Developed Correlations for Prediction of The Enthalpies of Saturated Vapor Liquid Coexisting Phases

Transcription

1 Nahrain University, College of Engineering Journal (NUCEJ) Vol.13 No.2, 2010 pp Developed Correlations for Prediction of he Enthalpies of Saturated Vapor Liquid Coexisting Phases Mahoud Oar bdullah Cheical Engineering Departent Nahrain University bstract In this study a new criteria is adopted by the use of Rectilinear Diaeter principle to express the saturated vapor and liquid enthalpies at the coexisting phases for pure substances and ixtures, this requires another relation that used to calculate enthalpies of vaporization at the corresponding teperatures. he relationships between saturated vapor enthalpy, saturated liquid enthalpy and pressure are investigated for any pure substances. hese investigations show that the saturated vapor and liquid enthalpies cannot be described by pressure as a siple general relation. New siple correlations are developed for pure substances and ixtures. hese correlations are based on the principle of Rectilinear Diaeter depending on the law of corresponding states, where D is introduced as a diensionless ter including the average reduced saturated enthalpies. he relations developed are: For Pure Substances For Mixtures D D he constants and are correlated by the following relations: b c he second relation developed by fitting of literature values using Maxiu Likelihood Principle for twenty pure substances out of thirty, and then applying the successfully to the reaining coponents, which indicates the generality of this correlation. hese correlations can be applied successfully up to the critical region. o use this ethod it requires the values of critical teperature and noral boiling point teperature, and any suitable correlation used to calculate enthalpies of vaporization. b c 2 Khalid Farhod Chasib l-jiboury Cheical Engineering Departent University of echnology he present new ethod of calculation of saturated enthalpies copares favorably with any equation of state, for exaple when using Lee Kesler equation of state. he coparison shows that the accuracy of the proposed ethod is better than that of Lee and Kesler at the saturation region. his is in addition to the fact that the present correlations are straight forward, easier, and sipler as copared with that of Lee and Kesler ethod. hese correlations can be further developed to be used directly for design purpose of distillation operations, and other processes that involves vaporization, condensation phenoena. Keywords: vapor-liquid Equilibria, Saturated Enthalpy, Rectilinear Diaeter principle Introduction Enthalpy is one of the ost iportant therodynaic properties of fluids, which can be explicitly defined for any syste by the atheatical expression; U PV 1 where U = internal energy, P = absolute pressure, V = volue. he units of all ters of this equation ust be the sae. hough enthalpy is often applied to a unit ass or to a ole, like volue and internal energy, enthalpy is an extensive property (i.e. depends on the quantity of the aterial involved); also it is a state function which is independent on the path followed. Enthalpy is used as a therodynaic property because the U + PV group appears frequently, particularly in probles involving a flow processes such as extraction, distillation, huidification, absorption, petroleu refinery processes, refrigeration cycles, power plants. NUCEJ Vol.13, No.2 Correlations for Prediction 116

2 For pure fluids the equation used to calculate enthalpy is not satisfactory when applied at the saturated region; in addition to the fact that for ixtures, ost of the equations used to estiate pure fluid saturated enthalpies can not be applied or if they applied the results is in an enorous deviation fro the experiental data. hus, in order to estiate the saturated vapor and liquid enthalpies for pure coponents and ixtures, it is essential to introduce a suitable correlations. It is the ai of this work to develop these correlations for prediction of coexisting saturated vapor and saturated liquid enthalpies for pure coponents and ixtures. Eploying the principle of rectilinear diaeter based on the law of corresponding states which greatly facilitate the procedure for correlating the enthalpies of saturated vapor and saturated liquid at the coexisting phases for pure coponents and ixtures within the teperature, using the reduced paraeters (reduced teperature, and reduced saturated enthalpies). he correlations developed by this ethod can be used for the prediction of saturated enthalpies at the coexisting vapor and liquid phases for pure coponents and ixtures, together with any other correlation used for the estiation of the latent heat of vaporization, without direct reference to the nature of the vaporization process, the agnitude and type of the interolecular forces, and the olecular structure. It should be pointed out, that serious efforts have been ade to get literature data for saturated vapor and saturated liquid enthalpies which are required in developing a generalized correlations for enthalpies of the saturated coexisting phases. Unfortunately, such data are liited in the literature, and are available only for a few coponents and ixtures. heory Rectilinear Diaeter Principle Many years ago Cailletat and Mathias [1] observed that for pure fluid the arithetic average of the densities of saturated vapor and saturated liquid is, to a close approxiation, a linear function of the teperature; this observation is known as the law (principle) of rectilinear diaeter. Won and Prausnitz [2] reported their observation that an analogous relation appears to exist for soe binary ixtures; when the arithetic average of the olar densities of the saturated liquid and the equilibriu saturated vapor is plotted against pressure at a constant teperature, a nearly straight line is obtained. In a two phase region for a binary ixture the two saturated phases at equilibriu have different copositions, thus at a constant teperature, 1 DV 2 Dsl( x) Dsv( x) P where P is the pressure, and are constants, D sl and D sv are the saturated liquid and vapor densities respectively, and x is ole fraction in the liquid phase. It follows that: D P C C 3 where subscript c refers to the critical state. he accuracy of the critical density deterined by eans of Rectilinear Diaeter Principle (RDP) has been seriously questioned owing to the relatively poor precision of the easured saturated densities close to the critical state. he rectilinear behavior observed for the various saturated properties in the coexistence phases leads one to believe that the rectilinear diaeter principle can be applied to other saturated properties (transport and therodynaic) for pure copounds and ixtures [3]. Developent of Correlations In order to develop a correlation for certain therodynaic property, one has to deterine all the paraeters that influence this property and then studies the influence of each of these paraeters on the property under consideration, using literature data available for this purpose. Saturated enthalpies of coexisting vapor and liquid phases are highly influenced by teperature, pressure, and volue. hus, the first approach is to plot saturated enthalpies of coexisting vapor and liquid phases against saturated pressure and against saturated teperature. hese procedures have been ade for twenty three pure copounds. Figure 1 shows a typical of these literature data of saturated enthalpies of coexisting vapor and liquid phases for ethane and propane (as non-polar hydrocarbons), and for freon-12 and aonia (as polar copounds) against saturated teperature. Figure 2 shows the sae property plotted against saturated pressure for the sae two groups copounds. 2 NUCEJ Vol.13 No.2 bdullah, l-jiboury 117

3 pplying he Rectilinear Diaeter Principle he arithetic average of saturated vapor and liquid enthalpies are plotted against saturated vapor teperatures and saturated pressures for twenty three pure coponents. Figures 3 and 4 show typical of these relations, naely ethane and propane (non-polar copounds), and aonia and refrigerant-12 (polar copounds). Figure (3) ypical relations between average saturated enthalpies and teperature Figure (1) ypical relations between saturated enthalpy and teperature Figure (4) ypical relations between average saturated enthalpies and pressure Figure (2) ypical relations between saturated enthalpy and pressure s seen fro figures 3 and 4 that soe of these relations are siple linear relations, this is especially true for enthalpy teperature relations for any non-polar copounds. hus it sees ore convenient to develop a siple generalized relations based on enthalpy teperature relations rather than enthalpy pressure relations. Developent of he Generalized Correlations of Rectilinear Diaeter Principle In order to develop generalized correlations of Rectilinear Diaeter Principle based on the law of corresponding state conditions, the average saturated enthalpies will be replaced by reduced average saturated enthalpies and the teperature will be replaced by reduced teperatures. NUCEJ Vol.13, No.2 Correlations for Prediction 118

4 Figure 5 represents the relations between average reduced saturated enthalpies and reduced teperatures for the four coponents (ethane, propane, aonia and freon-12). It is ore convenient to plot reduced average saturated enthalpies against (1- r ) rather than against r. his is because the reduced average saturated enthalpies at critical point can be obtained fro the intercept point only. Figure 6 represent the relations between average reduced saturated enthalpies and 1- r (which can be indicated as ), for these four coponents (ethane, propane, aonia and freon-12). Figure (5) ypical relations between average reduced saturated enthalpies and reduced teperature D 4 D sv sv sl sl b 1 6 C 2R 2R C b sv sv sl sl b sv, sl = saturated vapor and liquid enthalpies respectively R = gas constant. c = critical teperature. b = noral boiling point teperature. rb = reduced teperature at the noral boiling point. and are constants. Figures 7 through 11 show the relation between D and for twenty three pure coponents classified to the following groups of substances, as shown in table 1. he reason for dividing the average reduced saturated enthalpies of coexisting vapor and liquid at any teperature by the sae values at the noral boiling point teperature is to obtain a atheatical relation between constant and constant of eq.(4). When eq.(4) is applied at the noral boiling point teperature the following equations will result: D sv sv sl b sl b rb 7 rb 5 rb 1 rb 8 thus the following siple relation is obtained: rb 1 rb 9 Figure (6) ypical relations between average reduced saturated enthalpies and 1- r ttepts have been ade to relate the constant and of eq.(4) with the other paraeter characteristics of pure coponents such as acentric factor, Z c, P c, c and V c. No siple workable relation could be obtained. he fruitful result achieved, based on reduced rectilinear diaeter principle, is the relation obtained between D and. his relation is alost a linear relation for twenty three pure coponents, and can be represented by the following equations: NUCEJ Vol.13 No.2 bdullah, l-jiboury 119

5 Figure (7) Relation between D. and 1- r of group Figure (10) relation between D. and 1- r of group D Figure (8) Relation between D. and 1- r of group Figure (11) Relation between D. and 1- r of group E Values of the constant are plotted against the values of constant for the twenty coponents used in this investigation including (ethane, ethane, propane, n-butane, i-butane, nitrogen, oxygen, argon, chlorine, freon-11, freon-12, freon-13, freon-21, freon-22, ethyl chloride, benzene, octaflouro cyclobutane, 1,3- butadiene, water and aonia) [1-2, 4-6] as shown in figure 12 Figure (9) relation between D. and 1- r of group C NUCEJ Vol.13, No.2 Correlations for Prediction 120

6 he generalized atheatical relation between constant and constant can be represented by the following equation: Figure (12) relation between the constants and which appears at eq.(4) for the pure copounds Each of the eq.(9) and eq.(10) represents the relation between the constant and constant, fro the two equations, it can be recognized that and constants are independent of teperature. Solving these two equations siultaneously for twenty three pure coponents to obtain the values of and for each coponent. able (1) Groups of substances Extension of he Correlations to he Mixtures he generalized correlations for pure copounds can be extended to involve also ixtures. Saturated enthalpies data are rarely available in the literature especially for the saturated coexisting vapor and liquid phases for ixtures. Experiental data for saturated coexisting enthalpies of ixtures have been taken fro literature and soeties it was necessary to interpolate or extrapolate atheatically those data. he sae procedure that is used for pure copounds has been also used for ixtures with the exception of using ixing rules to calculate therodynaic properties of ixtures fro the therodynaic properties of pure coponents. hus, the procedure is as follows: D 11 where Z i i 12 i Z i i 13 i Z i = ole percent of coponent i in the feed. NUCEJ Vol.13 No.2 bdullah, l-jiboury 121

7 D where sv sl rb 14 sv sl b rb c 15 b sv sl = ddition of saturated vapor and liquid enthalpies at any teperature of the ixture. calculate enthalpies of vaporization); figures 13 through 22 are prepared to show the relation between the enthalpies of saturated vapor and saturated liquid with teperatures for a typical pure coponents using the data fro literatures and the developed ethod. sv sl b = ddition of saturated vapor and liquid enthalpies at the noral boiling point teperature of the ixture. c Z iz j cij 16 i j 0.5 cij ci cj 17 Figure (13) Enthalpy teperature diagra for chlorine i j Z iz j ij 18 ij Z k k 19 k Mt Z iz jmtij 20 i j Mt ij Z k Mtk 21 k b Z iz j bij 22 i j Figure (14) Enthalpy teperature diagra for nitrogen 0.5 bij bi bj 23 vb Z i vbi 24 i and constants are not functions of teperature as indicated before, so, they can be used at any teperature desired to calculate saturated coexisting vapor and liquid enthalpies. hus, using the sae procedure of calculations (o use this ethod it requires the values of critical teperature and noral boiling point teperature, and any suitable correlation used to Figure (15) Enthalpy teperature diagra for ethane NUCEJ Vol.13, No.2 Correlations for Prediction 122

8 Figure (16) Enthalpy teperature diagra for ethane Figure (19) Enthalpy teperature diagra for aonia Figure (17) Enthalpy teperature diagra for Freon-21 (CFCl 2 ) Figure (20) Enthalpy teperature diagra for water Figure (18) Enthalpy teperature diagra for Freon-22 (CF 2 Cl) Figure (21) Enthalpy teperature diagra for 1,3-butadine NUCEJ Vol.13 No.2 bdullah, l-jiboury 123

9 Figure (22) Enthalpy teperature diagra for Octafluoro Cyclobutane Figures 23 through 26 have been prepared to show the relation between the enthalpies of saturated vapor and saturated liquid with teperatures for typical syste ixtures using the data fro literatures and the developed ethod. Figure (25) Enthalpy teperature diagra for (79.1 percent) nitrogen oxygen syste Figure (23) Enthalpy teperature diagra for (90 percent) benzene hexadecane syste Figure (26) Enthalpy teperature diagra for (35.2 percent) benzene (51.5 percent) octanecane - tetraline syste Discussion he correlations of the coexisting saturated vapor and liquid enthalpies that developed in the proposed work are generally on the sae line adopted in the literature [4,5] which have reported the correlations of densities of saturated vapor and saturated liquid using the rectilinear diaeter principle and the law of corresponding states. he resulted correlation for pure substances in this study is: D 4 Figure (24) Enthalpy teperature diagra for (61.2 percent) pentane cyclohexane syste It should be entioned that, this relation has been extended to involve ixtures also, by eans of using suitable ixing rules. he resulted correlation for ixtures is: NUCEJ Vol.13, No.2 Correlations for Prediction 124

10 D 11 owever, the use of reduced rectilinear diaeter principle to deterine the saturated vapor, and saturated liquid enthalpies is a new approach which has not been used previously in the literature. Usually, the ethod adopted in literature to calculate saturated vapor and liquid enthalpies depended on direct use of the equations of state, ost of these equations of state are epirical equations, very coplicated, in addition to the fact that these equations of state are not suited to the saturated region which resulted in a large errors when applied to calculate enthalpies at this saturated region if they copares with experiental data. When using the available correlations in the literature to calculate the saturated enthalpies at coexisting vapor and liquid phases for pure substances and ixtures, the calculation ethods using either hand calculator or coputer progras are always associated with difficulties copares with the ethod developed in this study. Edister and Lee [6], used Redlich- Kwong equation of state to predict enthalpy using equation: R C Z 1.5 a 1 br a V 1 25 where a and b are constants. he ethod requires to obtain six values of constants in order to get the enthalpy, that akes it not siple ethod as that in the present work, they presented their results as graphs fro which reasonable accurate data can not be obtained. he correlations of this study are based on analyzing the relation between the saturated vapor, saturated liquid enthalpies and teperature, where the values of D and the values of are used as the basic generalized functions. he resulted odels of correlations are very siple and can be applied easily to hand calculator or using a coputer progra. hese correlations can be applied successfully up to the critical region and for soe of the exained substances and ixtures (especially the polar substances) the deviation of the experiental data increases with linear relation of eq.(4) and eq.(11). For pure substances, the D of saturated vapor enthalpy fall in range of % to % and the D of saturated liquid enthalpies fall in range of % to % for the corresponding 319 literature data point of saturated vapor and saturated liquid enthalpies. It is indicates also that the over all D of saturated vapor and saturated liquid enthalpies of these points are % and % respectively. For ixtures, the D of saturated vapor and saturated liquid enthalpies fall in range of % to % and % to % respectively, for the corresponding 165 literature data point of saturated vapor and saturated liquid enthalpies, and that the overall D of saturated vapor and saturated liquid enthalpies of these points are % and % respectively. he relationship of eq.(4) that resulted fro the general fitting of literature saturated vapor and saturated liquid enthalpies values at different teperatures is linear relation. his akes it possible to obtain one of the relations that relates the two constants and for each pure copound which are defined in ter of the corresponding critical teperature and noral boiling point teperature representing at the following equation: rb 1 rb 9 nother relation has been presented that relates the two constants and by the fitting of the literature values representing at the following equation: his relation together with relation of eq.(9) represent the ost accurate relations that correlate the two constants and. Only twenty (ethane, ethane, propane, n-butane, i-butane, nitrogen, oxygen, argon, chlorine, freon-11, freon-12, freon-13, freon-21, freon-22, ethyl chloride, benzene, octaflouro cyclobutane, 1,3- butadiene, water and aonia) out of thirty were used to obtain the constants and. the resulted absolute average deviation indicate that, these relations represent a generalized correlations for saturated coexisting vapor and liquid enthalpies. he reason for using the diensionless ter D is to obtain as uch as possible a linearized siple relation (eq.(4)), fro which the constants and were obtained when applied it at the noral boiling point. ccuracy of he Developed Correlations he correlation of the enthalpies of saturated vapor and liquid coexisting phases that 2 10 NUCEJ Vol.13 No.2 bdullah, l-jiboury 125

11 developed which are represented by the eq.(4) and eq.(11) have been applied for various pure copounds and ixtures. Experiental data of enthalpies of saturated vapor and liquid for these pure coponents and ixtures were obtained fro literature. he overall percent average absolute deviations of saturated vapor and saturated liquid enthalpies for 83 literature data point of these seven pure coponents are % and % respectively for the present work, and using the equation of state for exaple that for Lee and Kesler [7] ethod are % and % respectively. he overall percent average absolute deviations of saturated vapor and saturated liquid enthalpies for 58 literature data points of these six binary ixtures are % and % respectively, and that for Lee and Kesler ethod are % and % respectively. hese coparisons indicate that the present ethod of calculating the saturated vapor and saturated liquid enthalpies is straight forward, easy, and siple as copares with that of equation of state which involves coplexity and less accuracy. Conclusion New siple relations are developed for the prediction of the saturated vapor and saturated liquid enthalpies of pure substances and ixtures. hese relations are based on the rectilinear diaeter principle and the law of corresponding states. hey are: [ For Pure Copounds D 4 For Mixtures D 11 Where D represents the average reduced saturated enthalpies at any teperatures divided by the average reduced saturated enthalpies at the noral boiling point. he ters D and are defined as follows: D sv R C sv 2R b sl 2 5 sl b 1 6 C nd the constants and are defined in ters of critical teperature and noral boiling point teperature, using the following relations: 1 9 b c b c rb he average absolute deviations when using the above correlation in range fro % to % for saturated liquid enthalpies and range fro % to % for saturated vapor enthalpies for 319 data points of 23 pure substance. For ixtures the average percent absolute deviations range fro % to % for saturated liquid enthalpies, and in the range of % to % for saturated vapor enthalpies of 165 data points of 13 binary ixtures and 1 ternary ixture. hese correlations are applied successfully at the saturated region up to the critical teperature. hey copare favorably with any equation of state. Noenclature D verage bsolute Deviation, Constants of equation 3 and 4 average reduced saturated D enthalpies at any teperatures divided by the average reduced - saturated enthalpies at the noral boiling point Enthalpy J/g Mt Molecular Weight g/gol P Pressure bar R Gas Constants eperature K U Internal Energy J/g V Molar Volue c 3 /ol z Copressibilty factor Greek Letters Defined by equation 6 centric factor rb 2 10 NUCEJ Vol.13, No.2 Correlations for Prediction 126

12 SUSCRIPS b c i j r V sl sv Superscripts Noral boiling point Critical point Coponent i Coponent j Reduced value verage value Saturated liquid Saturated Vapor Mixture * Ideal gas state References [1] Cailletat, L. V., Mathias, P., Equation of State Methods for Coputing Phase Equilibria and Enthalpies, International Cheical Engineering, 21, 1, [2] Won, S. D., Prausnitz, J. M., herodynaic Function of Gases, vol. 2, utter Worths, London, [3] Das. R. and Read C. O., J of Che & Eng. Data, 18, 3, , [4] Vargaftik N.., able of herophysical Properties of Liquid and Gasas, 2nd edition, [5] Vasseran.., herophysical Properties of ir and Its Coponents, Nauka Press., [6] Edister, W.C., Lee,. K., pplied ydrocarbon herodynaics, vol 1, 2nd ed., Gulf Publishing Copany, [7] Lee,. I. and Kesler M. G., ICE J, 21, 510, ppendix Statistical Measureent nd nalysis of Dispersion o know the applicability and accuracy of any proposed correlation it is very iportant to know how this correlation fits the experiental data which is done by coparing the obtained results fro the proposed correlation with the experiental data. he various easureent of dispersion or variation are available, the ost coon being the verage bsolute Deviation. D D n i 1 M obsd i n M calcd i Where M is an intensive property and n is the nuber of data point. D NUCEJ Vol.13 No.2 bdullah, l-jiboury 127

13 D D b c Maxiu Likelihood Principle b c 2 Lee- Kesler Lee- Kesler NUCEJ Vol.13, No.2 Correlations for Prediction 128

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