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Interfacial behavior prediction of alcohol + glycerol mixtures using gradient theory
Chemical Physics 534 (2020) 110747
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Interfacial behavior prediction of alcohol + glycerol mixtures using gradient theory
T
Ariel Hernándeza a
Facultad de Ingeniería, Universidad Católica de la Santísima Concepción, Alonso de Ribera 2850, Concepción, Chile
ABSTRACT
This work has been dedicated to modeling the interfacial tension and interfacial density profiles of alcohol + glycerol mixtures in the temperature range of 293.2 K–333.2 K. The Peng-Robinson equation of state and quadratic mixing rule is applied to the liquid - vapor phase equilibrium calculations. The binary interaction parameters are obtained according to the experimental phase equilibrium data at 1 atm. The gradient theory is used as a predictive approach to describe the interfacial behavior of alcohol + glycerol mixtures. The results of this study show that the equation of state used is capable of simultaneously representing the phase equilibrium and interfacial behavior of the mixtures studied. Despite using a simple equation of state, the results in phase equilibrium and interfacial tension are good. The main advantage of using the gradient theory is to obtain results on the accumulation and non-accumulation of the components at the interface, where the alcohols accumulated in the interfacial region, whereas glycerol does not show surface activity, except in the mixture with methanol.
1. Introduction Glycerol has several different uses in medical, pharmaceutical (drugs) and personal care preparations (cosmetics and toothpastes), tobacco and food processing (as a food additive, solvent, sweentener or a component of food packaging materials) and as a raw material in different chemical industries, i.e, in the production of acetals, amines, esters and ethers, mono- and di-glycerides and urethane polymers [17]. Due to surplus glycerol derived from biodiesel production, a substantial decrease in the price of glycerol may be possible in the near future [39]. Although ethanol and glycerol are important solvents in pharmaceutical and food industries, experimental data on physical properties of these mixtures are very scarce [14]. Furthermore, a better knowledge of the interfacial properties of glycerol and its mixtures can facilitate the experiments using glycerol in two-phase flow studies [14] and it is relevant to mention that the interfacial tension plays an important role in gas-liquid separations [36]. The separation of alcohols from glycerol + alcohol mixtures and physical properties of these systems play an important role in the operation of distillation columns [13]. Some authors have obtained experimental information of phase equilibrium and theoretical modeling using equations of state (EOS). Oliveira et al. [30] obtained experimental information for five alcohol + glycerol mixtures and water + glycerol mixture at atmospheric pressure. The authors used the Cubis-Plus-Association EOS to model mixtures containing glycerol. Shimoyama et al. [35] obtained the vapor-liquid equilibria (VLE) for binary (alcohol + glycerol) mixtures in the temperature range of 493 K to 573 K and the experimental data were modelled with Peng-Robinson equation of state (PR-EOS).
Veneral et al. [39] obtained experimental vapor-liquid equilibrium data of several binary (alcohol + glycerol) mixtures and ternary (alcohol + glycerol + water) mixtures over the pressure range of 6.7 kPa to 66.7 kPa and the experimental data without NaCl were correlated using the UNIQUAC model. The interfacial tension of mixtures containing glycerol has also been measured by some authors. Alkindi et al. [1] obtained different physical properties, among them the interfacial tension, for the mixture ethanol + glycerol at 294 K. Erfani et al. [14] obtained the interfacial tension and interfacial composition of different binary mixtures of glycerol and alcohols. They used Peng-Soave-Redlich-Kwong EOS for phase equilibrium and interfacial tension calculations. Iqbal et al. [16] obtained interfacial tension measurements of glycerol with organic cosolvents at 298.15 K. In this work, the gradient theory (GT) is combined to PR-EOS to calculate both phase equilibrium equations and interfacial properties of the methanol + glycerol, ethanol + glycerol, 1-propanol + glycerol, 2propanol + glycerol and 1-butanol + glycerol mixtures in the temperature range of 293.2 K to 333.2 K. The GT was initially developed by van der Waals [37] and later reformulated by Cahn and Hilliard [3]. Numerous authors thereafter have utilized GT with different EOSs to describe the interfacial behavior of systems with different types of phase equilibria [4,6–9,22–24,26,28,34]. With respect to the mixtures used in this work, no author has used GT to model the interfacial behavior of the mixtures, in this way, no one has studied the possible adsorption or desorption of the components in the alcohol + glycerol mixtures. Due to the above, the present work arises in order to use GT (widely used theory), except in these mixtures, to study the prediction
E-mail address: [email protected]. https://doi.org/10.1016/j.chemphys.2020.110747 Received 13 January 2020; Received in revised form 21 February 2020; Accepted 9 March 2020 Available online 14 March 2020 0301-0104/ © 2020 Elsevier B.V. All rights reserved.
Chemical Physics 534 (2020) 110747
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Nomenclature List of symbols a a0 AAD b c kij m n nc P R T u x, y
cohesion parameter in the EOS Helmholtz energy density of the homogeneous system statistical deviation covolumen parameter in the EOS influence parameter interaction parameter for the quadratic mixing rule parameter of the thermal cohesive function number of points number of components absolute pressure universal gas constant absolute temperature auxiliary variable mole fraction of the liquid and vapor phases
Subscriptors c i , j, k, s
0 exp l ref theo v
of the theoretical model in the calculations of interfacial tension and the behavior of the components in the interfacial region.
between different molecules. In this work, the parameter of interaction was obtained by adjusting the experimental information of the equilibrium of phases.
2. Theory
2.2. van der Waals square gradient theory
2.1. The EOS model
Following the work of Cahn and Hilliard [3], the GT was generalized and modified by different authors ([2,4–6,11,12,21], among others). This theoretical approach is characteristic for describing of inhomogeneous fluid behavior through a continuous model of hypothetically homogeneous fluids. The GT proposes that under conditions with constant particle numbers, volume, and temperature, Helmholtz energy (A) can be mathematically described by Eq. (5):
In this work, Peng-Robinson equation of state [31] was selected for modeling phase equilibrium and interfacial behavior for different mixtures. This equation of state is expresses in terms of density by Eq. (1):
a 2 , (1 + d1 b )(1 + d2 b )
RT b
d1 = 1 +
2,
d2 = 1
2 (1)
A( ) =
where a and b are cohesive and covolume parameters, respectively. For each pure component they are given by Eq. (2):
(RTc, i )2 ai = 0.45724 Pc, i
RTc, i bi = 0.07780 Pc, i
i (T ),
= [1 + mi (1
T / Tc, i )]2 ,
(2)
mi = 0.37464 + 1.54226
i
0.26992
a0 = RT
2 i
(3)
a=
2
nc i j i, j = 1
ai aj (1
kij ),
b=
1
i=1
nc
cij (
i·
j)
dV
(5)
i, j = 1
0
P RT
1 2
d + ln
RT P ref
+
1
nc i ln i i=1
(6)
where is a reference pressure, which, in the current study, was 1 bar. For PR-EOS, the calculation of the Helmholtz energy density can be obtained using Eq. (7):
a0
nc
= i=1
nc i bi
1 2
P ref
where mi is a parameter that can be generalized in terms of the acentric factor ( i ) for each pure component. Eq. (2) is extended to mixtures using a mixing rule. In this work, Quadractic mixing rule (QMR) [19] is selected for calculating the cohesive and covolume parameters of the mixture. This rule proposes that the cohesive and covolume parameters of the mixture are defined from Eq. (4):
1
V
a0 ( ) +
where a0 ( ) is the Helmholtz energy density of a hypothetically homogeneous fluid with a uniform concentration, cij is the cross-influence parameter, nc is the number of mixture components, and i and j are concentrations of the species i and j, respectively. Furthermore, the Helmholtz energy density of the homogeneous system can be determined from any EOS model using Eq. (6) [41]:
where R is the universal gas constant, Tc, i and Pc , i are the critical properties of temperature and pressure, respectively, and i is the thermal cohesive function, defined as Eq. (3): i
equilibrium condition experimental liquid phase reference theoretical vapor phase
thermal cohesive function
(T )
1
critical condition species
Superscripts
Greek Letters
P=
relative adsorption isotherm chemical potential grand thermodynamic potential acentric factor molar concentration interfacial tension
µ
RT i ln
1 + (1 1 + (1 +
(4)
i
2 )b 2 )b
RT ln(1 RT ln
b )+ P ref RT
a ln 2 2b (7)
Concentration distribution in the interfacial region was determined through the Euler-Lagrange equation [4], which, in the case of a planar interface and density-independent influence parameter, is reflected by Eq. (8):
where ai and bi are the cohesive and covolume parameters of the pure components, respectively, i , j and are the densities in a specific phase of components i , j , and the mixture, respectively, nc is the number of components in the mixture, and kij is the parameter of interaction 2
Chemical Physics 534 (2020) 110747
A. Hernández nc
cij j=1
d2
j
µi0 ; i , j = 1, …, nc
= µi ( ) = µi ( )
dz 2
(8)
where µi ( ) is the chemical potential of component i and is the chemical potential of component i in the bulk equilibria phases. The boundary conditions for Eq. (8) are given by Eq. (9):
µi0
i |z =
=
,
i
i |z =
=
(9)
i
where and symbolize the homogeneous phases. The chemical potential of the pure component i in the mixture can be obtained using Eq. (10) [38]:
µi =
a0 i
(10)
T , V , nj
For PR-EOS, the calculation of the chemical potential of component i can be obtained by Eq. (11):
µi =
RT ln(1 bi
b )+
( ) 2a i
1 2 2b
1 + (1 2 )b 1 + (1 + 2 ) b
a ln 2 2 b2
ln
1 + (1 2 )b 1 + (1 + 2 ) b
RT ln
( ) + RT P ref RT i
a 2 ) b ][1 + (1 + 2 ) b ] b
bi [1 + (1
RT b )
+ bi (1
(11) Solving for concentration-space variables is complex as the boundary conditions had infinite limits. Therefore, this study used the concentration-space to concentration-concentration transformation, as proposed by Carey [4]. This transformation required solving the following system of ordinary differential equations (ODEs) given by Eq. (12): nc
cij i, j = 1
d d
i s
d
j
d
s
d d
Hs
k
Hk + 2
det( C )
s
d2 d
k 2 s
= 0;
k (12)
= 1, …, s, …, nc where s must be defined monotonic variable and thermodynamic potential defined as Eq. (13): nc
( ) = a0 ( )
i=1
is the grand
[ i µi0] + P 0
(13)
where is the pressure in the bulk equilibria phases. In Eq. (12), Hk is the kth component of the vector function H , which is defined as Eq. (14):
P0
T
H ( ) = det( C )· C 1·
T
,
=
, …, 1
(14)
nc
where C is the influence parameters matrix given by Eq. (15):
C=
c11
c1nc
cn c 1
cn c n c
(15)
The ODEs system given by Eq. (12) is subject to boundary conditions as shown in Eq. (16): k( s
)=
k
,
k( s
)=
(16)
k
Fig. 1. Schematic representation of concentration profiles for the mixtures. (•) VLE bulk densities, ( ) stationary points: A (adsorption) and D (desorption) of the species on the interfacial region.
When the cross-influence parameters obey the geometric mixing rule, i.e, cij = cii cjj , Eq. (12) reduces to Eq. (17):
css [µk ( )
µk0 ] =
ckk [µs ( )
µs0 ]; k = 1, 2, …, s
1, s + 1, nc
Nevertheless, some authors have developed correlations for the influence parameter [15,27,42]. According to Carey procedures, the influence parameter of the pure component (cii ) in the case of pure fluids (i = j ) and VLE is given by Eq. (18):
(17) where cii and cjj are the influence parameters of the pure components i and j, respectively. When the mentioned rule is used for cij , the gradient theory is used as a predictive approach, because some adjustment parameter is not required. The typical approach for calculating the influence parameter applies procedures suggested by Carey et al. [4–6] and Cornelisse et al. [7–9].
cii = 3
exp (T 0 ) 2
2
l v
2
( ) + P0 d
(18)
Chemical Physics 534 (2020) 110747
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Table 1 Physical properties of the pure components. Fluid
Tc (K )
Pc (bar )
Methanol Ethanol 1-Propanol 2-Propanol 1-Butanol Glycerol
512.5 514.0 536.8 508.3 563.0 850.0
80.84 61.37 51.69 47.64 44.14 75.00
Table 3 Binary interaction parameters of PR-EOS + QMR for the mixtures and statistical deviation in boiling temperature at 1 atm. 0.565831 0.643558 0.620432 0.666873 0.589462 0.512690
where exp (T 0) is the value of experimental tension at the boiling temperature T 0, v , l , and P 0 are the vapor density, liquid density and pressure of the phase equilibria, respectively. On the other hand, Eq. (19) is applied to determine the interfacial density profile of each component in the mixture:
z ( s ) = z ref +
s (z ) ref s
nc
1
cij
2
i, j = 1
d
i
d
s
d
j
d
s
d
s
Mixture
kij
AADT%
Methanol(1) + Glycerol(2) Ethanol(1) + Glycerol(2) 1-Propanol(1) + Glycerol(2) 2-Propanol(1) + Glycerol (2) 1-Butanol(1) + Glycerol(2)
−0.0108 0.0203 −0.0116 −0.0054 0.0046
0.3099 2.1076 1.3964 1.6757 0.8498
homogeneous phases and . Discretizing the auxiliary variable u and solving the system of Eqs. (21) and (22), can be obtained the dependence of the densities with the variable u. Using = 0.5, the interfacial density profile can be compute as shown in Eq. (24):
(19)
z i+1 = z i +
where z ref and sref are reference values. Finally, interfacial tension is compute by Eq. (20): nc
s
=
(2
s
)
cij i, j = 1
d d
i s
d d
j s
d
cii
i
(ui + 1
(20)
=u 2
i
ui
+ (1
)
2
[ (
i + 1)
0]
(24)
(22)
nc
ui ) { 2 [ ( i )
ij
=
z 0j
[ i (z )
nc
cii
+
2 [ (
i + 1)
0]
}
(25)
i
] dz +
+ z 0j
[ i (z )
i
] dz
(26)
where is the localization of the divide position relative to a species j and and are the bulk phases. From Eq. (26), z 0j is calculated considering that species j does not have adsorption along the interfacial
z 0j
(23)
i=1
0]
where n is the number of points used in the discretization of u. In this work, n = 1000 was used. Eqs. (19) or (24) allows quantification of the population of species at the interface, i (z ), from which the surface activity (adsorption/ desorption of species along the interface region), relative Gibbs adsorption isotherm and the interfacial tension can be calculated. Characteristic shapes of i (z ) profiles are shown in Fig. 1 obtained from the work of Mejía et al. [25]. From Fig. 1 it is seen that along the interface width the concentration profile may be monotonic (type a) or nonmonotonic (type b and c). It should be pointed out that the nonmonotonic behavior of the concentration profiles type b and c is interesting because it reflects the surface activity at the interface. Nonmonotonic profiles are characterized by stationary points which correspond to the interfacial point where adsorption (point A) or desorption (point D) of the species take place [25]. The accumulation of a species i at the interface region is characterized by the condition, d i /dz = 0 , and it may be positive or negative. When d 2 i / dz 2 < 0 , the positive surface activity or adsorption of species along the interface region is reflected, whereas d 2 i / dz 2 > 0 denotes negative surface activity or desorption of species along the interface region [29]. The relative Gibbs adsorption isotherm of a species i relative to a species j ( ij ) can be expressed by Eq. (26) [33,40]:
where nc is the number of components, u is an auxiliary variable that varies in the finite and bounded range [u , u ], and the parameter is given by Eq. (23):
=
ui + 1
i=0
(21)
cii µ1 ( );
0]
n 1
s
i=1
c11 µi ( ) =
[ (
i)
Finally, interfacial tension can be obtained by Eq. (25):
To use Eqs. (17), (19) and (20), the choice of the defined monotonic variable ( s ) is very important. In order to avoid this criterion, because the mixtures may have stationary points for both components in the density-density profile, several authors have introduced a new independent variable to define the density path profiles, which then is called as a patch function and it must be monotonic. Poser and Sanchez [32] were the first to suggest a form for the new auxiliary variable. Later, authors as Cornelisse [7], Kou et al. [18] and Liang et al. [20] have proposed new ways for this auxiliary variable in order to cover stability in the calculations. In this work, Kou et al. [18] technique was chosen because it is simple, moreover because presents the equations for interfacial tension and density profiles based on the new defined variable. This technique consists of applying GT predictively to a multicomponent system, adding an auxiliary variable u that is defined monotonic. The authors [18] propose to solve a system of nonlinear equations given by Eqs. (21) and (22): nc
2
In addition, u and u values can be determined using Eq. (21) in the Table 2 Influence parameters for the pure components at each temperature. Fluid
Methanol Ethanol 1-Propanol 2-Propanol 1-Butanol Glycerol
cii × 1019 (J m5 mol−2) 293.2 K
294 K
298.15 K
303.2 K
313.2 K
323.2 K
333.2 K
– 0.452 0.837 0.881 1.455 2.208
– 0.452 – – – 2.216
0.224 – 0.829 – – 2.216
– 0.467 0.866 0.926 1.482 2.262
– 0.487 0.898 0.948 1.498 2.341
– 0.510 0.933 0.972 1.486 2.385
– 0.537 0.982 0.987 1.458 2.455
4
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Table 4 Statistical deviation for interfacial tension for the mixtures at different temperatures. Mixture
Methanol(1) + Glycerol(2) Ethanol(1) + Glycerol(2) 1-Propanol(1) + Glycerol(2) 2-Propanol(1) + Glycerol(2) 1-Butanol(1) + Glycerol(2)
AAD % 293.2 K
294 K
298.15 K
303.2 K
313.2 K
323.2 K
333.2 K
– 4.777 6.358 3.735 2.971
– 5.811 – – –
8.121 – 2.205 – –
– 5.050 6.100 3.483 2.163
– 4.868 6.038 3.280 1.490
– 5.127 5.809 3.896 1.692
– 5.246 6.052 4.662 1.983
x1 projection for the mixture methanol(1) + glycerol(2) at Fig. 2. 298.15 K. ( ) Iqbal et al. [16], ( ) prediction obtained with GT + PREOS + QMR.
x1 projection for the mixture 1-propanol(1) + glycerol(2) at difFig. 4. ferent temperatures. ( ) 298.15 K from Iqbal et al. [16], ( , ) 313.2 K and 333.2 K from Erfani et al. [14], ( , , ) prediction at 298.15 K, 313.2 K and 333.2 K obtained with GT + PR-EOS + QMR.
x1 projection for the mixture ethanol(1) + glycerol(2) at different Fig. 3. temperatures. ( , , ) 293.2 K, 313.2 K and 333.2 K from Erfani et al. [14], (—–, , ) prediction at 293.2 K, 313.2 K and 333.2 K obtained with GT + PR-EOS + QMR.
x1 projection for the mixture 2-propanol(1) + glycerol(2) at difFig. 5. ferent temperatures. ( , , ) 293.2 K, 313.2 K and 333.2 K from Erfani et al. [14], (—–, , ) prediction at 293.2 K, 313.2 K and 333.2 K obtained with GT + PR-EOS + QMR.
region (Fig. 1 a), i.e, Eq. (26) is solved for the case that jj = 0 . When z 0j is fixed, the relative Gibbs adsorption isotherm of a species i relative to a species j is calculated from Eq. (26).
better represent the interfacial behavior of the mixture at the given temperature. The parameters obtained are in Table 2. The experimental interfacial tension for each temperature has been obtained from: Erfani et al. [14] for 293.2 K, 303.2 K, 313.2 K, 323.2 K and 333.2 K, Iqbal et al. [16] for 298.15 K, and Alkindi et al. [1] for 294 K.
3. Results and discussions 3.1. Pure fluids
3.2. Binary mixtures
All pure-component parameters of PR-EOS have been listed in Table 1. The pure component parameters have been taken from DIADEM [10]. The influence parameters of the pure components were obtained at each temperature using Eq. (18). This methodology was used in order to
3.2.1. Phase equilibrium calculations The accuracy of phase equilibrium calculations is very important for the successful description of thermophysical properties such as interfacial tension. In this work, PR-EOS + QMR was used to describe the 5
Chemical Physics 534 (2020) 110747
A. Hernández
x1 projection for the mixture 1-butanol(1) + glycerol(2) at different Fig. 6. temperatures. ( , , ) 293.2 K, 313.2 K and 333.2 K from Erfani et al. [14], (—–, , ) prediction at 293.2 K, 313.2 K and 333.2 K obtained with GT + PR-EOS + QMR.
Fig. 8. Dimensionless density profiles for the mixture ethanol(1) + glycerol(2) at 294 K. (—–, , ) density profile of ethanol at x1 = 0.1, x1 = 0.5 and x1 = 0.9 , (- - -, , ) density profile of glycerol at x1 = 0.1, x1 = 0.5 and x1 = 0.9 , (- – -, , ) z 0 value at x1 = 0.1, x1 = 0.5 and x1 = 0.9 , ( , ) vapor and liquid density.
boiling temperature ( AADT%) was calculated using Eq. (27):
AADT % =
100 n
n i=1
Tiexp Titheo Tiexp
(27)
where n represent number of points, exp represent experimental value and theo represent theoretical value. From Table 3 it can be seen that the theoretical model PREOS + QMR correctly adjusts the phase equilibrium for the studied mixtures. 3.2.2. Interfacial tension of alcohol + glycerol mixtures PR-EOS + QMR was able to correctly adjust the phase equilibrium, then it is expected that in combination with the gradient theory acceptable results in the interfacial tension are obtained. It is important to mention that to represent the inhomogenous phase (interfacial region) as well as possible, the homogeneous phases must be well represented, which has been obtained satisfactorily. The gradient theory was used as a predictive approach to model the interfacial behavior of the analyzed mixtures and in Table 4 have been listed the statistical deviation of interfacial tension for the mixtures. The absolute average deviation for interfacial tension ( AAD % ) was calculated using Eq. (28):
Fig. 7. Dimensionless density profiles for the mixture methanol(1) + glycerol (2) at 298.15 K. (—–, , ) density profile of methanol at x1 = 0.1, x1 = 0.5 and x1 = 0.9 , (- - -, , ) density profile of glycerol at x1 = 0.1, x1 = 0.5 and x1 = 0.9 , (- – -, , ) z 0 value at x1 = 0.1, x1 = 0.5 and x1 = 0.9 , ( , ) vapor and liquid density.
phase equilibrium of the following mixtures: methanol + glycerol, ethanol + glycerol, 1-propanol + glycerol, 2-propanol + glycerol and 1-butanol + glycerol. The binary interaction parameters of the PR-EOS + QMR have been listed in Table 3 and were determined using experimental phase equilibrium from Oliveira et al. [30]. The absolute average deviation in
AAD % =
6
100 n
n i=1
exp i
exp i
theo i
(28)
Chemical Physics 534 (2020) 110747
A. Hernández
Fig. 9. Dimensionless density profiles for the mixture 1-propanol(1) + glycerol (2) at 313.2 K. (—–, , ) density profile of 1-propanol at x1 = 0.1, x1 = 0.5 and x1 = 0.9 , (- - -, , ) density profile of glycerol at x1 = 0.1, x1 = 0.5 and x1 = 0.9 , (- – -, , ) z 0 value at x1 = 0.1, x1 = 0.5 and x1 = 0.9 , ( , ) vapor and liquid density.
Fig. 10. Dimensionless density profiles for the mixture 2-propanol(1) + glycerol(2) at 323.2 K. (—–, , ) density profile of 2-propanol at x1 = 0.1, x1 = 0.5 and x1 = 0.9 , (- - -, , ) density profile of glycerol at x1 = 0.1, x1 = 0.5 and x1 = 0.9 , (- – -, , ) z 0 value at x1 = 0.1, x1 = 0.5 and x1 = 0.9 , ( , ) vapor and liquid density.
From Table 4 it can be seen that the theoretical model GT + PREOS + QMR correctly predicts interfacial tension for 2-propanol + glycerol and 1-butanol + glycerol mixtures. The best results are obtained for the butanol + glycerol mixture. Also, Table 4 indicates that the maximum deviation (8.12% ) is reached for methanol + glycerol mixture at 298.15 K and the minimum deviation (1.49% ) is reached for 1-butanol + glycerol mixture at 313.2 K. The deficiency of the theoretical model for the prediction of the interfacial tension for the methanol + glycerol mixture is due to the inability of PR-EOS + QMR for the treatment of mixtures with associative interactions. A more robust equation of state could improve the results obtained in the interfacial tension for the methanol + glycerol mixture. Figs. 2–6 shows the interfacial tension for the mixtures studied. Fig. 2 shows that GT does not correctly predicts the interfacial tension for the methanol + glycerol mixture at 298.15 K and the theory underpredicts the interfacial tension for a molar fraction of alcohol greater than 0.4. Fig. 3 shows that GT underpredicts the interfacial tension for the ethanol + glycerol mixture at different temperatures analyzed for a molar fraction of ethanol below 0.6 and for a molar fraction of alcohol greater than 0.7 it is observed that GT correctly predicts the interfacial tension at different temperatures. Fig. 4 shows that GT does not correctly predicts the interfacial tension for the 1-propanol + glycerol mixture at 313.2 K and 333.2 K and for a molar fraction of alcohol greater than 0.7, the theory correctly predicts interfacial tension at 298.15 K, 313.2 K and 333.2 K. For this mixture, the best prediction is obtained at 298.15 K. Fig. 5 shows that GT underpredicts the interfacial
tension for the 2-propanol + glycerol mixture for a molar fraction of 2propanol less than 0.4, and for a molar fraction greater than 0.5, the theory correctly predicts the interfacial tension for the analyzed temperatures. Fig. 6 shows that GT is able to correctly represent the interfacial tension for the 1-butanol + glycerol mixture, however the best results were obtained at 313.2 K and 333.2 K. 3.2.3. Interfacial density profiles of alcohol + glycerol mixtures GT has the advantage of studying interfacial behavior, i.e, studying properties such as interfacial tension and interfacial density profiles. Figs. 7–11 shows the alcohol and glycerol profiles in the different mixtures at a given temperature and at different molar fractions of alcohol in the mixture, the localization of the divide position relative to glycerol (z 0 ) and the bulk phases (vapor and liquid density). Fig. 7 shows that methanol accumulated in the interfacial region due to a greater affinity for remaining in the interface than in the homogenous phase, while glycerol simultaneously showed accumulation and nonaccumulation at x1 = 0.5 and x1 = 0.9 . According to the literature, the density profile of methanol is type b and the density profile of glycerol is type a at x1 = 0.1 and type c at x1 = 0.5 and x1 = 0.9. It is also observed that as the molar fraction of methanol in the mixture increases, the surface activity of methanol decreases. In Figs. 8–11 it is observed that for the ethanol + glycerol at 294 K, 1-propanol + glycerol at 313.2 K, 2-propanol + glycerol at 323.2 K and 1-butanol + glycerol at 333.2 K mixtures, alcohol has stationary accumulation points in the density profiles (type b) for the three compositions analyzed and the surface 7
Chemical Physics 534 (2020) 110747
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mixture at 298.15 K, 2-propanol + glycerol and 1-butanol + glycerol mixtures for the entire temperature range. In spite of the simplicity of the mixing rule used in this work (quadratic mixing rule), the results obtained in the interfacial tension are good. Another advantage of having used the gradient theory was that it allowed obtaining the interfacial profiles for the components of the mixture, where relevant information was obtained about the behavior of alcohols and glycerol in the interfacial region. In all studied mixtures, alcohol exhibited a maximum stationary point in the density profile, i.e, positive surface activity (d 1 /dz = 0 ; d 2 1 / dz 2 < 0 in the interfacial region). This condition is attributed to the lower interfacial tension of the alcohol compared to glycerol, which implies its preference to locate itself in the interfacial region rather than the homogeneous phases, whereas for glycerol accumulation and non-accumulation (d 2 / dz = 0 ; d 2 2 / dz 2 < 0;d 2 2 / dz 2 > 0 in the interfacial region) was observed at the interface for the mixture with methanol at 298.15 K and a monotonic behavior in its density profile was obtained for the mixtures with ethanol, 1-propanol, 2-propanol and 1-butanol. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments A.H acknowledges the economic support given by the UCSC. References [1] A.S. Alkindi, Y.M. Al-Wahaibi, A.H. Muggeridge, Physical properties (density, excess molar volume, viscosity, surface tension, and refractive index) of ethanol+ glycerol, J. Chem. Eng. Data 53 (12) (2008) 2793–2796. [2] V. Bongiorno, L.E. Scriven, H.T. Davis, Molecular theory of fluid interfaces, J. Colloid Interface Sci. 57 (3) (1976) 462–475. [3] J.W. Cahn, J.E. Hilliard, Free energy of a nonuniform system. i. Interfacial free energy, J. Chem. Phys. 28 (2) (1958) 258–267. [4] B.S. Carey, The Gradient Theory of Fluid Interfaces. PhD thesis, University of Minnesota, 1979. [5] B.S. Carey, L.E. Scriven, H.T. Davis, Semiempirical theory of surface tensions of pure normal alkanes and alcohols, AIChE J. 24 (6) (1978) 1076–1080. [6] B.S. Carey, L.E. Scriven, H.T. Davis, Semiempirical theory of surface tension of binary systems, AIChE J. 26 (5) (1980) 705–711. [7] P.M.W. Cornelisse. The Gradient Theory Applied, Simultaneous Modelling of Interfacial Tension and Phase Behaviour. PhD thesis, Technische Universiteit Delft, 1997. [8] P.M.W. Cornelisse, C.J. Peters, J. de Swaan Arons, Application of the peng-robinson equation of state to calculate interfacial tensions and profiles at fluid interfaces, Precision Process Technology, Springer, 1993, pp. 651–660. [9] P.M.W. Cornelisse, C.J. Peters, J. de Swaan Arons, Simultaneous prediction of phase equilibria, interfacial tension and concentration profiles, Mol. Phys. 80 (4) (1993) 941–955. [10] T.E. Daubert, R.P. Danner, Physical and thermodynamic properties of pure chemicals. data compilation, Taylor & Francis, Bristol, PA., 1989. [11] H.T. Davis, L.E. Scriven, Gradient theory of fluid microstructures, J. Stat. Phys. 24 (1) (1981) 243–268. [12] H.T. Davis, L.E. Scriven, Stress and structure in fluid interfaces, Adv. Chem. Phys 49 (1982) 357–454. [13] B.R. Dhar, K. Kirtania, Excess methanol recovery in biodiesel production process using a distillation column: a simulation study, Chem. Eng. Res. Bull. 13 (2) (2009) 55–60. [14] A. Erfani, S. Khosharay, C.P. Aichele, Surface tension and interfacial compositions of binary glycerol/alcohol mixtures, J. Chem. Thermodyn. 135 (2019) 241–251. [15] J.M. Garrido, A. Mejía, M.M. Piñeiro, F.J. Blas, E.A. Müller, Interfacial tensions of industrial fluids from a molecular-based square gradient theory, AIChE J. 62 (5) (2016) 1781–1794. [16] M.J. Iqbal, M.A. Rauf, N. Ijaz, Surface tension measurements of glycerol with organic cosolvents, J. Chem. Eng. Data 37 (1) (1992) 45–47. [17] R.E. Kirk, Encyclopedia of Chemical Technology vol. 18, Wiley, 1982. [18] J. Kou, S. Sun, X. Wang, Efficient numerical methods for simulating surface tension of multi-component mixtures with the gradient theory of fluid interfaces, Comput. Methods Appl. Mech. Eng. 292 (2015) 92–106. [19] T.Y. Kwak, G.A. Mansoori, Van der waals mixing rules for cubic equations of state. applications for supercritical fluid extraction modelling, Chem. Eng. Sci. 41 (5) (1986) 1303–1309.
Fig. 11. Dimensionless density profiles for the mixture 1-butanol(1) + glycerol (2) at 333.2 K. (—–, , ) density profile of 1-butanol at x1 = 0.1, x1 = 0.5 and x1 = 0.9 , (- - -, , ) density profile of glycerol at x1 = 0.1, x1 = 0.5 and x1 = 0.9 , (- – -, , ) z 0 value at x1 = 0.1, x1 = 0.5 and x1 = 0.9 , ( , ) vapor and liquid density.
activity of the alcohol decreases as the molar fraction of the alcohol in the mixture increases. On the other hand, glycerol has a monotonic density profile (type a) throughout the molar fraction range. Finally, due to the simple EOS used and the inadequate representation of the cross association in the methanol + glycerol mixture, it was the cause for the glycerol to only have simultaneous adsorption and desorption in the methanol + glycerol mixture. 4. Concluding remarks The Peng-Robinson equation of state, coupled with the quadratic mixing rule, was applied to alcohol + glycerol (methanol + glycerol, ethanol + glycerol, 1-propanol + glycerol, 2-propanol + glycerol and 1-butanol + glycerol) mixtures in vapor-liquid equilibria at a pressure of 1 atm, in order to adjust the binary interaction parameter to be used in gradient theory to predict interfacial behaviour (interfacial tension and density profiles) of alcohol + glycerol mixtures at different temperatures. Boiling temperatures calculations for the alcohol + glycerol mixtures were obtained with all the AADT% values below 2.1% and the smallest deviation (0.3%) was obtained for the methanol + glycerol mixture. Interfacial tension calculations for the alcohol + glycerol were obtained with AAD % values between 1.5% (1-butanol + glycerol mixture at 313.2 K) and 8.1% (methanol + glycerol mixture at 298.15 K). The theoretical model was able to represent the variation of the interfacial tension with the molar fraction for the mixtures studied, getting good results as a predictive approach for 1-propanol + glycerol 8
Chemical Physics 534 (2020) 110747
A. Hernández [20] X. Liang, M.L. Michelsen, G.M. Kontogeorgis, A density gradient theory based method for surface tension calculations, Fluid Phase Equilib. (2016). [21] B.F. McCoy, H.T. Davis, Free-energy theory of inhomogeneous fluids, Phys. Rev. A 20 (3) (1979) 1201. [22] A. Mejía, Comportamiento interfacial de mezclas fluidas en equilibrio. PhD thesis, Universidad de Concepción, Chile, 2004. [23] A. Mejía, H. Segura, L. Vega, J. Wisniak, Simultaneous prediction of interfacial tension and phase equilibria in binary mixtures: an approach based on cubic equations of state with improved mixing rules, Fluid Phase Equilib. 227 (2) (2005) 225–238. [24] A. Mejía, H. Segura, J. Wisniak, I. Polishuk, Correlation and prediction of interface tension for fluid mixtures: an approach based on cubic equations of state with the wong-sandler mixing rule, J. Phase Equilib. Diffus. 26 (3) (2005) 215–224. [25] A. Mejía, H. Segura, J. Wisniak, I. Polishuk, Association and molecular chain length effects on interfacial behavior, Phys. Chem. Liq. 44 (1) (2006) 45–59. [26] C. Miqueu, Modélisation, á température et pression élevées, de la tension superficielle de composants des fluides pétroliers et de leurs mélanges synthétiques ou réels. PhD thesis, Université de Pau, Pau, France, 2001. [27] C. Miqueu, B. Mendiboure, A. Graciaa, J. Lachaise, Modelling of the surface tension of pure components with the gradient theory of fluid interfaces: a simple and accurate expression for the influence parameters, Fluid Phase Equilib. 207 (1) (2003) 225–246. [28] C. Miqueu, B. Mendiboure, C. Graciaa, J. Lachaise, Modelling of the surface tension of binary and ternary mixtures with the gradient theory of fluid interfaces, Fluid Phase Equilib. 218 (2) (2004) 189–203. [29] E.A. Müller, A. Mejía, Interfacial properties of selected binary mixtures containing n-alkanes, Fluid Phase Equilib. 282 (2) (2009) 68–81. [30] M.B. Oliveira, A.R.R. Teles, A.J. Queimada, J.A.P. Coutinho, Phase equilibria of glycerol containing systems and their description with the cubic-plus-association (cpa) equation of state, Fluid Phase Equilib. 280 (1–2) (2009) 22–29.
[31] D.Y. Peng, D.B. Robinson, A new two-constant equation of state, Ind. Eng. Chem. Fund. 15 (1) (1976) 59–64. [32] C.I. Poser, I.C. Sanchez, Interfacial tension theory of low and high molecular weight liquid mixtures, Macromolecules 14 (2) (1981) 361–370. [33] J.S. Rowlinson, B. Widom, Molecular Theory of Capillarity. The International Series of Monographs on Chemistry, Clarendon Press, Oxford, 1982. [34] M. Sahimi, B.N. Taylor, Surface tension of binary liquid–vapor mixtures: a comparison of mean-field and scaling theories, J. Chem. Phys. 95 (9) (1991) 6749–6761. [35] Y. Shimoyama, T. Abeta, L. Zhao, Y. Iwai, Measurement and calculation of vapor–liquid equilibria for methanol+ glycerol and ethanol+ glycerol systems at 493–573 k, Fluid Phase Equilib. 284 (1) (2009) 64–69. [36] S.R. Syeda, A. Afacan, K.T. Chuang, Effect of surface tension gradient on froth stabilization and tray efficiency, Chem. Eng. Res. Des. 82 (6) (2004) 762–769. [37] J.D. van der Waals, The thermodynamik theory of capillarity under the hypothesis of a continuous variation of density, Zeit Phys Chem. 13 (1894) 675–725. [38] H. Van Ness, M. Abbott, Classical Thermodynamics of Non-Electrolyte Solutions, McGraw-Hill, 1982. [39] J.G. Veneral, T. Benazzi, M.A. Mazutti, F.A.P. Voll, L. Cardozo-Filho, M.L. Corazza, R. Guirardello, J.V. Oliveira, Thermophysical properties of biodiesel and related systems. part i. vapour–liquid equilibrium at low pressures of binary and ternary systems involving methanol, ethanol, glycerol, water and nacl, J. Chem. Thermodyn. 58 (2013) 398–404. [40] B. Widom, Remarks on the gibbs adsorption equation and the van der waals, cahnhilliard theory of interfaces, Physica A 95 (1) (1979) 1–11. [41] J. Wisniak, A. Apelblat, H. Segura, Application of cubic equations of state to the fit of vapor pressures of pure components, Chem. Eng. Sci. 53 (4) (1998) 743–751. [42] Y.X. Zuo, E.H. Stenby, Calculation of interfacial tensions with gradient theory, Fluid Phase Equilib. 132 (1) (1997) 139–158.
9
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Interfacial behavior prediction of alcohol + glycerol mixtures using gradient theory
T
Ariel Hernándeza a
Facultad de Ingeniería, Universidad Católica de la Santísima Concepción, Alonso de Ribera 2850, Concepción, Chile
ABSTRACT
This work has been dedicated to modeling the interfacial tension and interfacial density profiles of alcohol + glycerol mixtures in the temperature range of 293.2 K–333.2 K. The Peng-Robinson equation of state and quadratic mixing rule is applied to the liquid - vapor phase equilibrium calculations. The binary interaction parameters are obtained according to the experimental phase equilibrium data at 1 atm. The gradient theory is used as a predictive approach to describe the interfacial behavior of alcohol + glycerol mixtures. The results of this study show that the equation of state used is capable of simultaneously representing the phase equilibrium and interfacial behavior of the mixtures studied. Despite using a simple equation of state, the results in phase equilibrium and interfacial tension are good. The main advantage of using the gradient theory is to obtain results on the accumulation and non-accumulation of the components at the interface, where the alcohols accumulated in the interfacial region, whereas glycerol does not show surface activity, except in the mixture with methanol.
1. Introduction Glycerol has several different uses in medical, pharmaceutical (drugs) and personal care preparations (cosmetics and toothpastes), tobacco and food processing (as a food additive, solvent, sweentener or a component of food packaging materials) and as a raw material in different chemical industries, i.e, in the production of acetals, amines, esters and ethers, mono- and di-glycerides and urethane polymers [17]. Due to surplus glycerol derived from biodiesel production, a substantial decrease in the price of glycerol may be possible in the near future [39]. Although ethanol and glycerol are important solvents in pharmaceutical and food industries, experimental data on physical properties of these mixtures are very scarce [14]. Furthermore, a better knowledge of the interfacial properties of glycerol and its mixtures can facilitate the experiments using glycerol in two-phase flow studies [14] and it is relevant to mention that the interfacial tension plays an important role in gas-liquid separations [36]. The separation of alcohols from glycerol + alcohol mixtures and physical properties of these systems play an important role in the operation of distillation columns [13]. Some authors have obtained experimental information of phase equilibrium and theoretical modeling using equations of state (EOS). Oliveira et al. [30] obtained experimental information for five alcohol + glycerol mixtures and water + glycerol mixture at atmospheric pressure. The authors used the Cubis-Plus-Association EOS to model mixtures containing glycerol. Shimoyama et al. [35] obtained the vapor-liquid equilibria (VLE) for binary (alcohol + glycerol) mixtures in the temperature range of 493 K to 573 K and the experimental data were modelled with Peng-Robinson equation of state (PR-EOS).
Veneral et al. [39] obtained experimental vapor-liquid equilibrium data of several binary (alcohol + glycerol) mixtures and ternary (alcohol + glycerol + water) mixtures over the pressure range of 6.7 kPa to 66.7 kPa and the experimental data without NaCl were correlated using the UNIQUAC model. The interfacial tension of mixtures containing glycerol has also been measured by some authors. Alkindi et al. [1] obtained different physical properties, among them the interfacial tension, for the mixture ethanol + glycerol at 294 K. Erfani et al. [14] obtained the interfacial tension and interfacial composition of different binary mixtures of glycerol and alcohols. They used Peng-Soave-Redlich-Kwong EOS for phase equilibrium and interfacial tension calculations. Iqbal et al. [16] obtained interfacial tension measurements of glycerol with organic cosolvents at 298.15 K. In this work, the gradient theory (GT) is combined to PR-EOS to calculate both phase equilibrium equations and interfacial properties of the methanol + glycerol, ethanol + glycerol, 1-propanol + glycerol, 2propanol + glycerol and 1-butanol + glycerol mixtures in the temperature range of 293.2 K to 333.2 K. The GT was initially developed by van der Waals [37] and later reformulated by Cahn and Hilliard [3]. Numerous authors thereafter have utilized GT with different EOSs to describe the interfacial behavior of systems with different types of phase equilibria [4,6–9,22–24,26,28,34]. With respect to the mixtures used in this work, no author has used GT to model the interfacial behavior of the mixtures, in this way, no one has studied the possible adsorption or desorption of the components in the alcohol + glycerol mixtures. Due to the above, the present work arises in order to use GT (widely used theory), except in these mixtures, to study the prediction
E-mail address: [email protected]. https://doi.org/10.1016/j.chemphys.2020.110747 Received 13 January 2020; Received in revised form 21 February 2020; Accepted 9 March 2020 Available online 14 March 2020 0301-0104/ © 2020 Elsevier B.V. All rights reserved.
Chemical Physics 534 (2020) 110747
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Nomenclature List of symbols a a0 AAD b c kij m n nc P R T u x, y
cohesion parameter in the EOS Helmholtz energy density of the homogeneous system statistical deviation covolumen parameter in the EOS influence parameter interaction parameter for the quadratic mixing rule parameter of the thermal cohesive function number of points number of components absolute pressure universal gas constant absolute temperature auxiliary variable mole fraction of the liquid and vapor phases
Subscriptors c i , j, k, s
0 exp l ref theo v
of the theoretical model in the calculations of interfacial tension and the behavior of the components in the interfacial region.
between different molecules. In this work, the parameter of interaction was obtained by adjusting the experimental information of the equilibrium of phases.
2. Theory
2.2. van der Waals square gradient theory
2.1. The EOS model
Following the work of Cahn and Hilliard [3], the GT was generalized and modified by different authors ([2,4–6,11,12,21], among others). This theoretical approach is characteristic for describing of inhomogeneous fluid behavior through a continuous model of hypothetically homogeneous fluids. The GT proposes that under conditions with constant particle numbers, volume, and temperature, Helmholtz energy (A) can be mathematically described by Eq. (5):
In this work, Peng-Robinson equation of state [31] was selected for modeling phase equilibrium and interfacial behavior for different mixtures. This equation of state is expresses in terms of density by Eq. (1):
a 2 , (1 + d1 b )(1 + d2 b )
RT b
d1 = 1 +
2,
d2 = 1
2 (1)
A( ) =
where a and b are cohesive and covolume parameters, respectively. For each pure component they are given by Eq. (2):
(RTc, i )2 ai = 0.45724 Pc, i
RTc, i bi = 0.07780 Pc, i
i (T ),
= [1 + mi (1
T / Tc, i )]2 ,
(2)
mi = 0.37464 + 1.54226
i
0.26992
a0 = RT
2 i
(3)
a=
2
nc i j i, j = 1
ai aj (1
kij ),
b=
1
i=1
nc
cij (
i·
j)
dV
(5)
i, j = 1
0
P RT
1 2
d + ln
RT P ref
+
1
nc i ln i i=1
(6)
where is a reference pressure, which, in the current study, was 1 bar. For PR-EOS, the calculation of the Helmholtz energy density can be obtained using Eq. (7):
a0
nc
= i=1
nc i bi
1 2
P ref
where mi is a parameter that can be generalized in terms of the acentric factor ( i ) for each pure component. Eq. (2) is extended to mixtures using a mixing rule. In this work, Quadractic mixing rule (QMR) [19] is selected for calculating the cohesive and covolume parameters of the mixture. This rule proposes that the cohesive and covolume parameters of the mixture are defined from Eq. (4):
1
V
a0 ( ) +
where a0 ( ) is the Helmholtz energy density of a hypothetically homogeneous fluid with a uniform concentration, cij is the cross-influence parameter, nc is the number of mixture components, and i and j are concentrations of the species i and j, respectively. Furthermore, the Helmholtz energy density of the homogeneous system can be determined from any EOS model using Eq. (6) [41]:
where R is the universal gas constant, Tc, i and Pc , i are the critical properties of temperature and pressure, respectively, and i is the thermal cohesive function, defined as Eq. (3): i
equilibrium condition experimental liquid phase reference theoretical vapor phase
thermal cohesive function
(T )
1
critical condition species
Superscripts
Greek Letters
P=
relative adsorption isotherm chemical potential grand thermodynamic potential acentric factor molar concentration interfacial tension
µ
RT i ln
1 + (1 1 + (1 +
(4)
i
2 )b 2 )b
RT ln(1 RT ln
b )+ P ref RT
a ln 2 2b (7)
Concentration distribution in the interfacial region was determined through the Euler-Lagrange equation [4], which, in the case of a planar interface and density-independent influence parameter, is reflected by Eq. (8):
where ai and bi are the cohesive and covolume parameters of the pure components, respectively, i , j and are the densities in a specific phase of components i , j , and the mixture, respectively, nc is the number of components in the mixture, and kij is the parameter of interaction 2
Chemical Physics 534 (2020) 110747
A. Hernández nc
cij j=1
d2
j
µi0 ; i , j = 1, …, nc
= µi ( ) = µi ( )
dz 2
(8)
where µi ( ) is the chemical potential of component i and is the chemical potential of component i in the bulk equilibria phases. The boundary conditions for Eq. (8) are given by Eq. (9):
µi0
i |z =
=
,
i
i |z =
=
(9)
i
where and symbolize the homogeneous phases. The chemical potential of the pure component i in the mixture can be obtained using Eq. (10) [38]:
µi =
a0 i
(10)
T , V , nj
For PR-EOS, the calculation of the chemical potential of component i can be obtained by Eq. (11):
µi =
RT ln(1 bi
b )+
( ) 2a i
1 2 2b
1 + (1 2 )b 1 + (1 + 2 ) b
a ln 2 2 b2
ln
1 + (1 2 )b 1 + (1 + 2 ) b
RT ln
( ) + RT P ref RT i
a 2 ) b ][1 + (1 + 2 ) b ] b
bi [1 + (1
RT b )
+ bi (1
(11) Solving for concentration-space variables is complex as the boundary conditions had infinite limits. Therefore, this study used the concentration-space to concentration-concentration transformation, as proposed by Carey [4]. This transformation required solving the following system of ordinary differential equations (ODEs) given by Eq. (12): nc
cij i, j = 1
d d
i s
d
j
d
s
d d
Hs
k
Hk + 2
det( C )
s
d2 d
k 2 s
= 0;
k (12)
= 1, …, s, …, nc where s must be defined monotonic variable and thermodynamic potential defined as Eq. (13): nc
( ) = a0 ( )
i=1
is the grand
[ i µi0] + P 0
(13)
where is the pressure in the bulk equilibria phases. In Eq. (12), Hk is the kth component of the vector function H , which is defined as Eq. (14):
P0
T
H ( ) = det( C )· C 1·
T
,
=
, …, 1
(14)
nc
where C is the influence parameters matrix given by Eq. (15):
C=
c11
c1nc
cn c 1
cn c n c
(15)
The ODEs system given by Eq. (12) is subject to boundary conditions as shown in Eq. (16): k( s
)=
k
,
k( s
)=
(16)
k
Fig. 1. Schematic representation of concentration profiles for the mixtures. (•) VLE bulk densities, ( ) stationary points: A (adsorption) and D (desorption) of the species on the interfacial region.
When the cross-influence parameters obey the geometric mixing rule, i.e, cij = cii cjj , Eq. (12) reduces to Eq. (17):
css [µk ( )
µk0 ] =
ckk [µs ( )
µs0 ]; k = 1, 2, …, s
1, s + 1, nc
Nevertheless, some authors have developed correlations for the influence parameter [15,27,42]. According to Carey procedures, the influence parameter of the pure component (cii ) in the case of pure fluids (i = j ) and VLE is given by Eq. (18):
(17) where cii and cjj are the influence parameters of the pure components i and j, respectively. When the mentioned rule is used for cij , the gradient theory is used as a predictive approach, because some adjustment parameter is not required. The typical approach for calculating the influence parameter applies procedures suggested by Carey et al. [4–6] and Cornelisse et al. [7–9].
cii = 3
exp (T 0 ) 2
2
l v
2
( ) + P0 d
(18)
Chemical Physics 534 (2020) 110747
A. Hernández
Table 1 Physical properties of the pure components. Fluid
Tc (K )
Pc (bar )
Methanol Ethanol 1-Propanol 2-Propanol 1-Butanol Glycerol
512.5 514.0 536.8 508.3 563.0 850.0
80.84 61.37 51.69 47.64 44.14 75.00
Table 3 Binary interaction parameters of PR-EOS + QMR for the mixtures and statistical deviation in boiling temperature at 1 atm. 0.565831 0.643558 0.620432 0.666873 0.589462 0.512690
where exp (T 0) is the value of experimental tension at the boiling temperature T 0, v , l , and P 0 are the vapor density, liquid density and pressure of the phase equilibria, respectively. On the other hand, Eq. (19) is applied to determine the interfacial density profile of each component in the mixture:
z ( s ) = z ref +
s (z ) ref s
nc
1
cij
2
i, j = 1
d
i
d
s
d
j
d
s
d
s
Mixture
kij
AADT%
Methanol(1) + Glycerol(2) Ethanol(1) + Glycerol(2) 1-Propanol(1) + Glycerol(2) 2-Propanol(1) + Glycerol (2) 1-Butanol(1) + Glycerol(2)
−0.0108 0.0203 −0.0116 −0.0054 0.0046
0.3099 2.1076 1.3964 1.6757 0.8498
homogeneous phases and . Discretizing the auxiliary variable u and solving the system of Eqs. (21) and (22), can be obtained the dependence of the densities with the variable u. Using = 0.5, the interfacial density profile can be compute as shown in Eq. (24):
(19)
z i+1 = z i +
where z ref and sref are reference values. Finally, interfacial tension is compute by Eq. (20): nc
s
=
(2
s
)
cij i, j = 1
d d
i s
d d
j s
d
cii
i
(ui + 1
(20)
=u 2
i
ui
+ (1
)
2
[ (
i + 1)
0]
(24)
(22)
nc
ui ) { 2 [ ( i )
ij
=
z 0j
[ i (z )
nc
cii
+
2 [ (
i + 1)
0]
}
(25)
i
] dz +
+ z 0j
[ i (z )
i
] dz
(26)
where is the localization of the divide position relative to a species j and and are the bulk phases. From Eq. (26), z 0j is calculated considering that species j does not have adsorption along the interfacial
z 0j
(23)
i=1
0]
where n is the number of points used in the discretization of u. In this work, n = 1000 was used. Eqs. (19) or (24) allows quantification of the population of species at the interface, i (z ), from which the surface activity (adsorption/ desorption of species along the interface region), relative Gibbs adsorption isotherm and the interfacial tension can be calculated. Characteristic shapes of i (z ) profiles are shown in Fig. 1 obtained from the work of Mejía et al. [25]. From Fig. 1 it is seen that along the interface width the concentration profile may be monotonic (type a) or nonmonotonic (type b and c). It should be pointed out that the nonmonotonic behavior of the concentration profiles type b and c is interesting because it reflects the surface activity at the interface. Nonmonotonic profiles are characterized by stationary points which correspond to the interfacial point where adsorption (point A) or desorption (point D) of the species take place [25]. The accumulation of a species i at the interface region is characterized by the condition, d i /dz = 0 , and it may be positive or negative. When d 2 i / dz 2 < 0 , the positive surface activity or adsorption of species along the interface region is reflected, whereas d 2 i / dz 2 > 0 denotes negative surface activity or desorption of species along the interface region [29]. The relative Gibbs adsorption isotherm of a species i relative to a species j ( ij ) can be expressed by Eq. (26) [33,40]:
where nc is the number of components, u is an auxiliary variable that varies in the finite and bounded range [u , u ], and the parameter is given by Eq. (23):
=
ui + 1
i=0
(21)
cii µ1 ( );
0]
n 1
s
i=1
c11 µi ( ) =
[ (
i)
Finally, interfacial tension can be obtained by Eq. (25):
To use Eqs. (17), (19) and (20), the choice of the defined monotonic variable ( s ) is very important. In order to avoid this criterion, because the mixtures may have stationary points for both components in the density-density profile, several authors have introduced a new independent variable to define the density path profiles, which then is called as a patch function and it must be monotonic. Poser and Sanchez [32] were the first to suggest a form for the new auxiliary variable. Later, authors as Cornelisse [7], Kou et al. [18] and Liang et al. [20] have proposed new ways for this auxiliary variable in order to cover stability in the calculations. In this work, Kou et al. [18] technique was chosen because it is simple, moreover because presents the equations for interfacial tension and density profiles based on the new defined variable. This technique consists of applying GT predictively to a multicomponent system, adding an auxiliary variable u that is defined monotonic. The authors [18] propose to solve a system of nonlinear equations given by Eqs. (21) and (22): nc
2
In addition, u and u values can be determined using Eq. (21) in the Table 2 Influence parameters for the pure components at each temperature. Fluid
Methanol Ethanol 1-Propanol 2-Propanol 1-Butanol Glycerol
cii × 1019 (J m5 mol−2) 293.2 K
294 K
298.15 K
303.2 K
313.2 K
323.2 K
333.2 K
– 0.452 0.837 0.881 1.455 2.208
– 0.452 – – – 2.216
0.224 – 0.829 – – 2.216
– 0.467 0.866 0.926 1.482 2.262
– 0.487 0.898 0.948 1.498 2.341
– 0.510 0.933 0.972 1.486 2.385
– 0.537 0.982 0.987 1.458 2.455
4
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Table 4 Statistical deviation for interfacial tension for the mixtures at different temperatures. Mixture
Methanol(1) + Glycerol(2) Ethanol(1) + Glycerol(2) 1-Propanol(1) + Glycerol(2) 2-Propanol(1) + Glycerol(2) 1-Butanol(1) + Glycerol(2)
AAD % 293.2 K
294 K
298.15 K
303.2 K
313.2 K
323.2 K
333.2 K
– 4.777 6.358 3.735 2.971
– 5.811 – – –
8.121 – 2.205 – –
– 5.050 6.100 3.483 2.163
– 4.868 6.038 3.280 1.490
– 5.127 5.809 3.896 1.692
– 5.246 6.052 4.662 1.983
x1 projection for the mixture methanol(1) + glycerol(2) at Fig. 2. 298.15 K. ( ) Iqbal et al. [16], ( ) prediction obtained with GT + PREOS + QMR.
x1 projection for the mixture 1-propanol(1) + glycerol(2) at difFig. 4. ferent temperatures. ( ) 298.15 K from Iqbal et al. [16], ( , ) 313.2 K and 333.2 K from Erfani et al. [14], ( , , ) prediction at 298.15 K, 313.2 K and 333.2 K obtained with GT + PR-EOS + QMR.
x1 projection for the mixture ethanol(1) + glycerol(2) at different Fig. 3. temperatures. ( , , ) 293.2 K, 313.2 K and 333.2 K from Erfani et al. [14], (—–, , ) prediction at 293.2 K, 313.2 K and 333.2 K obtained with GT + PR-EOS + QMR.
x1 projection for the mixture 2-propanol(1) + glycerol(2) at difFig. 5. ferent temperatures. ( , , ) 293.2 K, 313.2 K and 333.2 K from Erfani et al. [14], (—–, , ) prediction at 293.2 K, 313.2 K and 333.2 K obtained with GT + PR-EOS + QMR.
region (Fig. 1 a), i.e, Eq. (26) is solved for the case that jj = 0 . When z 0j is fixed, the relative Gibbs adsorption isotherm of a species i relative to a species j is calculated from Eq. (26).
better represent the interfacial behavior of the mixture at the given temperature. The parameters obtained are in Table 2. The experimental interfacial tension for each temperature has been obtained from: Erfani et al. [14] for 293.2 K, 303.2 K, 313.2 K, 323.2 K and 333.2 K, Iqbal et al. [16] for 298.15 K, and Alkindi et al. [1] for 294 K.
3. Results and discussions 3.1. Pure fluids
3.2. Binary mixtures
All pure-component parameters of PR-EOS have been listed in Table 1. The pure component parameters have been taken from DIADEM [10]. The influence parameters of the pure components were obtained at each temperature using Eq. (18). This methodology was used in order to
3.2.1. Phase equilibrium calculations The accuracy of phase equilibrium calculations is very important for the successful description of thermophysical properties such as interfacial tension. In this work, PR-EOS + QMR was used to describe the 5
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x1 projection for the mixture 1-butanol(1) + glycerol(2) at different Fig. 6. temperatures. ( , , ) 293.2 K, 313.2 K and 333.2 K from Erfani et al. [14], (—–, , ) prediction at 293.2 K, 313.2 K and 333.2 K obtained with GT + PR-EOS + QMR.
Fig. 8. Dimensionless density profiles for the mixture ethanol(1) + glycerol(2) at 294 K. (—–, , ) density profile of ethanol at x1 = 0.1, x1 = 0.5 and x1 = 0.9 , (- - -, , ) density profile of glycerol at x1 = 0.1, x1 = 0.5 and x1 = 0.9 , (- – -, , ) z 0 value at x1 = 0.1, x1 = 0.5 and x1 = 0.9 , ( , ) vapor and liquid density.
boiling temperature ( AADT%) was calculated using Eq. (27):
AADT % =
100 n
n i=1
Tiexp Titheo Tiexp
(27)
where n represent number of points, exp represent experimental value and theo represent theoretical value. From Table 3 it can be seen that the theoretical model PREOS + QMR correctly adjusts the phase equilibrium for the studied mixtures. 3.2.2. Interfacial tension of alcohol + glycerol mixtures PR-EOS + QMR was able to correctly adjust the phase equilibrium, then it is expected that in combination with the gradient theory acceptable results in the interfacial tension are obtained. It is important to mention that to represent the inhomogenous phase (interfacial region) as well as possible, the homogeneous phases must be well represented, which has been obtained satisfactorily. The gradient theory was used as a predictive approach to model the interfacial behavior of the analyzed mixtures and in Table 4 have been listed the statistical deviation of interfacial tension for the mixtures. The absolute average deviation for interfacial tension ( AAD % ) was calculated using Eq. (28):
Fig. 7. Dimensionless density profiles for the mixture methanol(1) + glycerol (2) at 298.15 K. (—–, , ) density profile of methanol at x1 = 0.1, x1 = 0.5 and x1 = 0.9 , (- - -, , ) density profile of glycerol at x1 = 0.1, x1 = 0.5 and x1 = 0.9 , (- – -, , ) z 0 value at x1 = 0.1, x1 = 0.5 and x1 = 0.9 , ( , ) vapor and liquid density.
phase equilibrium of the following mixtures: methanol + glycerol, ethanol + glycerol, 1-propanol + glycerol, 2-propanol + glycerol and 1-butanol + glycerol. The binary interaction parameters of the PR-EOS + QMR have been listed in Table 3 and were determined using experimental phase equilibrium from Oliveira et al. [30]. The absolute average deviation in
AAD % =
6
100 n
n i=1
exp i
exp i
theo i
(28)
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Fig. 9. Dimensionless density profiles for the mixture 1-propanol(1) + glycerol (2) at 313.2 K. (—–, , ) density profile of 1-propanol at x1 = 0.1, x1 = 0.5 and x1 = 0.9 , (- - -, , ) density profile of glycerol at x1 = 0.1, x1 = 0.5 and x1 = 0.9 , (- – -, , ) z 0 value at x1 = 0.1, x1 = 0.5 and x1 = 0.9 , ( , ) vapor and liquid density.
Fig. 10. Dimensionless density profiles for the mixture 2-propanol(1) + glycerol(2) at 323.2 K. (—–, , ) density profile of 2-propanol at x1 = 0.1, x1 = 0.5 and x1 = 0.9 , (- - -, , ) density profile of glycerol at x1 = 0.1, x1 = 0.5 and x1 = 0.9 , (- – -, , ) z 0 value at x1 = 0.1, x1 = 0.5 and x1 = 0.9 , ( , ) vapor and liquid density.
From Table 4 it can be seen that the theoretical model GT + PREOS + QMR correctly predicts interfacial tension for 2-propanol + glycerol and 1-butanol + glycerol mixtures. The best results are obtained for the butanol + glycerol mixture. Also, Table 4 indicates that the maximum deviation (8.12% ) is reached for methanol + glycerol mixture at 298.15 K and the minimum deviation (1.49% ) is reached for 1-butanol + glycerol mixture at 313.2 K. The deficiency of the theoretical model for the prediction of the interfacial tension for the methanol + glycerol mixture is due to the inability of PR-EOS + QMR for the treatment of mixtures with associative interactions. A more robust equation of state could improve the results obtained in the interfacial tension for the methanol + glycerol mixture. Figs. 2–6 shows the interfacial tension for the mixtures studied. Fig. 2 shows that GT does not correctly predicts the interfacial tension for the methanol + glycerol mixture at 298.15 K and the theory underpredicts the interfacial tension for a molar fraction of alcohol greater than 0.4. Fig. 3 shows that GT underpredicts the interfacial tension for the ethanol + glycerol mixture at different temperatures analyzed for a molar fraction of ethanol below 0.6 and for a molar fraction of alcohol greater than 0.7 it is observed that GT correctly predicts the interfacial tension at different temperatures. Fig. 4 shows that GT does not correctly predicts the interfacial tension for the 1-propanol + glycerol mixture at 313.2 K and 333.2 K and for a molar fraction of alcohol greater than 0.7, the theory correctly predicts interfacial tension at 298.15 K, 313.2 K and 333.2 K. For this mixture, the best prediction is obtained at 298.15 K. Fig. 5 shows that GT underpredicts the interfacial
tension for the 2-propanol + glycerol mixture for a molar fraction of 2propanol less than 0.4, and for a molar fraction greater than 0.5, the theory correctly predicts the interfacial tension for the analyzed temperatures. Fig. 6 shows that GT is able to correctly represent the interfacial tension for the 1-butanol + glycerol mixture, however the best results were obtained at 313.2 K and 333.2 K. 3.2.3. Interfacial density profiles of alcohol + glycerol mixtures GT has the advantage of studying interfacial behavior, i.e, studying properties such as interfacial tension and interfacial density profiles. Figs. 7–11 shows the alcohol and glycerol profiles in the different mixtures at a given temperature and at different molar fractions of alcohol in the mixture, the localization of the divide position relative to glycerol (z 0 ) and the bulk phases (vapor and liquid density). Fig. 7 shows that methanol accumulated in the interfacial region due to a greater affinity for remaining in the interface than in the homogenous phase, while glycerol simultaneously showed accumulation and nonaccumulation at x1 = 0.5 and x1 = 0.9 . According to the literature, the density profile of methanol is type b and the density profile of glycerol is type a at x1 = 0.1 and type c at x1 = 0.5 and x1 = 0.9. It is also observed that as the molar fraction of methanol in the mixture increases, the surface activity of methanol decreases. In Figs. 8–11 it is observed that for the ethanol + glycerol at 294 K, 1-propanol + glycerol at 313.2 K, 2-propanol + glycerol at 323.2 K and 1-butanol + glycerol at 333.2 K mixtures, alcohol has stationary accumulation points in the density profiles (type b) for the three compositions analyzed and the surface 7
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mixture at 298.15 K, 2-propanol + glycerol and 1-butanol + glycerol mixtures for the entire temperature range. In spite of the simplicity of the mixing rule used in this work (quadratic mixing rule), the results obtained in the interfacial tension are good. Another advantage of having used the gradient theory was that it allowed obtaining the interfacial profiles for the components of the mixture, where relevant information was obtained about the behavior of alcohols and glycerol in the interfacial region. In all studied mixtures, alcohol exhibited a maximum stationary point in the density profile, i.e, positive surface activity (d 1 /dz = 0 ; d 2 1 / dz 2 < 0 in the interfacial region). This condition is attributed to the lower interfacial tension of the alcohol compared to glycerol, which implies its preference to locate itself in the interfacial region rather than the homogeneous phases, whereas for glycerol accumulation and non-accumulation (d 2 / dz = 0 ; d 2 2 / dz 2 < 0;d 2 2 / dz 2 > 0 in the interfacial region) was observed at the interface for the mixture with methanol at 298.15 K and a monotonic behavior in its density profile was obtained for the mixtures with ethanol, 1-propanol, 2-propanol and 1-butanol. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments A.H acknowledges the economic support given by the UCSC. References [1] A.S. Alkindi, Y.M. Al-Wahaibi, A.H. Muggeridge, Physical properties (density, excess molar volume, viscosity, surface tension, and refractive index) of ethanol+ glycerol, J. Chem. Eng. Data 53 (12) (2008) 2793–2796. [2] V. Bongiorno, L.E. Scriven, H.T. Davis, Molecular theory of fluid interfaces, J. Colloid Interface Sci. 57 (3) (1976) 462–475. [3] J.W. Cahn, J.E. Hilliard, Free energy of a nonuniform system. i. Interfacial free energy, J. Chem. Phys. 28 (2) (1958) 258–267. [4] B.S. Carey, The Gradient Theory of Fluid Interfaces. PhD thesis, University of Minnesota, 1979. [5] B.S. Carey, L.E. Scriven, H.T. Davis, Semiempirical theory of surface tensions of pure normal alkanes and alcohols, AIChE J. 24 (6) (1978) 1076–1080. [6] B.S. Carey, L.E. Scriven, H.T. Davis, Semiempirical theory of surface tension of binary systems, AIChE J. 26 (5) (1980) 705–711. [7] P.M.W. Cornelisse. The Gradient Theory Applied, Simultaneous Modelling of Interfacial Tension and Phase Behaviour. PhD thesis, Technische Universiteit Delft, 1997. [8] P.M.W. Cornelisse, C.J. Peters, J. de Swaan Arons, Application of the peng-robinson equation of state to calculate interfacial tensions and profiles at fluid interfaces, Precision Process Technology, Springer, 1993, pp. 651–660. [9] P.M.W. Cornelisse, C.J. Peters, J. de Swaan Arons, Simultaneous prediction of phase equilibria, interfacial tension and concentration profiles, Mol. Phys. 80 (4) (1993) 941–955. [10] T.E. Daubert, R.P. Danner, Physical and thermodynamic properties of pure chemicals. data compilation, Taylor & Francis, Bristol, PA., 1989. [11] H.T. Davis, L.E. Scriven, Gradient theory of fluid microstructures, J. Stat. Phys. 24 (1) (1981) 243–268. [12] H.T. Davis, L.E. Scriven, Stress and structure in fluid interfaces, Adv. Chem. Phys 49 (1982) 357–454. [13] B.R. Dhar, K. Kirtania, Excess methanol recovery in biodiesel production process using a distillation column: a simulation study, Chem. Eng. Res. Bull. 13 (2) (2009) 55–60. [14] A. Erfani, S. Khosharay, C.P. Aichele, Surface tension and interfacial compositions of binary glycerol/alcohol mixtures, J. Chem. Thermodyn. 135 (2019) 241–251. [15] J.M. Garrido, A. Mejía, M.M. Piñeiro, F.J. Blas, E.A. Müller, Interfacial tensions of industrial fluids from a molecular-based square gradient theory, AIChE J. 62 (5) (2016) 1781–1794. [16] M.J. Iqbal, M.A. Rauf, N. Ijaz, Surface tension measurements of glycerol with organic cosolvents, J. Chem. Eng. Data 37 (1) (1992) 45–47. [17] R.E. Kirk, Encyclopedia of Chemical Technology vol. 18, Wiley, 1982. [18] J. Kou, S. Sun, X. Wang, Efficient numerical methods for simulating surface tension of multi-component mixtures with the gradient theory of fluid interfaces, Comput. Methods Appl. Mech. Eng. 292 (2015) 92–106. [19] T.Y. Kwak, G.A. Mansoori, Van der waals mixing rules for cubic equations of state. applications for supercritical fluid extraction modelling, Chem. Eng. Sci. 41 (5) (1986) 1303–1309.
Fig. 11. Dimensionless density profiles for the mixture 1-butanol(1) + glycerol (2) at 333.2 K. (—–, , ) density profile of 1-butanol at x1 = 0.1, x1 = 0.5 and x1 = 0.9 , (- - -, , ) density profile of glycerol at x1 = 0.1, x1 = 0.5 and x1 = 0.9 , (- – -, , ) z 0 value at x1 = 0.1, x1 = 0.5 and x1 = 0.9 , ( , ) vapor and liquid density.
activity of the alcohol decreases as the molar fraction of the alcohol in the mixture increases. On the other hand, glycerol has a monotonic density profile (type a) throughout the molar fraction range. Finally, due to the simple EOS used and the inadequate representation of the cross association in the methanol + glycerol mixture, it was the cause for the glycerol to only have simultaneous adsorption and desorption in the methanol + glycerol mixture. 4. Concluding remarks The Peng-Robinson equation of state, coupled with the quadratic mixing rule, was applied to alcohol + glycerol (methanol + glycerol, ethanol + glycerol, 1-propanol + glycerol, 2-propanol + glycerol and 1-butanol + glycerol) mixtures in vapor-liquid equilibria at a pressure of 1 atm, in order to adjust the binary interaction parameter to be used in gradient theory to predict interfacial behaviour (interfacial tension and density profiles) of alcohol + glycerol mixtures at different temperatures. Boiling temperatures calculations for the alcohol + glycerol mixtures were obtained with all the AADT% values below 2.1% and the smallest deviation (0.3%) was obtained for the methanol + glycerol mixture. Interfacial tension calculations for the alcohol + glycerol were obtained with AAD % values between 1.5% (1-butanol + glycerol mixture at 313.2 K) and 8.1% (methanol + glycerol mixture at 298.15 K). The theoretical model was able to represent the variation of the interfacial tension with the molar fraction for the mixtures studied, getting good results as a predictive approach for 1-propanol + glycerol 8
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