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Application of an equation of state incorporating association to alcohols up to decanol
Accepted Manuscript Application of an equation of state incorporating association to alcohols up to decanol Thomas A.S. Menegazzo, Acir M. Soares, Junior, Breno T. Mota, Nélio Henderson, Adolfo P. Pires PII:
S0378-3812(18)30437-0
DOI:
https://doi.org/10.1016/j.fluid.2018.10.015
Reference:
FLUID 11977
To appear in:
Fluid Phase Equilibria
Received Date: 22 May 2018 Revised Date:
11 October 2018
Accepted Date: 17 October 2018
Please cite this article as: T.A.S. Menegazzo, A.M. Soares Junior., B.T. Mota, Né. Henderson, A.P. Pires, Application of an equation of state incorporating association to alcohols up to decanol, Fluid Phase Equilibria (2018), doi: https://doi.org/10.1016/j.fluid.2018.10.015. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Application of an equation of state incorporating association to alcohols up to decanol Thomas A. S. Menegazzoᵃ, Acir M. Soares Juniorᵇ, Breno T. Motaᵇ, Nélio Hendersonᵇ and Adolfo P. Piresᵃ* b *
Universidade Estadual do Rio de Janeiro-UERJ, Nova Friburgo-RJ, Brazil Corresponding author
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ᵃLaboratory of Petroleum Engineering and Exploration, Universidade Estadual do Norte Fluminense-UENF, Macaé-RJ, Brazil
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Abstract The accurate prediction of fluid phase equilibria is essential to the design of any process in chemical industry. Most of the simulators used to model multiphase multicomponent phase equilibria use traditional cubic equations of state, like Peng-Robinson and Soave-RedlichKwong. However, self-associating substances, such as alcohols, are not precisely modeled with these equations of state. Different approaches have been proposed to overcome this limitation. In this work, we use the technique that separates the compressibility factor in two parts, a physical and a chemical one, and we compare the results with the PengRobinson equation of state and the Cubic Plus Association equation of state against experimental data for alcohols up to decanol. The parameters are estimated adjusting the proposed EOS results to two independent data set measurements, saturation pressures and saturated liquid molar volumes of the pure components, through multi-objective optimization. The results present excellent agreement for all alcohols modeled.
1. Introduction
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Keywords: Equation of State, Associating Substance, Multi-Objective Optimization.
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Alcohols are organic compounds used in several different industrial processes, such as fuel, as an organic solvent, a component of hydraulic fluids, or as industrial feedstock in the synthesis of organic compounds. In order to optimize industrial processes using alcohols, it is important to calculate precisely their thermodynamics properties, especially when there is a phase change or multiphase multicomponent equilibrium conditions along the process. A particular and important characteristic of this organic class of compounds is that they form hydrogen bonds, affecting its microscopic (molecular structure) and macroscopic (thermodynamic) properties. These components are called associating fluids. It is possible to calculate the phase equilibrium of systems containing non-polar or slightly polar substances using traditional cubic equations of state (EOS), but these models lose accuracy when there are polar or associating components in the mixture (SCHMIDT; WENZEL, 1980 [1]). Excess Gibbs energy and EOS models have been developed in order to predict the thermodynamic properties and phase equilibrium of associating substances, and different approaches are used to mathematically describe the association effects in those calculations. In the perturbation theory, the total energy of hydrogen bonding is calculated based on statistical mechanics. A potential function is used to calculate the energy bonding between
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two sites in this approach. The number of bonding sites is a characteristic of the molecule, and it is an input parameter of the model (ECONOMOU; DONOHUE, 1996 [2]). One of the most known and used models based on the perturbation theory is the StatisticalAssociating-Fluid-Theory (SAFT), which can be used for both pure associating fluids and mixtures containing associating components (CHAPMAN et al., 1989 [3]; HUANG; RADOSZ, 1990 [4]). The SAFT equation accounts for hard-sphere repulsions, chain connectivity, dispersion attractive forces and hydrogen bonding, and each of its terms is obtained from perturbation theory. Despite being accurate for actual fluids, SAFT does not account for polar interactions due to dipole and quadrupole moments. The SAFT-Dimer equation of state (GHONASGI; CHAPMAN, 1994a [5]) and the physical theory for fluids of small associating molecules (WALSH et al., 1992 [6]) try to overcome this limitation. Both models were developed based on the SAFT equation of state. The Cubic Plus Association (CPA) equation of state (KONTOGEORGIS et al., 2006 [30]) uses the SoaveRedlich Kwong equation of state and the Wertheim perturbation theory to model the thermodynamic behavior of mixtures containing associating and non-associating components. The lattice theory takes into account the number of bonds formed between segments of different molecules that are adjacent in a lattice. The number of bonds determines the extent of association, but not the number of oligomers. Lattice models have also been successfully used to model thermodynamic properties and phase behavior of mixtures containing this kind of chemical species. These models can be modified to include polar effects by the introduction of specific forces between adjacent sites (GUGGENHEIM, 1952 [7]; FLORY, 1953 [8]). In the chemical theory, it is assumed that hydrogen bonding generates new molecular complexes. In this approach, the fluid is composed by a mixture of different oligomers of the associating substance, and of any other substances. Along with the EOS, it is necessary to solve material balance equations taking into account the original molecule and its many oligomers. The amount of each oligomer depends on equilibrium, density, temperature and composition of the system. Analytic expressions for the fraction of monomers and oligomers can be obtained for pure associating fluids (CAMPBELL et al., 1992 [9]), or for mixtures containing only one associating fluid. Mixtures of many associating fluids require numerical solutions (ANDERKO, 1989 [10]; ECONOMOU et al., 1990 [11]). In this theory, the EOS can be written as:
(1) where and represent the “physical” part of the equation, which can be calculated from a traditional cubic equation of state (ANDERKO, 1989 [12]; ELLIOTT et al., 1990 [13]), and deals with the hydrogen bonding. In this paper, we present an association equation of state based on the chemical theory where the physical part is based on the Peng-Robinson equation of state (PENG; ROBINSON, 1976 [14]) and the associating part calculated through a linear association approach. It is able to calculate pure component properties with less adjusting parameters
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2. Thermodynamics Modelling
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and its mathematical form is simpler than other theories. This equation was adjusted to saturation pressure and saturated liquid molar volume data of ten pure alcohols. Results show that this model improves the correlation of pure alcohols saturated properties.
The second virial coefficient can be separated into a chemical and a physical part, based on its relationship with the dimerization constant (LAMBERT, 1953 [15]): (2)
(3)
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Substituting equation 2 in equation 3, we obtain:
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The virial equation truncated at the second term is given by:
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(4) The two terms on the right hand side of equation 4 can be named as the chemical and physical compressibility factor, respectively, and this equation can be rewritten as (ANDERKO, 1991 [16]): (5)
2.1 Physical Model
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In order to represent the physical part of the compressibility factor, , van der Waals type equations of state that do not account for association can be used. Many EOS have been created based on the van der Waals model, such as the Peng-Robinson and the SoaveRedlich-Kwong (SOAVE, 1972 [17]) equations. In this paper, the Peng-Robinson model was used to represent the physical part:
where a, b and
(6)
are given by:
(7)
(8)
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(9)
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2.2 Chemical Model
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The chemical compressibility factor is defined as the ratio between the true number of mols , which includes the new species created through association, and the total number of mols if the system had no association, :
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(10) At this point, expressions to determine and , which describe the characteristics of the association between molecules, such as the number of bonds, are necessary. There are different models to evaluate this ratio, such as the one-site model for carboxylic acids (IKONOMOU; DONOHUE, 1986 [18]), or the three-site model for water (HUANG; RADOSZ, 1990 [4]). These models are developed for a specific geometry of association, number of bonds, and strength of association. Alcohols form chain oligomers in the form of dimers, trimers etc. In this work the infinite equilibrium scheme is used, it is also called the two-site model (IKONOMOU; DONOHUE, 1986 [18]):
(11) (12) (13)
(14) was considered zero, so the chemical reaction constant, K, is defined as:
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In this work,
(15) From equation 11 it is possible to determine the chemical part contribution to the system pressure:
(16) 2.3 Association Equation of State
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Rewriting equation 5 in terms of pressure, we have
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(17) and applying equations 6 and 16 in equation 5, the association equation of state becomes:
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(18) In this equation of state, the critical properties , and the acentric factor used to calculate parameters a and b of pure components in the physical part of the equation of state should account only for the monomers of the mixture, and they must be replaced by apparent critical properties , and . These apparent properties can be determined from experimental saturated data of the pure components. Besides the apparent properties, there are also two other parameters to be determined, the association reaction enthalpy ( ) and entropy ( ). 2.4 Equilibrium Calculations
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The fugacity coefficient of a pure substance can be calculated by:
(19)
From the definitions of P and Z in equations 5 and 17 we have:
(20)
and
(21)
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Equation 20 can be separated into:
(22) The physical part of the fugacity coefficient for the Peng-Robinson equation of state is given by:
(23) Applying equation 16 in equation 22, we obtain:
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After integration we have:
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(24) Equation 24 is the chemical part contribution for the fugacity coefficient of an association component. Figure 1 presents a simplified diagram for saturation pressure calculation of a pure substance using the Association Equation of State for a fixed set of parameters.
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Figure 1 - Diagram for saturation pressure calculation of a pure substance using the Association Equation of State 3. EOS Parameters Estimation
There are 5 parameters to be determined in the associating equation of state: , , , ( ) and ( ). The characteristic parameters in most EOS are exclusively tuned to the phase equilibrium data in order to obtain the best possible representation of vapor-liquid
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3.1 General Aspects of the Optimization Given a function follows
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, a single-objective optimization problem can be formulated as
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where
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equilibrium (VLE). Although VLE predictions are of primary importance for practical purposes, it is found that these fitted parameters sometimes give very poor estimates for other thermophysical properties. In this work we seek for improvement in predictions of phase equilibrium estimating EOS parameters using multi-objective optimization (MOO) as sugested by Punnapla, Vargas and Elkamel (RANGAIAH; PETRICIOLET, 2013 [19]) to adjust the EOS parameters to two independent experimental data sets: saturation pressures and saturated liquid molar volumes.
(25)
is the set of constraints that, in general, can be of the form (26)
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where and are real functions of real variables. A multi-objective problem is one that involves the simultaneous optimization of two or more objective functions, , , etc. Thus, for a set with functions, the multiobjective optimization problem can be described as follows:
(27)
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The problem shown in Eq. (27) is often written in the following summarized form: (28)
where denotes a mapping of into . Optimization problems involving multiple and conflicting objectives have been traditionally handled by combining the objectives into a scalar function, and next solving the equivalent single-optimization problem to identify the best-compromise solution. That approach presents some serious disadvantages, and the major problems are its subjectivity (e.g. in choosing weights) and the fact that it hides the competition among the conflicting criteria (EFSTRATIADIS; KOUTSOYIANNIS, 2010 [20]). Multi-objective optimization has gained interest in engineering applications recently. It is applied to numerous areas such as process design, biotechnology, petroleum refining, pharmaceuticals and polymerization, (BHASKAR; GUPTA; RAY, 2000 [21]).
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The major difference of the MOO approach when compared to the combination schemes, usually referred to as aggregating functions optimization, is a clear focus on the inherent multi-objective nature of the model parameters calculation (GUPTA; SOROOSHIAN; YAPO, 1998 [22]). In MOO the conflicting objectives ( ), represented by a vector of objective functions with "m" elements and a control vector with "n" parameters are solved simultaneously. As the criteria are conflicting, there is no unique point that optimizes all of them simultaneously. In that case we look for acceptable trade-offs according to the fundamental concept of Pareto optimality. To solve the model calibration problem by MOO one should follow three steps: 1- Specify the objective functions. Define functions length (L) to measure how far the predictions are from experimental data; 2- Estimate the location of the Pareto solution set. Minimize the objective functions accordingly to Pareto optimum criteria; 3- Choose a specific solution among optimal solutions. In our problem, in order to determine the difference between calculated and experimental data, the Relative Mean Square Error was chosen for experimental saturated pressure and saturated liquid molar volume:
(29)
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(30) Multi-objective particle swarm optimization (MOPSO) is a relative popular and easy to use algorithm (COELLO; LECHUGA, 2002 [23]) based on the Kennedy and Eberhart (1995 [24]) PSO metaheuristic. In this work we used a MOPSO based on the modification presented in Coello, Pulido and Lechuga (2004 [25]), with the constriction factor ( χ ) suggested by Clerc and Kennedy (2002 [26]) to assure the method convergence and a nonlinear inertia factor ( ω ) introduced by Chatterjee and Siarry, (2006 [27]). The later was included to enhance a global exploration capability at the beginning of the process and a local one for the last iterations (REZAEE; JORDEHI; JASNI, 2013 [28]). Both parameters, ω and χ , are used to set particle velocity at time t=t+1: r r r r r r v (t + 1) = χ [ω I ⋅ v (t ) + C1 ⋅ φ1 ( x pbest (t ) − x (t )) + C 2 ⋅ φ 2 ( x gbest (t ) − x (t ))] (31) where ( itermax − iter )q ωI = ⋅ (ω max − ω min ) + ω min q ( itermax )
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Table 1 presents the parameters used in the optimization procedure.
C1 , C2
Constriction factor Inertia weight Nonlinear modulation index Number of cycles
χ
ω max , ωmin q
itermax
2.05, 2.05 0.7298 0.9, 0.4
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Learning factor
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Table 1 – Standard parameters set used for MOPSO Parameter Value Number of particles NMAX 100 Size of repository NREP 100 Number of divisions 30, 30 F1 , F2
1.2 100
4. Results and Discussion
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Finally, parameters were picked up from an interest region around the point that optimizes the sum OF1+OF2. To choose a single solution, it was observed the parameter consistency within carbon chain results.
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The experimental thermodynamic data (saturation pressure and saturated liquid and vapor volumes) range used for the optimization procedure can be found in Table 2 (SMITH, 1986 [29]).
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Table 2 – Equilibrium experimental data (saturation pressure and saturated volumes) Alcohol Number of points Temperature range (K) Methanol 39 212-497 Ethanol 35 231-503 Propanol 29 280-517 Butanol 31 295-548 Pentanol 29 278-508 Hexanol 17 310-428 Heptanol 15 343-445 Octanol 31 296-549 Nonanol 15 366-481 Decanol 27 301-526 As the apparent critical properties and acentric factor refer to a hypothetical non-associating alcohol, we adopted the acentric factor of the alkane with the same carbon number as the apparent acentric factor for the alcohol. So, our model has only four adjusting parameters: apparent critical temperature and pressure, and standard enthalpy and entropy of association. These parameters were adjusted to correlate the experimental saturation pressure and liquid
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b)
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a)
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molar volume of the alcohols with the results obtained using the AEOS, the CPA and the Peng-Robinson equation. The four parameters estimated for each component were picked up from a relatively narrow optimum region, where the Relative Mean Square Deviation presents the smallest values for saturation pressure and saturated liquid molar volume. As the objective functions are conflicting, any attempt to reduce the deviation of one of these variables increases the deviation of the other. Figure 2 shows the Pareto front for methanol in this narrow optimum region together with each of four estimated parameters. Any set of parameters values within this range is an optimal solution and yields equivalent results. This procedure was followed for all the components. In order to select a specific point, an additional physical criteria was used; each point was chosen in order to keep the critical temperature rising and critical pressure decreasing with increasing total number of carbons of the alcohols.
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d) Figure 2 – Pareto´s Front for methanol plotted with estimated parameters
The AEOS parameters chosen for the alcohols up to decanol are presented in Table 3.
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Table 3 – Association Equation of State parameters Alcohol
(K)
(MPa)
396.4
84.75 0.011
-25192
-95.1
Ethanol
433.9
61.49 0.099
-26444
Propanol
492.5
51.33 0.152
-27663
Butanol
536.8
44.32 0.200
-28470
Pentanol
547.0
37.36 0.252
-28032
Hexanol
552.2
32.24 0.300
-29489
Heptanol
590.6
29.69 0.350
-29117
-101.4
Octanol
642.3
28.23 0.399
-29625
-109.0
Nonanol
657.8
25.80 0.445
-29038
-105.5
Decanol
674.2
24.10 0.490
-29702
-105.7
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Methanol
-99.9
-106.1 -108.7 -102.5 -100.9
Table 4 and Figure 3 show the ratio between the calculated parameters and the experimental critical temperature and pressure. Except for the critical temperature ratio for methanol, all the relations are quite similar, with a slight increase of the critical temperature ratio with carbon number.
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Table 4 – Ratios between apparent and actual critical properties Substance
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Methanol Ethanol Propanol Butanol Pentanol Hexanol Heptanol Octanol Nonanol Decanol
77.3% 84.4% 91.8% 95.3% 93.0% 90.4% 93.3% 98.4% 98.0% 98.4%
104.7% 100.0% 99.2% 100.2% 95.6% 92.5% 94.3% 98.7% 99.2% 101.7%
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Figure 3 - Ratio between apparent and experimental critical data
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Table 5 displays the average relative error between experimental data and calculated values for saturated liquid molar volume and vapor pressure for the AEOS, CPA and PengRobinson equations of state. It can be seen that the AEOS greatly improves the data correlation for pure alcohols when compared to Peng-Robinson, and its results are only slightly worse than those obtained with the CPA. While Peng-Robinson equation of state leads to average errors greater than 10% for saturation pressure for most components, and a large variation in saturated liquid molar volumes (between 0.7 and 20.6%), the greatest average errors for the saturation pressure and saturated liquid molar volume for the AEOS are below 4.0% for methanol, and for most of the alcohols are smaller than around 2.0%.
PR
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Alcohol
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Table 5 – Average relative errors Saturated Liquid Molar Saturation Pressure (%) Volume (%) AEOS
CPA
PR
AEOS
CPA
Methanol
11.6
3.7
1.7
20.6
3.4
0.4
Ethanol
4.0
2.9
0.4
10.3
2.8
0.9
Propanol
6.1
2.0
1.2
4.6
1.6
0.3
Butanol
10.6
1.3
0.5
3.4
1.6
0.6
Pentanol
22.5
0.9
3.0
0.7
0.6
0.8
Hexanol
30.9
0.6
0.8
0.9
0.2
1.0
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21.4
1.0
0.3
1.0
0.2
0.4
Octanol
39.8
0.8
1.3
1.0
0.9
0.9
Nonanol
15.5
0.8
0.3
0.7
0.5
0.6
Decanol
49.4
2.3
6.5
2.9
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Heptanol
1.1
0.9
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Figure 4 compares the adjusted apparent critical pressures to experimental data. As expected, the apparent critical pressure decreases with increasing alcohol carbon number. The apparent critical temperature also exhibited the same general behavior of the correlated experimental data, increasing with carbon number (Figure 5).
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Figure 4 – Experimental and apparent critical pressure
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Figure 5 – Experimental and apparent critical temperature behavior
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Figures 6 and 7 present the standard association enthalpy and entropy for each alcohol. It is interesting to note that there is an approximately linear behavior for the association enthalpy, while the standard association entropy does not present a clear pattern relating its value with the carbon number.
Figure 6 - Standard association enthalpy behavior
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Figure 7- Standard association entropy behavior
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Figures 8 and 9 present experimental saturation pressure and saturated liquid molar volume data and calculated values. As we can see, both results present excellent agreement for all alcohols analyzed in this work.
Figure 8 - Experimental and calculated vapor pressure
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Figure 9 - Experimental and calculated saturated liquid molar volume
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Figures 10 to 19 present the relative deviation between each calculated and experimental results for the ten alcohols analyzed in this paper. As we can observe, the deviations at each point are consistent with the average deviation, the relative error is greater than 5% for only a few points, and they appear very close to the critical point or at low temperatures.
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Figure 10 – Relative deviation of experimental points for methanol
Figure 11 - Relative deviation of experimental points for ethanol
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Figure 12- Relative deviation of experimental points for propanol
Figure 13 - Relative deviation of experimental points for butanol
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Figure 14 - Relative deviation of experimental points for pentanol
Figure 15 - Relative deviation of experimental points for hexanol
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Figure 16 - Relative deviation of experimental points for heptanol
Figure 17 - Relative deviation of experimental points for octanol
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Figure 18 - Relative deviation of experimental points for nonanol
Figure 19 - Relative deviation of experimental points for decanol
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5. Conclusions
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In this paper, we present a new equation of state developed for self-associating components based on the chemical theory. The model is composed by the Peng-Robinson equation of state representing the physical part, and an infinite linear association expression for the chemical part. The parameters of the resulting Association Equation of State (AEOS) are the apparent critical pressure and temperature, the apparent acentric factor, and the standard enthalpy and entropy of association. The equivalent alkane (same carbon number) acentric factor was used as the apparent acentric factor for the AEOS, while the remaining parameters were obtained through a Multi Objective Optimization Procedure (MOOP) based on the Particle Swarm Optimization (PSO) technique. The experimental data used to determine the parameters were the saturated data (vapor pressure and liquid molar volume) of ten alcohols (methanol to decanol). The apparent parameters follow the same behavior and are similar to experimental data, and both the standard enthalpy and entropy of association are physically meaningful. The average deviations for the saturated data were smaller than 4% for all the components analyzed, and smaller than 2% for eight of them. The results obtained here were compared to CPA equation of state and presented similar performance. Acknowledgements
References
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We acknowledge funding from the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) in the form of a scholarship.
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[30] KONTOGEORGIS, G.M.; MICHELSEN, M.L; FOLAS, G.K.; DERAWI, S.; SOLMS, N.; STENBY, E.H. 2006. Ten years with the CPA (Cubic-Plus-Association) equation of state. Part 1. Pure compounds and self-associating systems. Industrial & Engineering Chemistry Research, v.45, n14, p.4855-4868.
S0378-3812(18)30437-0
DOI:
https://doi.org/10.1016/j.fluid.2018.10.015
Reference:
FLUID 11977
To appear in:
Fluid Phase Equilibria
Received Date: 22 May 2018 Revised Date:
11 October 2018
Accepted Date: 17 October 2018
Please cite this article as: T.A.S. Menegazzo, A.M. Soares Junior., B.T. Mota, Né. Henderson, A.P. Pires, Application of an equation of state incorporating association to alcohols up to decanol, Fluid Phase Equilibria (2018), doi: https://doi.org/10.1016/j.fluid.2018.10.015. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Application of an equation of state incorporating association to alcohols up to decanol Thomas A. S. Menegazzoᵃ, Acir M. Soares Juniorᵇ, Breno T. Motaᵇ, Nélio Hendersonᵇ and Adolfo P. Piresᵃ* b *
Universidade Estadual do Rio de Janeiro-UERJ, Nova Friburgo-RJ, Brazil Corresponding author
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ᵃLaboratory of Petroleum Engineering and Exploration, Universidade Estadual do Norte Fluminense-UENF, Macaé-RJ, Brazil
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Abstract The accurate prediction of fluid phase equilibria is essential to the design of any process in chemical industry. Most of the simulators used to model multiphase multicomponent phase equilibria use traditional cubic equations of state, like Peng-Robinson and Soave-RedlichKwong. However, self-associating substances, such as alcohols, are not precisely modeled with these equations of state. Different approaches have been proposed to overcome this limitation. In this work, we use the technique that separates the compressibility factor in two parts, a physical and a chemical one, and we compare the results with the PengRobinson equation of state and the Cubic Plus Association equation of state against experimental data for alcohols up to decanol. The parameters are estimated adjusting the proposed EOS results to two independent data set measurements, saturation pressures and saturated liquid molar volumes of the pure components, through multi-objective optimization. The results present excellent agreement for all alcohols modeled.
1. Introduction
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Keywords: Equation of State, Associating Substance, Multi-Objective Optimization.
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Alcohols are organic compounds used in several different industrial processes, such as fuel, as an organic solvent, a component of hydraulic fluids, or as industrial feedstock in the synthesis of organic compounds. In order to optimize industrial processes using alcohols, it is important to calculate precisely their thermodynamics properties, especially when there is a phase change or multiphase multicomponent equilibrium conditions along the process. A particular and important characteristic of this organic class of compounds is that they form hydrogen bonds, affecting its microscopic (molecular structure) and macroscopic (thermodynamic) properties. These components are called associating fluids. It is possible to calculate the phase equilibrium of systems containing non-polar or slightly polar substances using traditional cubic equations of state (EOS), but these models lose accuracy when there are polar or associating components in the mixture (SCHMIDT; WENZEL, 1980 [1]). Excess Gibbs energy and EOS models have been developed in order to predict the thermodynamic properties and phase equilibrium of associating substances, and different approaches are used to mathematically describe the association effects in those calculations. In the perturbation theory, the total energy of hydrogen bonding is calculated based on statistical mechanics. A potential function is used to calculate the energy bonding between
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two sites in this approach. The number of bonding sites is a characteristic of the molecule, and it is an input parameter of the model (ECONOMOU; DONOHUE, 1996 [2]). One of the most known and used models based on the perturbation theory is the StatisticalAssociating-Fluid-Theory (SAFT), which can be used for both pure associating fluids and mixtures containing associating components (CHAPMAN et al., 1989 [3]; HUANG; RADOSZ, 1990 [4]). The SAFT equation accounts for hard-sphere repulsions, chain connectivity, dispersion attractive forces and hydrogen bonding, and each of its terms is obtained from perturbation theory. Despite being accurate for actual fluids, SAFT does not account for polar interactions due to dipole and quadrupole moments. The SAFT-Dimer equation of state (GHONASGI; CHAPMAN, 1994a [5]) and the physical theory for fluids of small associating molecules (WALSH et al., 1992 [6]) try to overcome this limitation. Both models were developed based on the SAFT equation of state. The Cubic Plus Association (CPA) equation of state (KONTOGEORGIS et al., 2006 [30]) uses the SoaveRedlich Kwong equation of state and the Wertheim perturbation theory to model the thermodynamic behavior of mixtures containing associating and non-associating components. The lattice theory takes into account the number of bonds formed between segments of different molecules that are adjacent in a lattice. The number of bonds determines the extent of association, but not the number of oligomers. Lattice models have also been successfully used to model thermodynamic properties and phase behavior of mixtures containing this kind of chemical species. These models can be modified to include polar effects by the introduction of specific forces between adjacent sites (GUGGENHEIM, 1952 [7]; FLORY, 1953 [8]). In the chemical theory, it is assumed that hydrogen bonding generates new molecular complexes. In this approach, the fluid is composed by a mixture of different oligomers of the associating substance, and of any other substances. Along with the EOS, it is necessary to solve material balance equations taking into account the original molecule and its many oligomers. The amount of each oligomer depends on equilibrium, density, temperature and composition of the system. Analytic expressions for the fraction of monomers and oligomers can be obtained for pure associating fluids (CAMPBELL et al., 1992 [9]), or for mixtures containing only one associating fluid. Mixtures of many associating fluids require numerical solutions (ANDERKO, 1989 [10]; ECONOMOU et al., 1990 [11]). In this theory, the EOS can be written as:
(1) where and represent the “physical” part of the equation, which can be calculated from a traditional cubic equation of state (ANDERKO, 1989 [12]; ELLIOTT et al., 1990 [13]), and deals with the hydrogen bonding. In this paper, we present an association equation of state based on the chemical theory where the physical part is based on the Peng-Robinson equation of state (PENG; ROBINSON, 1976 [14]) and the associating part calculated through a linear association approach. It is able to calculate pure component properties with less adjusting parameters
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2. Thermodynamics Modelling
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and its mathematical form is simpler than other theories. This equation was adjusted to saturation pressure and saturated liquid molar volume data of ten pure alcohols. Results show that this model improves the correlation of pure alcohols saturated properties.
The second virial coefficient can be separated into a chemical and a physical part, based on its relationship with the dimerization constant (LAMBERT, 1953 [15]): (2)
(3)
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Substituting equation 2 in equation 3, we obtain:
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The virial equation truncated at the second term is given by:
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(4) The two terms on the right hand side of equation 4 can be named as the chemical and physical compressibility factor, respectively, and this equation can be rewritten as (ANDERKO, 1991 [16]): (5)
2.1 Physical Model
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In order to represent the physical part of the compressibility factor, , van der Waals type equations of state that do not account for association can be used. Many EOS have been created based on the van der Waals model, such as the Peng-Robinson and the SoaveRedlich-Kwong (SOAVE, 1972 [17]) equations. In this paper, the Peng-Robinson model was used to represent the physical part:
where a, b and
(6)
are given by:
(7)
(8)
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(9)
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2.2 Chemical Model
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The chemical compressibility factor is defined as the ratio between the true number of mols , which includes the new species created through association, and the total number of mols if the system had no association, :
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(10) At this point, expressions to determine and , which describe the characteristics of the association between molecules, such as the number of bonds, are necessary. There are different models to evaluate this ratio, such as the one-site model for carboxylic acids (IKONOMOU; DONOHUE, 1986 [18]), or the three-site model for water (HUANG; RADOSZ, 1990 [4]). These models are developed for a specific geometry of association, number of bonds, and strength of association. Alcohols form chain oligomers in the form of dimers, trimers etc. In this work the infinite equilibrium scheme is used, it is also called the two-site model (IKONOMOU; DONOHUE, 1986 [18]):
(11) (12) (13)
(14) was considered zero, so the chemical reaction constant, K, is defined as:
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In this work,
(15) From equation 11 it is possible to determine the chemical part contribution to the system pressure:
(16) 2.3 Association Equation of State
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Rewriting equation 5 in terms of pressure, we have
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(17) and applying equations 6 and 16 in equation 5, the association equation of state becomes:
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(18) In this equation of state, the critical properties , and the acentric factor used to calculate parameters a and b of pure components in the physical part of the equation of state should account only for the monomers of the mixture, and they must be replaced by apparent critical properties , and . These apparent properties can be determined from experimental saturated data of the pure components. Besides the apparent properties, there are also two other parameters to be determined, the association reaction enthalpy ( ) and entropy ( ). 2.4 Equilibrium Calculations
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The fugacity coefficient of a pure substance can be calculated by:
(19)
From the definitions of P and Z in equations 5 and 17 we have:
(20)
and
(21)
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Equation 20 can be separated into:
(22) The physical part of the fugacity coefficient for the Peng-Robinson equation of state is given by:
(23) Applying equation 16 in equation 22, we obtain:
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After integration we have:
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(24) Equation 24 is the chemical part contribution for the fugacity coefficient of an association component. Figure 1 presents a simplified diagram for saturation pressure calculation of a pure substance using the Association Equation of State for a fixed set of parameters.
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Figure 1 - Diagram for saturation pressure calculation of a pure substance using the Association Equation of State 3. EOS Parameters Estimation
There are 5 parameters to be determined in the associating equation of state: , , , ( ) and ( ). The characteristic parameters in most EOS are exclusively tuned to the phase equilibrium data in order to obtain the best possible representation of vapor-liquid
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3.1 General Aspects of the Optimization Given a function follows
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, a single-objective optimization problem can be formulated as
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where
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equilibrium (VLE). Although VLE predictions are of primary importance for practical purposes, it is found that these fitted parameters sometimes give very poor estimates for other thermophysical properties. In this work we seek for improvement in predictions of phase equilibrium estimating EOS parameters using multi-objective optimization (MOO) as sugested by Punnapla, Vargas and Elkamel (RANGAIAH; PETRICIOLET, 2013 [19]) to adjust the EOS parameters to two independent experimental data sets: saturation pressures and saturated liquid molar volumes.
(25)
is the set of constraints that, in general, can be of the form (26)
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where and are real functions of real variables. A multi-objective problem is one that involves the simultaneous optimization of two or more objective functions, , , etc. Thus, for a set with functions, the multiobjective optimization problem can be described as follows:
(27)
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The problem shown in Eq. (27) is often written in the following summarized form: (28)
where denotes a mapping of into . Optimization problems involving multiple and conflicting objectives have been traditionally handled by combining the objectives into a scalar function, and next solving the equivalent single-optimization problem to identify the best-compromise solution. That approach presents some serious disadvantages, and the major problems are its subjectivity (e.g. in choosing weights) and the fact that it hides the competition among the conflicting criteria (EFSTRATIADIS; KOUTSOYIANNIS, 2010 [20]). Multi-objective optimization has gained interest in engineering applications recently. It is applied to numerous areas such as process design, biotechnology, petroleum refining, pharmaceuticals and polymerization, (BHASKAR; GUPTA; RAY, 2000 [21]).
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The major difference of the MOO approach when compared to the combination schemes, usually referred to as aggregating functions optimization, is a clear focus on the inherent multi-objective nature of the model parameters calculation (GUPTA; SOROOSHIAN; YAPO, 1998 [22]). In MOO the conflicting objectives ( ), represented by a vector of objective functions with "m" elements and a control vector with "n" parameters are solved simultaneously. As the criteria are conflicting, there is no unique point that optimizes all of them simultaneously. In that case we look for acceptable trade-offs according to the fundamental concept of Pareto optimality. To solve the model calibration problem by MOO one should follow three steps: 1- Specify the objective functions. Define functions length (L) to measure how far the predictions are from experimental data; 2- Estimate the location of the Pareto solution set. Minimize the objective functions accordingly to Pareto optimum criteria; 3- Choose a specific solution among optimal solutions. In our problem, in order to determine the difference between calculated and experimental data, the Relative Mean Square Error was chosen for experimental saturated pressure and saturated liquid molar volume:
(29)
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(30) Multi-objective particle swarm optimization (MOPSO) is a relative popular and easy to use algorithm (COELLO; LECHUGA, 2002 [23]) based on the Kennedy and Eberhart (1995 [24]) PSO metaheuristic. In this work we used a MOPSO based on the modification presented in Coello, Pulido and Lechuga (2004 [25]), with the constriction factor ( χ ) suggested by Clerc and Kennedy (2002 [26]) to assure the method convergence and a nonlinear inertia factor ( ω ) introduced by Chatterjee and Siarry, (2006 [27]). The later was included to enhance a global exploration capability at the beginning of the process and a local one for the last iterations (REZAEE; JORDEHI; JASNI, 2013 [28]). Both parameters, ω and χ , are used to set particle velocity at time t=t+1: r r r r r r v (t + 1) = χ [ω I ⋅ v (t ) + C1 ⋅ φ1 ( x pbest (t ) − x (t )) + C 2 ⋅ φ 2 ( x gbest (t ) − x (t ))] (31) where ( itermax − iter )q ωI = ⋅ (ω max − ω min ) + ω min q ( itermax )
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Table 1 presents the parameters used in the optimization procedure.
C1 , C2
Constriction factor Inertia weight Nonlinear modulation index Number of cycles
χ
ω max , ωmin q
itermax
2.05, 2.05 0.7298 0.9, 0.4
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Learning factor
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Table 1 – Standard parameters set used for MOPSO Parameter Value Number of particles NMAX 100 Size of repository NREP 100 Number of divisions 30, 30 F1 , F2
1.2 100
4. Results and Discussion
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Finally, parameters were picked up from an interest region around the point that optimizes the sum OF1+OF2. To choose a single solution, it was observed the parameter consistency within carbon chain results.
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The experimental thermodynamic data (saturation pressure and saturated liquid and vapor volumes) range used for the optimization procedure can be found in Table 2 (SMITH, 1986 [29]).
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Table 2 – Equilibrium experimental data (saturation pressure and saturated volumes) Alcohol Number of points Temperature range (K) Methanol 39 212-497 Ethanol 35 231-503 Propanol 29 280-517 Butanol 31 295-548 Pentanol 29 278-508 Hexanol 17 310-428 Heptanol 15 343-445 Octanol 31 296-549 Nonanol 15 366-481 Decanol 27 301-526 As the apparent critical properties and acentric factor refer to a hypothetical non-associating alcohol, we adopted the acentric factor of the alkane with the same carbon number as the apparent acentric factor for the alcohol. So, our model has only four adjusting parameters: apparent critical temperature and pressure, and standard enthalpy and entropy of association. These parameters were adjusted to correlate the experimental saturation pressure and liquid
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b)
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a)
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molar volume of the alcohols with the results obtained using the AEOS, the CPA and the Peng-Robinson equation. The four parameters estimated for each component were picked up from a relatively narrow optimum region, where the Relative Mean Square Deviation presents the smallest values for saturation pressure and saturated liquid molar volume. As the objective functions are conflicting, any attempt to reduce the deviation of one of these variables increases the deviation of the other. Figure 2 shows the Pareto front for methanol in this narrow optimum region together with each of four estimated parameters. Any set of parameters values within this range is an optimal solution and yields equivalent results. This procedure was followed for all the components. In order to select a specific point, an additional physical criteria was used; each point was chosen in order to keep the critical temperature rising and critical pressure decreasing with increasing total number of carbons of the alcohols.
c)
d) Figure 2 – Pareto´s Front for methanol plotted with estimated parameters
The AEOS parameters chosen for the alcohols up to decanol are presented in Table 3.
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Table 3 – Association Equation of State parameters Alcohol
(K)
(MPa)
396.4
84.75 0.011
-25192
-95.1
Ethanol
433.9
61.49 0.099
-26444
Propanol
492.5
51.33 0.152
-27663
Butanol
536.8
44.32 0.200
-28470
Pentanol
547.0
37.36 0.252
-28032
Hexanol
552.2
32.24 0.300
-29489
Heptanol
590.6
29.69 0.350
-29117
-101.4
Octanol
642.3
28.23 0.399
-29625
-109.0
Nonanol
657.8
25.80 0.445
-29038
-105.5
Decanol
674.2
24.10 0.490
-29702
-105.7
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Methanol
-99.9
-106.1 -108.7 -102.5 -100.9
Table 4 and Figure 3 show the ratio between the calculated parameters and the experimental critical temperature and pressure. Except for the critical temperature ratio for methanol, all the relations are quite similar, with a slight increase of the critical temperature ratio with carbon number.
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Table 4 – Ratios between apparent and actual critical properties Substance
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Methanol Ethanol Propanol Butanol Pentanol Hexanol Heptanol Octanol Nonanol Decanol
77.3% 84.4% 91.8% 95.3% 93.0% 90.4% 93.3% 98.4% 98.0% 98.4%
104.7% 100.0% 99.2% 100.2% 95.6% 92.5% 94.3% 98.7% 99.2% 101.7%
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Figure 3 - Ratio between apparent and experimental critical data
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Table 5 displays the average relative error between experimental data and calculated values for saturated liquid molar volume and vapor pressure for the AEOS, CPA and PengRobinson equations of state. It can be seen that the AEOS greatly improves the data correlation for pure alcohols when compared to Peng-Robinson, and its results are only slightly worse than those obtained with the CPA. While Peng-Robinson equation of state leads to average errors greater than 10% for saturation pressure for most components, and a large variation in saturated liquid molar volumes (between 0.7 and 20.6%), the greatest average errors for the saturation pressure and saturated liquid molar volume for the AEOS are below 4.0% for methanol, and for most of the alcohols are smaller than around 2.0%.
PR
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Alcohol
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Table 5 – Average relative errors Saturated Liquid Molar Saturation Pressure (%) Volume (%) AEOS
CPA
PR
AEOS
CPA
Methanol
11.6
3.7
1.7
20.6
3.4
0.4
Ethanol
4.0
2.9
0.4
10.3
2.8
0.9
Propanol
6.1
2.0
1.2
4.6
1.6
0.3
Butanol
10.6
1.3
0.5
3.4
1.6
0.6
Pentanol
22.5
0.9
3.0
0.7
0.6
0.8
Hexanol
30.9
0.6
0.8
0.9
0.2
1.0
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21.4
1.0
0.3
1.0
0.2
0.4
Octanol
39.8
0.8
1.3
1.0
0.9
0.9
Nonanol
15.5
0.8
0.3
0.7
0.5
0.6
Decanol
49.4
2.3
6.5
2.9
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Heptanol
1.1
0.9
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Figure 4 compares the adjusted apparent critical pressures to experimental data. As expected, the apparent critical pressure decreases with increasing alcohol carbon number. The apparent critical temperature also exhibited the same general behavior of the correlated experimental data, increasing with carbon number (Figure 5).
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Figure 4 – Experimental and apparent critical pressure
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Figure 5 – Experimental and apparent critical temperature behavior
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Figures 6 and 7 present the standard association enthalpy and entropy for each alcohol. It is interesting to note that there is an approximately linear behavior for the association enthalpy, while the standard association entropy does not present a clear pattern relating its value with the carbon number.
Figure 6 - Standard association enthalpy behavior
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Figure 7- Standard association entropy behavior
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Figures 8 and 9 present experimental saturation pressure and saturated liquid molar volume data and calculated values. As we can see, both results present excellent agreement for all alcohols analyzed in this work.
Figure 8 - Experimental and calculated vapor pressure
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Figure 9 - Experimental and calculated saturated liquid molar volume
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Figures 10 to 19 present the relative deviation between each calculated and experimental results for the ten alcohols analyzed in this paper. As we can observe, the deviations at each point are consistent with the average deviation, the relative error is greater than 5% for only a few points, and they appear very close to the critical point or at low temperatures.
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Figure 10 – Relative deviation of experimental points for methanol
Figure 11 - Relative deviation of experimental points for ethanol
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Figure 12- Relative deviation of experimental points for propanol
Figure 13 - Relative deviation of experimental points for butanol
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Figure 14 - Relative deviation of experimental points for pentanol
Figure 15 - Relative deviation of experimental points for hexanol
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Figure 16 - Relative deviation of experimental points for heptanol
Figure 17 - Relative deviation of experimental points for octanol
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Figure 18 - Relative deviation of experimental points for nonanol
Figure 19 - Relative deviation of experimental points for decanol
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5. Conclusions
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In this paper, we present a new equation of state developed for self-associating components based on the chemical theory. The model is composed by the Peng-Robinson equation of state representing the physical part, and an infinite linear association expression for the chemical part. The parameters of the resulting Association Equation of State (AEOS) are the apparent critical pressure and temperature, the apparent acentric factor, and the standard enthalpy and entropy of association. The equivalent alkane (same carbon number) acentric factor was used as the apparent acentric factor for the AEOS, while the remaining parameters were obtained through a Multi Objective Optimization Procedure (MOOP) based on the Particle Swarm Optimization (PSO) technique. The experimental data used to determine the parameters were the saturated data (vapor pressure and liquid molar volume) of ten alcohols (methanol to decanol). The apparent parameters follow the same behavior and are similar to experimental data, and both the standard enthalpy and entropy of association are physically meaningful. The average deviations for the saturated data were smaller than 4% for all the components analyzed, and smaller than 2% for eight of them. The results obtained here were compared to CPA equation of state and presented similar performance. Acknowledgements
References
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We acknowledge funding from the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) in the form of a scholarship.
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