I wish to publish my paper on The International Journal of Thermophysics. A Practical Method to Calculate Partial Properties from Equation of State

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1 I wsh to publsh my paper on The Internatonal Journal of Thermophyscs. Ttle: A Practcal Method to Calculate Partal Propertes from Equaton of State Authors: Ryo Akasaka (correspondng author) 1 and Takehro Ito 2 Afflatons: 1: Faculty of Humantes, Kyushu Lutheran College, Kurokam, Kumamoto, , Japan, FAX: , e-mal: akasaka@klc.ac.jp 2: Graduate School of Integrated Scence and Art, Unv. East Asa, 2-1 Ichnomyagakuen- Mach, Shmonosek, Yamaguch, , Japan 1

2 Abstract: Ths paper presents a mathematcal method to calculate partal propertes from a cubc equaton of state. Partal propertes are very mportant characterstcs n the analyss of solutons, and are generally determned by expermental procedures. The presented method makes the predcton of these propertes possble wthout any expermental manpulaton. Although everythng used n the method comes from well-known thermodynamc relatons, ths method provdes useful examples of calculaton of partal propertes from an equaton of state snce such examples are scarce n the lterature. The Patel-Teja equaton of state s employed n ths paper. If an approprate mxng rule s gven, ths equaton of state s able to descrbe the thermodynamc behavor of hghly nondeal mxtures. Ths paper shows the dervatons of the partal molar volume, the partal molar enthalpy and the partal molar entropy from the equaton of state. The chemcal potental and actvty coeffcent are obtaned from these propertes. For acetonemethanol mxture, the calculatons of the partal propertes are demonstrated by applyng the presented method. Key Words: Actvty coeffcent, Chemcal potental, Equaton of state, Mxng rule, Partal property 2

3 Man Text: 1 Introducton Partal molar propertes are very mportant characterstcs n the analyses of mxtures. These propertes are descrbed by the followng formula: m = [ ] (Nm) N P, T, N j (1) where m s the partal molar property of speces n a mxture, m the molar property, N the total number of moles of the mxture, and N the number of moles of speces. The generc symbol m may stand for the partal molar volume v, the partal molar enthalpy h, the partal molar entropy s, the partal molar Gbbs energy (often refferred to as the chemcal potental µ ), and so on. These are response functons representng the change of total property Nm due to addton at constant T and P. If an equaton of mxture property as a functon of composton s avalable, the expressons of the partal propertes can always be derved from the equaton by drect applcaton of Eq. (1). The lterature has descrbed graphcal methods to evaluate the partal property from the relaton between mxture property and composton obtaned expermentally. These methods are smple but not sutable for ndustral use, snce an equaton or graph representaton of mxture property as a functon of composton s not always avalable for the mxture of nterest. Ths paper presents a more practcal and unversal method usng an equaton of state (EOS). Ths method makes the predcton of the partal propertes possble wthout any expermental manpulaton. Although everythng used n the method comes from well-known thermodynamc relatons, ths method provdes useful examples of the calculaton of partal propertes usng EOS snce such examples are scarce n the lterature. Any type of EOS s applcable, n prncple, to the presented method. Ths paper focuses on a cubc EOS (van der Waals-Type EOS). The cubc EOS coupled wth an approprate mxng rule s probably the most extensvely used approach for modelng the vapor-lqud equlbrum (VLE) of mxtures. Snce van der Waals developed the frst practcal cubc EOS n 1873, many dfferent cubc EOSs have been proposed to mprove the orgnal van der Waals equaton. The most successful modfcatons were made by Soave (SRK) [1] and Peng-Robnson (PR) [2]. In spte of ther 3

4 smple equaton forms contanng only two substance dependent parameters, the SRK and PR EOS can correlate the vapor pressures of hydrocarbons very well. These EOSs are wdely used n the refnery and gas processng ndustres for the predcton of VLE for mxtures contanng hydrocarbon components. However, the two-parameter cubc EOSs lke SRK and PR are scarcely used for the analyses of solutons, snce these EOSs generally yeld unsatsfactory predctons for lqud densty. Patel-Teja (PT) [3] developed a cubc EOS by ntroducng a thrd parameter to mprove ths shortcomng of the two-parameter EOS. The three-parameter PT EOS s capable of provdng satsfactory predctons for both vapor pressure and densty even for polar substances. If an approprate mxng rule s gven, the PT EOS can correlate the VLE of mxtures well, even though the mxtures contan polar components. For ths reason, n ths paper the PT EOS was chosen for the dervaton of expressons of partal propertes. 2 Dervaton of partal propertes from EOS 2.1 EOS and mxng rule The PT EOS s represented n the followng form: P = RT v b a[t] v(v + b) + c(v b) (2) where P s the pressure, T the temperature, v the molar volume, and R the unversal gas constant. When c = b, Eq. (2) reduces to the PR EOS, and when c = 0, t reduces to the SRK EOS. Substance dependent parameters a, b and c are determned from the crtcal pressure P c, the crtcal temperature T c and the crtcal volume v c by applyng the followng equatons. a[t] = Ω a R 2 T 2 c P c α[t R ] (3) b = Ω b RT c P c c = Ω c RT c P c (4) (5) 4

5 where Ω c = 1 3ζ c (6) Ω a = 3ζ 2 c + 3(1 2ζ c )Ω b + Ω 2 b + 1 3ζ c (7) ζ c = P c v c RT c (8) and Ω b s the smallest postve root of the cubc equaton: Ω 3 b + (2 3ζ c)ω 2 b + 3ζ2 cω b ζ 3 c = 0 (9) Instead of lettng ζ c have a value equal to the expermental value of the crtcal compressblty factor, an arbtrary value s used. In fact, ζ c s treated as an emprcal parameter. For non-polar substances, t s correlated wth the acentrc factor ω by the followng equaton [3]. ζ c = ω ω 2 (10) For α[t R ], the same functon of reduced temperature as the SRK and PR EOS s used. α = [ 1 + F(1 T 0.5 R )] 2 (11) The parameter F s nserted to obtan better agreement n vapor pressure calculaton wth the expermental value. For non-polar substances, t s also correlated wth ω [3]. F = ω ω 2 (12) Equaton (2) can be used for the calculaton of mxture propertes f the parameters a, b, and c are replaced by the mxture parameters a m, b b, and c m calculated usng a mxng rule. An applcaton of any type of mxng rule to the PT EOS s possble. For smplcty, ths paper uses 5

6 the followng mxng rule (the classcal var der Waals mxng rule). a m = x x j a j j (13) a = a, a jj = a j, a j = (1 k j ) a a j (14) b m = x b (15) c m = x c where x s the mole fracton of speces n the mxture, and k j s the bnary nteracton parameter obtaned by fttng EOS predctons to measured phase equlbrum and volumetrc data. (16) 2.2 Partal propertes n bnary mxtures Due to more convenent for the dervaton from an EOS, the rght hand sde of Eq. (1) s arranged as follows. [ ] ( ) ( ) (Nm) m m m = = m + N = m + N N P, T, N j N P, T, N j x and P, T, x j ( ) x N P, T, N j (17) ( ) x N P, T, N j = [ ] (N /N) = N N N P, T, N j N 2 = 1 x N (18) Therefore, m = m + (1 x ) m x (19) where ( ) m m x = x P, T, x j (20) By applyng an EOS, m and m x are calculated. As used here, n the followng part of ths paper, the parameter wth subscrpt x, e.g. A x, B x, means the dfferentaton wth respect to x at constant T, P, and x j. 6

7 2.3 Partal volume The expresson of the partal molar volume v s obtaned by substtutng m = v = ZRT/P n Eq. (19). v = RT P [Z + (1 x )Z x ] (21) where Z x = ( ) Z x P, T, x j (22) For the calculatons of Z x, t s useful to apply the alternatve EOS form represented as the cubc equaton for Z. The PT EOS can be wrtten n the form: Z 3 + (C 1)Z 2 + (A 2BC B 2 B C)Z + (B 2 + BC AB) = 0 (23) where A = ap/(r 2 T 2 ), B = bp/(rt), and C = cp/(rt) for pure substances, and A = a m P/(R 2 T 2 ), B = b m P/(RT), and C = c m P/(RT) for mxtures. Dfferentaton of Eq. (23) wth respect to x at constant T, P, and x j yelds: Z x = β 1Z 2 + β 2 Z + β 3 β 4 (24) where β 1 = C x (25) β 2 = A x + B x (2B + 2C + 1) + C x (2B + 1) (26) β 3 = A x B B x ( A + 2BC + C) BC x (B + 1) (27) β 4 = 3Z 2 + 2(C 1)Z + A B(B + 2C + 1) C (28) 2.4 Partal enthalpy The molar enthalpy of a mxture h can be expressed n the form: h(t, P, x) = h R (T, P, x) + [ h IGM (T, x) h IGM (T 0, x) ] + h IGM (T 0, x) (29) 7

8 where x means all but one of the speces mole fractons,.e. x = x 1, x 2,..., x n 1. The superscrpts IGM and R ndcate deal gas mxture property and resdual property, respectvely. T 0 s a reference temperature. Dfferentaton of Eq. (29) wth respect to x at constant T, P, and x j yelds the partal molar enthalpy h. Thus, h (T, P, x) = h R (T, P, x) + [ h IGM (T, x) h IGM (T 0, x) ] + h IGM (T 0, x) (30) For deal gas mxture, h IGM (T, x) = h IG (T) (31) and h IGM (T, x) h IGM (T 0, x) = h IG T (T) h IG (T 0 ) = T 0 cp IG (T) dt (32) where the superscrpt IG ndcates deal gas property, and c IG p deal gas. Therefore, T h (T, P, x) = h R (T, P, x) + T 0 s the sobarc heat capacty of cp IG (T) dt + h IG (T 0 ) (33) Substtutng m = h R n Eq. (19) gves: h R = h R + (1 x ) h R x (34) The resdual enthalpy h R s defned as h R = RT(Z 1) + v ( ) P T T v, x Applyng the PT EOS to Eq. (35) yelds: P dv (35) h R RT = (Z 1) + A TA M ln J K (36) 8

9 where M = B 2 + C 2 + 6BC J = 2Z + B + C M K = 2Z + B + C + M ( ) A P am = (RT) 2 T x (37) (38) (39) (40) Dfferentaton of Eq. (36) wth respect to x at constant T, P, and x j yelds: h R x RT = 1 RT ( ) h R x P, T, x j = Z x + (A x TA x)m (A TA )M x M 2 ln J K + A TA M [ Jx J K x K ] (41) 2.5 Partal entropy The molar entropy of a mxture s can be expressed n the form: s(t, P, x) = s R (T, P, x) + [ s IGM (T, P, x) s IGM (T 0, P 0, x) ] + s IGM (T 0, P 0, x) (42) where P 0 s a reference pressure. Dfferentaton of Eq. (42) wth respect to x at constant T, P, and x j yelds the partal molar enthalpy s. s (T, P, x) = s R (T, P, x) + [ s IGM (T, P, x) s IGM (T 0, P 0, x) ] + s IGM (T 0, P 0, x) (43) For deal gas mxture, s IGM (T, P, x) = s IG (T, P) R ln x (44) and s IGM (T, P, x) s IGM (T 0, P 0, x) = s IG (T, P) s IG (T 0, P 0 ) = T T 0 c IG p T dt R ln P P 0 (45) Therefore, T s (T, P, x) = s R (T, P, x) + T 0 c IG p T dt R ln P P 0 + s IG (T 0, P 0 ) R ln x (46) 9

10 Substtutng m = s R n Eq. (19) gves: s R = s R + (1 x ) s R x (47) The resdual entropy s R s defned as s R = R ln Z + v [ ( ) P R T v v Applyng the PT EOS to Eq. (48) yelds: ] dv (48) s R R = ln(z B) TA M ln J K (49) Dfferentaton of Eq. (49) wth respect to x at constant T, P, and x j yelds: s R x R = 1 R ( ) s R x = Z x B x Z B P, T, x j T ( A x M A ) M x ln J K TA M M 2 ( Jx J K x K ) (50) 2.6 Chemcal potental, actvty coeffcent and fugacty The chemcal potental µ, the actvty coeffcent γ and the fugacty f are drectly calculated from an EOS. Alternatvely, as shown below, these are also calculated from the partal molar enthalpy h and the partal molar entropy s. Dfferentaton = h Ts wth respect to x at constant T, P, and x j yelds: µ = h T s (51) From Eqs. (30) and (43), µ (T, P, x) = h R T s R + [ h IG (T) h IG (T 0 ) ] T [ s IG (T, P) s IG (T 0, P 0 ) ] + RT ln x (52) Here, for smplcty, h IG (T 0 ) = 0 and s IG (T 0, P 0 ) = 0. For deal mxture, the chemcal potental, the partal molar enthalpy, and the partal molar 10

11 entorpy are µ IM h IM s IM = h IM = h T s IM = s R ln x (53) (54) (55) where the superscpt IM ndcates deal mxture. Substtutng Eqs. (54) and (55) to (53) yelds: µ IM (T, P, x) = h (T, P, x) T s (T, P, x) + RT ln x = h R (T, P, x) T sr (T, P, x) + [ h IG (T) h IG (T 0 ) ] T [ s IG (T, P) s IG (T 0, P 0 ) ] + RT ln x (56) From Eqs. (52) and (56), the actvty coeffcent γ s RT ln γ (T, P, x) = µ µ IM = h R h R T ( s R s R ) (57) For deal gas mxture, µ IGM h IGM s IGM = h IGM T s IGM (T) = h IG (T) h IG (T 0 ) (59) (T, P) = [ s IG (T, P) s IG (T 0, P 0 ) ] R ln x (60) (58) From Eqs. (52), and (58) to (60), the fugacty f s RT ln f (T, P, x) = µ µ IGM = h R T s R (61) 3 Examples For acetone-methanol mxture, the partal propertes are calculated usng the equatons derved n the prevous secton. Due to strong nondealty of acetone and methanol, the correlaton of ζ c and F by Eqs. (10) and (12) are not adequate. The specfc values of these parameters are gven n Table 1. Table 2 shows the denstes and vapor pressures of acetone and methanol calculated usng the PT EOS wth the parameters gven n Table 1. For comparson, the values obtaned from 11

12 the experment [4] and calculated usng the PR EOS are also shown n the table. The PT EOS s able to correlate the denstes and vapor pressures of acetone and methanol wth hgh accuracy. Fgure 1 shows the sobarc vapor-lqud equlbrum of acetone-methanol mxture at 1 bar. The PT EOS coupled wth the classcal van der Waals mxng rule represents ths VLE successfully. The partal molar propertes and the chamcal potentals at 1 bar and 50 C are shown n Fgure 2 to 5. The parameters A x, B x and C x are calculated as follows: A x = 2xA 1 + 2(1 2x)(1 k j ) A 1 A 2 2(1 x)a 2 (62) B x = B 1 B 2 (63) C x = C 1 C 2 (64) where x ndcates the mole fracton of acetone ([mol(c 3 H 6 O)/mol]), and the subscrpt 1 and 2 mean acetone and methane, respectvely. The partal molar volumes are almost ndependent of the mole fracton. In contrast, the partal molar enthalpes and the partal molar entropes are strongly affected by the mole fracton. Fgure 6 shows the actvty coeffcents calculated usng the PT EOS and the Wlson correlaton [5]. The PT EOS coupled wth the classcal van der Waals mxng rule presents satsfactory agreement n the range of mole fracton from 0.2 to 0.8 mol(c 3 H 6 O)/mol. 4 Concluson The method to calculate of partal propertes usng a cubc equaton of state was presented. By applyng ths method, t s possble to predct partal propertes wthout any expermental manpulaton. The Patel-Teja equaton of state coupled wth the classcal van der Waals mxng rule was employed n ths paper, and the expressons of the partal molar volume, the partal molar enthalpy and the partal molar entropy were derved from ths equaton of state. Addtonally, ths paper showed that the chemcal potental, the actvty coeffcent and the fugacty can be calculated from the partal molar enthalpy and the partal molar entropy. For acetone-methanol mxture, the calculatons of partal propertes and actvty coeffcents were demonstrated by applyng the presented method. 12

13 Table 1: The specfc values of ζ c and F used n the the Patel- Teja equaton of state for acetone and methanol ζ c F Reference acetone [4]* methanol [3] * calculated for ths work usng the reference 13

14 Table 2: Denstes and vapor pressures of acetone and methanol calculated usng the Patel-Teja equaton of state Denstes (1 bar) acetone T ρ exp ρ PT δ PT ρ PR δ PR methanol T ρ exp ρ PT δ PT ρ PR δ PR T=[ C], ρ=[m 3 /kmol], δ X = (ρx ρ exp )/ρ exp 100 Vapor pressures acetone T (P s ) exp (P s ) PT δ PT (P s ) PR δ PR methanol T (P s ) exp (P s ) PT δ PT (P s ) PR δ PR T=[ C], P s =[bar], δ X = [(Ps ) X (P s ) exp ]/(P s ) exp

15 66 64 VLE data at 1 bar 62 T [ o C] k j x [mol(c 3 H 6 O)/mol] Fg. 1: 15

16 9 8 v 1 v, v [ 10-5 m 3 /mol] v 4 v x [mol(c 3 H 6 O)/mol] Fg. 2: 16

17 h 1 h, h [kj/mol] h -30 h x [mol(c 3 H 6 O)/mol] Fg. 3: 17

18 s, s [kj/(mol K)] s 1 s -0.1 s x [mol(c 3 H 6 O)/mol] Fg. 4: 18

19 8 6 µ 1 IM µ 1 IGM 4 2 µ 1 g g IM g IGM g, µ [kj/mol] µ 2 IM µ 2 µ 2 IGM x [mol(c 3 H 6 O)/mol] Fg. 5: 19

20 γ 1 ln γ 0.4 γ 1,wlson γ γ 2,wlson x [mol(c 3 H 6 O)/mol] Fg. 6: 20

21 Fgure 1: Isobarc vapor-lqud equlbrum of acetone-methanol mxture at 1 bar - Sold lnes and dotted lnes represent VLE results calculated usng the Patel-Teja equaton of state wth k j = and k j = 0.0, respectvely. Expermental VLE data are from Ref. [5]. Fgure 2: Molar volume and partal molar volumes of acetone-methanol mxture at 1 bar and 50 C Fgure 3: Molar enthalpy and partal molar enthalpes of acetone-methanol mxture at 1 bar and 50 C Fgure 4: Molar entropy and partal molar entropes of acetone-methanol mxture at 1 bar and 50 C Fgure 5: Molar Gbbs free energy and chemcal potentals of acetone-methanol mxture at 1 bar and 50 C Fgure 6: Actvty coeffcents of acetone-methanol mxture at 1 bar and 50 C - Sold lnes are the result calculated usng the Patal-Teja equaton of state, and dotted lnes are the predctons usng the Wlson correlaton wth the parameters of Ref. [5] 21

22 References [1] G. Soave: Chem. Eng. Sc., 27: 1197 (1972). [2] D. Peng and D. B. Robnson: Ind. Eng. Chem. Fundam., 15, 1: 59 (1976). [3] N. C. Patel and A. S. Teja: Chem. Eng. Sc., 37, 3: 463 (1982). [4] R. H. Perry et al: Perry s chemcal engneerngs handbook seventh edton (McGraw- Hll, 1997). [5] K. Och and B. C. Y. Lu: Flud Phase Equlbra, 1(1977),