Kyushu University Global COE Program
Journal of Novel Carbon Resource Sciences, Vol. 5, pp. 19-22, Feb. 2012

Correlation of Vapor-Liquid Equilibria ofSupercritical Methanol + Glycerol SystemYongfu Liu*1, Ryosuke Kusano*1, Yoshio Iwai*2*1Graduate School of Engineering, Kyushu University*2
Faculty of Engineering, Kyushu University(Received June 27, 2011; accepted February 6, 2012)

The α function in the original Peng-Robinson equation of state was modified and expressed as a temperaturedependent form. The vapor-liquid equilibria for methanol + glycerol system were correlated at high temperatures andpressures with the original Peng-Robinson equation of state with conventional mixing rules (PR EOS), the modified
Peng-Robinson equation of state with conventional mixing rules (PR-M EOS), and the modified Peng-Robinson
equation of state with Wong-Sandler mixing rules (PR-M-WS). The PR-M EOS greatly enhances the correlation
accuracy both in liquid and vapor phases compared with PR EOS. In particularly, the correlation accuracy of liquid
phase has been further improved by PR-M-WS.

1. Introduction

Biodiesel is produced by transesterification oftriglycerides from oils and fats with an excess of alcohol1).
For the production of biodiesel fuels, supercriticalmethanol method is superior to the conventional alkaline
and acid catalyst methods. Supercritical methanol canbe conducted without catalyst. It requires very short
reaction time2).
The separation of products from biodiesel productions
requires the knowledge of phase equilibria of binary andmulti-component mixtures containing methanol, glycerol and/or fatty esters. In the previous work of the authors the vapor-liquid equilibria for methanol + methyl
laurate and methanol + glycerol systems were measured from 493 to 573K. Cubic equations of states withseveral mixing rules have been applied to correlate theexperimental data. However, good correlated results are
not obtained in vapor phase.
In order to describe the phase behavior for methanol +glycerol system near the critical temperature of methanol, Peng-Robinson equation of state (PR EOS) is moresuitable than the other cubic equations of states. It is difficult to accurately correlate both liquid and vapor phases
at the same time. In this work, the authors proposed tomodify the α function in the original Peng-Robinson
equation of state as temperature-dependent form (PRM EOS). Furthermore, PR-M EOS with Wong-Sandler
mixing rules (PR-M-WS) were also applied to calculate
the vapor-liquid equilibria of methanol + glycerol systemat high temperatures and high pressures. NRTL model
was used to calculate the excess Gibbs free energy GE
in Wong-Sandler mixing rules. The binary parameter τijwere optimized based on minimizing an objective function.

RTa			−Vm − b Vm (b + Vm ) + b(Vm − b)(1)

a=0.45724R 2Tc2 				
αPc

(2)

b=0.0778RTc 					
Pc

(3)α = (m(1 − TR ) + 1) 2 				(4)m = −0.26992ω 2 + 1.54226ω + 0.37464(5)

TTc 				

(6)

TR =

where p is the pressure, R is the gas constant, T is thetemperature, and Vm is the molar volume. The parameters
a and b are the energy and size parameters, respectively.
The ω is the acentric factor. Subscript c means the critical property.

2.2 Modified α function

In order to improve the correlation accuracy in vapor
phase for methanol + glycerol system, The α function
was modified3). In this work, the α function is explainedby the following equation.
αi = (0.468992+ 0.350664) 		
TR,i − 0.3087282

The Peng-Robinson equation of state is explained asfollows.

(7)

The equation of state used Eq. (7) instead of Eq. (4) iscalled as PR-M EOS in this work.

2.3 Parameters of pure compounds

2. Calculation model

2.1 Peng-Robinson Equation of State

p=The molecular weights Mw, acentric factors ω and thecritical temperatures Tc and pressures Pc of the pure substances focused in this work are obtained from aspen
plus database and are listed in Table 1.

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Table 1 Critical properties and acentric factors of substances
Substance

MW

Tc[K]

Pc[MPa]ωMethanol

32.04

512.5

8.084

0.5658Glycerol

92.09

850

7.5

0.51272.4.1 Conventional mixing rules
The following van der Waals one-fluid mixing rules arefrequently used to correlate the vapor-liquid equilibria.
cam = ∑∑xi x j ai a j (1 − kij ) 			

(8)

i =1 j =1cbm = ∑xi bi 					(16)Where xexp and yexp are the literature data in liquid andvapor phases, respectively, the superscript cal represents
the calculated results, NP is the number of pressure data at
each temperature, NT is the number of temperature data.

2.4 Mixing rulesc ∑ | yiexp − yical | d ( y) = ∑  / NT 		NP
T(9)

i =1

3. Results and discussion

3.1 pVmT relationship of methanol

The pressures at different temperatures of methanol
were calculated by PR EOS and PR-M EOS at the constant molar volume 0.0002823 m3/mol (density = 113.5kg/m3)4). The results are shown in Figure 1. The bothmodels give almost identical trends; however, the deviation by PR-M EOS is slightly larger than that by PR EOS.

where k ij is the binary interaction parameter.

122.4.2 Wong-Sandler mixing rules
The Wong–Sandler mixing rules were developed inorder to reproduce the composition dependences of thesecond virial coefficients accurately. Using WS mixing rules, the energy and size parameters, a and b in PR
equation of state can be obtained as follows:

Pressure [MPa]10c ccai a jb + bj
xaGEbm = ∑∑xi x j [ i(1 − kij )] / [1 − ∑ i i −]−RTCRT2i =1 j =1i =1 bi RT

86            (10)cx a G E 			am = bm (∑ i i +)Ci =1 biln ( 2 − 1)
C== −0.623225
2

where GE is the excess Gibbs free energy. In this work,
GE was calculated by NRTL model.
The NRTL model is given byc∑ j =1τ jiG ji x j 			
GE= ∑xi
RT i =1 ∑ c Gki xkc(12)k =1G ji = exp(−α jiτ ji ) 			

(13)

where, αij is the non-randomness parameter which isfixed as 0.3 in this work, τij are the binary parameters.

2.5 Objective function

The objective function used to optimize the binaryparameters is explained by the following equation.
(14)
F (kij ) = d ( x) × d ( y ) 			
where d(x) and d(y) represent the deviation functions inliquid and vapor phases, respectively.
 ∑ | xiexp − xical | 		d ( x) = ∑  / NTNP
T

4(11)

(15)

500

520
540
Temperature [K]560

Fig. 1 Pressures of methanol at constant molar volume0.0002823 m3/mol.
(●)Literature data4).
(——) Calculated results by PR EOS.
(——) Calculated results by PR-M EOS.

3.2 Saturated vapor pressures

PR EOS and PR-M EOS were adopted to calculate
the saturated vapor pressures of methanol and glycerol.
The results are shown in Figures 2 and 3. From Figure
3, the results by PR-M EOS are obviously lower thanthe experimental data for glycerol. It means that theα function of PR-M EOS gives stronger molecularinteractions of glycerol in pure liquid glycerol. The
reasons why the molecular interactions of glycerol inpure liquid glycerol should be stronger are explained in
the following section.

3.3 Binary vapor- liquid equilibria

The binary parameters k ij and τij were optimized to give
good correlated results of the vapor- liquid equilibria formethanol + glycerol system. The literature data2) wereused for the optimization of binary parameters. The objective function used in this work is presented in Eq. (14)- Eq. (16). The binary parameters are shown in Table 2.

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Correlation of Vapor-Liquid Equilibria of Supercritical Methanol + Glycerol Systemborder and that of right slope represents vapor phase border. In other words, the vapor and liquid phases coexist
in the region between both slopes. As shown in Table 3and Figure 4, it is difficult to satisfy both liquid and vapor
phase by using PR EOS with conventional mixing rules,
although the binary parameter was optimized. Specially,
the calculated results of mole fractions of methanol in vapor phase are low compared with the experimental data.
In other words, the calculated mole fractions of glycerol
in vapor phase are higher compared with the experimental data. The results cannot be improved even if thebinary parameter was optimized. This means that themolecular interactions of glycerol may become stronger
for the presence of methanol in the liquid phase. Table
3 and Figure 5 show that the results by PR-M EOS withconventional mixing rules are much better than those of
PR EOS with conventional mixing rules. The calculated
results in vapor phase are improved by PR-M EOS withconventional mixing rules. This means that the α function of PR-M EOS gives stronger molecular interactionsof glycerol, and the mole fractions of glycerol are reducedin vapor phase. The hydrogen bond formations between
glycerol and methanol molecules may be the reasons forthe stronger molecular interactions of glycerol in liquidmethanol + glycerol compared with those in pure liquidof glycerol. The α function should be modified to explain
the strong interaction of glycerol in methanol + glycerol
system, even if the calculated results of saturated vapor
pressures for glycerol are low compared with the experimental data for pure glycerol.
From Table 3 and Figure 6, the results by PR-M-WS arealmost the same as those by PR-M EOS with conventional
mixing rules in vapor phase. However, the results by PRM-WS are improved compared with those by PR-M EOS
with conventional mixing rules in liquid phase. Specially,
the calculated results in liquid phase at 493K by PR-MWS are much improved compared with those by PR-M
EOS.

8Saturated Vapor Pressure [MPa]

7654
3
2
1420

440460480
Temperature [K]500Fig. 2 Saturated vapor pressures of methanol.
(●)Literature data4).
(——) Calculated results by PR EOS.
(——) Calculated results by PR-M EOS.

Saturated Vapor Pressure [MPa]

30000

25000

20000

15000

10000

5000

0400

450500550
Temperature [K]600Fig. 3 Saturated vapor pressures of glycerol.
(●)Literature data5).
(——) Calculated results by PR EOS.
(——) Calculated results by PR-M EOS.

Table 3 Deviations between literature and calculated
results for methanol + glycerol system

Table 2 Binary parameters for methanol + glycerolsystem
kij

τij

PR-EOS withconventional mixing rule-0.027—

PR-M EOS withconventional mixing rule-0.035—

PR-M EOS with
Wong-Sandler mixing rule-0.035

τ1, 2 =6.41 - 0.01144T
τ2, 1 =-0.025

EOS and mixing rule

The calculated results by using PR EOS withconventional mixing rules, PR-M EOS with conventional
mixing rules and PR-M-WS are listed in Table 3,and shown in Figures 4, 5 and 6, respectively. The left
slope of each the same color pair represents liquid phase- 21 -PR-EOS withconventional
mixing rule

PR-M withconventional
mixing rule

δx

δy

δx

δy

δx

δy493

4.93

0.41

2.57

0.04

0.64

0.04

523

3.56

1.09

2.78

0.26

1.74

0.27

543

3.41

1.51.86

0.69

2.48

0.68

573

3.01

2.55

2.14

1.11

1.98

1.11

Σ

14.91

5.56

9.35

2.09

6.84

2.1

T[K]

δ ( x) =100∑ | xiexp − xical |NPδ ( y) =PR-M EOS with
Wong-Sandler
mixing rule100∑ | yiexp − yical | ,NP

Np is number pressure data at each temperature.

1212101088Pressure [MPa]Pressure [MPa]Journal of Novel Carbon Resource Sciences, Vol.5, Feb. 2012

64400.3

0.4

0.5
0.6
0.7
0.8
molar fraction of methanol x, y

0.9

20.3

1.0

Fig. 4 Vapor-liquid equilibria for methanol + glycerol system
calculated by PR EOS with conventional mixing rule.
(●,●,●,●) Literature data at 493, 523, 543, 573K,respectively2).
(——, ——, ——, ——) Calculated results by
PR EOS with conventional mixing rule at 493,523, 543, 573K, respectively.
12Pressure [MPa]

0.4

0.5
0.6
0.7
0.8
molar fraction of methanol x, y0.9

1.0

Fig. 6 Vapor-liquid equilibria for methanol + glycerolsystem calculated by PR-M EOS with WongSandler mixing rule.
(●,●,●,●) Literature data at 493, 523, 543, 573K,respectively2).
(——, ——, ——, ——) Calculated results by
PR-M EOS with Wong-Sandler mixing rule at493, 523, 543, 573K, respectively.

4. Conclusion

The authors proposed a new equation for α function in Peng-Robinson equation. The α function gives
strong interaction for glycerol. The vapor-liquid equilibria for methanol + glycerol system are correlated by PR
EOS, PR-M EOS, and PR-M-WS. The PR-M EOS withoptimized binary interaction parameter greatly enhanced
the correlation accuracy both in liquid and vapor phases.
In particularly, the correlation accuracy of liquid phasehas been further improved by using Wong-Sandler
mixing rules. It can be applied to explore the operating
conditions in the reactor and the separation process of
biodiesel production plants.

1086

4

2
0.3

6References
0.4

0.5
0.6
0.7
0.8
molar fraction of methanol x, y0.91)

1.0

Fig. 5 Vapor-liquid equilibria for methanol + glycerol
system calculated by PR-M EOS with conventional mixing rule.
(●,●,●,●) Literature data at 493, 523, 543, 573K,
respectively2).
(——, ——, ——, ——) Calculated results by
PR-M EOS with conventional mixing rule at493, 523, 543, 573K, respectively.

2)
3)
4)
5)

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