a. Simple fluids with spherical molecules ω=0 as b. Quantum gases having ω<0 as He, H 2, and c. Other nonpolar fluids which are having ω>0 as

Transcription

1 EDICTION AND COELATIONS OF ESIDUAL ENTOY OF SUEHEATED AO FO UE COMOUNDS of. D. Mahmoud O. Abdullah, D. Samad T. Najim, Assist. Let. Shahad Z. Atta,, Chemial Eng. Depatment, ollege of engineeing, Nahain Univesity Abstat The alulation of the entopy (S) fo gases is essential fo desn poess equipments whih involve the alulation of the heat and wok equiements fo industial poesses. edition of auate values of esidual entopy (S ) is neessay step fo the alulation of the entopy. In ode to obtain auate values of (S ), attention has been oiented to alulate it using an Equation Of State (EOS(. In this pape impotant equations of state as Lee-Kesle and iial equation tunated to seond and to thid tems wee used to alulate the esidual entopy fo pue supeheated vapo. In addition to those equations, ubi equations of state epesented by Soave-edlih-Kwong (SK), and eng-obison equations wee also used. These EOS wee tested fo the available 79 expeimental data points of 0 pue supeheated vapo ompounds ( pue nonpola ompounds + 6 pue pola ompounds). The Aveage Absolute Deviation (AAD) fo S of 79 expeimental data points of the all 0 pue ompounds (nonpola and pola) when using equations of Lee-Kesle, eng-obinson, iial tunated to seond and to thid tems, and Soave-edlih-Kwong wee.059,.589,.9686, 5.050, and.08 J/mol.K espetively. It was found fom these esults that the Lee-Kesle equation was the best (moe auate) one ompaed with the othes, but this equation is sometimes not vey pefeable, beause it needs moe time and not easy to apply as ubi equations of state. Noted that SK equation was the losest one in its auay to that of the Lee-Kesle equation in alulating the esidual entopy S of supeheated vapo, but it was developed pimaily fo alulating vapo-liquid equilibium and to oveome this poblem, effots wee dieted towad possibility of modifying SK equation to inease its auay in pediting the esidual entopy as muh as possible. The modifiation was made by edefining the paamete α in SK equation to be a funtion of edued pessue, aenti fato, and polaity fato fo pola ompounds in addition to be oinally funtion of edued tempeatue and n paamete whih is also funtion of aenti fato by using statistial methods. This oelation is as follows: n , T T This new modified oelation deeases the deviations in esults obtained by using SK equation in alulating S when ompaing with the expeimental data. The AAD fo 79 expeimental data points of 0 pue ompounds is.08 J/mol.K while it beomes.6 J/mol.K afte modifiation. Thus SK equation afte this modifiation gives moe auate esults fo esidual entopy of supeheated vapo of pue 0 ompounds than the est of equations mentioned above. Keywods: entopy, esidual entopy, supeheated vapo, equation of state, edued tempeatue, edued pessue, aenti fato, and polaity fato,. Intodution Themodynamis has been alled by many the siene of enegy and entopy. Howeve, unlike enegy, the wod entopy is seldom head in eveyday onvesation, enegy and entopy play impotant oles in themal systems engineeing []. Although thee ae many ways to intodue the onept of entopy, the simplest is just to deal with its utility; namely: a mathematial tool to desibe the dietion in whih things atually ou and if it ous spontaneously o not, and it is one of the themodynami popeties of fluids that ae essential fo desn poess equipment that alulates the heat and wok equiements of industial poess, and also the analysis of the pefomane of ompessos o expandes equies knowledge of the entopy behavio. Neithe enegy no entopy an be measued dietly on enegy o entopy mete, so values ae usually expessed in elation to an abitay efeene state by depending on the expeimental data of anothe popety that an be measued expeimentally as Tempeatue and essue that denoted by T and espetively [, ]. One of the impotant ways to obtain the entopy data fo pue substanes at vaious states is the expeimental data usually available in gaphial o tabula foms, but fo gaphial it is a moe ompliated method to patial use ompaing with the popety tables that povide vey auate infomation about the popeties simply, but they ae vey bulky and vulneable to typogaphial eos [] and many times some intepolations between two pessues o tempeatues is needed to obtain the value of a themodynami popety as entopy at etain point. Also fo the phase of supeheated vapo it is so diffiult eahing the onditions

2 of hh pessues o tempeatues fo many ompounds in laboatoy, thus a moe patial and desiable appoah a elation based on Equation Of State (EOS). ue gases ae ategoized into []:. Nonpola gases whih inlude: a. Simple fluids with spheial moleules ω=0 as Agon, Kypton, b. Quantum gases having ω<0 as He, H, and. Othe nonpola fluids whih ae having ω>0 as Benzene, opane.. ola gases that ould be subdivided into: a. Non-hydogen bonding ompounds suh as Ketones, and Aldehydes, and b. Hydogen bonding ompounds (a bond foms between the H atom attahed to Oxygen atom in one moleule with the Oxygen atom of anothe moleule) suh as Alohols, and Wate. In addition to aentiity, the pola ompounds ae haateized by the pesene of dipole moment aises fom positive and negative hages that ae pesent in the moleule. Thee is no peise eommended method fo alulating entopy o esidual entopy fo supeheated vapo. This wok involves studying the deviation in alulated entopy values fom its atual values (obtained by available expeimental data fo diffeent ompounds: pola and nonpola gases) then stating whih method is moe suitable than the othes.. EOS (Models) The most onvenient method of epesenting the popeties o the behavio of a substane, is by a mathematial expession; that is, an equation whih epesent the (--T) behavio of a fluid. A geneal fom of suh an expession known as EOS is: f,,t 0 (). Soave-edlih-Kwong (SK) Equation Soave [5, 6] in 97 intodued a modifiation on the edlih-kwong (K) equation of state ; this modifiation has been suessful in extending the appliability of K equation to be applied with hh auay fo wide ange of non-pola and slhtly pola omponents. The tempeatue dependent tem a/t 0.5 of the K equation was alteed to inlude both the tempeatue and the aenti fato by Soave; the SK equation is [7]: T a () b b Whee the fato α is an empiial funtion detemined fom the vapo-pessue data of pue hydoabons. The SK equation fo esidual entopy is: S BD B lnz B ln () aa Z The ubi fom in tems of ompessibility fato is: Whee Z BZ A B BZ AB 0 () A 0.77 T a 0.77 T 0.5 n T B (5) T T b (6) (7a) n (7b) and D na T (8). eng-obinson () Equation The equation of eng and obinson in 976 [8] is stutually athe simila to the SK and, like the SK, equies only the itial onstants and the aenti fato fo its appliation fo a pue fluid. This equation of state was developed pimaily fo vapo liquid equilibium peditions. eng-obinson modified the standad fom as follows [7]: T a (9) b b b b The SK equation fo esidual entopy is: S BD Z.B lnz B ln.88 a A Z 0.B (0) The ubi fom in tems of ompessibility fato is: Z BZ A B BZ AB B B 0 () Whee A 0.57 T a 0.57 T 0.5 n T B () T T b () n ()

3 also, D na T (5). Lee-Kesle Equation Lee and Kesle in (975) [9] developed an analytial oelation, based on itze s thee-paamete oesponding states piniple [0] to povide ineased auay and oveing the whole ange of T and of patial inteest in hydoabon poessing. It is to be noted that the oinal oelations by itze et al., wee limited to edued tempeatues above 0.8. itze et al. oelations fo the ompessibility fato of a fluid whose aenti fato is ω is given by the following equation: 0 Z Z Z (6) Whee Z (0) is the ompessibility fato of a simple fluid and Z () oets Z (0) fo the effets of nonspheial intemoleula foes (pimaily dispesion and ovelap). Z (0) and Z () ae assumed funtions of T and. Howeve, Lee and Kesle found that the ompessibility fato of any fluid is a funtion of the ompessibility of a simple fluid (Z (0) ), the ompessibility of a efeene fluid (Z () ), and the aenti fato, whee Z (0) and Z () ae funtions of T and and the oelation of Lee and Kesle takes the fom: 0 0 Z Z Z Z (7) Whee ω =0.978 and it is the aenti fato fo efeene fluid, and the oetion tem Z () in eq. (6) is obviously equivalent to (Z () - Z (0) ) ω this expession is onvenient sine both Z () and Z (0) ae given by the same equation with, howeve, diffeent onstants. Lee and Kesle hose n-otane as the heavy efeene fluid sine it is the heaviest hydoabon fo whih thee ae auate (--T) and enthalpy data ove a wide ange of onditions [9, ]. The funtion fo both the simple fluid Z (0) and the efeene fluid Z () ae deived though a ombination of expeimental data and a edued fom of the modified Benedit-Webb-ubin as ited in [9, and ef.] equation of state sheduled in table. Z T with a diffeent set of onstants that ae B C D 5 T exp.. (8) Table Constants fo Calulating equation (8) [9] onstants b b b b d 0 d 0 β γ Simple Fluid (0) efeene Fluid () b b b B b (9) T T T C (0) T T d D d () T Fo alulating Z fo the fluid of inteest given at T and, fist the appopiate values of T (T/T ) and (/ ) ae alulating by using itial popeties of the fluid. Fom the simple fluid onstants in table and eq. (8) solving fo whih is not the oet edued volume fo the fluid of inteest, but athe a pseudoedued volume by the tial and eo method when is defined as ( /T ), whih an be onsideed the initial guess fo the alulation, o fom the fist equality of eq. (8) the initial guess an be taken as: T Z () the pevious equation was depended by aul and Fanis [] in pepaing thei ompute pogam fo the tables of Lee and Kesle. Afte tial and eo alulation the obtained value of = (0) fo simple fluid and when employed in the fist equality of eq. (8), Z (0) is alulated fo simple fluid. This poess is then epeated using the efeene fluid onstants with the same T and values of the fluid of inteest to find = () and Z () fo the efeene fluid. Finally, with Z (0) fom the fist alulation aul and Fanis [] took the initial guess fo Z of eq. () equal to 0. but in this eseah whih onentates on the vapo phases espeially supeheated vapo the initial Z value taken equal to beause fo vapo phases at hh edued tempeatues and pessues the Z value often moe than unity [].

4 and Z () fom the seond, the ompessibility fato Z fo the fluid of inteest is detemined fom eq. (7) [9]. The esidual entopy is deived fom eq. (8): S S ln ln * Z b b b T T T d E () 5 5 Whee: E exp () T Afte detemining (0) and Z (0) fo the simple fluid at the T and appopiate fo the fluid of inteest, and employing eq. () with the simple fluid onstants in table, (S-S )/ is alulated. This tem epesents [(S-S )/] (0) in this alulation and Z in eq. () is Z (0). Then, when epeating the same alulation, using the same T and and the () values of and Z () fo the efeene fluid whih also detemined peviously, but employing the efeene fluid onstants fom table. With these, eq. () allows the alulation of [(S-S )/] (). Now, detemining the esidual entopy funtion fo the fluid of inteest fom: S S S S ln SS * SS ln * 0 ln * 0 ln * (5). iial Equation The viial equation of state, also alled the viial expansion, is the most inteesting and vesatile of the equations of state whih ae used to desibe the (--T) popeties of a fluid and its impotane due to that it has a sound theoetial basis. It is a polynomial seies in pessue o in invese volume whose oeffiients ae funtions only of T fo a pue fluid. iial oeffiients ae lassified into many tunated foms aoding to the ode of the tem seies. The onsistent foms fo the initial tems ae: CT DT B T Z T (6a) B C D... (6b) B C D (6) The oeffiient B o B' is alled the seond viial oeffiient, C o C' is alled the thid viial oeffiient, and so on. In patie, sine not all of the oeffiients of the viial seies ae known, and only data of the seond viial oeffiients ae plentiful in the liteatue, tems above the thid viial oeffiient ae aely used in hemial themodynamis and the seies is usually limited in patie up to modeate pessues. Howeve, the advantages of the viial equation ould be ineased if quantitative infomation wee available on the thid viial oeffiient [,, 5]... elations between the viial oeffiients The viial expansion fo is: CT DT T B T (7) The oeffiients of the expansion in pessue ae elated to the oeffiients of the expansion in density (/) as follows [6]: B B TB B (8a) T C T C B C B C (8b) T The fist step of the deivation of these elations is by solving the oinal viial expansion fo above then equating the two viial expansions, and substituting this expession fo into the pessue fom-side of esulting equation to obtain: B C BT BTB C T (9) Both sides of equation (9) ae powe seies in / (thid and hhe powes of / have been omitted beause the seond powe is the hhest powe that ae using by the ommon efeenes). Sine the two powe seies must be equal, the oeffiients of eah powe of / must be the same on both sides this ompaison povides the elations between the oeffiients [5]... Seond iial Coeffiient Coelation of seond viial oeffiient of both pola and nonpola systems is pesented by [, 5]. B B Z B (0) T

5 Tsonopoulos oelation fo B: T 0 B B B () B 0.5 () 8 T T T T B () 8 T T T.. Thid iial Coeffiient At hh pessues -above 500 ka- equations. (6a, b, and ) may be tunated afte thee tems []: B C Z B C () Obey-ea oelation fo C: T 0 C C C (5) C (6) T T C (7) T T T T By using the esidual popeties, the final expession of the esidual entopy afte deivation an be expessed as []: S S db dt dc C T B dt db BT dt (8) T The aim of the pesent wok is to alulate the esidual entopy by using (Lee-Kesle, eng- obinson, iial tunated to B o to C tems, and Soave-edlih-Kwong equations) by detemining the deviation fom the atual esidual entopy using statistial methods to modify the best equation depending on the shape of patile (ω) in addition to the polaity fato (χ) fo the pola gases in ode to ome out with an equation that pedits the esidual entopy fo diffeent types of supeheated vapo of pue gases with hh ageement with expeimental data.. Aenti fato In 955 itze [0] obseved that the edued vapo pessues of moleules with aenti foe fields ae lowe than that of simple fluids; and the diffeene is geate fo moleules of geate aentiity. itze noted that all vapo pessue data fo the simple fluids (A, K, and Xe) lie on the same line when plotted as log 0 sat vs. /T and that the line passes though log 0 sat = -.0 at T = 0.7. This is illustated in F.. Fue Appoximate tempeatue dependene of the edued vapo pessue [5]. Data fo othe fluids define othe lines whose loation an be fixed in elation to the line of simple fluids. Theeupon, itze defined the aenti fato ω (a thid paamete) of a substane by [5]: s T 0. 7 log 0 (9) Theefoe ω an be detemined fo any fluid fom T,, and a single vapo-pessue measuement was made at T = 0.7 whih is nea the nomal boiling point of most substanes, so the impotane of hoosing T = 0.7 that was adopted by itze not only povides numeial sat simpliity (log 0 = - fo simple fluids) but also benefiial beause vapo pessue data ae most ommonly available at pessue nea atmosphei [5].. olaity Fato of Halm and Stiel Beause the vapo pessue fomed the basis fo the definition of the aenti fato, this popety was hosen as the stating point fo the extension of the nomal fluid appoah to pola fluids to obtaining polaity fato χ whih is an empiial paamete fo pola substanes simila to the aenti fato fo nomal fluids. The fato χ is defined to be zeo fo nomal fluids and an be expessed fo pola fluids as follows: log s T (0) 5

6 alues of χ an be obtained fom the liteatues fo some pola ompounds[7, 8]. 5 Calulation of Entopy fo Supeheated egion The alulation of entopy fo supeheated egion needing fou steps in a alulational path leading fom an initial to a final state of a system as obtained in equation below: T dt S Sv C ln S S () T T Thus, in F.. the atual path fom state to state the dashed line is eplaed by a fou-step alulational path, these steps visualize as the sum of fou hanges epesented by the sequene of isothemal and isobai steps: S v v S S S S S S S S S () Step v : The tansfomation of satuated liquid at (T, ) to satuated vapo at T and : S v S Sv Satuated entopy of vapo an be alulated by onveting satuated liquid at efeene T and to satuated atual gas at the same T and by using the entopy of vapoization at the nomal boiling point ΔS v afte saling it with the efeene tempeatue by Watson elation [5] H H v v T T 0.8 () In ef. [] some of the bette estimation methods wee tested fo seveal diffeent ompounds as hydoabons, alohols, ae gases, oxides and othe pola ompounds and obtaining useful ompaison between these methods (Giaalone, iedel, Chen, and etee) and expeimental values of ΔH v. This ompaison shows that the aveage absolute peentage eo of Giaalone, iedel, Chen, and etee methods ae.8,.8,.7, and.6 espetively. Theefoe the pesent investation employs the moe auate one etee method eq. () to alulate ΔH v at the nomal boiling point and saled it with eq. () H vb T T b 0. ln T T b b Tb. () to obtain ΔH v alulating ΔS v tempeatue. at the efeene tempeatue, then by dividing ΔH v by the efeene Fue Calulational path fo entopy hange. Step v : A hypothetial poess that tansfoms a eal gas into an ideal gas at T and by using suitable esidual entopy of an equation of state. v S S S Step : Changes in the ideal-gas state fom (T, ) to (T, ). Fo this poess: T S S S T C dt T ln In the pesent wok C p is alulated by: C p a (5) 0 at at at at This polynomial besides it povides simpliity in use, it also oves good ange of tempeatues. The data elated to eq. (5) fo some pue ompounds ae listed in (Appendix A- setion ) of ef. [7]. Step : Anothe hypothetial poess that tansfoms the ideal gas bak into a eal gas at T and : S S S Theefoe, Equation () is esults of the totality of the entopy hanges fo the above fou steps. 6. Seleting the Optimum EOS fo the esent Wok [9] The onflit between auay and simpliity is a b dilemma in the development of an equation of state. Despite the wide use of hh-speed omputes, the 6

7 simpliity is still hhly desied fo easy and unequivoal appliations of the equation to omplex poblems. So, fo most alulations, the empiial appoah is often bette than the moe omplex theoetial appoah in view of auay as well as minimum data equiements. Often, viial equation whih tunates to seond o to thid tem an only be onsideed useful fom the fist fom and employed in the pesent wok. In addition to the Soave- edlih-kwong equation (SK) and eng-obinson equation () as the models of ubi equations of state fom the seond fom whih have valuable appliations in most ommon use today, Lee and Kesle equation epesents although it is inluding some omplexity the easiest example of the thid fom that was also used in the pesent wok. The Aveage Absolute Deviation fo esidual entopy AAD in (J/mol.K), whih is defined as follows: (S expeimental S alulated) AAD (6) n AAD is onsideed as a fato fo ompaison between the diffeent methods that wee used fo alulating atual esidual entopy of supeheated vapo fo diffeent ompounds. It is alulated in (J/mol.K) beause when it is alulated in (J/g. K) was being vey deimal and the impaity of deviations of using the equations being not peeptible. Also, the dimensionless deviation an be obtained by dividing AAD by (gas onstant) whih denoted by AAD/. In addition, the Aveage Absolute eentage Deviation fo entopy AAD% is defined as follows: (Sexp eimental Salulated) /Sexpeimental 00% AAD% n (7) whih is onsideed as a fato fo ompaison between the diffeent methods that wee assoiated in detemining the atual entopy of supeheated vapo fo diffeent ompounds. 7. esults and Disussion [9] 7. Appliation of the EOS fo Compounds 7.. Classifiation of the Appliation of EOS into egions Five diffeent equations of state wee applied fo alulating esidual entopy (S ) in ompaison fo all expeimental data of pue ompounds that suppoted the pesent investation as expessed ealie. To be insht of the peision of these equations with the ange of T and, summay of esults lassified into thee egions is pesented as follows: egion ) T <, and < : inluding ompounds involving 859 expeimental data points (9 nonpola ompounds with expeimental data points and 5 pola ompounds with 8 expeimental data points) fom the 79 expeimental data points of the all 0 ompounds. egion ) T >, and < : inluding the 0 ompounds involving 50 expeimental data points ( nonpola ompounds with 9 exp. data points and 6 pola ompounds with 580 exp. data points) fom the all 79 exp. data points of the all 0 ompounds. egion ) T >, and > : inluding ompounds involving expeimental data points (0 nonpola ompounds with 08 expeimental data points and pola ompounds with expeimental data points) fom the all 79 expeimental data points of the all 0 ompounds. Although this egion epesents the supeitial egion, but the knowledge of the auay of employing these equations in ompaison with the auay of the modified equation in the pesent wok fo this egion is advantageous fo the some available expeimental data. Table The existene of the ompounds in egions. Non pola egions Non pola egions ompounds ompounds Agon,, n-hexane,, Methane, n-heptane,, Oxygen, n-otane,, Nitogen, 5 Ethane, ola egions ompounds 6 Cylopopa efeant, ne, 7 opane,, Isopentane, 8 Aetylene, Ammonia, 9 Neopentan efeant, e 5a, 0 Benzene, 5 efeant a,, Cabon dioxide, 6 Wate,, 7.. Total egion This egion epesents all thee egions and onsists of 0 ompounds involving 79 expeimental data points 7

8 ( nonpola ompounds with 660 expeimental data points and 6 pola ompounds with expeimental data points). 7. Modifiation of EOS [9] 7.. Seleting the Optimum EOS fo the Modifiation Although Lee-Kesle equation poved to be bette than Soave-edlih-Kwong, eng-obinson, and iial equations fo the pedition of esidual entopy of supeheated vapo fo the most of ompounds that used in this investation, but it is moe useful if the eos an be edued to value less than those obtained with Lee- Kesle equation. The modifiation of any equation is usually done fo the equation that poved to be the most auate. The moe auate one is Lee-Kesle equation, but it is vey diffiult to modify it, so Soave-edlih-Kwong equation whih is the neaest one in auay to the Lee-Kesle equation was seleted fo modifiation. 7.. Modifiation of Soave-edlih-Kwong Equation [9] Soave-edlih-Kwong equation was deived mainly to alulate vapo-liquid equilibia, so all attention was onentated on that pupose in its deivation, and theefoe, thee is still oom fo impoving it fo supeheated vapo. The modifiation would be based on modifying α paamete of Soave equation whih is funtion of edued tempeatue and also on aenti fato whih is inluded in paamete n: 0.5 n Soave eq. paamete (7a) T Fues to 5 show lealy that the values of pessues o edued pessues influene on the value of α although the tempeatue is onstant. Thus α in Soave equation an be onsideed a funtion of tempeatue, pessue and aenti fato and its equation would be witten as [9]: n new fom of α paamete (8a) g g 5 g gt g T (8b) The oeffiients of this equation had been detemined by using statistial methods. These oeffiients wee alulated with the aid of ompute pogam on non-linea 8 estimation of statistia softwae that fitting to minimize the eo obtained fo alulating new α fo two seleted ompounds. Table Coeffiients of equation (8b). oeffiient value oeffiient value g g g g g In the pesent wok n-otane and wate wee used in the fitting. By tying many diffeent equations, it was found that equation (8b) was the optimum equation fo pediting γ with lowest eo. The oeffiients of eq. (8b) ae listed in table. The modifiation was made by ompaing the expeimental values of esidual entopy of supeheated vapo with the values alulated by Soave equation whih was obtained by using all oinal paametes of Soave equation exept γ paamete whih is inseted in the ompute pogam empiially and emain vaying until the deviation between the inseted value of γ with the alulated value by Soave equation fo eah expeimental data point- appoahed to zeo (% eo of (S exp.-s al.) ). The new equation of α gives a hhe auay whee the oveall AAD was.6 J/mol.K fo esidual entopy and the (AAD%) was.08 fo entopy, fo the all ompounds studied in the pesent wok. The main eason fo hoosing the mentioned two ompounds (n-otane, and wate) in the fitting of the expeimental data due to thei moleula natue that n- otane epesents the nomal nonpola gases whih was onsideed having the hhest ω > 0 in ompaison with the othe used ompounds in the pesent wok and thus to be able of ontolling the othe ompounds of less ω and has no polaity popeties found (χ = 0). On the othe hand wate epesents the pola ompounds whih have in addition to ω > 0 the polaity popeties χ > 0 and also it has the hhest χ among othes. The only way fo obtaining the expeimental values of the paamete γ is by empiially tial and eo method whih needs geat time to obtain any value. Theefoe the modifiation was limited to only these mentioned two types of gases whih

9 wee used to pedit the new α paamete equation that gives moe auay in alulating S and S fo all data points (79) expeimental data points fo the all 0 nonpola and pola gases employed in the pesent wok Expeimental values Soave equation esent wok Fue The elation between the values of paamete α and fo n-otane at T = expeimental values Soave equation This wok Fue 5 The elation between the values of paamete γ and fo wate at T =.9 Table a Summay of appliation of EOS though egions individually. Equations used egion at (T < and < ) No. of points in egion is (859) AAD fo S AAD/ fo S (J/mol.K) AAD% fo S L-K iial tunated to B) iial tunated to C) S--K This wok Expeimental values Soave equation This wok Fue The elation between the values of paamete γ and fo n-otane at T = expeimental values Soave equation This wok Fue The elation between the values of paamete α and fo wate at T =.9 9 Equations used egion at (T > and < ) No. of points in egion is (50) AAD fo S AAD/ fo S (J/mol.K) AAD% fo S L-K iial tunated to B) iial tunated to C) S--K This wok egion at (T > and > ) No. of points in egion is () Equations used AAD fo S (J/mol.K) AAD/ fo S AAD% fo S L-K iial tunated to B) iial tunated to C) S--K This wok

10 Table b Summay of appliation of EOS though whole egions. (fom T < & < to T > & > ) Equations used 8. Disussion All egions (,, and ) No. of points fo all egions (79) AAD fo S AAD/ fo S (J/mol.K) The entopy of a pue ompound annot be dietly measued but is alulated fom othe popeties. It is a funtion of both tempeatue and pessue; in geneal, it ineases with the inease in tempeatue and deeases with the inease in pessue at onstant tempeatue. At zeo pessue, all gases behave ideally, and the ideal gas state entopy is emaining dependent on pessue. The esidual entopy tem S-S, is the diffeene between the entopy of a ompound at etain and T and that of an ideal gas state at the same onditions. In the absene of --T data fo the ompounds of inteest, o if the data do not ove the onditions unde whih engineeing alulations ae to be made, genealized oelations whih expess Z as a funtion of T and have been found to be of geat value in estimating esidual popeties as esidual entopy whih ae based on a modified theoy of oesponding states. AAD% fo S L-K iial tunated to B) iial tunated to C) S--K This wok The usual method available fo pediting the esidual entopy of supeheated vapo fo pue ompounds is by employing the equation of state. In the pesent wok the equations of state employed wee: L-K, -, S--K, and iial tunated to seond and to thid tems equations. It is well known that the evaluation of any oelation o pedition method is done by ompaison of the alulated values fo any equation used with those of the expeimental data whih nomally limited and mostly not oveing wide ange of tempeatues and pessues fo any etain ompound. The deviation between the expeimental data and esults of pedition o oelation method detemines the auay of the method and this auay in the pesent wok was epesented as mentioned in the pevious setion by AAD% fo entopy and AAD J/mol.K fo esidual entopy. Tables a and b show the AAD% and the AAD in J/mol.K fo the alulated entopy and alulated esidual entopy espetively of the supeheated vapo fo nonpola and pola ompounds as ompaed with expeimental values. 8. Compaison of the esults with the Expeimental Data [9] Compaing the esults that ae shown in Tables a and b, indiate that the L-K equation gives hhe auay fo pediting S as ompaed with the -, iial tunated to seond o thid tems, and S--K equations using the expeimental data points of the pesent study. The AAD% fo alulating the entopy of supeheated vapo fo nonpola ompounds of 660 data points ae.89%,.97%,.98%,.57%, and.799% when using L-K, -, iial tunated to seond o thid tems, S--K espetively, while the AAD fo alulating the esidual entopy of supeheated vapo of these ompounds by using these equations ae.677,.9,.978,.950, and.7665 J/mol.K espetively. Futhe moe the AAD% fo alulating the entopy of supeheated vapo fo 6 pola ompounds of data points ae.096%,.999%,.8575%,.98, and.66% when using L-K, -, iial tunated to seond o thid tems, S--K espetively, while the AAD fo alulating the esidual entopy of supeheated vapo of these ompounds by using these equations ae.7,.0867,.956, 5.597, and.659 J/mol.K espetively. The AAD% fo alulating the entopy of supeheated vapo fo the all ompounds of 79 data points ae.75%,.0%,.505%,.50%, and.8850% when using L-K, -, iial tunated to seond o thid tems, S--K espetively, while the AAD fo alulating the esidual entopy of supeheated vapo of these ompounds by using these equations ae.059,.589,.9686, 5.050, and.08 J/mol.K espetively. 8. The Modified Equation [9] Although of the bette esults wee obtained by L-K equation, but this equation is sometimes not vey pefeable, beause it needs moe time and not easy to apply as ubi equations of state. S--K equation showed

11 also a vey good auay losest to that of the L-K equation in alulating the esidual entopy S of supeheated vapo. Thus effots wee dieted to modify S--K equation to inease its auay as muh as possible and to be moe auate than the oinal Soave- edlih-kwong and even the L-K equation. This may be done by using a statistial pogam and statistial methods that give the suitable fom of oelation. Many attempts wee done to develop S--K equation to this pupose, and the best oelation of this modifiation was obtained in setion.6 whih poved a vey good auay fo most ompounds unde study. The modifiation was applied fo 0 pue ompounds of 79 expeimental data points (nonpola and pola ompounds). It edued the AAD% and AAD of Soave- edlih-kwong equation fo 660 expeimental data points of nonpola ompounds fom.799% to 0.959% and fom.7665 to.87 J/mol.K espetively. While it edued the AAD% and AAD of Soave-edlih- Kwong equation fo expeimental data points of 6 pola ompounds fom.66% to.70% and fom.659 to.999 J/mol.K espetively. On the othe hand it edued the AAD% and AAD of Soave-edlih-Kwong equation fo all the 0 pue ompounds unde study fom.8850% to.08% and fom.08 to.6 J/mol.K espetively as shown in tables - to -. Examples of nonpola ompounds: fo otane the AAD% and AAD by using SK equation wee.0085% and.060 J/mol.K espetively, while they wee 0.7% and.8 J/mol.K espetively when using the modified SK equation. Fo nitogen the AAD% and AAD by using SK equation wee 0.08% and J/mol.K espetively, while they wee 0.95% and J/mol.K espetively when using the modified SK equation. Examples of pola ompounds: fo wate the AAD% and AAD by using SK equation wee 5.695% and 6.56 J/mol.K espetively, while they wee.59% and.08 J/mol.K espetively when using the modified SK equation. Fo efeant (a) the AAD% and AAD by using SK equation wee.007% and J/mol.K espetively, while they wee.97 % and.75 J/mol.K espetively when using the modified SK equation. Fo pola ompounds when applying the modified SK equation without onsideing polaity effet tem the deviation fom the expeimental data was moe than that when onsideing polaity effet tem as shown in Table -5. Although this inease was not vey snifiant, but in patial use moe auay is desiable. The inteesting featues of the developed equation in the pesent wok fo alulating the esidual entopy ae:. It is a athe simple equation that ahieved good esults fo both nonpola and pola ompounds.. It needs only well-known popeties of pue ompounds (T,, ω, and fo pola ompounds χ) fo eah ompound. The new oelation gives vey good auay fo alulating S of the ompounds shown in Table - by ompaison with the expeimental data appoximately ove the whole tempeatue and pessue anges tested. Whee the onditions tested fo tempeatue up to T > and fo pessue aised fo some ompounds up to >. Fues to 5 show the elation eithe between esidual entopy with pessue at onstant tempeatue o between esidual entopy with tempeatue at onstant pessue fo the esults obtained using this new method of oelation and othe equations used ompaing with the expeimental data. Tables a and b show the ompaison of esults of deviations fom the expeimental data of n-otane and wate when using Lee-Kesle, eng-obinson, Soave- edlih-kwong equations of state, and the new oelation of this wok. Although viial equation was employed in this study but its esults showed ahh deviations fom the expeimental values obtained ompaing with deviations of othe equations that employed. 9. Conlusions. Diffeent equations of state wee used to pedit the esidual entopy of supeheated vapo fo pue ompounds. They ae Lee-Kesle, eng-obinson, Soave-edlih-Kwong and iial tunated to seond and to thid tems equations of state. The esults indiate that Lee-Kesle equation is the most auate equation among these five equations, SK equation is the losest one in

12 its auay to the Lee-Kesle, and the viial equation (tunated to seond o to thid tems) gave hhest deviations fom the expeimental values whih posibed the need to listing its esults in tables.. New modifiation was made by edefining the paamete α in Soave equation to be a funtion of edued pessue, aenti fato, and polaity fato fo pola ompounds in addition to be oinally funtion of edued tempeatue and n paamete whih is also funtion of aenti fato by using statistial methods. This oelation is as follows: n T T The esults of this oelation wee ompaed with the esults obtained fom Lee-Kesle, eng-obinson, iial tunated to seond and to thid tems, and Soave- edlih-kwong methods. a.the AAD of 660 expeimental data points of pue nonpola ompounds obtained fom this oelation was.87 J/mol.K in ompaison with those obtained fom Lee-Kesle, eng-obinson, iial tunated to seond and to thid tems, and Soave-edlih-Kwong methods wee.677,.9,.978,.950, and.7665 J/mol.K espetively. b.the AAD of expeimental data points of 6 pue pola ompounds obtained fom this oelation was.999 J/mol.K in ompaison with those obtained when using the same equations above wee.7,.0867,.956, 5.597, and.659 J/mol.K espetively..the AAD of 79 expeimental data points of all the 0 pue ompounds obtained fom this oelation was.6 J/mol.K in ompaison with those obtained fom Lee-Kesle, eng-obinson, iial tunated to seond and to thid tems, and Soave-edlih-Kwong methods wee.059,.589,.9686, 5.050, and.08 J/mol.K espetively. Aknowledgement To all who ty to undestand the philosophy of entopy and intepet why it intudes in evey thing of ou lifes. Nomenlatue aiable Notations A, B, Constants used in the ubi EOS, eq.() and eq. (), B, B', Seond iial Coeffiient, m /mol, b,, d, Coeffiients of eq.(), C, C', Thid iial Coeffiient, Cm 6 /mol, C p, Heat apaity at onstant pessue, J/mol.K, g,g,..g 5, Coeffiients of eq. (8b), n, Constant used in the ubi EOS, eq.() and eq. (),, essue, ka,, Univesal gas onstant, J/mol.K,, S, Entopy, J/mol.K, T, Tempeatue, K,, olume,m, Z, Compessibility fato. Abbeviations AD, absolute deviations fo S,J/(mol.K), AD%,absolute peentage deviation fo S, AD/, absolute deviation fo S divided by, AAD, Aveage Absolute Deviation fo S,J/(mol.K) AAD%, Aveage Absolute eentage Deviation fo S, AAD/, Aveage Absolute Deviation pe fo S, EOS, Equation Of State, LK, Lee-Kesle equation, M.wt., Moleula weht, eng- obinson equation, K, edlih-kwong equation, ef., efeene, SK, Soave-edlih-Kwong equation. Geek Lettes α, Constant Used in ubi equations of state, β, aamete used in equation (-57)., γ, Constant used in equation (8), γ, aamete used in equation (8a), ω, Aenti fato.,χ, olaity fato of eq. (0). Supesipts, Ideal gas,, esidual, sat., Satuated state, (0), Simple Fluid Equation (7), (), efeene Fluid Equation (7). Subsipts b, Boiling point,, Citial popety, al., Calulated value, exp.,=,expeimental value, g, Gas state, l, Liquid state,, edued popety, efeene fluid, v, vapoization.

13 efeenes. Moan, M. J., Shapio, H. N., Munson, B.., and De Witt, D.., Intodution to Themal Systems Engineeing John Wiley and Sons, In., (00).. Cengel, Y. A., and Boles, M. A., Themodynamis: An Engineeing Appoah d Edition, M Gaw Hill, (998).. Moan, M. J., and Shapio, H. N., Fundamentals of Engineeing themodynamis th Edition, John Wiley and Sons, In., (000).. Tsonopoulos, C., An Empiial Coelation of the Seond iial Coeffiients AIChE J., ol. 0, No., (97): Soave, G., Equilibium Constants fom a Modified edlih-kwong Equation of State Chem. Eng. Si., ol. 7, (97) pp Soave, G., Impovement of the an De Waals Equation of State Chem. Eng. Si., ol. 9, No., (98) pp Assael, M. J., Tusle, J.. M., and Tsolakis, T. F., Themophysial opeties of Fluids (996). 8. eng, D. Y., obinson, D. B., A New Two- Constant Equation of State Ind. Eng. Chem. Fundam., ol. 5, 59 (976). 9. Lee, B. I., and Kesle, M. G., A Genealized Themodynami Coelation Based on Thee- aamete Coesponding States AIChE J., ol., No., (975): itze, K. S., Lipmann, D. Z., Cul,. F., J., Huggins, C. M., and etesen, D. E., The olumeti and Themodynami opeties of Fluids - II. Compessibility Fato, apo essue and Entopy of apoization Ameian Chemial Soiety, 77, (955). Edmiste, W. C., and Lee, B. IK., Applied Hydoabon Themodynamis ol., nd Edition, Gulf ublishing Company (98).. aul, B. J., and Fanis, O., Compute Aided Chemial Themodynami of Gases and Liquids Models ogams John Wily and Sons (985).. oling, B. E., ausnitz, J. M., and O Connell, J.., The opeties of Gases and Liquids 5th Edition, MGaw-Hill (00).. Santis,. D., and Gande, B., An Equation fo editing Thid iial Coeffiients of Nonpola Gases AIChE J., ol. 5, No. 6, (979): Smith, J. M., an Ness, H. C., and Abbott, M. M., Intodution to Chemial Engineeing Themodynamis 6 th Edition, MGaw-Hill (00). 6. Eubank,. T., and Angus, S., Tunation of iial Equations in T Data edution Jounal of Chemial and Engineeing Data, ol. 8, No., (97): Halm, L.., and Stiel, L. I., A Fouth aamete fo the apo essue and Entopy of apoization of ola Fluids AIChE J.,, 5 (967). 8. Kukolja, M. D., and Gozdani, D. K., New alues of the olaity Fato J. Seb. Chem. So., ol. 65,, (000): AL-Najja, Sh. Z., "edition And Coelations Of esidual Entopy Of Supeheated apo Fo ue Compounds" M. S. Thesis, Nahain Univesity (009).

14 أ. د. هحوىد عورعبذ هللا, تنبؤ وعالقات عاهةلقيوة اإلنتروپي الوتبقي للبخارالوحوص للوركبات النقية د. سرهذ طالب نجن, م. م. شهذ زهير,, جاهعة النهرين, كلية الهنذسة, قسن الهنذسة الكيوياوية عطا الخالصة ان التنبؤب بقلمة ؤص الةؤمةمص ل مؤراپ ا)نتپ ؤ )entopy أسقسؤؤ ؤ ا ةؤقم تةؤ ةت ال ثؤرام مةؤف أنحؤق تسؤؤحت ؤؤ تمرةؤر ت بؤؤقم المپاپة الشغل ل ث ةص الةنقاةص ت ثل ثپ ص قة ص ا)نتپ ال تبم خ ة ح ص جرا مسقب قة ؤص ا)نتپ ؤ ل غؤق.امم ب ؤق أن المؤةت ال ختبپةؤص لؤ ل قؤر ت جؤ ؤر ؤرا مؤر ر لؤرپجقم المؤپاپة ال ؤغ لرالؤص الثشؤ الةص لةبخؤپة ال م ةؤص مؤر ر لؤبثا ال پتبؤقم الغؤق.ام الشؤقلثص مؤ االهت قت الى مسقبحق بقستخرات ثقرالم المقلصم هؤ الرپاسؤص تؤت ا سؤتخرات ثؤقرالم ح ؤص )متسؤقب قة ؤص ا)نتپ ؤ ال تبمؤ لةبخؤپة ال م ةؤص النمةؤص نحؤق ثؤقرالم Lee-Kesle ال م اص لمرةن پة لثالثص مر ر پةم بقال ق ص الى استخرات ال ثقرالم التتثةبةص ل مقلؤص ال ت ث ؤص ب ثؤقرلت Soave-edlih-Kwong 7 iial پتؤب نمؤ رةؤپ ق بؤ لثشؤپةن پتؤب نمؤ.eng-obinson جپبم ه ال ثقرالم ا ؤى 6 نم ؤص ختبپةؤص ل بخؤقپ ال م ؤ 56 م 56 م ؤح لالنمؤپار ل ثشؤپةن پتؤب تتؤل ؤر 6 نم ؤص ختبپةؤص لت ؤ ال ثؤقرالم هؤ پتب نمؤ ق بؤ (م تؤقن ال ثؤرل ال ا ى الت ال م ال تبم 055 م ج ل\ لمت ڤن ل مراپ ا)نتپ 505 م 65 م النتقلج ان ثقرلص Lee-Kesle ه االقل انمپار ان النتقلج ال ختبپةص انر مقپنتحق ببمةص ال ثقرالم لتنحق رةؤپ ض ؤ ص االسؤتخرات قر جر ن ت ؤق ص ا لؤى ت نحؤق لةسؤم سؤح ص ؤ الت بةؤح ت ؤق هؤ المؤقل ؤ ال ثؤقرالم التتثةبةؤص ل مقلؤصم بثؤر أمةقن تثةپة لمقجتحق الى قم أتثپ ن رةپهق بؤق) مسقب مراپ ا)نتپ ؤ ال تبمؤ لةبخؤپة ال م ةؤص لتنحؤق ؤ پم رقتحق ل ثقرلص Lee-Kesle المةص ا ن ثقرلص SK تقنم ه األقپب ا تقنةؤص تثؤرة حق ل.ةؤقرة رقتحؤق لؤم ال مؤق الم بجتجؤق ل تغ ب ا ى ه ال سألص بؤ اسقسق ل مسقبقم ال تث مص بقلت ا.ن بةن مقلت البخقپ السقلل ل ل ال تبم لةبخپة ال م ةص قرپ ا) تقنم مسقب مراپ ا)نتپ تؤ ل اق ؤل الم بةؤص ل پتبؤقم الم بةؤص ؤغ ال نمؤ تؤت تثؤرةل ثقرلؤص SK بجاؤقرة تثپةؤر ال تغةؤپ α ؤ ال ثقرلؤص لةتؤ ن رالؤص ل ال تغةپ n لح ال ثقرلص بجستخرات ال پح ا)مةقلةص ت ق ة 7 ن بق) ق ص لت ن اسقسق رالص لرپجص المپاپة ال نم ةص اق ل الال پت.ةص ال n , T T ه الثالقص الجرةرة ال ثرلص ق م ا)نمپا قم نتقلج ثقرلص Soave انر مقپنتحق بقلنتقلج ال ختبپةص مةف ان ال ثرل ال ؤح لالنمؤپار لثشپةن پتب نم تتل ر 6 نم ص ختبپةص قبل التثرةل ه 055 م ج ل\ لمت ڤن بةن ق اةبح م ج ل\ لمت ڤن لمة ؤص ا)نتپ ؤ ال تبم ر التثرةلم ب ل تث ثقرلص SK بثر ه ا التثرةل نتقلج أتثپ رقص لمسقب قة ص ا)نتپ ال تبمؤ لةبخؤپة ال م ةؤص ل پتبؤقم النمةؤص ؤن بقق ال ثقرالم ال ت پةم

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