional parameter .Source: Eng lezos et al . (1990a ) .

Table 14.3 Parameter E stimates for the Carbon Dioxide-n-Hex ane SystemParameter V a lu e

k b = 0 . 1 9 7 7kc= -1 .0699k b = 0 . 1 3 4 5k c = -0.7686

Standard Deviation0.03340 .16480.03750 . 17 7 5

Object ive Funct ionI mp l i c i t LS(Equa t ion 14.23)

Simplif ied C LS(Eq. 14.24& 14.26)

14.2.7.6 The Carbon Dioxide-n-Hexane SystemThis system i l lustrates the use of s impl i f ied constrained least squares

(CLS) estimation. In Figure 14.3, the experimental data by Li et al . (1981)together with th e calculated phase diagram for the system carbon dioxide-n-hexane are shown. The calculations were done by using th e best set ofinteraction parameter values obtained by impl ic i t LS estimation. Theseparameter values together with standard deviations are given in Table 14.3. Thevalues of the other parameters (k a, k d) were equal to zero. As seen from Figure14.3, er roneous l iquid phase separation is predicted by the EoS in the h ighpressure region. Subsequently , constrained least squares estimation (CLS) w asperformed by m i n i m i z i n g Equa t ion 14.24 sub ject to the constraint of Equat ion14.26. In other words by satisfying the l iquid phase stability criterion at allexper imental points . The new parameter est imates are also given in Table 14.3.These estimates were used for the re-calculation of the phase diagram, which isalso shown in Figure 14.3. As seen, the EoS no longer predicts erroneous l iquidphase sp l i t t ing in the h igh p ressure region . T he parameter est imates ob ta ined b y



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248 C h a p t e r 14

the const ra ined L S est imation should suffice for engineer ing type calcula t ionss ince the EoS can now adequate ly represent the correct phase behavior of thesystem.

C Oo.!ft )3C OW0>

21 08

6

4 -2

Calculations withLS parameters\Calculations withCLS parameters

Data by Li et al. (1981)

0.0 0.2 0.4 0.6 0.8x, y Carbon Dioxide

1.0

Figure 143 Vapor-liquid equilibrium data and calculated values for the carbondioxide-n-hexane system. Calculations were done using interactionparameters from implicit and constrained least squares (LS)estimation, x and y are the mole fractions in the liquid and vaporphase respectively [reprinted from the Canadian Journal of ChemicalEngineering with permission].

14.2.7.7 The Propane-Methanol SystemThis system represents the case where the structural inadequacy of thethermodynamic model (EoS)is such that the overal l fit is s imp ly unacceptablew h e n the EoS is forced to predict the correct phase behavior by using the CLSparameter estimates. In particular, it has been reported by Englezos et al.(1990a) that when the Trebble-Bishnoi EoS is utilized to model the phasebehav io r of the propane-methanol system (data b y Gal ive l -Solas t iuk et al. 1986)erroneous l iqu id phase splitting is predicted. Those calculations were performedusing th e best set of LS parameter estimates, which are given in Table 14.4.Fol lowing th e steps outl ined in the systematic approach presented in an earlier

section, simplified C LS estimation was performed (Englezos et al. 1993;Englezos and Kalogerakis, 1993). The CLS estimates are also given in Table14.4.



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P a ra me te r Est imat ion in C u b i c E q u a t i o n s of State 249

Table 1 4 . 4 Parameter Estimates for the Propane-Methanol SystemParameter Value

k a = 0.1531k d = -0 .2994k a = 0 . 1 1 9 3k d = - 0 . 3 1 9 6

Standard Deviation0.01130.02800.03050 .0731

Object ive FunctionImpl i c i t L S(Equa t ion 14.23)

Simplified C LS(Equat ion 1 4 . 2 4 & 14.26)Source: Englezos et al. (1993).

(0a5ak.U)a >

Computat ions wi th LS p a r a m e t e r sO

T = 3 4 3 . 1 K

C o mp uta t i o ns withC L S p a r a m e t e r s

D a t a b y Gal ive l -Solast iuk et al . ( 1 9 6 6 )0.4 0.6

x, y Propane1 . 0

Figure 14.4 Vapor-liquid equilibrium data an d calculated values for thepropane-methanol system [reprinted from the CanadianJournal of Chemical Engineering with permission].

N e x t , t h e V E E w a s calculated using these parameters and the resul tstogether with the experimental data are shown in Figure 1 4 . 4 . The erroneousphase behavior has been suppressed. However, the deviations between th eexperimental data and the EoS-based calculated phase behavior are excessivelylarge. In this case, the overall fit is j udged to be unacceptable and one shouldproceed and search fo r more suitable mixing rules. Schwartzentruber et al.(1987) also modeled this system and encountered the same problem.



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250 C h a p t e r !4

Table 14.5 Parameter Estimates for the DiethylamineWater SystemParameterValues

k a = -0.2094k d = -0.2665k a = -0 .8744k d = 0.9808k a = -0.7626k d = 0.6952

StandardDevia t ion0 .01830 .02790 .08140.48460 . 1 1 4 30 .4923

Covar iance

0 .0005

0.0364

0 .0551

Object ive FunctionImpl ic i t LS(Equat ion 14.23)

Simpl i f ied C L S(Equa t ion 1 4 . 2 4 & 14.26)

Const ra ined LS(Equation 14.24 & 14.28)

Source: Englezos and Kalogerakis (1993).

14.2.7.8 The Diethylamine-W ater SystemCop p and Everet (1953) have presented 33 exper imenta l V LE data pointsat three temperatures. The diethylamine-water system dem onstrates the prob lemthat may arise when using the simplified constrained least squares estimationdue to inadequate n u m b e r of data. In such case there is a need to interpolate th edata poin ts and to perform the m inim iza t ion subjec t to constraint of Equa t ion

14.28 instead of Equation 14.26 (Englezos and Kalogerakis, 1993). First,unconstrained LS estimation was performed b y using the objective functiondefined by Equat ion 14.23. The parameter values together with their standarddeviat ions that were obtained are shown in Table 14.5. The covariances are alsog iven in the table . The other parameter values are zero.Using the values (-0 .2094, -0.2665) for the parameters k a and k d in the EoS,th e stability function, c p , was calculated at each exper imenta l VLE point. Them i n i m a of the stability function at each isotherm are shown in Figure 14.5. Thestability function at 311.5 K is shown in F igure 14.6. A s seen, c p becomesnegative. This indicates that the EoS predicts th e existence of two l iquid phases .This is also evident in Fig. 14.7 where th e EoS-based V LE calculations at 311.5K are shown.Subsequently, constrained LS estimation was performed by min imiz ing theobjective function given by Equation 14.24 subject to the constraint described inEquation 14.26. The calculated parameters are also shown in Table 14.5. Them i n i m a of the stability function were also calculated and they are shown inFigure 14.8. A s seen, c p is pos i t iv e at all the exper imenta l condit ions. H o w e ve r ,the new VLE calculations indicate in Figure 14.9 that the EoS predictserroneous l iquid phase separation. This result prompted th e calculation of thestability function at pressures near the vapor pressure of water where it wasfound that q > becomes negat ive . Figure 14.8 shows the stability functionbecoming negat ive .



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Parame t e r Est imat ion in C u b i c E q u a t i o n s of State 251

20

cooc3*-fI151050-5-10

-15-20

A Unconstrained LS Constrained LS (simplified) Constrained LS

310 315 320T e m p e r a t u r e

325 330

Figure 14.5 The minima of the stability function at the experimentaltemperatures for the diethylamine-water system [reprintedfrom Computers & Chemical Engineering with permissionfrom Elsevier Science].

:= 0oc

.Qc oHh-*C/3 -10-

-15

Unstable LiquidPhase

Calculated usingLS parametersTemperature = 3 1 1 , 5 K

0.01 002 0.03 0.040.05Pressure (MPa)

0 .06

f-'igure 14.6 The stability function calculated with interaction parametersfrom unco nstrained least squares estimation.C i h 2001 b T l & F i G LLC


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252 C h a p t e r 14

0.06

r eQ.05tnQ J11

EoS prediction usingLS parameters

Temperature = 311.5 K0.000.0 0.2 0.4

Diethylamine Mole FractionFigure 14.7 Vapor-liquid equilibrium data and calculated values for the diethyla-

mine-water system. Calculations were done using parameters fromunconstrained LS estimation [reprinted from Computers & ChemicalEngineering with permission from Elsevier Science].

Calculated by EoS usingsimplified CIS parameters

0.00 0,01 0.02 0.03 0.04 0.05 0,06Pressure (MPa)

Figure 14.8 The stab ility function calculated with interaction parameters fromsimplified constrained LS estimation.



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P a r a m e t e r Es t i m a t i o n in C u b i c E q u a t i o n s of State 253

0.060.05-

'&9 0.04-1

0.00

Eo S prediction usingsimplified CLS parameters

Temperature = 311.5 K0,0 0.2 0.6 0.8 1.0

Diethylamine Mole FractionFigure 14.9 Vapor-liquid equilibrium data and calculated values for the diethyl-amine-water system. Calculations were done using parameters fromsimplified CLS estimation [reprinted from Computers & Chemical

Engineering with permissionfrom Elsevier Science].20IcZ5

* 1115

Calculated by EoS usingCLS parameters

Temperature = 3 1 1 . 5 K

0.00 0,01 0.02 0.03 0.04 0.05 0.06Pressure (MPa)

Figure 14.10 The stability function calcu lated w ith interaction parameters fromconstrained LS estimation [reprinted from Comp uters & ChemicalEngineering with permissionfrom Elsevier Science].Copyright 2001 by Taylor & Francis Group, LLC


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254 Chapter 14

006

C O0.Iol

Temperature = 311.5 K0,01-0.00

Figure

0,0 0,2 0.4 0.6 0.8Diethylamine Mole Fraction

14.1! Vapor-liquid equilibrium data and calculated values for the diethyl-amine-water system. Calculations were done using parametersfrom CLS estimation [reprinted from Computers & ChemicalEngineering with permission from Elsevier Science].

Therefore, although the stability function was found to be positive at all theexperimental conditions it becomes negative at mole fractions between 0 and thefirst measured data point . Obviously , i f there were additional data available inth i s reg ion , the s impl i f ied constrained L S method that w as fo l lowed abovewould have yielded interaction parameters that do not result in prediction offalse l iquid phase spli t t ing.Fol lowing the procedure described in Section 14.2.5.2 the region overwhich the stability function is examined is extended as follows. A t eachexperimental temperature, a large n u mbe r of mole fraction values are selectedand then at each T, x poin t the values of the pressure, P, corresponding to V L B ,are calculated. This calculat ion is performed by using cubic spl ine interpolationof the experimental pressures. For this purpose the subroutine I C SSC U from th eIMS L library w as used. The interpolation was performed only once and prior toth e parameter estimation by using 100 mole fraction values selected between 0 .and 0.05 to ensure a close examinat ion of the stability function. During th eminimizat ion, these va lues of T, P and x together with the given V LB datawere used to compute the stability function and find its m i n i m u m value. Thecurrent estimates of the parameters were employed at each minimizat ion step.



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P a r a m e t e r Est imat ion in Cubic E q u a t i o n s of State 255

N ew interact ion paramete r values were obtained and they are also shown inTable 14.5. These values are used in the EoS to calculate th e stability functionand the calculated results at 311.5 K are shown in Figure 14.10.As seen from th e figure, the stability function does not become negative atany pressure when the hydrogen sulfide mo le fraction lies anywhere between 0and 1 . The phase diagram calculat ions at 311.5 K are shown in Figure 14 .11 . A sseen, th e correct phase beha vior is now p redicted by the EoS.The improved method guarantees that the EoS will calculate th e correctV LB not only at the experimental data b ut also at any other point that belongs toth e same isotherm. The question that arises is what happens at temperaturesdifferent than the exper imenta l . A s seen in Figure 14.10 the m i n i m a of thestability function increase monotonical ly with respect to temperature. Hence, itis safe to assume that at any temperature between the lowest and the highest one,th e EoS predicts the correct behavior of the system. However , below th em i n i m u m experimental temperature, it is l ikely that the EoS wil l predicterroneous liquid phase separation.The above observation provides useful information to the experimenterwhen investigating systems that exhib i t vapor- l iquid- l iquid equ i l i b r ium. Inparticular, it is desirable to obtain VEE measurements at a temperature near th eone where th e third phase appears. Then by performing CLS estimation, it isguaranteed that the EoS predicts complete misc ib i l i ty everywhere in the actualtwo phase region. It should be noted, however, that in general the m i n i m a of thestability function at each temperature might not change monotonical ly . This isth e case with th e C O 2-n-Hexane system where it is risky to interpolate forintermediate temperatures. Hence , VLB data should also be collected atintermediate temperatures too.

14.3 P A R A M E T E R E S T I M A T I O N U S I NG TH E ENTI R E B I N A R YP H A S E E Q U I L I B R I U M D A T AIt is kn o wn that in five of the six principal types of binary fluid phase

equil ibrium diagrams, data other than VEE may also be available for a particularbinary (van Konynenburg and Scott, 1980). Thus, th e entire database may alsocontain V E 2E , V L ^ E , V L i L 2 E , and L^E data. In this section, a systematicapproach to uti l ize the entire phase equi l ib r ium database is presented. Thematerial is based on the wo rk of Eng lezos et al. (1990b; 1998)

14.3 .1 The Objective FunctionLet us assume that for a binary system there are available N , V E , E , N 2

V E 2E , N 3 E |E 2 E and N 4 V L , L 2 E data points . The light l iquid phase is E ; and L2is th e heavy one. Thus, the total number of available data is N = N i + N 2+ N 3+ N 4 .Gas-gas equi l ib r ium type of data are not included in the analysis because theyCopyright 2001 by Taylor & Francis Group, LLC


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256 C h ap te r 14

are beyond the range of most practical applications. A n impl ic i t objectivefunct ion tha t p rov ides an appropr ia te measu re of the abi l i ty of the EoS torepresent all these types of equi l ib r ium data is the fol lowing

(14.35)

where the vectors r, are the residuals and R j are suitab le weight ing matrices. Theresiduals are based on the iso-fugacity phase equilibrium cri ter ion. When twophase data ( V L ] E , or V L 2E or L ,L 2E ) are considered, the residual vector takesthe form

/f7a-/f,P (14 .36)

where f a and f P are the fugaci t ies of c o m p o n e n t j ( j = l or 2) in phase a (V aporor Liqu id ] ) and phase [3 ( L i q u i d ] or Liqu id 2 ) respectively for the i* expe r imen t .When three phase data ( V L , L 2 E ) are used, the residuals become

r, =l n f , v - l n f , L ll n f 2 v - l n f 2 L ll n f , v - l n f , L 2l n f 7 v - l n f ? L 2

(14 .37)

The residuals are functions of temperature, pressure, composition and theinteraction parameters. These functions can easily be derived analytical ly forany equation of state. A t equi l ib r ium the value of these residuals should be equalto zero. However, when th e measurements of the temp erature, pressure and molefractions are introduced into these expressions the result ing values are not zeroeven if the EoS were perfect. The reason is the random exper imenta l errorassociated with each measurement of the state variables.The values of the elements of the weight ing matrices R ; depend on the typeof estimation method being used. Wh en the residuals in the above equations canb e assumed to be independent, normal ly distributed with zero mean and thesame constant variance, Least Squares (LS) estimation should be performed. Inthis case, the weight ing matrices in Equation 14.35are replaced by the identitymatrix I . M a x i m u m l ikel ihood (ML) estimation should be appl ied when the EoSis capable of calculating the correct phase behav ior of the system within th eexperimental error. Its application requires the knowledge of the measurement



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P a r a m e t e r Es t imat ion in C u b i c E q u a t i o n s of State 257

errors fo r each var iable (i.e. temperature, pressure, and mole fractions) at eachexper imenta l data point . Each of the weight ing matr ices i s replaced by theinverse of the covariance matr ix of the res iduals . T he e lements of the covar iancematrices are computed by a first order variance approximation of the residualsand from th e var iances of the errors in the measu remen ts of the T, P andcom position (z) in the coexist ing phases (Englezos et al. 1990a;b).The optimal parameter values are found by the Gauss -Newton methodusing Marquardt 's modification. The estimation problem is of considerablemagni tude, especially for a mult i-parameter EoS, when th e entire fluid phaseequil ibr ium database is used. Usual ly , there is no pr ior information about theability of the EoS to represent the phase behavior of a particular system. It ispossible that conve rgence of the minimiza t ion of the objective function given b yE q . 14.35 wil l b e diff icul t to achieve due to structural inadequacies of the EoS.Based on the systematic ap proach for the treatment of V L B data of Englezos etal. (1993) the fol lowing step-wise methodology is advocated for the treatment ofth e entire phase database: Cons ider each type of data separately and estimate the best set of interactionparameters by Least Squares . If the estimated best set of interaction parameters is found to be different foreach type of data then it is rather mean ing l e s s to correlate the entiredatabase s imultaneously . One may proceed, however , to find the parameterset that correlates th e m a x i m u m n u m b e r of data types. If the estimated best set of interaction parameters is found to be the samefor each type of data then use the entire database and perform least squaresestimation. C omput e the phase behavior of the system and compare with the data. If the

f i t is exce llent then proceed to max imum l ikelihood parameter estimation. If the fit is s imply acceptable then the LS estimates suffice.

14.3.2 Covariance Matrix of the ParametersThe elements of the covariance matr ix of the parameter estimates arecalculated when the minimizat ion algorithm has converged with a zero value forthe Marquardt 's directional parameter . The covariance matrix of the parametersCOV(k) is determined using Equation 11.1 where th e degrees of freedom used

i n the calculation of d ^ r are(d . f . ) = 2 (N , + N 2 + N 3 )+ 4 N 4 ~ p (14.38)



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258 C h a p t e r 14

14 .3 .3 Numerical ResultsData for the hydrogen sulfide-water and the methane-n-hexane binarysystems were considered. The first is a type III system in the binary phasediagram classification scheme of van Konynenburg and Scott. Experimental data

from Selleck et al. (1952) were used. Carroll and Mather (1989a; b) presented anew interpretation of these data and also new three phase data. In this work , onlythose V L B data from Selleck et al. (1952) that are consistent with the new datawere used. Data for the methane-n-hexane system are available from Poston andM c K e t t a ( 1966) and L in et al. (1977) . This is a type V system.It was shown by Englezos et al. (1998) that use of the entire database can bea str ingent test of the correlational ability of the EoS and/or the m i x i n g rules. A naddi t ional benef i t of using a ll types of phase equ i l i b r ium data in the parameterestimation database is the fact that the statistical propert ies of the estimated

parameter values are usual ly improved in terms of their standard deviation.14 .3 .3 .1 The Hydrogen Sulf ide-Water System

The same tw o interaction parameters (k a, k d) were found to be adequate tocorrelate th e V E E , EEE and V E L E data of the H 2S -H 2O system. Each data setwas used separately to est ima te the parameters b y im p l ic i t least squares (ES) .

Table 14.6 Parameter Estimates for the Hydrogen Sulfide-Water SystemParameterValues

k a = 0.2209k d = -0.5862k a = 0 . 1 9 2 9k d = -0.5455k a = 0.2324

k d = -0.6054k a = 0.2340k d = - 0 . 6 1 7 9k a = 0 . 2 1 0 8

k d = -0.5633

StandardDeviation0 .01200.03060 . 0 1 4 90.03660 .02320.05810 .01320 .03330 .00030 .0013

Database

V E 2E

E ,E 2 E

V E , E 2 EV E 2 E & E , E 2 E& V E , E 2 EV E 2 E & E , E 2 E& V E , E 2 E

Object ive FunctionImpl ic i t E S

( E q n 14.35 with R,=I )Implici t ES

( E q n 14.35 with R ;= I)I mp l i c i t E S(Eqn 14.35 with R;=I )I mp l i c i t E S

( E q n 14.35 with R j = I )Impl ic i t M E(Equat ion 14.35 with

R ^ C O V ' C e , ) )Note: E! is the l ight l iquid phase and E 2 th e heavy.Source: Englezos et al . (1998).



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P a r a m e t e r Est imat ion in C u b i c E q ua t i ons o f State 259

The values of these parameter est imates together with th e standarddeviations are shown in Table 14.6. Subsequently , these two interactionparameters were recalculated using the entire database. T he algori thm easi lyconverged with a zero va lue for Marquard t ' s directional parameter and theestimated values are also shown in the table. The phase behavior of the systemw as then computed using these interaction parameter values and it was found toagree well with the experimental data. This allows ma x imu m l ikelihood (ML)estimation to be performed. The ML estimates that are also shown in Table 14.6were utilized in the EoS to illustrate the computed phase behavior . Thecalculated pressure-composi t ion diagrams corresponding to V L B and LLE dataat 344.26 K are shown in F igures 14.12 and 14.13. There is an exce l len tagreement between the va lues compu ted by the EoS and the exper imental data.The Trebble-Bishnoi EoS was also found capable of representing well the threephase equ i l i b r ium data (Englezos et al. 1998).

14.3 .3 .2 The Methane-n-Hexane SystemThe proposed methodology was also followed and the best parameterestimates for the various types of data are shown in Table 14.7for the methane-n-hexane system. A s seen, the parameter set (k a, k d) w as found to be the best tocorrelate the VL 2E, the L,L 2E and the V L 2 L iE data and another (k a, k j , ) for theV L , E data.

Table 14.7 Parameter Estimates for the Methane-n-Hexane SystemParameterValues

k a = -0.0793k d = 0.0695k a = - 0 . 1 5 0 3k b = -0 .2385k a = -0 .0061

k d = 2 2 3 6k a = 0 . 0 1 1 3k d = 3 1 8 5

k a = -0.0587k d = 0.0632

StandardDevia t ion0 . 0 10 20 . 0 12 00 . 0 2 7 70 .04030 . 0 0 1 80 . 0 1 1 60 . 0 0 8 40 .02540 . 0 0 750 . 0 1 0 4

Used DataV L 2E

V L , E

L ,L 2 E

V L , L 2 EVLjE&L^E

& V L , L 2 E

Object ive Funct ionImpl i c i t L S

(Eqn 14.35with R , = I )Implicit LS(Eqn 14.35 with Rj = I )Impl ic i t LS

(Eqn 14.35with R , = I )Impl i c i t LS

(Eqn 14.35 with R, = I )Implicit LS

(Eqn 14.35 with R, = I )Source: Englezos et al. (1998).



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260 C h ap te r 14

3

_ 4 'to"Q. 3 -0){/) 0 -(/>C DICL 1 -

n -

IData by Selleck et al. (1 952) ]I ".,II "Temperature = 344.26 Ktp f: ;

0.0 0.2 0.4 0.6 0.8Hydrogen Sulfide Mole Fraction

1 .0

Figure 14.12 VLE data and calculated phase diagram for the hydrogensulfide-water system [reprinted from Industrial EngineeringChemistry Research with permission from the AmericanChemical Society].

C OQ _0)t/3tf)C U

1 1 . 0 -

9.0

7.0-

5.00.0

Data by Selleck et al. (1952)

Temperature = 344.26 K

0.2 0.4 0.6 0.8 1.0

Figure 14.13Hydrogen Sulfide Mole Fraction

LLE data and calculated phase diagram for the hydroge nsill fide-water system [reprinted from Industrial EngineeringChemistry Research with permission from the AmericanChemical Society].



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P a r a m e t e r Est imat ion in C u b i c E q u a t i o n s of State 2 61

Phase equi l ib r ium calculations using th e best set of interaction parameterestimates for each type of data revealed that the EoS is ab le to represent wel lonly the V L 2 E data. The EoS was not found to be able to represent the L 2 L iEand the V L ^ L j E data. A l t h o u g h the EoS is capab le of represent ing on ly part ofth e fluid phase equ i l i b r ium d iagram, a single set of va lues for the bes t set ofinteraction parameters w as found by using al l but the V L t E data. The values forthis set (k a, k d) were calculated by least squares and they are also given in Table14.7. Using the estimated interaction parameters phase equi l ibr ium computat ionswere performed. It was found that the EoS is able to represent th e V L 2Ebehav io r of the methane-n-hexane system in the temperature r ange of 198.05 to444.25 K reasonably wel l . Typ ica l resu l ts together with the exper imental data at273 .16 and 444.25 K are shown in Figures 14.14and 14.15 respect ively .However , the EoS was found to be unable to correlate the entire phase behaviori n th e temperature range of 195.91 K (Upper Critical Solution Temperature) and182.46K (Lower Critical Solution Temperature) .

14.4 P A R A M E T E R E S T I M A T I O N U S I NG B I N A R Y C R I T I C A L P O I N TD A T AG ive n th e inherent robustness and efficiency of implicit methods, their usew as extended to include critical point data (Englezos et al. 1998). Illustration of

the method was done with the Trebble-B ishnoi EoS (Trebble and Bishnoi , 1988)and th e Peng-Robinson E oS (Peng and Robinson, 1976) with quadratic mixingrules and temperature- independent interaction parameters . The methodology,however , is not restricted to any particular EoS /mixing rule .Prior work on the use of critical point data to estimate binary interactionparameters employed th e minimiza t ion of a summation of squared differencesbetween experimental and calculated critical temperature and/or pressure(Equa t ion 14.39). Dur ing that m i n i m i z a t i o n the EoS uses the current parameterestimates in order to compute the critical pressure and/or th e criticaltemperature. However, the initial estimates are often away from th e optimumand as a consequence , such i terat ive computa t ions are diff icul t to conve rge andthe overall computat ional requirements are significant.14.4.1 The Objective Function

It is assumed that there are available N C P experimental binary critical pointdata. These data include values of the pressure, P c, th e temperature, T c, and themole fraction, x c, of one of the components at each of the critical points for thebinary mixture . The vector k of interaction parameters is determined by fittingthe EoS to the critical data. In explicit formulations th e interaction parametersare obtained by the minimiza t ion of the fo l lowing least squares objectivefunction:



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262 Chapter 14

CL53Wto

Data by Lin el al. (1977)

Temperature = 273.16 K

0.0 0.2 0.4 0.6 0.8Methane Mole Fraction

Figure 14.14 VLE data and calculated phase diagram for the methane-n-hexane system [reprinted from Industrial Engineering ChemistiyResearch with permission from the American Chemical Society].

C Oa.10503C D

10"8-6-4 -2-0

Temperature=444.25 K

Data by Poston and M cKetta (1966)0,0 0.2 0.4 0.6

Methane Mole Fraction0.8

Figure 14.15 VLE data and calculated phase diagram for the methane-n-hexanesystem.Copyright 2001 by Taylor & Francis Group, LLC


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P a r a m e t e r Es t i m a t i o n in Cubic E q ua t i ons o f State 263

NCp NCPf-Pccf )2 +q2 -Tccfc)2 (14.39)

T he values of q, and q 2 are usual ly taken equal to one. Hissong and Kay(197 0) assigned a value of four for the ratio o f q 2 over q ^The critical temperature and/or th e critical pressure are calculated bysolving the equations that define the critical point given below (Model l andReid , 1983)

ox, (14.40a)

f \d / n f j

V X l(14.40b)

Because the current estimates of the interaction parameters are used whensolving the above equations , convergence problems are often encountered whenthese estimates are far from their optimal values. I t is therefore desirable tohave, especially for mult i -parameter equations of state, an efficient and robustestimation procedure. Such a procedure is presented next .A t each point on the critical locus Equations 40a and b are satisfied whenth e true values of the binary interaction parameters and the state variables, T c, P cand x c are used. As a result , fo l lowing an implici t formulation, one may attemptto minim ize the fo l lowing residuals .

f d l n f i (14 .41a)

(14.41b)

Furthermore, in order to avoid any iterative computations for each criticalpoint , we use the experimental measurements of the state variables instead oftheir unknown t rue values. In the above equations r, and r2 are residuals whichcan be easily calculated for any equation of state.Based on the above residuals the following object ive function isformulated:



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264 C h a p t e r 14

NCPS ( k ) = i-jXn (14 .42)

where r=(rb r2)T . The optimal parameter values are found b y using the Gauss-Newton-Marquard t minimiz ation algorithm. For LS estimation, R ;= I is used,however , for ML estimation, the weighting matrices need to be computed. Thisis done using a first order variance approximation.T he elements of the covar iance matrix of the parameter estimates arecalculated when the m inim iza t ion algori thm has converged with a zero value forthe Marquardt ' s direct ional parameter . The covar iance matr ix of the parametersCOy(k) is determined u s i n g Equa t ion 11 .1 where the degrees of freedom usedin th e calculation of d are

(d.f.) = 2N C P - p (14.43)

14.4.2 Numerical ResultsFive critical points for the methane-n-hexane system in the temperaturerange of 198 to 273 K measured by Lin et al. (1977) are available. Byemploying the Trebble-Bishnoi EoS in our critical point regression least squaresestimation method, the parameter set (k a, k t , ) was found to be the opt imal one.Convergence from an initial guess of (k a,k b=0.001, -0 .001) was achieved in sixiterations. The estimated values are given in Table 14 .8 .A s seen, coincidentally, this is the set that was optimal for the V L i E data.The same data were also used with th e impl ic i t least squares optimizationprocedure but by using the Peng-Robinson EoS.The results are also shown inTable 14.8 together with resul ts for the CH 4 -C 3H S system. O ne interactionparameter in the quadrat ic mixing rule for the attractive parameter was used.Temperature-dependent values estimated from V LE data have also been

reported by Ohe (1990) and are also given in the table. As seen from the table,we did not obtain opt imal parameters values same as those obtained from thephase equi l ib r ium data. This is expected because the EoS is a semi-empir icalmodel and different data are used for the regression. A perfect thermodynamicmodel would be expected to correlate both sets of data equally well . Thus,regression of the crit ical po in t data can prov ide addi t ional informat ion abou t thecorrelational and predictive capability of the EoS/mixing rule.In pr inc ip le , one may c o m b i n e equ i l i b r ium and crit ical data in one database

fo r the parameter estimation. From a numer ical implementa t ion point of viewthis can easily be done with th e proposed estimation methods. However , it wasnot done because it puts a t remendous demand in the correlational abi l i ty of theEoS to describe all the data and it wil l be s imply a computational exercise.



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P a r a m e t e r E s t i ma t i on i n C u b i c E q u a t i o n s of State 265

Table 14.8 Interaction Parameter Values from Binary Critical Point Data

System

CH4-nC 6H 1 4CH4-nC 6H 1 4CH4-C 3H 8

Reference

Lin et al.(1977)Lin et al.(1977)

Reamer etal. (1950)

TemperatureRange (K)

1 9 8 - 2 7 3

1 9 8 - 2 7 3277 .6-327 .6

Parameters Values UsingCri t ical

Point Datak a =-0.1 3970.015k b=-0 .52510.035k j j = -0 .05070.014k i j= 0 . 1 0 62

0 .0 13

V L B Data

seeTable 14.7k0.0269kj j=0.0462bkj j=0.0249ki j=0.0575

b

EoS

Trebble-BishnoiPeng-Rob insonPeng-

RobinsonInteraction parameter values reported b y Ohe (1990) .Source: Englezos et al. (1998) .

Table 14.9 Vapor-Liquid Equilibrium Data for the Methanol (l)-Isobutane( 2 ) System at 133.0C

Pressure(MPa)0.8861.1361.6332.5863.6623.7233.9203.9593.9513 .8163.7423.6513.5913.529

Liqu id Phase Mole Fraction(X2)0 .0000

0 . 0 1 1 20 .03430 .12290 . 4 1 0 70 .46710 .72440 .8048 30.8465b0.91300.94520 .97820.98971.0000

V apor Phase Mole Fraction( y 2 )0 .0000

0 .17900 .38150 .61690.73340.75670.78680.80480.84650.91300.93690 .97110.98331.0000

aThis is an azeotropic composi t ion .b This is a critical point composi t ion.Source: Leu and Robinson (1992).



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266 C h a p t e r 14

1 4 . 5 P R O B L E M SThere are numerous sources of phase equi l ibr ium data available that serveas a database to those developing or improv ing equa t ions of state. References tothese databases are widely available. In addition, new data are added mainly

through the Journal of Chemical and Engineering Data and the j ou rna l FluidPhase Equilibria. Next , we give data for two systems so that the reader m aypractice the estimation methods discussed in this chapter .

14.5.1 Data for the Methanol-Isobutane SystemLeu and Robinson (1992) reported data for this binary system. The datawere obtained at temperatures of 0 . 0 , 5 0 . 0 , 100.0, 125.0, 1 3 3 . 0 and 1 5 0 . 0 C . A teach tempera ture the v a p o r and l iquid phase mole fractions of isobutane weremeasured at different pressures. The data at 1 3 3 . 0 and 1 5 0 . 0 are given in Tables

1 4 . 9 and 1 4 . 1 0 respectively. The reader should test if the Peng-Robinson and theTrebble-Bishnoi equations of state are capable of describing the observed phasebehav iour. First , each isothermal data set should be examined separately.Table 14.10 VaporLiquid Equilibrium Data for the Methanol (1)Isobutane

(2 ) System at 150.0 "CPressure

( M P a )1 . 3 3 61 . 5 2 01 . 9 8 22.4483.5554.4244 .5424.6234.679

Liquid Phase Mole Fraction(X2)

0 .0000 .00920.02850.05660 .15500.32790 .48760 .57240.6254 a

V a p o r Phase M ole Fraction( y 2 )0 .0000

0 .12090 . 2 8 1 90 .41690.61960.66010.64890 .64000.6254

a Th is is a critical point composit ion.Source'. Leu and Robinson (1992) .

14 .5 .2 Data for the Carbon Dioxide-Cyclohexane SystemShibata and Sandier (1989) reported V LE data for the C O 2-cyclo-hexane and N 2-cyclohex ane systems. The data for the C O 2-cyc lohexane systemare given in Tables 1 4 . 1 1 and 14.12. The reader should test if the Peng-



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Parameter Es t i m a t i o n in C u b i c E q ua t i ons of S tate 267

Robinson and the Trebb le -B ishno i equations of state are capable of descr ibingthe observed phase behav io r . First, each isothermal data set should be examinedseparately.

Table 14.11 VaporLiquid Equilibrium Data for the Carbon Dioxide (1 )-Cydohexane (2 ) System at 366.5 K

Pressure( M P a )0 . 1 7 11 . 7 5 23.4695 . 1 9 16.8868.63710.36012.51112.800

Liqu id Phase M ole Fract ion( x , )

0 . 0 0 0 00 .09590 . 1 68 10 .26570 .34840.45980.56980 .75260 .7901

Vapor Phase Mole Fraction( y O0 .00000.89840 .93740.94950 .95150.95000.94020.89770 .8732

Source: Shibata and Sandier (1989).

Table 14.12 Vapor-Liquid E quilibrium Data for the Carbon Dioxide (1 )-Cydohexane (2 ) System at 4 1 0 . 9 K

Pressure( M P a )0 .4431 . 7 1 83.4445 . 1 5 16.9288 . 6 5 110.40212.05613.814

Liqu id Phase Mole Fraction( x , )

0.00000.06710 . 147 00.20500.28870.35780 .43290 .51650 . 6 10 7

Vapor Phase Mole Fraction( y O0 .00000.71660.82950.85930.88500.88610.87770.86360.8320

Engineers (Chemical Industries) by Peter Englezos [Cap 14]

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