Appl. Math. Inf. Sci. 6-3S, 777-78 (01 777 Evaluation of Cubic EOS Modls in nar Critical Rgions of n-alkans Jibom Kim 1 and Joonhyon Jon 1, 1 Dpt. of Information and Communication Enginring, Dongguk Univrsity, Soul, Kora Corrsponding author: mmory@dongguk.du Applid Mathmatics & Information Scincs An Intrnational Journal @ 01 NSP Natural Scincs Publishing Cor. Rcivd 01 Jun. 011; Rvisd 1 Jul. 01; Accptd 11 Sp. 01 Abstract: Th purpos of this papr is to valuat th prdiction accuracy and thoroughnss of four wll-known Cubic Equation Of Stat (CEOS modls, which ar calld Png-Robinson (PR, Modifid Png-Robinson (MPR, Modifid Rdlich-Kwong (MRK and Gnralizd van dr Waals (GvdW, to rprsnt prssur of tn pur n-alkan molculs (with a gnral formula C n H n+, n=1,,3,...,8,10,1 nar thir critical rgions. Th prformanc valuation rsults show that in th critical rgion nar th critical point, all four CEOS modls hav a prdiction accuracy that givs rsults quantitativly similar to th xprimntal vapor-prssur data, whil in th critical rgion far from th critical point, MPR and MRK giv mor accurat prdictions of vapor prssur than PR and GvdW. In contrast, byond th critical rgion, GvdW provids much mor accurat and rliabl prdictions of prssur than th othr CEOS modls. Espcially, it is found that GvdW accuratly rproducs th xprimntal prssur data in and byond th critical rgion as th molcular siz of th n-alkans incrass. Th prformanc valuation of th CEOS modls through this approach givs an additional insight into th physical significanc of an accurat prdiction in and byond th critical rgion for pur alkan molculs with linar carbon chains. Kywords: Equation of stat, Critical point, Critical rgion, Suprcritical rgion, EOS valuation. 1 Introduction Th stimation of quilibrium phas bhavior of complx mixturs ovr broad tmpratur and prssur rangs is an important problm in chmical procss dsign. Major rsarch has bn conductd in ths phas quilibria studis, whrby an Equation Of Stat (EOS, which is a thrmodynamic quation dscribing th stat of mattr undr a givn st of physical conditions, has bn stablishd by van dr Waals [1. Th van dr Waals EOS has mad significant contributions to th dvlopmnt of novl Cubic EOS (CEOS modls [ as proposd by Rdlich-Kwong [3, Soav [4 and Png-Robinson [5. Ths famous CEOS modls wr univrsally usd in calculating fluid thrmodynamic proprtis du to thir simpl form, fw paramtrs and analytical solutions [6. Howvr, thir simplifid original forms had limitd applicability du to thir inaccurat liquid dnsitis, poor rprsntations in th nar-critical rgion and unrliability for fluids with larg molculs. Th thrmo-physical proprtis of pur matrials nar thir critical points ar indispnsabl in dscribing fluid phas bhavior, prdicting physical proprtis, dvloping EOS modls, and so on. Many rsarchrs faild to masur proprtis in th critical rgion dirctly, bcaus th dnsity fluctuations hav a significant influnc on fluid proprtis [7. Schmidt and Wnzl concludd svral yars ago that accurat rprsntation of saturatd fluid dnsitis in th critical rgion always lads to a loss of accuracy in dscribing saturatd vapor dnsitis and vic vrsa [8. In ordr to improv th vapor-prssur prdiction of Nonhydrocarbon and Hydrocarbon Compounds, Chorng t al. [9, 10 has dvlopd modifid Png- Robinson and Rdlich-Kwong CEOS modls using a modifid alpha function (which is a gnralizd tmpratur and acntric factor dpndnt function
778 Jibom Kim and Joonhyon Jon : Evaluation of Cubic EOS Modls in nar... of th attractiv trm. Rcntly, a gnralizd van dr Waals (GvdW CEOS for fluid stats was proposd [11. It has a highly simplifid Ditrici's form of P = [RT/(V-b - a(b/v c that is ralizd by a nw modl potntial construction that dscribs intrmolcular intractions. On th basis of th modl potntial construction, it is shown that th paramtrs a, b, and c hav physical intrprtations as an intrnal prssur, a void volum, and a dimnsionlss valu that rprsnts an inharmonic intrmolcular cll potntial, rspctivly. Howvr, thr is still controvrsy as to whthr GvdW can accuratly rprsnt saturatd fluid dnsitis in th nar-critical rgion [1. It is also unclar if it can b applid to th thr othr CEOS modls, namly, Png-Robinson (PR, Modifid Png-Robinson (MPR and Modifid Rdlich- Kwong (MRK. Th main focus of this papr is to valuat th prformanc of th four CEOS modls in and byond th critical rgion of pur molculs. For this purpos, ths modls ar applid to tn pur n- alkan molculs: mthan, than, propan, butan, pntan, hxan, hptan, octan, dcan and dodcan. Ths pur molculs hav uniqu lctronic, optical and physic-chmical proprtis bcaus of thir linar carbon chains. It is wll known that most EOS modls hav difficultis in making accurat prdictions of prssur nar thir critical rgions. Rsults and Discussions.1 Application of EOS modls to n-alkans Evr sinc van dr Waals proposd th CEOS modl in th form of P = [RT/(V-b - (a/v, svral modifications of it hav takn plac [1. Out of many volutions, as mntiond bfor, PR, MPR, MRK and GvdW hav bn th most succssful in vapor-liquid quilibria calculations of polar as wll as non-polar proprtis of fluids. Th major modifications hav bn as follows: modifications of th scond or attractiv trm or and/or th introduction of tmpratur dpndnc on th paramtr a or us of th xprimntal critical comprssibility factor Z cc for th paramtrs a, b and c [5, [9, [10, [11. Tabl 1 and show th basic forms of th four CEOS modls and rlatd symbols, rspctivly. Also, Tabl 3 indicats th critical proprtis of th tn pur molculs [13. Tabl 4-7, basd on Tabl 3, dnot th calculatd paramtrs of ach CEOS usd to apply th four CEOS modls to th tn pur molculs. Tabl 1 : Th basic forms of th four CEOS modls usd in th comparison PR(Png Robinson Equation of Stat [5 RT a V b V bv b R Tc a 0.457535, RTc b 0. 077796 and th alpha function is, 0.5 ( 1(1 Tr 0.37464 1.546 0.699 T Tr Tc MPR(Modifid Png Robinson Equation of Stat [9 RT a V b V bv b R Tc a 0.457355891, RTc b 0.0777960739039 and th alpha function is, a( T ( T a( Tc (0 (1 (0 ( (0 0.99615 0.79615 0.40119[1 Tr T r 9.98471 1.98471 0.04955[1 Tr (1 T r acntric factor dfind at T r = 0.7 MRK(Modifid Rdlich Kwong Equation of Stat [10 RT a V b V ( V b R Tc a 0.474803354, RTc b 0.086640349965 and th alpha function is, a( T ( T a( Tc (0 (1 (0 ( (0 1.30000 1.10000 0.441411[1 Tr T r 10.318.3178 0.03580[1 Tr (1 T r acntric factor dfind at T r = 0.7 GvdW(Gnralizd van dr Waals Equation of Stat [11 RT a V b b V c Z 4 cc Z 1, c1 cc c 1 a P, c c 1 c 1 b V c c 1 Tabl : List of symbols usd in CEOS modls in Tabl 1 Symbol Maning of symbol a, b, c Paramtrs for EOS (PR, MPR, MRK and GvdW c
Jibom Kim and Joonhyon Jon : Evaluation of Cubic EOS Modls in nar... 779 P R T V α ω c r Prssur Gas constant Tmpratur Molar volum Alpha function Acntric factor Subscript for critical proprty Subscript for rducd proprty Tabl 3 : Critical proprtis of th tn pur molculs usd in this work Formula ω M/W (g/mol Z T c P c V c cc (K (bar (cm 3 /mol CH 4 0.011 16.043 0.88 190.56 45.99 98.6 C H 6 0.099 30.069 0.84 305.33 48.7 145.6 C 3H 8 0.15 44.096 0.80 369.83 4.48 0.48 C 4H 10 0.199 58.13 0.74 45.1 37.97 54.9 C 5H 1 0.49 7.148 0.31 469.70 33.71 310.98 C 6H 14 0.305 86.175 0.70 507.8 30.4 369.56 C 7H 16 0.351 100.01 0.98 540.13 7.31 431.90 C 8H 18 0.369 114.8 0.97 569.3 4.978 486.8 C 10H 0.484 14.81 0.49 617.70 1.030 609.76 C 1H 6 0.573 170.334 0.4 658.10 18.170 751.86 Tabl 4 : Paramtrs usd in PR for individual molculs Molcul Formula a(t c (bar b (cm 3 Mthan CH 4 49543.899 6.77575 Ethan C H 6 6046009.089 40.48711 Propan C 3H 8 10180936.5 56.9834 Butan C 4H 10 15059090.04 7.430948 Pntan C 5H 1 0706413.93 90.15350 Hxan C 6H 14 688435.9 108.645 Hptan C 7H 16 337675.5 17.69507 Octan C 8H 18 41057075.48 147.47888 Dcan C 10H 5738648.04 189.98979 Dodcan C 1H 6 7539154.68 34.7659 Tabl 5 : Paramtrs usd in MPR for individual molculs Molcul Formula a(t c (bar b (cm 3 Mthan CH 4 493610.683 6.775301 Ethan C H 6 604051.785 40.48749 Propan C 3H 8 101747.78 56.98395 Butan C 4H 10 1504933.40 7.431017 Pntan C 5H 1 069860.93 90.153587 Hxan C 6H 14 6866639.34 108.646 Hptan C 7H 16 33704650.3 17.69519 Octan C 8H 18 410300.33 147.53810 Dcan C 10H 5734890.78 189.98997 Dodcan C 1H 6 7534178.55 34.768 Tabl 6 : Paramtrs usd in MRK for individual molculs Molcul Formula a(t c (bar b (cm 3 Mthan CH 4 331335.186 9.81961 Ethan C H 6 5648856.103 45.085046 Propan C 3H 8 951166.546 6.698699 Butan C 4H 10 14069881.71 80.665365 Pntan C 5H 1 1934641.63 100.4073 Hxan C 6H 14 51185.05 10.57173 Hptan C 7H 16 3151169.09 14.17 Octan C 8H 18 38360099.70 164.453 Dcan C 10H 5361685.80 11.58905 Dodcan C 1H 6 7043917.04 60.91066 Tabl 7 : Paramtrs usd in GvdW for individual molculs Molcul Formula c a (bar b (cm 3 Mthan CH 4 1.74973 1697.069580 6.3716 Ethan C H 6 1.70435 1853.59399 37.900089 Propan C 3H 8 1.69561 1638.07556 51.578754 Butan C 4H 10 1.687634 1480.64174 65.0978 Pntan C 5H 1 1.671981 1347.804443 78.00554 Hxan C 6H 14 1.665549 18.683716 9.93816 Hptan C 7H 16 1.65497 11.487305 106.5407 Octan C 8H 18 1.637403 1057.753540 117.49171 Dcan C 10H 1.616515 9.695930 143.68490 Dodcan C 1H 6 1.616339 796.895000 177.31190. Comparison nar th critical-point rgion This study mad a comparison of four CEOS modls, which ar PR, MPR, MRK and GvdW. It was prformd by valuating th prdiction accuracy of ths modls in th nar-critical rgions of th aformntiond tn pur alkan molculs. All of th highly accurat xprimntal data on th pur molculs usd in this work was obtaind from th National Institut of Standards and Tchnology (NIST [13. Th prformanc valuation of th CEOS modls is prformd in and byond th critical rgion of th rducd volum rang 0.7 < V r < 1. (: th rducd tmpratur is thn T r = 1, whr 0.7 < V r < 1.0 and 1.0 < V r < 1. corrspond to th suprcritical and critical rgions, rspctivly. Figur 1 shows th prdiction rsults obtaind from applying th four CEOS modls to th tn pur n-alkans in th nar-critical rgion. Th solid lin indicats th xprimntal data givn by NIST and th " " symbol is usd to mark th xprimntal critical point on th solid lin. From ths rsults, it is clar that in th calculation volum rang byond th critical rgion, GvdW provids much mor rliabl and accurat prdiction of suprcritical prssur than th othr thr CEOS modls. Whras, in th critical rgion, th prdition graphs of PR and MPR ar gnrally closr to th xprimntal curvs than MRK and GvdW. Espcially, th prdictd lin obtaind from
780 Jibom Kim and Joonhyon Jon : Evaluation of Cubic EOS Modls in nar... GvdW xhibits th largst rprsntation rror in th critical rgion far from th critical point, although it is most xact in th rgion nar th xprimntal critical point. Howvr, GvdW shows mor improvd accuracy in and byond th critical rgion as th molcular siz of th alkans incrass (i.., as th numbr of carbons in th carbon backbon incrass and thir boiling and mlting points also incras. Tabl 8 shows th avrag absolut dviation prcntag (AAD% btwn th xprimntal and prdictd data in and byond th critical rgions. Basd on th prformanc valuation rsults so far, th prdiction accuracy of th four CEOS modls to rprsnt vapor prssur in th critical rgion for pur n-alkans is similar and lis within allowabl rror limits, whil for suprcritical prssur byond th critical rgion, th prdiction accuracy of GvdW is much bttr than th othr CEOS modls. This is bcaus PR, MPR and MRK wr ralizd by introducing an alpha function, which is a linar function of th acntric factor at a constant rducd tmpratur, whrby paramtr a is introducd to improv th prdiction accuracy from th tripl point to bfor th critical point. On th othr hand, GvdW was optimizd by introducing th critical comprssibility factor Z cc into thr paramtrs, which can b physically intrprtd as an intrnal prssur, a void volum, and a dimnsionlss valu that rprsnts an inharmonic intrmolcular cll potntial. (b (c (d (a (
Jibom Kim and Joonhyon Jon : Evaluation of Cubic EOS Modls in nar... 781 (f (g (h (i (j Figur 1: Th CEOS-valuation rsults nar th critical rgion of pur n-alkan molculs: (a Mthan, (b Ethan, (c Propan, (d Butan, ( Pntan, (f Hxan, (g Hptan, (h Octan, (i Dcan, (j Dodcan Tabl 8 : Avrag absolut dviation prcntag (AAD% of prdicting alkan prssurs in th rang of 0.7 < V r < 1. from CEOS modls Rang formula GvdW PR MPR MRK (V r CH 4 C H 6 C 3H 8 C 4H 10 C 5H 1 C 6H 14 C 7H 16 C 8H 18 C 10H C 1H 6 0.7 ~ 1.0 1.378E00 4.094E00 4.471E00 15.43E00 1.0 4.377E-.113E-1 3.749E- 4.886E-1 1.0 ~ 1. 1.698E00 5.166E-1 3.86E-1 3.87E-1 0.7 ~ 1.0 1.706E00 7.690E00 8.105E00 4.44E00 1.0 1.150E-1 1.64E-1 9.509E- 7.995E-1 1.0 ~ 1. 1.041E00.094E-1 5.183E- 3.3E-1 0.7 ~ 1.0 1.715E00 8.03E00 8.435E00 3.93E00 1.0 1.005E-1 1.315E-1 1.331E-1 9.759E-1 1.0 ~ 1. 1.60E00 1.953E-1 4.037E- 3.738E-1 0.7 ~ 1.0 1.654E00 9.488E00 9.914E00 7.30E00 1.0 7.037E-3 9.516E- 1.748E-1 1.149E00 1.0 ~ 1. 9.193E-1 3.37E- 1.35E-1 3.800E-1 0.7 ~ 1.0.066E00 13.36E00 13.81E00 35.61E00 1.0 5.34E- 1.807E-.601E-1 1.554E00 1.0 ~ 1. 4.318E-1 7.83E- 1.154E-1 1.486E-1 0.7 ~ 1.0 1.504E00 14.55E00 15.01E00 38.85E00 1.0 8.417E-.073E-1 7.50E- 1.56E00 1.0 ~ 1. 1.37E00 6.6E- 1.014E-1 3.945E-1 0.7 ~ 1.0 9.15E-1 16.8E00 16.74E00 64.4E00 1.0 1.38E- 3.413E-1 6.98E-1.313E00 1.0 ~ 1. 9.948E-1 1.103E-1.918E-1 4.004E-1 0.7 ~ 1.0 9.067E-1 17.86E00 18.65E00 46.19E00 1.0 1.150E-1 4.065E-1 8.561E-1.997E00 1.0 ~ 1..567E-1 1.71E-1 1.40E-1 1.086E-1 0.7 ~ 1.0 7.57E-1.14E00.6E00 53.18E00 1.0.869E- 9.375E-1 1.5E00 4.354E00 1.0 ~ 1. 1.643E00 7.558E-1 9.17E-1 1.07E00 0.7 ~ 1.0 1.865E00 30.37E00 30.89E00 70.16E00 1.0 4.74E-1 8.8E-1 1.143E00 4.4E00 1.0 ~ 1. 1.75E00.45E-1 4.53E-1 5.48E-1
78 Jibom Kim and Joonhyon Jon : Evaluation of Cubic EOS Modls in nar... 3 Conclusion Hrin w hav dscribd th prformanc valuation of four famous CEOS modls in th nar-critical rgions of tn pur n-alkan molculs with linar carbon chains. It has bn shown that th prdiction accuracy of all four CEOS modls to rprsnt th vapor prssur in th critical rgion is similar and lis within allowabl rror limits. Rmarkably, GvdW givs much mor accurat and rliabl prdiction of suprcritical prssur than th othr thr CEOS modls, and it rprsnts an improvd accuracy in and byond th critical rgion as th molcular siz of th n-alkans incrass. Acknowldgmnts This work was supportd by th Dongguk Univrsity Rsarch Fund of 01 Rfrncs [1 D. Johanns and V. D. Waals, On th Continuity of th Gasous and Liquid Stats(doctoral dissrtation, Univrsitit Lidn, (1873. [ J. O. Valdrrama, Th Stat of th Cubic Equations of Stat, Ind. Eng. Chm. Rs., (003, 4(8, 1603-1618. [3 O. Rdlich and J. N. S. Kwong, On th Thrmodynamics of Solutions, V. An Equation of Stats. Fugacitis of Gasous Solutions. Chm. Rv., (1949, 44(1, 33-44. [4 G. Soav, Equilibrium Constants from a Modifid Rdlich-Kwong Equation of Stat, Chm. Eng. Sci., (197, 7, 1197-103. [5 D. Y. Png and D. B. Robinson, A Nw Two-Constant Equation of Stat, Ind. Eng. Chm. Fundamn., (1976, 15, 59-64. [6 P. Prrot, A to Z of Thrmodynamics, Oxford Univrsity Prss, (1998. [7 A. N. Vulfson and N. A. Skibitskaya, On th Us of th Hnry Law for th Dscription of th Sorption of Mthan and Ethan by Asphaltns undr Suprcritical Tmpratur and High Prssur Conditions, IEEE Russian J. Phys. Chm. B., (009, 3(7, 108-1089. [8 G. Schmidt and H. Wnzl, A modifid van dr Waals typ quation of stat, Chm. Eng. Sci., (1980, 35(7, 1503-151. [9 H. T. Chorng, E. C. John and R. C. John, A nw gnralizd alpha function for a cubic quation of stat Part 1. Png-Robinson quation, Fluid Phas Equlib., (1995, 105(1, 49-59. [10 H. T. Chorng, E. C. John and R. C. John, A nw gnralizd alpha function for a cubic quation of stat Part. Rdlich-Kwong quation, Fluid Phas Equlib., (1995, 105(1, 61-69. [11 L. Sukba, J. Joonhyon, K. Wonsoo and C. Tong-Sk, A Nw Modl Approach for th Nar-Critical Point Rgion: 1. Construction of th Gnralizd van dr Waals Equation of Stat, J. Phys. Chm. B., (008, 11(49, 1575-15741. [1 W. Yong, H. Guo-bo and G. Chang-hua, Saturatd Proprtis Prdiction in Critical Rgion by a Quartic Equation of Stat, Bull. Chm. Soc. Ethiop., (011, 5(, 87-94. [13 NIST, Chmistry Wb Book, National Institut of Standards and Tchnology, http://wbbook.nist.gov/ chmistry/, (011. Jibom Kim was born in Inchon, Kora in 198. H rcivd th B.S. and M.S dgrs in th dpartmnt of information and communication nginring, Dongguk Univrsity in Soul, Kora, in 008 and 010, rspctivly. H is currntly working to gt Ph.D. dgr in th Dpartmnt of Information and communication Enginring, Dongguk Univrsity in Soul, Kora. His rsarch intrsts includ lithiumammonia and calcium-ammonia solutions, and th thory and application of Equation of Stat (EOS. Joonhyon Jon was born in Soul, Kora in 1961. H rcivd th B.S. dgr in lctronic nginring from Dongguk Univrsity, Kora, in 1984 and th M.S and Ph.D. dgrs in lctric and lctronic nginring from Kora Advancd Institut of Scinc Tchnology, Kora, in 1986 and 1991, rspctivly. Sinc 00, h has bn an Associat Profssor in th Dpartmnt of Information and communication Enginring, Dongguk Univrsity in Soul, Kora. H was a Rsarch Scintist at Kora Tlcom Rsarch Cntr from 1991 to 1999. His currnt rsarch intrsts ar in th ara of thrmolctric matrials, in particular, mtal-ammonia and mtal-amin solutions, thrmolctric cll dsign, th thory and application of Equation of Stat (EOS, and th application of thrmolctric clls.