Genealized Vapo Pessue Pedition Consistent with Cubi Equations of State Laua L. Petasky and Mihael J. Misovih, Hope College, Holland, MI Intodution Equations of state may be used to alulate pue omponent vapo-liquid equilibium popeties suh as vapo pessue, heat of vapoization, liquid density, and vapo density. The standad appoah equies oupling the EOS with a phase equilibium iteion suh as fee enegy, hemial potential, o fugaity. The esulting equations ae nonlinea and must be solved by numeial methods. An altenative appoah is appliable to ubi EOS suh as the ommonly used Soave-Redlih-Kwong and Peng-Robinson equations. Equilibium popeties may be expliitly expessed as powe seies in edued tempeatue o elated funtions. These esults ae moe onvenient than numeial alulations, but ae subjet to tunation eo in many patial situations. Results fom suh a powe seies method fo the SRK equation wee used to geneate genealized dimensionless vapo pessue equations whih wee extensions of the ommonly used Antoine equation but valid ove wide tempeatue anges. The substane-speifi adjustable onstants of the vapo pessue equations wee expessed as funtions of the aenti fato of the substane and its itial tempeatue and pessue. The deviations between these esults and the exat vapo pessue peditions fom the SRK equation wee quantitatively haateized. VLE fom Cubi Equations of State Cubi equations of state of the Van de Waals type ae widely used in hemial engineeing patie beause they povide a easonable balane between auay and simpliity. Two ommon examples ae the Soave-Redlih-Kwong [1] and Peng-Robinson [2] equations, shown below as Eqs. (1) and (2), RT asrk ( T ) P = V bsrk V ( V + bsrk ) (1) RT a ( T ) P = V b V ( V + b ) + b ( V b ) (2) In these equations, the attative funtion a EQ (T) has the fom a 1/ 2 2 EQ ( T ) = a, EQ[1 + f EQ ( ω )(1 T )] (3) with the itial value of
2 2 a0, EQR T a, EQ = (4) P and the aenti fato funtions given by 2 f SRK ( ω) = 0.480 + 1.574ω 0.176ω (5) 2 f ( ω) = 0.37464 + 1.54226ω 0.26992ω (6) The oupied volume paamete b is given by b EQ b0, EQRT = (7) P Values of the dimensionless numeial onstants in Eqs. (4) and (7) ae a 0,SRK = 0.42748, a 0, = 0.45724, b 0,SRK = 0.08664, and b 0, = 0.07780. Like many equations of state, SRK and may be used to alulate vapo pessue and othe pue omponent vapo-liquid equilibium popeties. Doing this equies oupling the EOS with a phase equilibium iteion suh as fee enegy, hemial potential, o fugaity. Eqs. (8) and (9) give expessions fo the pue omponent fugaity oeffiient fo SRK and. asrk ln φ SRK = z 1 ln z ln (1 h) ln (1 + h) (8) b RT SRK a 1+ h(1 + 2) ln φ = z 1 ln z ln (1 h) ln 2 2b RT 1+ h(1 2) (9) PV z = RT (10) beq h = V (11) When an equilibium state exists, Eq. (1) gives thee eal oots fo volume fom SRK. The smallest and lagest of these ae liquid and vapo volumes, espetively. The fugaity oeffiients alulated by Eq. (8) will be equal when the liquid volume is substituted and when the vapo volume is substituted. Fo, Eqs. (2) and (9) ae used. A numeial algoithm is equied to solve eithe of these sets of nonlinea equations. Powe Seies Methods fo Cubi Equation VLE A fomal poedue [3] is available to expess esults fo phase densities and vapo pessue of oexisting liquid and vapo phases as analyti powe seies in tempeatue. The appoah begins by witing edued deviation vaiables fo phase densities ( ρ L and ρ V ) and tempeatue ( T) about the itial point. These vaiables ae defined as
M M ΔM = M 1 = (12) M with M epesenting eithe ρ L, ρ V, o T. The esulting fomal powe seies ae j= 0 j P = A ( ΔT ) (13) i= 1 j L i 2 Δρ = B ( ΔT ) (14) i= 1 i V i i 2 Δρ = ( 1) B ( ΔT ) (15) i Afte equilibium onstaints ae applied to these seies, the oeffiients A j and B i ae obseved to be funtions of the aenti fato funtion f EQ (ω) whih depend upon the ubi EOS being applied. They ae substane-dependent sine they ontain the aenti fato funtion. Fo SRK and, A j is given by a polynomial of degee j in the aenti fato funtion, B i is given by a polynomial of degee (i-1) in the aenti fato funtion fo even i, and by a simila polynomial multiplied by the squae oot of (1+ f EQ (ω)) fo odd i. Expessions fo these polynomials have been tabulated. [4,5] The utility of this method stems fom the onveniene of diet omputation of oexisting phase popeties without need fo auxiliay equilibium iteia suh as fugaities. The weaknesses of the method aise fom being based upon a seies expansion about the itial point. In patie, the infinite seies given by Eqs. (13) to (15) must be tunated. Few tems ae equied nea the itial point, but this is the situation whee ubi EOS ae least auate. At modeately high o modeate tempeatues, whee the equations have easonable auay, many seies tems ae needed to pevent tunation eo. The magnitude of tunation eo also depends upon the popety (phase density o vapo pessue) and vaies with aenti fato. In geneal, tunation eos ae lagest fo vapo density and smallest fo liquid density, they ae lage fo the equation as ompaed to SRK, and they inease with ineasing aenti fato. Genealized Antoine Vapo Pessue Funtions Deived fom Powe Seies In pevious wok, a genealized fom of the Antoine vapo pessue equation was developed. In the taditional Antoine equation, dimensional onstants ae used to fit vapo pessue data ove a tempeatue ange. B ln P = A (16) T + C This was adapted by witing the Antoine equation in edued vaiables as Eq. (17) whee the onstants ae expessed as funtions of the aenti fato.
ln P B = A (17) T + C Results fo the funtions A, B, and C fo the SRK equation ae available fo edued tempeatues in the ange 0.6 < T < 1.0, and fo the equation in the ange 0.7 < T < 1.0. [6] A sample of typial esults is shown in Table 1. As is the ase with the taditional Antoine equation, the ange of auay is limited -- in this example to edued tempeatues between 0.70 and 0.84. Figue 1 illustates the auay of the method. Eah uve in Figue 1 epesents a diffeent aenti fato. Within the speified tempeatue ange (0.70 < T < 0.84) the vapo pessues pedited by Eq. (17) math the exat SRK vapo pessues to within less than 0.1 peent deviation fo all aenti fatos between -0.12 and +0.72. Table 1. Antoine Constant Funtions fo the SRK Equation fo 0.70 < T < 0.84. A = 4.4401 + 2.2128f SRK (ω) - 0.53518f SRK (ω) 2 + 0.17368f SRK (ω) 3-0.018512f SRK (ω) 4 B = 5.0075 + 1.2494f SRK (ω) - 0.78155f SRK (ω) 2 + 0.32010f SRK (ω) 3-0.047601f SRK (ω) 4 C = 0.124652-0.273702f SRK (ω) + 0.0750076f SRK (ω) 2-0.0137818f SRK (ω) 3 + 0.00145038f SRK (ω) 4 0.10% Deviation between Eq. (24) and exat SRK vapo pessue 0.08% 0.06% 0.04% 0.02% 0.00% -0.02% -0.04% -0.06% -0.08% 0.7 0.72 0.74 0.76 0.78 0.8 0.82 0.84 Redued Tempeatue, T w=-0.12 w=-0.08 w=-0.04 w=0 w=0.04 w=0.08 w=0.12 w=0.16 w=0.2 w=0.24 w=0.28 w=0.32 w=0.36 w=0.4 w=0.44 w=0.48 w=0.52 w=0.56 w=0.6 w=0.64 w=0.68 w=0.72 Figue 1. Peentage Deviation between Vapo Pessue Pedited by Genealized Antoine Equation, Eq. (17), and Exat SRK Vapo Pessue fo Vaious Aenti Fatos.
In the uent wok, the method was adapted and applied to genealized vesions of extended foms of the Antoine equation, given by Eqs. (18) and (19). ln P B = A + K1 lnt (18) T ln P B = A + K 4T + K 5 lnt (19) T + C Analogous to Eq. (17), the onstants in Eqs. (18) and (19) wee polynomials of the aenti fato funtion f SRK (ω). Some examples of these funtions ae shown in Tables 2 and 3. Like Eq. (17), the genealized extended Antoine equation given by Eq. (18) was apable of estimating vapo pessues within less than 0.1 peent deviation fom the exat SRK values fo the edued tempeatue ange 0.6 < T < 1.0. Also like Eq. (17), thee distint sets of Antoine onstant funtions wee needed to span this tempeatue ange. One advantage of Eq. (18) was that the Antoine onstant funtions wee ubi polynomials in the aenti fato funtion f SRK (ω), wheeas Eq. (17) equied quati (4 th degee) polynomials to ahieve the same auay of pedition. Table 2. Extended Antoine Constant Funtions in Eq. (18) fo the SRK Equation fo the Tempeatue Range 0.60 < T < 0.72. A = 2.8278 + 4.8255 f SRK (ω) + 2.7682 f SRK (ω) 2-0.057280 f SRK (ω) 3 B = 2.8422 + 4.8991 f SRK (ω) + 2.8244 f SRK (ω) 2-0.069460 f SRK (ω) 3 K 1 = 1.1068-2.2926 f SRK (ω) - 3.2566 f SRK (ω) 2 + 0.13569 f SRK (ω) 3 Table 3. Extended Antoine Constant Funtions in Eq. (19) fo the SRK Equation fo the Tempeatue Range 0.60 < T < 1.0. A = 2.5713 + 3.8253 f SRK (ω) + 0.56359 f SRK (ω) 2 + 0.42434 f SRK (ω) 3-0.011805 f SRK (ω) 4 B = 4.2778 + 6.9822 f SRK (ω) + 6.8127 f SRK (ω) 2-0.57592 f SRK (ω) 3-0.027427 f SRK (ω) 4 C = 0.031642 + 0.00022474 f SRK (ω) - 0.00095525 f SRK (ω) 2 + 0.00074762 f SRK (ω) 3-0.00017628 f SRK (ω) 4 K 4 = 1.5758 + 2.9403 f SRK (ω) + 6.0484 f SRK (ω) 2-0.98555 f SRK (ω) 3-0.014798 f SRK (ω) 4 K 5 = -1.5345-6.4179 f SRK (ω) - 12.447 f SRK (ω) 2 + 1.5234 f SRK (ω) 3 + 0.036455 f SRK (ω) 4 Eq. (19) was moe omplex than Eqs. (17) and (18) beause it ontained five Antoine onstant funtions instead of thee. The advantage of Eq. (19) was that a single set of Antoine onstant funtions, shown in Table 3, was able to estimate SRK vapo pessues within the speified deviation fo the entie edued tempeatue ange 0.60 < T < 1.0, as Figue 2 veifies.
Deviation between Eq. (19) and Exat SRK Vapo Pessue 0.10% 0.08% 0.06% 0.04% 0.02% 0.00% -0.02% -0.04% -0.06% -0.08% -0.10% 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 Redued Tempeatue T -0.12-0.08 0.72 0.68 0.64 0.6 0.56 0.52 0.48 0.44 0.4 0.36 0.32 0.28 0.24 0.2 0.16 0.12 0.08 0.04 0-0.04 Figue 2. Peentage Deviation between Vapo Pessue Pedited by Genealized Extended Antoine Equation, Eq. (19), and Exat SRK Vapo Pessue fo Vaious Aenti Fatos. Wok ontinues on adapting these methods to othe extended vesions of the Antoine equation, espeially those ommonly used in poess simulation softwae, to othe ubi EOS suh as, and to additional tempeatue anges. Conlusions Results fo a genealized dimensionless fom of two vesions of an extended Antoine equation wee pesented. These wee shown to epodue the vapo pessue peditions of the SRK ubi equation of state to within 0.1 peent deviation ove the tempeatue ange 0.60 < T < 1.0 fo all aenti fatos between -0.12 and +0.72. Aknowledgements This mateial is based upon wok suppoted by the Mihigan Spae Gant Consotium and by the Booksta Faulty Development Fund. Any opinions, findings, and onlusions o eommendations expessed in this mateial ae those of the authos and do not neessaily eflet the views of these institutions.
Nomenlatue A Constant in Eq. (16), Antoine vapo pessue equation A Funtion of aenti fato in Eqs. (17) to (19), genealized Antoine vapo pessue equations A j tempeatue oeffiients in Eq. (13) fo edued pessue a EQ (T) enegy paamete in attative tem a,eq enegy paamete in attative tem at the itial point a 0,EQ dimensionless numeial oeffiient of enegy paamete B Constant in Eq. (16), Antoine vapo pessue equation B Funtion of aenti fato in Eqs. (17) to (19), genealized Antoine vapo pessue equations B i tempeatue oeffiients in Eqs. (14) and (15) fo edued density deviation vaiables b EQ exluded volume paamete in equation of state b 0,EQ dimensionless numeial oeffiient of exluded volume paamete C Constant in Eq. (16), Antoine vapo pessue equation C Funtion of aenti fato in Eqs. (17) and (19), genealized Antoine vapo pessue equations f EQ (ω) quadati funtion of aenti fato h atio of exluded volume paamete to mola volume K 1 Funtion of aenti fato in Eq. (18), genealized Antoine vapo pessue equation K 4 Funtion of aenti fato in Eq. (19), genealized Antoine vapo pessue equation K 5 Funtion of aenti fato in Eq. (19), genealized Antoine vapo pessue equation M geneal themodynami state popety (T, ρ L o ρ V ) in Eq. (12) P absolute pessue R gas onstant T absolute tempeatue V mola volume z ompessibility Geek lettes ΔM edued deviation vaiable of state popety M (T, ρ L o ρ V ) defined by M 1 ρ mola density φ EQ fugaity oeffiient fo equation of state ω aenti fato Subsipts EQ itial (tempeatue, pessue, o mola density) paamete o oeffiient applying to equation of state EQ (SRK o ) edued (tempeatue, pessue, o mola density) Supesipts L V liquid (mola density o mola volume) vapo (mola density o mola volume)
Refeenes [1] Peng, D.-Y. D.B. Robinson. Ind. Eng. Chem. Fundam. 15. 59-64. 1976. [2] Soave, G. Chem. Engng. Si. 27. 1197-1203. 1972. [3] Senges, J.V., J.M.H. Levelt Senges. In Coxton, C.A. (Ed.). Pogess in Liquid Physis. 103-174. Wiley: New Yok. 1978. [4] Singley, C.D. I.P. Buns. M.J. Misovih. Fluid Phase Equilib. 137. 99-109. 1997. [5] Shaih, D.A. A.R. Runge. S.A. Gangloff. M.J. Misovih, manusipt in pepaation fo Computes and Chemial Engineeing. 2008. [6] DeDoes, A.J. M.D. Goetz. M.J. Misovih, A Simple Appoah to Genealized Vapo Pessue Pedition. Chem. Eng. Pog. 103(1). 39-44. 2007.