an analytical method of predicting lee-kesler interaction parameters

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An Analytical Method of Predicting Lee-Kesler-PlikkerBinary Interaction Coefficients-Part I: For Non-Polar Hydrocarbon Mixtures

SolomonD.LabinovTlm-modynamicCenterKiev?Ukraine

JamesR andOak Ridge National LaboratoryOak Ridge, TN

Resented at theTwelfth Symposhm onThemphysical PropertiesJune 1 9 2 4 ,1 9 9 4Boulder, Colorado


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Solomon D. Labinog, James R. Sand

Paper presented at the Tbelfth Symposium on ThennophysicalProperties, une 19-24 1994 Boulder,Colorado.7

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Guest scientist at Oak Ridge National Laboratolyfrom the ThermodynamicsCenter,Kiev,Ukraine.Oak Ridge National LaboratolyEnergy DivisionP . 0 Box 2008, Building 3147, MS 070Oak Ridge,Tennessee 37831-6070Author towhom correspondenceshould be addressed.

DISCLAIMERThis report was prepared as an a w u n t of work sponsored by an agency of the United StatesGovernment. Neither the United S tates Government nor any agency thereof, nor any of theiremployees, makes any warranty, express or implied, or assumes any legal liability or responsi-bility for the accuracy, completeness, or usefulness of any information, apparatus, product, orprocess disclosed, or represents that its use would not infringe privately owned rights. Refer-ence herein to any specific commercial product, process, or service by trade name, trademark,manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recom-mendation, or favoring by the United States Government or any agency thereof. The viewsand opinions of authors expressed herein do not necessarily state or reflect those of theUnited States Government or any agency thereof.


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1. INTRODUCTIIONIn 1975 Lee and Kesler proposed an equation of state for pure non-polar

substances and described mixing rules for calculating pressure-volume-temperature(p-v-t) and thermodynamic properties of mixtures, but binary interaction coefficientswere not mentioned in this original work. In order to calculate the pseudocriticaltemperature of mixtures, the authors proposed the well known equation:

where: T- is the pseudocritical temperature of a mixture, K;

Vcij = (1/8) Vr V p ;Tcij= Pd J ;

V,, Vciare critical molar volumes of the components i and j correspondingly,cm3/gm01;T,, Tciare critical temperatures of the components i and j correspondingly,K; xi, X,are molar fractions of the components i and j correspondingly; xii equals one.

In 1978 V. J. Pliicker, H. Knapp, and J. Prausnitz extended this work bypublishing work applying the Lee and Kesler equation of state to phase equilibriumanalysis in mixtures of non-polar substances with a considerable difference in critical

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parameters? For this purpose the authors estabbhed that equation (1) did not agreewith the experimental data, and they offered another formula for the pseudocriticaltemperature of a mixture:

q is an empirical coefficientand K~ is a binary interaction coefficient IC) that doesnot depend on composition, pressure and temperature. The authors assigned auniversal numerical value of 0.25 to q which they obtainedby processing experimentaldata. The K~~~~ was considered to be a fitting parameter determined by processingexperimental data exclusively. Numerical valuesof s were calculated by the authors

for approximately one hundred binary combinations of non-polar components, and anempirical correlation of these q ' s for mixtures of hydrocarbons and other non-polarmolecules was established as a function of the parameter (V, Td)/(Vd Td).Henceforth, whenever the Lee and Kesler(LK)quationof state was used for mixtureanalysis, the mixing rule indicated by equation (2) and x i s determined by processingavailable experimental data were used for the thermodynamic property calculations.

In this manner the main advantage of the LK quation, which is the ability tocalculate properties of mixtures by making use of critical parameters of purecomponents, was compromised. The original LK equation provided the opportunity

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2 INTERAcIlON OEFFI IENT CALCULATIONIn this work an attempt is made to predict the values analytically, to

develop a fundamentally grounded method for the value prediction, and toestimate the importance of factors that impact the accuracy of the prediction. Themost simple assumption that makes it possible to obtain the K~~ value is:

T- = Th (3)From this expression we have to find different values of K~~~~ for each value of molarconcentration,x , and then to average them. Table I1presents the xij- values fromreference 2 and the K ~ ~ ~ . ~ ~alues found from equation (3); the formula used foraveraging is:

0.9Kijo25ak = ( c '(ir+ 2y11 ;x 4 . 1 (4)

K~~values were determined over a range of molar compositions from 0.1 to 0.9 in 0.1increments; the sum of values obtained was increasedby two because xiiaZr = 1whenx = 0, and x = 1.0. Table I1 shows that the calculated values track the experimentalvalues well, but, naturally, differ from them, because formula (1) is not sufficientlyaccurate when mixtures of substanceswith large differences in critical parameters isencountered. It is evident that the multiplier, K , needed for Tciin equation 1)differsmore from 1.0 with greater differences in critical parameters of mixture components.The equations given in reference 1 were used to find the value of the multiplier.

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Mixing rules of the pseudocritical parameters for a binary mixture assume the form:

2- = 0.2905 - 0.085 ,

s 9)- 5.92714 6.09648/Tk+ 1.28862hTk - 0.16934nLa = 15.2518 - 15.6875/Tk - 13.47211nTk + 0.4357 ;where:

k = l a w ep, = critical pressure, atmT b = the saturatedtemperature under 1 atm,K.w = the acentric. factorZ- = the pseudocritical COmpressiWity factor of the mjxture

Tbr =Tdrc

= L? i f q = 1.0

Then, knowing that In PIbr= In(1) - InP, = -InPo and using the following functions:A = 15.2518 - 15.6875fI'bf - 13.4 721h(Tbf) + 0.43577Th6,B = -5.92714 + 6.O9648flbr+ 1.288621n(Th) - 0.169347T;

and using equations (7) and (9) equation (10) can be derived.

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where:P M = the pseudocritical pressure of a mixture, atm,= f,[(TJI JJ for a mixture

= fA( I flJJ for amixturek i KBDixA, = f,[(TflJ,] for a component 14 = f,[(TJI J.J for a component 2B, = f& flJ,] for a component 1= f&Tfl&J for a component 2

(TDJmks the pseudoparameter of a mkture, and the functions of fi and fi are thoseshown for A and B above. Equation (10) fits the hypothesis that the LK equation isbased on: that a mixture is a pseudosubstance. Using equation (10) the followingmaybe determined:

where: R = 8204, gas law constant atma3~mole-oIc]).If equation (11) is set equal to equation (9, n equation wit two unknowns,

(TJTJmi. nd rcGt0, results. The dependenceof (TdT,),, on x is definedby the changeof two parameters: T and Tad both are pseudoparameters. The dependence ofTCmkn molar composition is given by equation (5) ; an equation similar to equation( 5 ) Cart be written for the pseudoparameter Tam&:

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where: = l/ d,,,, bzare normal boiling temperatures of the components1 and 2. Setting equation (11)equal to equation ( 5 ) and implementing equation (12),will result in an equation with only one unknom wijl. . With these uijla0 alues, the aWzvalues may be calculated by setting equation (11) equal to equation (2) for each valueof x and subsequent averaging in accordance with equation (4).

3. RESULTSIn Figure 1, the xiiO.% values are presented, which have been obbined by

processing experimental data formixtures of hydrocarbons containing com ponen ts with2-9 carbon atoms in their structure, and the xiio= values obtained by the methoddescribed above? When the K ~ ~ ~ . ~ ~alues were obtained, equation 8) was used todetermine ZcmirFor some mixtures, the experimental data show values that slightlydiffer from calculated values. his can be explained with th e he lp of Figure 2. Thisfigure shows that the actual 2 alues of components and, consequently, Z-) aresometimes different from the values obtained from equation (8).

T h e maximum divergence between the calculated and experimentd 5 valuesthat is K ~ ~ ~ ~ ~ )as approximately & 0.4 with an average divergence f 0.1%,which

may be cons idered quite satisfactory see Figure 1).

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In Figure 3. the change of the calculated (IJIJli. values is plotted against thepseudocritical volume of the mixture, Vd for ethane-propane and ethane-nonanemixtures. The change of the (TOc) d u e against Vc or pure hydrocarbons C, Cis also shown. The functions almost coincide which confirms the validity of thehypothesis descriiing a mixture as a pseudosubstance.

4. CONCLUSIONSThe method outlined above makes it possible to obtain binary interaction

coefficients values, K ~ ,or the LK quationsof statewit an average deviationof 0.1using the critical parameters of mixture components, and their normal boilingtemperatures, T,,. he method does not have any restrictions imposed by the natureof the components and may be recommended as a general method for calculating theK~~values when theLK equation of state is applied to mixtures of non-polar substances.

5. ACKNOWLEDGEMJ3UT3Research sponsored by the Office of Building Technologies, U.S.Department

of Energy under contract No. DE-AC05-840R21400 ith Oak Ridge NationalLaboratory, managed by Martin Marietta Energy Systems, Inc.REFERENCES1.2.3.

B.I. Lee and M.G.Kesler, IChE J., 21: 150 1975).U. Pliicker, H. Knapp, and J.M. Prausnitz, Int. Chem. Proc., 7:324 (1978).S.W.WaIas,Phase Equilibria in ChemicalE n g k e h g , Butterworth Publishers,Boston (1985).

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REFERENCES1.

2.3.

B.I. Lee and M.G. esler, AIChE J., 21: 150 (1975).U. Pliicker, 3. Knapp, and J.M. Prausnitz, Int. Chem. Proc., 7: 324 (1978).S.W. alas, Phase EquiZibria in Chemical Engineering, Buttenvorth Publishers,Boston (1985).

4. M. Benedict, G.B. Webb, and LC. Rubin, Chem. Eng. Phys., 10:747 (1942).5. ICs. Pitzer and G O Hultgren, J. Am. Chem. Soc., @: 4793 (1958).6 . K. Striim, State and 7kansp rt Prop of High Temperature WorkingF W ndNonazeotropic Mixture: h B, & C, Final Report, E A nnex XIII,

Chalmers University of Technology,Giiteborg, Sweden, 1992.

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Table L VariatMlls in LReKesler-Plikker InteractionCoefIicient from Experimental DataMixture

R-22/R-114

R-22/R-152a

AuthorHacksteinKruseStr6mValtzRadermacher~

LameStrumValtzKruseRadermacher~LavueStramKruseRadermacher

q Y)0.9630.9750.9790.973

1.0131.014-----e--

----0.9751.0420.9730.9971.0961.0461.038

----I0.97

Table IL Comparison of 1IF;rCperimental to CalculatedLed ble r-P Bc ker heraction CoefficientsUsing Equation (3)Mixture

ethane & ethane & ethane & ethane & ethane & ethane & ethane &propane n-butane n-pentane n-hexane n-heptane n-octane n-nonanec, c c4 c, c6 c,c, c 8 c,c,

1.01 1.029 1.064 1.106 1.143 1.165 1.2141.0113 1.0333 1.0605 1OB73 1.1147 1.1417 1.1676

KijarpKijuic


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1.30

1.25=5s

1 oo

p . . ...

0

s

0.46 a


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0.60.5

30.4

2 .30

ef5 0.2IL

w0 0.1

, - . .... .

.

I I I t0.24 0.25 0.26 0.27 0.28 0.29 0.30COMPRESSIBIUP/FACTOR (Zc )


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FIGURE CAPTIONScpart I

Figure 1. Comparison of experimental to calculated interaction coefficients for mixtures ofethane with longer, straight-chain alkanes through n-nonane, and relative deviationof experimental versus calculated K ~ S : - C ~ . ~;0 K ~deviation. + - ~ , , ~;

Figure 2.Experimental and calculated compressibility factors, 2 obtained from equation(8), plotted against the acentric factor, a or methane (1) through n-decane IO):0 - xperimental; alculated.

Figure 3. The reduced boiling temperature, Tdr, of pure substances C -C and thereduced pseudo-boiling temperature, (TJIJra for mixturesof ethanehonane andethane/propane plotted against critical volumes (VeVc J:+ - TJIJ- for ethanehonane; 0 - TbT,),, for ethane/propane;- br for C,-C,

an analytical method of predicting lee-kesler interaction parameters

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