Modified Redlich-Kwong and Peng-Robinson Equations of State for Solubility Calculation of Solid Compounds in Supercritical Carbon dioxide

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1 Indan Journal of Scence and Technology, Vol 9(16), DOI: /jst/2016/v916/52344, Aprl 2016 ISSN (Prnt) : ISSN (Onlne) : Modfed Redlch-Kwong and Peng-Robnson Equatons of State for Solublty Calculaton of Sold Compounds n Supercrtcal Carbon doxde Hossen Rostaman and Mohammad Nader Lotfollah * Faculty of Chemcal, Petroleum and Gas Engneerng, Semnan Unversty, Semnan, Iran; mnlotfollah@semnan.ac.r Abstract The modfed Redlch-Kwong (βrk) and modfed Peng-Robnson (βpr) Equatons of State (EoSs) combned wth the van der Waals mxng rule (vdw0) have been proposed to evaluate the solubltes of ten sold compounds n supercrtcal CO 2. These ten sold compounds are ncludng Methmazole, Ascorbc acd, Ascorbyl palmtate, Propyl gallate, Asprn, Fluoranthene, Trclocarban, Hnoktol, Phenol, and Clmbazole. In order to recognze the accuracy of the proposed EoSs, the results of these models have been compared wth the calculaton results of SRK and PR equatons of state n combnaton wth the vdw1 and the Wong-Sandler (WS) mxng rules and other models reported n the lterature. The results showed that the proposed models are well for solublty calculaton of these ten sold compounds n supercrtcal CO 2 and the βrk EoS n combnaton wth the vdw0 mxng rule led to better correlaton (AARD = 5.1%) compared wth other ones. Keywords: βpr-eos, βrk-eos, Equaton of State, Sold Solublty, Supercrtcal CO 2, WS Mxng Rule 1. Introducton Supercrtcal fluds can be used as the promsng solvents n many applcatons such as separaton, purfcaton and partcle szng of pharmaceutcals, cosmetcs, food supplements, natural products, etc. In the supercrtcal processes, the solublty of the solutes n supercrtcal flud s a key parameter for desgnng optmzed operatng condtons. The expermental solubltes of the sold compounds and ther mxtures n supercrtcal fluds are lmted due to the dffcultes of expermental measurements and also tme-consumng and costly nature of these measurements. Therefore, t s desrable to develop the predctve and relable methods for estmatng the solublty of sold compounds n supercrtcal fluds. One way to acheve ths am s usng Equaton of the State (EoSs). The cubc EoSs are flexble and relable accordng to ther accuracy. Although consderable progresses n the development of equatons of state were reported n the lterature, the applcaton of the EoSs s stll lmted because of ther complexty. Addtonally, the sememprcal models do not have theoretcal bass, but they are wdely used n ndustral and engneerng applcatons due to ther smplcty In recent years, some of researchers have worked on the predcton of sold solublty n supercrtcal fluds by usng cubc Equatons of the State (EOSs). Khamda et al. 1 nvestgated the cefxme trhydrate and oxymetholone solubltes n supercrtcal carbon doxde. They also used sem-emprcal correlaton and the Peng-Robnson Equaton of State (PR EOS) for modelng of these solubltes. Park et al. 2 nvestgated the equlbrum solubltes of two bocdes, clmbazole, and trclocarban n supercrtcal carbon doxde. Subsequently, they appled PR EOS and quas-chemcal nonrandom lattce flud model for these systems. Chen et al. 5 reported the expermental solubltes of cnnamc acd, phenoxyacetc acd and 4-methoxyphenylacetc acd n supercrtcal carbon doxde. In order to model these solubltes, they also used Soave-Redlch-Kwong (SRK) and Peng- Robnson (PR) equatons of state. De Zord et al. 6 studed the solublty behavor of pharmaceutcal compounds * Author for correspondence

2 Modfed Redlch-Kwong and Peng-Robnson Equatons of State for Solublty Calculaton of Sold Compounds n Supercrtcal Carbon doxde contanng antoxdants, antbotcs, sterods and antnflammatory n supercrtcal fluds. They used a model based on actvty coeffcents and they determned the parameters of the model as a functon of the pharmaceutcal compound propertes. Housandokht et al. 3 appled varous modfed Peng-Robnson equatons of state to model the solublty of sold compounds n supercrtcal carbon doxde. They also determned nteracton parameters for these systems. Cheng et al. 7 nvestgated the solublty of ergosterol n supercrtcal carbon doxde. They used the Peng-Robnson Equaton of State (EOS) n combnaton wth the one-parameter and two-parameter van der Waals mxng rules to ft the expermental solublty data. Splots et al. studed the predcton of the lqud and sold aromatc hydrocarbons solublty n supercrtcal CO 2 wth the Lnear Combnaton of the Vdal and Mchelsen (LCVM) and Modfed Huron-Vdal Two (MHV2) models. Yazdzadeh et al. [4] appled the Peng-Robnson (PR) and the Esmaelzadeh-Roshanfekr (ER) (EoSs) n combnaton wth Wong-Sandler (WS), the Covolume Dependent (CVD) and the van der Waals one (vdw1) and two (vdw2) flud mxng rules and the Van-Laar excess Gbbs energy (G ex ) model to model the solubltes of sold compounds n supercrtcal carbon doxde. In ths work, the modfed RK (brk) and modfed PR (bpr) equatons of state n combnaton wth the van der Waals zero (vdw0) mxng rule were proposed for calculaton of sold solubltes n supercrtcal carbon doxde. The Soave-Redlch-Kwong (SRK) and Peng- Robnson (PR) (EoSs) n combnaton wth the van der Waals one (vdw1) and Wong-Sandler (WS) mxng rules were also appled for modelng the solubltes of sold compounds n supercrtcal carbon doxde. To dentfy the advantages of the new proposed models n predctng solubltes of sold compounds n supercrtcal carbon doxde, the results of proposed models were compared wth the results of the conventonal SRK and PR Equatons of State (EoSs). 2. Thermodynamc model 2.1 Calculaton of Solublty To determne the sold solublty n supercrtcal flud, the thermodynamc equlbrum s used as follows: f PureSold = f (1) Supercrtcal where PureSold f s the fugacty of each pure solute and Supercrtcal f s the fugacty of the solute n supercrtcal flud. In ths study, the followng assumptons were consdered to obtan the requred expresson for performng the phase equlbrum calculatons: 1. The solublty of supercrtcal flud n the sold phase s neglected. 2. The pure sold fugacty s consdered to be equal to the fugacty of the solute n the mxture. 3. The molar volume of the sold phase s constant. 4. The sold phase fugacty coeffcent n saturaton condton s consdered to be unty. Consderng these assumptons, Equaton (1) can be expressed as follows: é S Sat v ( P-P Sat ) ù (2) P exp = yjp RT êë úû where P and T are pressure and temperature, v denotes the molar volume, R s the unversal gas constant. Sat stands for saturaton. y and j are mole fracton and fugacty coeffcent of the sold solute n supercrtcal phase, respectvely. The saturaton vapor pressures at dfferent temperatures were gven n Table The Modfed Redlch-Kwong Equaton of State (βrk) The RK EOS 14 can be wrtten as follows: RT a P = v- b T vv ( + b) r (3) The energy parameter (a) and volume parameter (b) are obtaned from the crtcal propertes. The crtcal propertes of pure fluds were lsted n Table 2. 2 ( RTc ) a = (4) Pc RTc b = (5) P c In ths nvestgaton, a new functonalty for energy parameter of Redlch-Kwong EoS as a functon of temperature and pressure smlar to the work of Hedaryan and Jarrahan 15 and smlar to our prevous work 20 was proposed to evaluate the solubltes of ten sold compounds n supercrtcal CO 2. RT ba P = - v- b vv ( + b) (6) 2 Vol 9 (16) Aprl Indan Journal of Scence and Technology

3 Hossen Rostaman and Mohammad Nader Lotfollah Table 1. Vapor pressures of the solds used n ths study at dfferent temperatures Compound Sublmaton Vapor (Pa) Unt References A B C 1. Trclocarban bar 2. Hnoktol bar [19] 3. Phenols Pa [1] 4. Clmbazole bar Temperature (K) 5. Methmazole T = K T = K T = K Pa [17] Ascorbc acd T= Pa [10] Ascorbyl T= Pa [10] palmtate Propyl gallate T = K T = K - Pa [10] (adjustable parameter) 9. Asprn T = K T = K T = K Pa [11] 10. Fluoranthene T = K T = K T = K Pa [12] In whch β s a temperature dependant parameter that can be expressed n terms of reduced temperature as follows: b b + b T 1 2 r = ` (7) 1 - b3tr In whch refer to CO 2 or solute. Thus, β 11 -β 13 and β 21 - β 23 are related parameters to solute and CO 2, respectvely. It worth notng that the b functon (ncludng three parameters for each compound) used n ths work s dfferent wth the β functon (ncludng sx parameters for each compound) used n the work of Hedaryan and Jarrahan 15. Therefore, not only our β functon s a new functon but also our applcaton s dfferent and the proposed models were used for solublty calculaton The Modfed Peng-Robnson Equaton of State (βpr) The PR EOS 14 s expressed as: RT a P = - v- b v + 2bv-b 2 2 () where a shows the energy parameter and b denotes the volume parameter. The PR EOS parameters are defned n terms of crtcal propertes as follows: 2 ( RTc ) a= a( T) Pc (9) RTc b = P (10) c The other parameters of PR EOS are expressed as follows: a ( T) = (1 + m(1 - T r )) (11) 2 ì w w w 0.49 m = ï í ï ïî > w w w w 0.49 (12) where ω shows the acentrc factor.the subscrpts c and r are related to the crtcal and reduced propertes, respectvely. In ths work, the modfed verson of PR EoS s also suggested for determnng the phase equlbrum. RT ba P = - v- b v( v+ b) + b( v-b) (13) Smlar to Equaton (7), a temperature dependant expresson n terms of reduced temperature s consdered for b functon. In ths work, the van der Waals and Wong Sandler mxng rules were used for the sold-supercrtcal equlbrum calculatons. Vol 9 (16) Aprl Indan Journal of Scence and Technology 3

4 Modfed Redlch-Kwong and Peng-Robnson Equatons of State for Solublty Calculaton of Sold Compounds n Supercrtcal Carbon doxde Table 2. Crtcal propertes of the chemcals used n ths study Compound Molecular weght (g/mol) T C (K) P C (bar) ω V m (cm 3 /mol) References 1. Methmazole [17] 2. Ascorbc acd [10] 3. Ascorbyl palmtate [10] 4. Propyl gallate [10] 5. Asprn [11] 6. Fluoranthene [12] 7. Trclocarban Hnoktol [19] 9. Phenols [1] 10. Clmbazole The van der Waals mxng rule s expressed as follows a = åå ayy (14) m j j j a = aa (1- k ) (15) b j j j = å by (16) m where y denotes the mole fracton of component n supercrtcal phase. If the parameter k j was taken as zero, the mxng rule was denoted as the vdw0 nstead of vdw1 mxng rules. The Wong Sandler mxng rule s wrtten as follows, Q bm = 1 - D (17) am = RTDbm (1) æ b ö n whch, Q= åå yy j ç a- çè RT ø (19) D E = b RT + W j j å ya G RT (20) n whch, æ b ö é a aj ùæ1 k ö j a b b - - = - + j- ú èç RT ø ê j RT RT ç 2 ë úè û ø (21) In ths study, the van-laar actvty model 4 was appled for calculatng the excess Gbbs energy. 3. Results and Dscusson In ths work, the solubltes of ten sold compounds ncludng Methmazole, Ascorbc acd, Ascorbyl palmtate, Propyl gallate, Asprn, Fluoranthene, Trclocarban, Hnoktol, Phenol and Clmbazole n supercrtcal CO 2 were modeled. The expermental solubltes of solds were obtaned from the lterature 2,9 13,19. In order to model the solubltes of these solds n supercrtcal CO 2, the Peng Robnson and the SRK equatons of state (EOSs) combned wth the van der Waals (vdw1) and Wong Sandler (WS) mxng rules were used. To obtan the bnary nteracton parameters and the parameters of the model for van der Waals (vdw1) and Wong Sandler (WS) mxng rules, the parameters were obtaned va regresson wth the expermental data through the mnmzaton of an objectve functon. The average absolute relatve devaton percent (AARD%), defned by the followng expresson: 100 y,exp - y, calc AARD = å (22) N y,exp n whch N represents the number of expermental ponts, y,exp s the expermental solublty data and y,exp represents the calculated solublty. The average absolute relatve devatons percent (AARD%), optmzed model and bnary nteracton parameters were represented n Table 3. Fgs. 1-2 compare the calculated solublty results by the sets of,, and wth the expermental data for Phenol and Trclocarban compounds, respectvely. One can see that the performance of WS mxng rule s much better than vdw1 mxng rule. Therefore, the combnaton of the SRK and the PR EOSs wth the WS mxng rule s more sutable for modelng the solubltes of these ten solds n supercrtcal CO 2. 4 Vol 9 (16) Aprl Indan Journal of Scence and Technology

5 Hossen Rostaman and Mohammad Nader Lotfollah Table 3. The results of PR and SRK EOSs n combnaton wth the vdw1 and WS mxng rules Compound T(K) P(bar) ND References Mxng rule Model parameters AARD% 1. Trclocarban 2. Hnoktol 3. Asprn [19] [19] [19] [19] [11] [11] [11] [11] PR- WS A j A j k j Vol 9 (16) Aprl Indan Journal of Scence and Technology 5

6 Modfed Redlch-Kwong and Peng-Robnson Equatons of State for Solublty Calculaton of Sold Compounds n Supercrtcal Carbon doxde 4. Methmazole 5. Phenol 6. Ascorbc acd 7. Ascorbyl palmtate. Propyl gallate [9] [9] [9] [9] [13] [13] [13] [13] [10] [10] [10] [10] [10] Vol 9 (16) Aprl Indan Journal of Scence and Technology

7 Hossen Rostaman and Mohammad Nader Lotfollah 9. Clmbazole 10. Fluoranthene Total [12] [12] [12] [12] Fgure 1. The expermental and calculated solubltes of Phenol n supercrtcal CO2 at T = 309 K. Fgure 2. The expermental and calculated solubltes of Trclocarban n supercrtcal CO2 at T = 333 K. Vol 9 (16) Aprl Indan Journal of Scence and Technology 7

8 Modfed Redlch-Kwong and Peng-Robnson Equatons of State for Solublty Calculaton of Sold Compounds n Supercrtcal Carbon doxde Subsequently, to nvestgate the performance of the proposed EOSs (brk and bpr EOSs) n combnaton wth the smple mxng rule of vdw0, these proposed equatons were appled to model the solubltes of these ten solds n supercrtcal CO 2. The results of Average Absolute Relatve Devatons Percent (AARD%) and the model parameters were reported n Table 4. Fgures 3 4 show the calculated solublty results by the sets of,, βpr-vdw0 and βrk-vdw0 for Phenol and Trclocarban compounds, respectvely. models for seven compounds ncludng Methmazole, Ascorbc acd, Ascorbyl palmtate, Propyl gallate, Asprn, Fluoranthene and Phenol were compared wth the results of Esmaelzadeh-Roshanfekr (ER) equaton of state n combnaton wth vdw1, vdw2, CVD and WS mxng rules [4]. The results of AARD% are presented n Table 5. It can be concluded that the proposed models performed better than the results of Esmaelzadeh-Roshanfekr (ER) equaton of state n combnaton wth vdw1, vdw2, CVD and WS mxng rules. Fgure 3. The expermental and calculated solubltes of Phenol n supercrtcal CO2 by usngthe proposed and other models at T = 309 K. As t s shown n Table 4, the accuracy of the proposed models s much better than the combnaton of SRK and PR EOSs wth vdw1 mxng rule, even better than combnaton of SRK and PR EOSs wth WS mxng rule. The calculaton results of the models demonstrated that the brk and bpr EoSs are capable of modelng the solubltes of these ten sold n supercrtcal CO 2 wthout usng the complcated mxng rule. Therefore, brk and bpr EOSs n combnaton wth the smple mxng rule (vdw0) are relable methods for determnng the phase equlbrum of (sold + supercrtcal CO 2 ) systems. In order to nvestgate the valdty of the proposed EOSs, these proposed models were compared wth the models reported n lterature. Frst, the results of the proposed Fgure 4. The expermental and calculated solubltes of Trclocarban n supercrtcal CO2 by usng the proposed and other models at T = 333 K. The results of the proposed models (βpr-vdw0 and βrk-vdw0) for three compounds ncludng Propyl gallate, Methmazole and Asprn were also compared wth the results of regular soluton model (One-parameter and Two-parameter) and Two commonly used sememprcal equatons (Chrastl and Mendes-Santago and Teja equatons) n Table 6. In comparson wth the results of these models 16, the present models (βpr-vdw0 and βrk-vdw0) perform better than regular soluton and sem-emprcal models. It s found that the present model s relable for solublty calculatons of these ten solds n supercrtcal carbon doxde. Vol 9 (16) Aprl Indan Journal of Scence and Technology

9 Hossen Rostaman and Mohammad Nader Lotfollah Table 4. The results of βrk and βpr EOSs n combnaton wth the vdw0 mxng rule Compound T EOS β Parameters AARD β 11 β 12 β 13 β 21 β 22 β 23 % 1.Clmbazole βpr βrk βpr βrk βpr βrk βpr βrk Trclocarban βpr βrk βpr βrk βpr βrk βpr βrk Hnoktol βpr βrk βpr βrk βpr βrk βpr βrk Phenol 309 βpr βrk βpr βrk βpr βrk βpr βrk Asprn 30 βpr βrk βpr βrk βpr βrk βpr βrk Propyl gallate 313 βpr Vol 9 (16) Aprl Indan Journal of Scence and Technology 9

10 Modfed Redlch-Kwong and Peng-Robnson Equatons of State for Solublty Calculaton of Sold Compounds n Supercrtcal Carbon doxde Table 5. The comparson of dfferent models4 wth the present model Model βpr βrk ER-WS ER-CVD PR-CVD ER-vdW2 PR-vdW2 ER-vdW1 AARD% Table 6. The comparson of dfferent models16 wth the present model Compound βrk βpr βrk βpr βrk Ascorbc acd 313 βpr βrk Ascorbyl palmtate 313 βpr βrk Fluoranthene 30 βpr βrk βpr βrk βpr βrk βpr 7. βrk βpr ΒRK Methmazole 30 βpr βrk βpr βrk βpr βrk βpr 5.3 Total ΒRK 5.1 AARD% Regular Soluton Models Sem-Emprcal Models Present Model One-Parameter Two-Parameter Chrastl Mendez-Santago and Teja βrk βpr Propyl gallate Methmazole Asprn Conclusons In ths nvestgaton, the modfed RK (brk) and the modfed PR (bpr) equatons of state n combnaton wth the vdw0 mxng rule were used to determne the solubltes of ten sold compounds n supercrtcal CO 2. The optmzed parameters of the proposed models were determned and reported. Subsequently, the results of these models were compared wth the SRK and PR EOSs n combnaton wth VdW1 and WS mxng rules and the other appled models n the lterature. It s demonstrated that the relatve error (AARD%) between expermental 10 Vol 9 (16) Aprl Indan Journal of Scence and Technology

11 Hossen Rostaman and Mohammad Nader Lotfollah data and the calculated results by the proposed model s less than 5.1% ndcatng that the proposed models n ths work has hgher precson than the models n the lterature. 5. References 1. Khamdaa M, Hossen MH, Rezaee M. Measurement and correlaton solublty of cefxme trhydrate and oxymetholonen supercrtcal carbon doxde (CO 2 ). Journal of Supercrtcal Fluds. 2013; 73: Park CI, Shn MS, Km H. Solublty of clmbazole and trclocarban n supercrtcal carbon doxde: Measurement and correlaton. J Chem Therm. 2009; 41(1): Housandokht MR, Haghgh B, Bozorgmehr MR. A comparson among three equatons of state n predctng of some solds n supercrtcal carbon doxde. Kor J Chem Eng. 2007; 24(1): Yazdzadeh M, Eslammanesh A, Esmaelzadeh F. Thermodynamc modelng of solubltes of varous sold compounds n supercrtcal carbon doxde: Effects of equatons of state and mxng rules. Journal of Supercrtcal Fluds. 2011; 55(3): Chen YP, Chen YM, Tang M. Solubltes of cnnamc acd, phenoxyacetc acd and 4-methoxyphenylacetc acd n supercrtcal carbon doxde. Flud Phase Equl. 2009; 275(1): De Zord N, Kkc I, Moneghn M, Solnas D. Solublty of pharmaceutcal compounds n supercrtcal carbon doxde. Journal of Supercrtcal Fluds. 2012; 66: Cheng SH, Yang FC, Yang YH, Hu CC, Chang WT. Measurements and modelng of the solublty of ergosterol n supercrtcal carbon doxde. Journal of the Tawan Insttute of Chemcal Engneers. 2013; 44(1): Splots N, Magoulas K, Tassos D. Predcton of the solublty of aromatc hydrocarbons n supercrtcal CO 2 wth EoSGE models. Flud Phase Equl. 1994; 102(2): Yamn Y, Arab J, Asghar-khav M. Solubltes of phenazopyrdne, propranolol, and methmazole n supercrtcal carbon doxde. J Pharmaceut Bomed Anal. 2003; 32(1): Cortes A, Kkc I, Aless P, Turto G, Garner S. Effect of chemcal structure on the solublty of antoxdants n supercrtcal carbon doxde: Expermental data and correlaton. Journal of Supercrtcal Fluds. 1999; 14(2): Huang Z, Lu WD, Kaw S, Chew YC. Solublty of asprn n supercrtcal carbon doxde wth and wthout acetone. J Chem Eng Data. 2004; 49: Barna L, Blanchard J M, Rauzy E, Berro C. Solublty of fluoranthene, chrysene, and trphenylene n supercrtcal carbon doxde. J Chem Eng Data. 1996; 41(6): Gonzalez JG, Molna MJ, Rodrguez F, Mrada F. Solubltes of phenol and pyrocatechol n supercrtcal carbon doxde. J Chem Eng Data. 2001; 46(4): Danesh A. PVT and phase behavour of petroleum reservor fluds. Amsterdam: Elsever; Hedaryan E, Jarrahan A. Modfed Redlch-Kwong equaton of state for supercrtcal carbon doxde. Journal of Supercrtcal Fluds. 2013; 1: Su CS, Chen YP, Correlaton for the solubltes of pharmaceutcal compounds n supercrtcal carbon doxde. Flud Phase Equl. 2007; 254(1 2): Housandokht MR, Bozorgmehr MR. Calculaton of solublty of methmazole, phenazopyrdne and propranolol n supercrtcal carbon doxde. Journal of Supercrtcal Fluds. 200; 43(3): Huang C, Tang M, Tao W, Chen Y. Calculaton of the sold solubltes n supercrtcal carbon doxde usng a modfed mxng model. Flud Phase Equl. 2001; 179(1 2): Lm J, Km H, Cho HK, Shn MS. Solublty of hnoktol n supercrtcal fluds; measurement and correlaton. Kor J Chem Eng. 2011; 2(12): Rostaman H, Lotfollah MN, A new smple equaton of state for calculatng solublty of solds n supercrtcal carbon doxde, perod. Polytech. Chem. Eng. 2015; 59(3): Vol 9 (16) Aprl Indan Journal of Scence and Technology 11