Thermodynamics of Phase and Chemical Equilibrium in the Processes

Article pubs.acs.org/IECR

Thermodynamics of Phase and Chemical Equilibrium in the Processes of Biodiesel Fuel Synthesis in Subcritical and Supercritical Methanol Vladimir Anikeev,* Denis Stepanov, and Anna Yermakova Boreskov Institute of Catalysis SB RAS, Novosibirsk, Russia S Supporting Information *

ABSTRACT: Thermodynamic data (critical parameters, enthalpy, entropy) were calculated for the monoglycerides, diglycerides, and triglycerides of fatty acids (palmitic, oleic and linoleic), free fatty acids, and methyl esters of fatty acids, which are involved in the transesterification of vegetable oils with lower alcohols. Phase diagrams of the reaction mixtures were calculated and reaction conditions providing supercritical state of the reaction mixture were selected for transesterification of triglycerides in various lower alcohols. Chemical equilibrium of the methanol transesterification of simple and mixed fatty acid triglycerides was studied in a wide range of temperatures, pressures, and ratios of initial components, including the region of supercritical state of the reaction mixture. Calculation of the phase equilibrium and equilibrium composition of the reaction products obtained by transesterification of triglycerides with lower alcohols allowed choosing the optimal reaction conditions for practical application.

1. INTRODUCTION The technology of motor fuel synthesis from vegetable oils or animal fats was developed quite a long time ago. Such fuels are classified as a synthetic fuel called biodiesel (BD) and, in contrast to diesel fuel produced from petroleum, are virtually free from aromatic and sulfur-containing compounds, which decrease the content of hazardous substances in combustion products. The main advantages of BD fuel as compared to similar fuels produced from petroleum are well-known1−3 and require no special discussion here. The term “biodiesel” refers to fatty acid esters (FAE), which are commonly obtained by the transesterification of vegetable oils  triglycerides (TG) of fatty acids (1) or esterification of free fatty acids (FFA) (2) with lower alcohols. Natural oils produced from vegetable feedstock (oil-bearing crops) contain ca. 97% of various triglycerides  fatty acid esters, diglycerides and monoglycerides, and FFA. Along with simple triglycerides, vegetable oil also consists of mixed triglycerides containing bases of different fatty acids.4 PFATG + 3CH3OH ⇆ 3PFAME + glycerol (1) PFA + CH3OH ⇆ PFAME + H2O

in oil transesterification, they provide only a very low reaction rate.5 Transesterification in supercritical lower alcohols is a promising new approach to the development of efficient and relatively inexpensive processes for synthesis of biodiesel from vegetable oils, animal fats, and food industry waste. Such studies are reported in the literature, for example.10−14,23 The synthesis of biodiesel by this method is performed most often with methanol. The application of supercritical alcohols for transesterification of vegetable feedstock makes it possible to run the process without homogeneous catalysts and, thus, use a lowergrade feedstock; to attain a 90%−98% conversion of the starting material at a much shorter time of contacting with the reaction mixture, which allows the use of the flow-type reactors and hence significantly raises the process productivity; to avoid wastewater that earlier resulted from washing the reaction products for removal of homogeneous alkaline or acid catalysts; all this essentially reduces the cost of biodiesel. It should be noted that transesterification of triglycerides in supercritical alcohols, in distinction to a similar reaction with participation of homogeneous catalysts, proceeds selectively to yield glycerol and esters of fatty acids and gives only a small amount of side products. The end product of transesterification reaction, biodiesel, must comply with the requirements to diesel fuel; in this connection, it is necessary to separate methyl esters from methanol (which is taken in excess) and glycerol, and to perform washing and drying of esters, which leads to a large amount of waste.1 A general problem in the synthesis of biodiesel under conventional and supercritical conditions is

(2)

where PFATG represents the triglyceride of palmitic fatty acid; PFAME denotes that methyl ester of palmitic fatty acid; PFA is palmitic fatty acid. Transesterification of triglycerides with alcohol is conventionally performed in the presence of homogeneous1,5,6 and sometimes heterogeneous7−9 acid or base catalysts; in recent years, subcritical and supercritical lower alcohols have been used for the purpose, both with and without heterogeneous catalyst.10−22 In transesterification of vegetable oils catalyzed by alkalis, the main disadvantage is sensitivity to water and the presence of free fatty acids in the feedstock. As for acid catalysts © 2012 American Chemical Society

Received: Revised: Accepted: Published: 4783

October 19, 2011 February 26, 2012 March 5, 2012 March 5, 2012 dx.doi.org/10.1021/ie202379u | Ind. Eng. Chem. Res. 2012, 51, 4783−4796

Industrial & Engineering Chemistry Research

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and ΔSf°(298.15 K), heat capacity as a function of temperature Cp(T), and acentric factor ω. For some simple compounds, such data can be found in reference books; however, for the majority of complex organic compounds, thermodynamic parameters should be calculated using appropriate semiempirical and empirical methods. 2.1. Determination of Critical Parameters for Individual Substances. By now, numerous group methods have been suggested for determination of critical parameters of individual substances and their boiling points. In the present work, we compare data on critical parameters and boiling points of triglycerides, fatty acids and esters of fatty acids that were calculated using different group methods developed by Lydersen, Thodos, and Riedel,33 Joback,34 and a method reported in ref 35. For the majority of compounds under consideration, it is impossible to determine the indicated parameters in the experiment, because of thermal instability of the compounds (at temperatures above 250 °C, virtually all fats decompose to form glycerol, acrolein, etc.). Some Internet sources provide experimental values of the boiling points for one or two triglycerides, but these values were obtained at low pressures (∼10 Torr). Thermodynamic parameters calculated by different methods were compared with the available literature data (see, for example, refs 27 and 32). Knowing the critical temperature of a compound, it is possible to calculate its boiling point using a relationship such as

related with a large amount of glycerol produced by the reaction: it cannot be used in pure form and should be converted into useful products. Besides, under supercritical conditions of biodiesel synthesis, esterification of FFAs, which are a low-percentage component of vegetable oil, yields water, which should be removed from biodiesel fuel. Natural vegetable oils are the most promising feedstock for synthesis of biodiesel fuel. Virtually all mass-produced vegetable oils are a mixture of various triglycerides (simple and mixed): esters of fatty acids (96%−98%), monoglycerides and diglycerides, free fatty acids. Transesterification of fatty acid triglycerides with lower alcohols usually proceeds in three steps via sequential substitution of one or two alkyl groups of fatty acid by hydroxyl groups to form di- and monoglycerides, which can be present in the reaction products together with FAE. In the general case, such transformations can be presented by the scheme depicted in eq 3: 1. R1R2R3TG + CH3OH ⇆ R2R3DiG + MER1FA 2. R2R3DiG + CH3OH ⇆ R3MonoG + MER2FA 3. R3MonoG + CH3OH ⇆ MER3FA + glycerol

(3)

Here, R1, R2, and R3 correspond to alkyl groups of three different fatty acids, for example, palmitic, oleic, or linoleic. For R1 ≠ R2 ≠ R3, the transformations of such mixed triglyceride via this scheme will include 12 reactions. For R1 = R2 ≠ R3, the number of reactions will be reduced to 7. A comparison of the reaction scheme shown earlier by eq 1 with the scheme depicted in eq 3 gives grounds to conclude, for example, that equilibrium compositions for the reaction calculated by the single- or three-step schemes may strongly differ. Although publications on the synthesis of biodiesel via transesterification of vegetable oils in alcohols are numerous, only some works consider the phase equilibrium of reaction mixtures,24−32 with no studies on the chemical equilibrium of FFA esterification and TG transesterification. This may be related with the absence of reliable thermodynamic data for monoglycerides, diglycerides, and triglycerides, FFA and esters of fatty acids. In this paper, the objective of the present work is to calculate the thermodynamic parameters (critical parameters, formation entropy and enthalpy) for all the reactants using various methods, compare the data obtained, and select the most correct data by comparing with available results of experiments. The found values of thermodynamic parameters will allow calculating the phase equilibrium for chosen mixtures and the chemical equilibrium for transesterification of simple and mixed TG with lower alcohols in a wide range of pressure, temperature and ratios of initial reactants. Calculation will be made to determine the reaction conditions providing a maximum yield of the target product: esters of fatty acids.

Tboil = ΘTcr

(4)

where Θ depends on the structure of molecule, polarity, dipole moment, etc. Riedel, Lydersen. and Jadoul33 suggested some dependences for the calculation of Θ from the fractions of atoms and groups of atoms. On the other hand, Meissner33 proposed a formula for determination of the boiling point without knowing the critical temperature of a compound: Tboil =

637RD1.47 + B Pch

(5)

where B is the constant, RD the molar refraction, and Pch the parachor, the value of which can also be found using the group methods of Eisenlohr, Vogel, and Soegden,33 as well as those of others. As an example of calculation, Table 1 lists the values of Tcr, Pcr, and Tboil for oleic fatty acid triglyceride (OFATG) obtained Table 1. Critical Temperature, Pressure, and Boiling Point for OFATG Calculated by Different Methods Method

2. CALCULATION OF THERMODYNAMIC AND PHYSICOCHEMICAL DATA FOR INDIVIDUAL COMPOUNDS To calculate the equilibrium constant of chemical reaction and its variation in a wide range of temperature and pressure, it is necessary to acquire thermodynamic and physicochemical data for individual compoundsreaction participants, namely: critical parameters Tcr and Pcr, boiling point Tboil, standard values of the formation enthalpy and entropy, ΔHf°(298.15 K)

parameter

ref 35

Riedel33

Tboil Tcr Pcr

854 K 1038 K 0.47 MPa

743.5 K 1245 K 0.46 MPa

Thodos33

literature data

945.6 K

827.4 K32 947 K,27 977.9 K32 0.48 MPa,27 0.33 MPa32

by us by three different group methods; a comparison with the recent literature data is presented. One may see that three methods give virtually identical values of critical pressure. The Riedel method33 provides a simple calculation procedure, so it was chosen for the determination of Pcr. As seen from Table 1, different methods used for calculation of critical temperature give a considerable spread of values. In 4784

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the literature,27,32 we found only the critical parameters and standard enthalpy formation for OFATG, which allowed us to estimate the calculation deviation. According to Table 1, the closest calculated value is obtained by Thodos’s method; therefore, this method was assumed to be the most accurate one and used to calculate the critical temperature for all of the chosen substances. However, when applied to the calculation of Tboil, this method introduces an essential error. With the known values of intensive thermodynamic parameters, the acentric factor can be calculated using the formula suggested in ref 36: 3 Θ ω = log(Pcr) −1 7 1−Θ

T where Θ = boil Tcr

complex organic molecules was calculated by various methods. The calculated results are presented in Table 2 of the Supporting Information. From these results, it can be seen that the MINDO method overestimates the heat of formation of the complex acids and ester glycerides considerably. Therefore, the thermodynamic characteristics (standard formation enthalpy and entropy, coefficients for heat capacity as a function of temperature) in this work were calculated using the Joback group contribution method. Results of the calculation are presented in Table 3 of the Supporting Information.

3. PHASE EQUILIBRIUM IN TRANSESTERIFICATION OF TRIGLYCERIDES OF FATTY ACIDS WITH METHANOL Under normal conditions (temperature 10−30 °C, atmospheric pressure), it is very difficult to dissolve triglycerides of fatty acids, products of their transesterification, and fatty acids in lower alcohols (particularly methanol).40,41 These immiscible liquids can form homogeneous single-phase mixtures with the properties close to those of compressed liquids only at elevated pressure and temperature. Such conditions generally correspond to the supercritical state of the mixture. Advantages of such state of the reaction mixture for chemical transformations manifest themselves mainly when the reaction runs entirely in the supercritical region near the critical point of the reaction mixture. Hence, for the experiments on the transesterification of triglycerides with supercritical lower alcohols, it is necessary, on the one hand, to know the temperature T0 and pressure P0 in the reactor at which the initial mixture with a fixed alcohol/ oil ratio is known to be in a supercritical state. As the triglycerides convert during transesterification, the reaction mixture changes its composition: glycerol and esters of fatty acids are produced. Thus, on the other hand, conditions should be created at which the reaction mixture of a varying composition will be under supercritical conditions, too. To meet the above listed requirements, we calculated the phase equilibrium for the mixtures containing both the initial substances (lower alcohols, triglycerides) and the reaction products (lower alcohols, triglycerides, glycerol, esters of fatty acids). 3.1. Calculation of Phase Diagrams and Critical Parameters. Critical temperature and pressure values for OFATG mixtures of different compositions with three lower alcohols were calculated before calculation of the phase diagrams. Results are presented in Table 4 in the Supporting Information. It is seen that the same increase in the mole fraction of alcohol causes a more pronounced decrease in the critical temperature for the methyl alcohol−OFATG system. State of the mixtures is presented most comprehensively by their phase diagrams with a fixed position of the critical point. To illustrate the calculation, Figure 1 shows in P−T coordinates the phase diagrams of two binary mixtures: mixture No. 1 (methanol−PFA) and mixture No. 2 (methanol−PFATG). The methanol content in these mixtures was set equal to 95 mol %. The chosen mixtures simulate the initial reaction mixtures in the synthesis of methyl esters of fatty acids and transesterification of fatty acid triglycerides with methanol. Calculation was made using the earlier developed multifunctional program Phasediagram.42 The calculation methods and algorithms used also were described in our articles.43−46 Curves 1 and 2 in Figure 1 are called the binodal lines. Dots on the curves indicate the critical points for mixtures No. 1 and No. 2 of a specified composition. Outside the region bounded

(6)

The calculated parameters Tcr, Pcr, Tboil, and ω for monoglycerides, diglycerides and triglycerides, FFA and esters of fatty acids, which are the initial substances and products of transesterification further used to calculate the phase state and chemical equilibrium of the chosen reactions, are presented in Table 1 in the Supporting Information. 2.2. Calculation of Standard Values of the Formation Enthalpy, Gibbs Energy, and Entropy. To calculate the equilibrium of any chemical reaction, it is necessary to know the values of the standard formation enthalpy and entropy (at 298 K) or the Gibbs formation energy of individual substances involved in the reaction, as well as their temperature dependences. Virtually for all the compounds under consideration, numerical values of these specific thermodynamic parameters are absent in the literature; thus, they should be calculated using the known methods. We chose the methods of quantum and statistical mechanics,37,38 semiempirical method of the MINDO type,39 and the group methods,33,35 in particular, the Joback method.34 Quantum-statistical methods for calculating thermodynamic properties are applicable for the molecules with the known quantum-mechanistic models. The methods are based on the calculation rotationally oscillatory, electronic, and nuclear states of the molecule taking into account mutual influence of different energy levels. Exact calculation of diatomic and triatomic molecules using this method presents no difficulties, whereas in the case of multiatomic molecules, only approximate calculations can be performed. Approximate quantum-statistic methods allow calculation of physicochemical and thermodynamic characteristics of complex molecules such as fatty acid esters. The MINDO (Modified Intermediate Neglect of Differential Overlap) is one of such approximate methods in which difficult integrals are replaced by some empirical expressions. For example, two-electron Coulomb integrals are calculated by the Ohno−Klopman equation; resonance integrals are calculated at the Mullikentype approximation using empirical dimensionless factor characterizing the considered types of atoms. Interaction integrals are estimated using various approximate expressions. The Joback additive group contribution methods26 are widely applied for calculating thermodynamic properties of complex multicenter molecules. In these methods, a complex molecule of known structure is considered as a combination of simple functional groups, and its thermodynamic properties are calculated as a sum of the selected parameters of each group. In order to select and validate the method for calculating standard enthalpy and entropy of formation of individual compounds (reaction participants), the heat of formation of 4785



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Figure 2. Critical curves for the mixtures under consideration: Curve 1 represents mixture No. 1; Curve 2 represents mixture No. 2. A indicates pure methanol; B1 indicates pure palmitic acid; and B2 indicates pure triglyceride of palmitic acid. Dots on the curves correspond to the maximum values of critical pressure.

Figure 1. Phase diagrams on the P−T plane. Curve 1 denotes mixture No. 1; Curve 2 represents mixture No. 2. Dots correspond to critical points of the mixtures.

by binodal lines, there are single-phase mixtures: to the left of the critical point, there is a compressed liquid, to the right, there is a vapor/gas. Within the region bounded by binodal lines, the mixture is separated into two phases. Noteworthy is a difference in the extent of two-phase regions for mixtures No. 1 and No. 2, which is naturally caused by different properties of the mixtures being compared. For these mixtures, the liquidphase lines virtually coincide, whereas positions of the gasphase lines differ considerably. Coordinates of critical points for the chosen mixtures are as follows: • Mixture No. 1: Tcr = 562.3 K, Pcr = 10.4 MPa • Mixture No. 2: Tcr = 666.6 K, Pcr = 12.9 MPa

Table 2. Separation of Mixture No. 1 into Equilibrium Phases (P = 8.1 MPa) T = 550 K component composition [%] CH3OH C16H32O2 phase yield [mol %] molar volume [cm3/mol]

Evidently, changes in the component composition of the mixture or in the quantitative content of components alter the phase boundaries, i.e., the contours of binodal lines; accordingly, this changes the coordinates of critical points. Figure 2 shows the critical parameters (positions of the critical point) for binary mixtures No. 1 and No. 2 versus their composition (the so-called “critical curves”), the content of a component varying from 0% to 100%. The presence of a critical pressure maximum on the curves (10 mol % PFA for mixture No. 1 and 5.5 mol % PFATG for mixture No. 2) is typical of the binary systems with a pronounced difference in critical parameters of individual components.47 Because of chemical transformations of triglyceride during transesterification with alcohol, its concentration decreases, and reaction productsglycerol and esters of fatty acidsare produced. Quantitative and qualitative changes in the reaction mixture composition are reflected both in the parameters of critical point and in the shape of phase diagram. Calculation was made for two mixtures with the composition corresponding to 50% and 100% conversion of triglyceride in methanol. According to the calculation, during the transformation of triglyceride, the reaction mixture remains in the supercritical region if the reaction temperature and pressure were chosen equal to the critical values for the initial mixture. The above examples clearly demonstrate that calculation of the reaction mixture phase state in each particular case should accompany the choice of conditions for a planned experiment and the processing of experimental data. Tables 2 and 3 present

T = 590 K

phase 1

phase 2

phase 1

phase 2

84.9 15.1 26.0 188.8

98.6 1.4 74.0 370.5

73.6 26.4 7.1 247.3

96.6 3.4 92.9 447.8

Table 3. Separation of Mixture No. 2 into Equilibrium Phases (P = 11.1 MPa) T = 600 K component composition [%] CH3OH C51H98O6 phase yield [mol %] molar volume [cm3/mol]

T = 700 K

phase 1

phase 2

phase 1

phase 2

92.7 7.3 63.2 307.4

98.9 1.1 36.8 324.1

84.9 15.1 19.3 388.1

97.4 2.6 80.7 467.1

the typical compositions of two equilibrium phases obtained by separation of the initial mixtures No. 1 and No. 2 (see the composition indicated above) into two equilibrium phases. The separation pressure and temperature were chosen to arrange the points within the respective regions limited by binodal lines. Judging from the molar volumes (the last row in Tables 2 and 3), both equilibrium phases can be considered to be compressed liquids. In both cases, one phase consists mainly of methanol, with small admixtures of high-molecular-weight organic compounds. Another, more-dense phase (with a smaller molar volume) has quite a large amount of high-molecular-weight organic compounds. Temperature elevation considerably increases the yield of phase 2, consisting mainly of methanol. 4786



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reaction j. Dimensionality of Xj is “mole”, since thermodynamic calculations commonly employ extensive values. Introduction of the reaction coordinates reduces the calculation of equilibrium to finding a minimum Gibbs energy as a function of X. Considering expression 11, eq 7 takes the form:

4. CHEMICAL EQUILIBRIUM, CALCULATION MODELS, AND METHODS Calculation of chemical equilibrium of the reaction mixtures is presented as a problem of minimizing the free Gibbs energy.48 For a mixture of components with molar numbers ni, the total Gibbs energy (Gmix) at specified T and P in the equilibrium state should take a minimum value: N

Gmix =

∑ niμi → min

(7)

yieq =

(13)

eq

ni

eq

∑iN= 1 ni

(14)

As follows from eq 13, in view of eqs 8, 9, 11, and 14, for each reaction j in the equilibrium state, the following equality should be fulfilled:

(9)

N

Fj ≡

∑ zjiμi i=1 N

≡

(10)

∑ zji(μi0 + RT(ln yi + ln Fi + ln P)) = 0

i=1

j = 1, 2 , ..., NR

are specified in the reaction system. Here, Z denotes the matrix of stoichiometric coefficients with dimensions NR × N, where NR is the number of chemical reactions, and N is the number of components. B = [B1, B2, ..., BN]T denotes the vector of molecular components involved in the reactions. Hereinafter, the superscript “T” indicates the transposition of vector/matrix. Elements zji of matrix Z are negative for the initial substances and positive for the reaction products. Conservation laws in the system (balance of chemical elements) serve as limitations for the solution of problem 7 and can be represented by the following equations:

n = n 0 + ZTX

(12)

where μ is the N-dimensional vector with elements μi prescribed by the formula described by eq 8. Given that rows of matrix Z are linearly independent, i.e., only the stoichiometrically independent reactions are taken into account in the system, eq 13 gives NR coupling equations per NR unknown values of reaction coordinates Xeq satisfying the conditions of equilibrium. The equilibrium molar composition of the mixture neq at specified T and P values is determined unambiguously by substituting Xeq to eq 11, with the mole fractions of components in the equilibrium mixture being calculated as

(8)

where P is the total pressure, and Fi(T, P, y) is the fugacity coefficient accounting for a deviation in the properties of reaction medium from those of ideal gas/liquid. Assume that determinate stoichiometric reactions, represented by a general formula

Z×B=0

j=1

∂Gmix ∂Gmix ∂n = = Zμ = 0 ∂X ∂n ∂X

where μi° is the chemical potential of pure component i in its standard state, which coincides with the Gibbs energy Gi(T,P) of one mole of a pure substance calculated at the process temperature T and bearing pressure P = 1 atm. These values are commonly indicated in thermochemical handbooks, but their lack of data for fatty acids, triglycerides, and transesterification products makes it necessary to calculate the enthalpy and entropy using the Joback group method.34 The fugacity of the mixture component (f i(T, P, y)) is a function of temperature, pressure, molar composition of the mixture (vector y), critical parameters of individual components, and their acentric factors. It is calculated using the cubic Redlich−Kwong−Soave (RKS) equation of state.36 The fugacity of component i of the mixture is presented as a product: fi (T , P , y) = Pyi Fi(T , P , y)

∑ zjiXj) → min

Here, zji (j = 1, 2, ..., NR; i = 1, 2, ..., NS) are the coefficients of stoichiometric matrix Z. Conditions of equilibrium (minimum of the function Gmix (eq 12)) at specified T and P, in terms of variables X, are written as

where N is the number of chemical components in the reaction mixture, and μi (J/mol) is the chemical potential of component i. The chemical potential of components μi is determined by a relationship ⎛ ∂G ⎞ μi ≡ ⎜ mix ⎟ = μ°i (T , P = 1) + RT ln fi (T , P , y) ⎝ ∂ni ⎠

∑

μi(ni° +

i=1

i = 1, 2 , ..., N

i=1

NR

N

Gmix =

(15)

The first summand in eq 15 is the so-called “thermodynamic” equilibrium constant of the jth reaction KTj has a negative sign; it is dependent only on the temperature and nature of the chemical compounds: −ln KTj ≡

1 RT

ln KTj ≡ −

(11)

N

∑ z jiμi0

or

i=1

1 RT

N

∑ zjiΔGi f (T, P = 1) (16)

i=1

where

where n0 and n denote, respectively, N-dimensional vectors of initial and current molar numbers of the components. The new variables X = [X1, X2, ..., XNR]T are called the reaction coordinates.49 A corresponding value Xj is assigned to each

f

f

f

ΔGi (T , P = 1) = ΔHi (T , P = 1) − T ΔSi (T , P = 1) (17) 4787



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constants kj+ and kj− by the corresponding functions of concentrations of substances involved in the reaction:

The second summand in eq 15 determines the equilibrium constant KYj, expressed in terms of mole fractions of the components:

+ + − − − R+ j = k j φj (y)R j = k j φ j (y)

N

ln KYj =

∑ zji ln yi i=1

Here, kinetic constants are called fictitious because they are unknown and are prescribed arbitrarily. Their numerical values affect only the nature and length of “unsteady” section of the solution, which is not essential for the final solution. Apparently, in equilibrium state at t → ∞, derivatives in eq 21 vanish, and concentrations take on the stationary equilibrium values, which are determined by the initial conditions yi = yi0 at t = 0, other things being equal. Functions φj+(y) and φj−(y) in eq 22 are expressed by the law of mass action: φj+ = ∏iyi|zji| for all the components with stoichiometric coefficients zji < 0, and φj− = ∏iyizji for the components with stoichiometric coefficients zji > 0. Taking into account eq 22 and the relationship described by eq 20b, the dynamic problem takes the form

(18)

The third summand in eq 15 accounts for nonideality of the medium: N

ln KFj =

∑ zji ln Fi(T , P , y) i=1

(19)

The fourth summand, σj ln P accounts for the explicit effect of the total pressure on the equilibrium state (via Le Chatelier’s N principle), where σj = ∑i=l zji. In the above equations, zji (j = 1, 2, ..., NR, i = 1, 2, ..., NS) denotes the coefficients of stoichiometric matrix Z. According to 15 and taking into account eqs 16, 18, and 19, a ratio of different factors determining thermodynamic equilibrium of each reaction j can be presented as KYjKFjP σj KTj

dyi dt

=1

(20b)

In each particular case, eqs 20a and 20b make it possible to estimate the effect of temperature and pressure on the shift of chemical equilibrium. For the reactions proceeding without changes in the number of moles, σj = 0, pressure has no explicit effect on the equilibrium yield of products. In this case, the pressure effect manifests itself only implicitly, in terms of fugacity coefficients (see eq 19). The fugacity coefficients in compressed liquids and supercritical fluids are generally much less than unity, in distinction to “ideal” approximation, where Fi(T, P,y) = 1. Hence, according to eq 20b, other things being equal, nonideality of the reaction medium results in shifting the equilibrium toward the reaction products. 4.1. A Method for Solving the Mathematical Model Equations. In the general case, problem 15 can be solved by any appropriate method applicable to a set of nonlinear algebraic equations.49 However, the use of any well-known gradient method is limited, in some cases, by the so-called “rigidity” of a problem. The function described as eq 15 does not admit zero or negative (nonphysical) values of concentration in sequential steps of the iteration process, thus strongly complicating the iteration process by requirements to step adjustment. Zero values for initial approximations are also inadmissible. These difficulties took place when applying the known Newton gradient method.50 So, in the present work, the problem was solved by the adjustment method. The essence of the method consists of formulating and solving the so-called “dynamic” problem: dyi dt

∑ zji(R +j − R −j ) j=1

Rj+

∑ j=1

⎝

̃ j(T , P , y) × P σ ⎞ φ−j (y) × k Φ ⎟ + ⎟ φj (y) × KTj ⎠

5. CALCULATION OF CHEMICAL EQUILIBRIUM FOR THE SYNTHESIS OF FATTY ACID METHYL ESTERS Calculation of the phase equilibrium for complex mixtures describes their actual state with good accuracy, whereas calculation of the chemical equilibrium in similar systems is much less predictive. The reason is that chemical reactions rarely attain true equilibrium under real conditions, so the composition of products is determined to a great extent by kinetic factors, i.e., by the reaction rate. However, an increase in the reaction time, in some cases, provides higher conversions and brings together the theoretical and actual compositions of the products. Calculation of the equilibrium composition of a mixture is particularly important in the case of multistep reactions. Thus, for transesterification of fatty acid triglycerides with lower alcohols, it is essential to determine the values of equilibrium

NR

=

=

⎛

+⎜ zjik+ j φj ⎜1 −

The KTj constants are independent of current values of the concentrations. At each temperature, they are calculated by eq 16, using the thermodynamic data. The form of function K̃ Φj(y) is determined by the expressions given by eq 18, whereas their numerical values are determined by current nonequilibrium values of the concentrations yi(t). In an equilibrium state, when the current value of the vector becomes equal to y = yeq, the derivatives in 23, in virtue of eq 20a, will vanish. A set of equations is integrated numerically over the range of 0−tk at a specified temperature. The k+j values are prescribed arbitrarily; at a given k+j , the tk value is selected so that the sum N of derivatives ∑i=1 dyi/dt (eq 23) will vanish with a prescribed −10 accuracy (10 −10−15). This is the “stationary point” of the dynamic problem and the solution of the equilibrium problem. The right parts of the bracketed expression in eq 23 vanish. Evidently, the stationary point of dynamic problem coincides with the solution of problem 15, i.e., satisfies the condition of a minimum free Gibbs energy. Numerical integration of a set of eqs 23 was performed by the ROW4A method.51,52 Calculation of the chemical equilibrium for the chosen reactions is considered below.

KTj KFjP σj

NR

(23)

(20a)

or KYj =

(22)

(21)

Rj−

where and are the rates of elementary gross reactions, both direct and inverse, as the products of “fictitious” kinetic 4788



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calculated upon attaining an equilibrium (i.e., at y = yeq). They illustrate the effect of reaction mixture nonideality on the chemical equilibrium. According to formula 20b, the corresponding values of KY(T, P, y = yeq) are determined by the values of KY(T, P, y = yeq); so, it is evident that the concentrations of substances also vary along these curves. It is seen that, at the chosen values of P and T, nonideality has a positive effect on a shift of equilibrium toward the end products, since KY(T, P, y = yeq) < 1 and, hence, KY(T, P, y = yeq) > KT. The presence of minima on the curves in Figure 4 is explained by a complicated nonlinear dependence of the fugacity coefficient on the temperature and concentrations. To conclude this example, the equilibrium conversions in esterification of fatty acids by the scheme represented by eq 3, as functions of temperature, are presented in Figure 5. The

constants for all intermediate reactions, since the equilibrium constant of the overall reaction is only their product. If one or several constants of the intermediate reactions are much higher than others, this indicates a considerable content of intermediate compounds in the final equilibrium products of the reaction. 5.1. Calculation of Chemical Equilibrium for Fatty Acid Esterification with Methanol. Let us consider the esterification of three fatty acidsPFA, OFA, and LFAwith methanol, according to the scheme represented by eq 3. For these reactions, as seen from the phase diagram in Figure 1, the area of practical interest is the region on a T−P plane, where the pressure exceeds 10.1 MPa at a temperature varying in the range of 600−650 K. The equilibrium constants KT(T), as functions of temperature, were calculated by eq 16 for each of the reactions (see Figure 3). It is seen that the reversibility of

Figure 3. Equilibrium constants for the reactions of the scheme represented by eq 3.

Figure 5. Equilibrium conversion of initial substances in transformations 1−3 in the scheme represented by eq 3, as a function of temperature. P = 11.1 MPa. Numbering of the curves corresponds to that of the reactions.

the reactions increases with temperature, with PFA esterification being the most reversible among them. The values of KT(T) calculated in the ideal gas approximation are necessary to calculate the KY constant using the formula described by eq 20b. Figure 4 shows the values of coefficients KF(T, P, y) calculated at P = 11.2 MPa that are typical of this series of reactions. Values of KF(T, P, y) at each temperature T were

calculation was made for a stoichiometric composition of the initial mixture, taking into account its nonideality at P = 11.2 MPa. 5.2. Calculation of Chemical Equilibrium for SingleStep Transesterification of Simple Triglycerides of Fatty Acids with Methanol. Here, we consider the single-step transesterification of three triglycerides of fatty acidsPFATG, OFATG, and LFATG with methanol, according to transformation 1 in the scheme represented by eq 3. Similar to the first example, Figure 6 depicts temperature dependences of the equilibrium constant KT(T), calculated using eq 16 for each of the tested reactions. As follows from Figure 6, reversibility is most pronounced for the transesterification of LFATG. Equilibrium conversions of the chosen triglycerides versus temperature are shown in Figure 7. Calculation was made also for a stoichiometric composition of the initial mixture, taking into account its nonideality at P = 13.2 MPa. The following examples illustrate data on the effect of a methanol excess and total pressure on shifting the chemical equilibrium of a chosen reaction. The calculation was made for LFATG. 5.2.1. Effect of Methanol Excess. Temperature dependences of equilibrium conversion of the chosen triglyceride calculated at different ratios of methanol to initial triglyceride are shown in Figure 8. The calculation was made at a constant pressure (P = 13.2 MPa). As seen from the figure, at the reaction temperature below 625 K, an excess of methanol in the reaction

Figure 4. Coefficient KF(T, P, y = yeq) versus temperature. P = 11.1 MPa. Numbering of the curves corresponds to that of reactions in the scheme represented by eq 3. 4789



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superheated vapor, the reaction mixture acquiring properties of ideal gas with the constant KF ≅ 1. However, as noted above, this temperature region has no practical value for the process under consideration. 5.2.2. Effect of Total Pressure. For the reactions proceeding without changes in the number of moles (i.e., at σj = 0 (formula 20b)), pressure has no explicit effect on the equilibrium yield of products. Total pressure may affect a shift of the chemical equilibrium only in the case of pronounced nonideality of the system, leading to changes in the function KF(T, P, Y) on pressure. Thus, an increase in the pressure from 15.2 MPa to 40.5 MPa at the reaction temperatures above 640 K increases the conversion of chosen TG only by a few percent (Figure 9). Figure 6. Equilibrium constants for transformations 1−3 in the scheme represented by eq 3. Stoichiometric composition of the initial mixture.

Figure 9. Effect of total pressure on the equilibrium conversion versus temperature: Curve 1, 5.2 MPa; Curve 2, 20.3 MPa; Curve 3, 25.3, Curve 4, 30.4; and Curve 5, 40.5 MPa. Determined using transformation 3 in the scheme described by eq 3. Calculation was made at a methanol:LFATG ratio of 10:1.

At the reaction temperatures below 640 K, variation in pressure has no effect on the conversion of triglyceride. This result is essential for practical implementation of the reaction, because it allows one to choose the lower pressure limit with no decrease in conversion of the initial triglyceride. 5.3. Calculation of Chemical Equilibrium for Stepwise Transesterification of Simple Triglycerides of Fatty Acids with Methanol. As noted above, the transesterification of fatty acid triglycerides with lower alcohols generally proceeds via a three-step scheme of transformations via sequential substitution of one or two alkyl groups of fatty acid by hydroxyl groups with the formation of diglycerides and monoglycerides by the scheme described by eq 3. For simple triglycerides, all alkyl groups R1, R2, and R3 are identical and correspond to one of the fatty acids. This section of the work presents calculation data on the chemical equilibrium for the transesterification of simple triglycerides − PFATG, OFATG, and LFATG proceeding in three steps by the scheme described by eq 3. The equilibrium compositions for the stepwise transesterification of TG were calculated by a procedure thoroughly that has been described in Section 4.1. Note that equilibrium compositions were calculated for a stoichiometric ratio of ° = 1 and ymethanol ° = 3 in components of the initial mixture at yTG the ideal gas approximation, where the corresponding fugacity coefficients were taken to be equal to unity. Figures 10a, 10b, and 10c show temperature dependences of the equilibrium compositions for stepwise transesterification of TG calculated with the obtained constants. As follows from the

Figure 7. Equilibrium conversion of triglycerides versus temperature. P = 13.2 MPa. Lines 1−3 correspond to numbering of the transformations 1−3 in the scheme represented by eq 3.

medium provides almost 100% conversion of triglyceride in the equilibrium state. At higher temperatures, conversion starts to decline. Such behavior can be explained by methanol transformation at temperatures of >625 K into a strongly

Figure 8. Effect of methanol excess on the equilibrium conversion of TG in transformation 3 of the scheme described by eq 3 versus temperature. The CH3OH/TG ratios were 10/1 (Curve 1), 20/1 (Curve 2), 30/1 (Curve 3), and 40/1 (Curve 4). 4790



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calculation is made for a single-step reaction, monoglycerides and diglycerides are absent in the equilibrium composition of products, which gives an overrated yield of fatty acid esters. Note that calculation of the chemical equilibrium for stepwise transesterification of fatty acid triglycerides with supercritical methanol makes it possible to find the reaction conditions that provide a minimum yield of monoglycerides and diglycerides and a maximum yield of the target product. 5.4. Calculation of Chemical Equilibrium for Stepwise Transesterification of Mixed Triglycerides of Fatty Acids with Methanol. To examine the chemical equilibrium of stepwise transesterification with methanol, two mixed triglycerides were chosen: PPOFAG, which is comprised of two radicals of palmitic (R1) and one radical of oleic (R2) fatty acids, and OLLFAG, which has two radicals of linoleic (R1) and one radical of oleic (R2) fatty acids. A general trivial scheme of PPOFAG and OLLFAG transformations is presented in Figure 11.

Figure 11. Schematic of PPOFAG and OLLFAG transformations. A1− A5 represent complex triglycerides, diglycerides, and monoglycerides, whereas A6 denotes glycerol. (See eq 24 for the designations.)

A schematic of PPOFAG and OLLFAG transformations is shown as eq 24. Here, the arrows show the directions of reversible reactions. The composition of reactants for reaction Ai (i = 1−6) is indicated in square brackets: ⎡ R1 ⎤ ⎢ ⎥ A1 = ⎢ R1 ⎥ ⎢ ⎥ ⎣ R2 ⎦

Figure 10. Stepwise transesterification of TG. (a) Equilibrium composition of the reaction products in transesterification of PFATG versus temperature (1, PFATG; 2, PFADiG; 3, PFAMonoG; 4, PFAME; 5, methanol; 6, glycerol. (b) Equilibrium composition of the reaction products in transesterification of OFATG versus temperature (1, OFATG; 2, OFADiG; 3, OFAMonoG; 4, OFAME; 5, methanol; 6, glycerol). (c) Equilibrium composition of the reaction products in transesterification of LFATG versus temperature (1, LFATG; 2, LFADiG; 3, LFAMonoG; 4, LFAME; 5, methanol; 6, glycerol).

⎡ R1 ⎤ ⎡ R1 ⎤ ⎢ ⎥ ⎢ ⎥ A2 = ⎢ R1 ⎥ A3 = ⎢ R2 ⎥ ⎢⎣ ⎥ ⎢⎣ ⎥ OH ⎦ OH ⎦

⎡ R1 ⎤ ⎡OH ⎤ ⎡OH ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ A 4 = ⎢OH ⎥ A5 = ⎢OH ⎥ A 6 = ⎢OH ⎥ ⎢⎣OH ⎥⎦ ⎢⎣OH ⎥⎦ ⎢⎣ R2 ⎥⎦

(24)

Here, A1−A5 are the corresponding complex triglycerides, diglycerides, and monoglycerides, whereas A6 is glycerol. For convenience, stoichiometric reactions of TG transformations can be presented as follows:

calculated chemical equilibrium of stepwise transesterification of TG (see Figures 10a, 10b, and 10c), at low temperatures (up to ∼450 K), a complete conversion of all TG takes place in all cases. Intermediate products of the reaction are the corresponding monoglycerides and diglycerides. In the case of LFATG, at low temperatures, the intermediate product is represented mainly by LFADiG, whereas at high temperatures, there remains a substantial amount of nonconverted LFATG, together with LFAMonoG. A comparison of the composition of products obtained in approximation to the single-step and stepwise transesterification of the same TG with methanol clearly demonstrates the necessity of calculating the chemical equilibrium of this reaction by the stepwise scheme of TG transformations. When

1. A1 + methanol ⇆ A2 + R2 ester 2. A1 + methanol ⇆ A3 + R1 ester 3. A2 + methanol ⇆ A 4 + R1 ester 4. A3 + methanol ⇆ A 4 + R2 ester 5. A3 + methanol ⇆ A5 + R1 ester 6. A 4 + methanol ⇆ A 6 + R1 ester 7. A5 + methanol ⇆ A 6 + R2 ester

(25)

The list of components for the system described by eqs 25 includes nine substances (NS = 9) involved in seven reactions (NR = 7). 4791



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characteristic temperature dependences of ln KT,j are wellapproximated by the third-order polynomials:

In distinction to simple transformation schemes, in complex schemes, the equilibrium constants of multiroute reactions may be inter-related. Thus, the transformation scheme in Figure 11 consists of two closed cycles: the first cycle being formed by reactions 1−4 in the system described by eqs 25, and the second one by reactions 4−7 in the same system (eqs 25). Reaction 4 is a “shared edge” of the two cycles. Because of the existence of the closed cycles in the scheme, rows of the stoichiometric matrix written for the reactions are not linearly independent. Hence, the equilibrium constants of the reaction are also not independent; there are some relationships between them. Revealing such relationships, one can decrease the number of routes in a chosen reaction scheme when calculating the chemical equilibrium and retain the list of components as well as their equilibrium distribution. Such relationships can be found using the known formalism reported in ref 53. To find an explicit relationship between equilibrium constants Ky,j, a complete set of linearly independent solutions for a system of linear equations is needed:53 x×Z=0

ln KT,j = a0, j + a1, jT + a2, jT 2 + a3, jT 3

Coefficients of the polynomials given in eq 32, found using the Forsythe method, are listed in Table 5 of the Supporting Information. The equilibrium composition was calculated using reduced versions of PPOFAG and OLLFAG transformations, where reaction 4 of the reaction set described by eqs 25 was neglected. Thus, the number of reactions diminishes by one, to give NR = 6. Numbering of the reactions shown in Figure 11 remains the same. 5.4.2. Reaction Coordinates. Essential information about the completeness of the reactions under equilibrium conditions can be obtained from the reaction coordinates X jeq, calculated at a point of detailed equilibrium at specified temperature and pressure. Material balance equations can be represented by a set of linear algebraic equations in matrix form:

(26)

n eq − n 0 = ZTX eq

where x = [x1, x2, ..., xNR] is the row vector of desired solutions (eq 26) with the dimension NR = 7; Z is the stoichiometric matrix constructed for reactions 1−7 constituting eq 25 with the dimension (NR × NS). A solution set for eq 26 is unique to an accuracy of a constant factor. As a result, for cycles 1, 2, 3, and 4, the following was found: x1 = 1, x2 = −1, x3 = 1, x4 = −1. Thus, for a cycle of reactions 1, 2, 3, and 4 from eqs 25, ln KY,1 − ln KY,2 + ln KY,3 − ln KY,4 = 0

(27)

KY,1KY,3 KY,2

(28)

By analogy, for a cycle of reactions 4, 5, 6, and 7 from eqs 25, KY,4 =

KY,5KY,7 KY,6

(29)

From eqs 28 and 29, it follows that, in the further calculation, reaction 4 can be eliminated from the reaction set given as eqs 25. Elimination of this reaction will have no effect on the equilibrium concentrations of components. From eqs 27 and 28, it follows also that KY, Σ = KY,1KY,3KY,6 = KY,2KY,5KY,7

(30)

According to eq 30, in the calculation of the single-step transformation of initial mixed triglyceride, for example, PPOFAG + 3Methanol ⇆ 2PFAME + OFAME + glycerol

(33)

where ZT is the transposed stoichiometric matrix Z with dimensions (NS × NR), NR = 6 after eliminating the fourth reaction from eqs 25; Xeq = [X1eq, X 1eq , ...,XNeqR ]T. A solution set for eq 33 for unknown Xeq can be obtained easily if the number of rows in matrix ZT is limited by the number of “key” components (NSK = 6) denoted in the scheme of Figure 11, with methanol and esters of fatty acids being excluded from the list. Thus, a reduced version of ZT with dimensions (NSK × NR) is a nonsingular square matrix. The solution was obtained by the method using the LU decomposition of matrix ZT.49 5.4.3. Effect of Temperature on the Chemical Equilibrium of Transesterification of Mixed Triglycerides. The temperature effect on the chemical equilibrium of chosen reactions was studied under ideal and nonideal conditions at a stoichiometric composition of the initial mixture TG:CH3OH = 1:3. 5.4.4. Transformations of PPOFAG. Temperature dependences of the reaction coordinates in equilibrium and equilibrium concentrations in the ideal approximation are presented in Figures 12a and 12b, respectively. As follows from Figures 12a and 12b, PPOFAG is converted virtually completely over the entire temperature range. A sum of coordinates for reactions 1 and 2 is close to unity, which indicates a complete conversion of the initial PPOFAG. At low temperatures (300−350 K), using two parallel routes (1, 3, and 6, and 2, 5, and 7 from eqs 25), the reaction proceeds mainly by the first route. The reaction runs almost to completion: a theoretical 75% yield of the target products OFAME + PFAME is attained, with accumulation of a small amount of PFAMonoG. At elevated temperature, routes 2 and 5 become more pronounced, whereas route 1 diminishes, reaching a small maximum. At a maximum temperature of 650 K, a substantial buildup of monoglycerides is observed. 5.4.5. Transformations of OLLFAG. The calculation data obtained for transformations of OLLFAG are presented in a similar form (see Figures 13a and 13b). It is seen that the reaction follows mainly the route 1, 3, 6 over the entire temperature range. OLLFAG is converted almost completely. At low temperatures, the predominant formation of LLFADiG is observed. As the temperature increases, the coordinate of

or

KY,4 =

(32)

(31)

the total equilibrium constant of reaction 31 is not dependent on a real reaction route, because it gives no information on the intermediate products. 5.4.1. Calculation of Equilibrium Constants, Equilibrium Composition, and Reaction Coordinates. The equilibrium constants KT,j in the ideal approximation over the temperature range of 300−650 K were calculated by eq 16. The 4792



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Figure 12. Transformations of PPOFAG. (a) Temperature dependence of the reaction coordinates (numbering of the curves corresponds to that for the reactions of the scheme described in Figure 11. (Reaction 4 was excluded from the list of reactions.) (b) Temperature dependence of the equilibrium concentrations (Curve 1, PPOFAG; Curve 2, PPFADiG; Curve 3, POFADiG; Curve 4, PFAMonoG; Curve 5, OFAMonoG; Curve 6, OFAME; Curve 7, PFAME; and Curve 8, glycerol.

Figure 13. (a) Transformations of OLLFAG. Temperature dependence of the reaction coordinates. Numbering of the curves corresponds to that for the reactions in the scheme described in Figure 11. (Reaction 4 was excluded from the list of reactions.) (b) Temperature dependence of the equilibrium concentrations. Curve 2, LLFADiG; Curve 3, LOFADiG; Curve 4, LFAMonoG; Curve 5, OFAMonoG; Curve 6, OFAME; Curve 7 , LFAME; and Curve 8, glycerol.

reaction 3, leading to the formation of LFAMonoG, sharply increases. Accordingly, the equilibrium concentration of LFAMonoG and LFAME also increases. The concentrations of LFAME has a stable value of 25% over the entire temperature range, because of the predominant occurrence of the reaction via the route 1, 3, 6. 5.5. Calculation of Chemical Equilibrium for Transesterification of Mixed Triglycerides of Fatty Acids with Supercritical Methanol. Supercritical conditions are characterized by the critical temperature, critical pressure, and concentration of initial reactant in a solvent that creates a supercritical medium. In this case, solvent can be represented by methanol with the critical parameters Tcr = 512.6 K and Pcr = 8.1 MPa. At a large excess of methanol, critical parameters of the reaction mixture should be close to the critical parameters of methanol. To select a region of temperatures and pressures at which the reaction mixture can pass to a supercritical state, we calculated critical parameters of the initial reaction mixtures comprising PPOFAG and methanol (mixture No. 1) or OLLFAG and methanol (mixture No. 2), respectively, at different methanol concentrations in the initial mixture. The calculation was made using the method reported in our earlier work.54 Results are presented in Table 4.

According to Table 4, as the methanol concentration in the reaction mixture increases, its critical parameters approach those of pure methanol. The range of operating temperatures and pressures for both mixtures were chosen at a specified methanol/triglyceride ratio (equal to 50:1). For mixture No. 1, P = 10.1 MPa and T = 520−600 K. This corresponds to the reduced parameters Pred = P/Pcr = 1.17 and Tred = T/Tcr = 0.98−1.09 K. Respectively, for mixture No. 2, we chose P = 12.1 MPa, T = 560−620 K, Pred = 1.11, and Tred = 0.97−1.07 K. When the process is performed under supercritical conditions, one may expect a strong effect of reaction mixture nonideality on the shift of equilibrium and the equilibrium distribution of the reaction products. The reaction completeness can be characterized by the yield of glycerol. Theoretically, a maximum yield of glycerol after the removal of an methanol of excess is equal to 25%. Such yield implies also a maximum yield of the total esters (75%) with the 2:1 ratios of PFAME:OFAME and LFAME:OFAME, respectively, for the transformations of PPOFAG and OLLFAG according to the scheme described in Figure 11. The calculation results are presented in Figures 14 and 15, for transformations of PPOFAG and OLLFAG, respectively. 4793



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Table 4. Calculated Critical Parameters of the Reaction Mixtures Comprising PPOFAG and Methanol (Mixture No. 1) or OLLFAG and Methanol (Mixture No. 2) Mixture No. 1

Mixture No. 2

CH3OH/TG ratio [mol/mol]

CH3OH concentration [mol %]

Tcr [K]

Pcr [MPa]

Tcr [K]

Pcr [MPa]

20:1 30:1 40:1 50:1 60:1

95.2 96.8 97.6 98.0 98.4

569.9 548.4 537.7 532.5 527.4

9.63 9.14 8.83 8.65 8.48

683.3 624.8 593.4 577.4 562.1

12.77 12.22 11.44 10.91 10.31

Only the substances whose concentrations exceed 2% are depicted here. As seen from Figure 14, when transformations of PPOFAG occur under supercritical conditions and in an excess of

Figure 15. Transformations of OLLFAG. Temperature dependence of the equilibrium yield of reaction products after the removal of methanol excess from the products. P = 12.2 MPa. (Curve 1, LFAMonoG; Curve 2, OFAME; Curve 3, LFAME; and Curve 4, glycerol.) Figure 14. Transformations of PPOFAG. Temperature dependence of the equilibrium yield of reaction products after the removal of methanol excess from the products. P = 10.1 MPa. Curve 1, OFAME; Curve 2, PFAME; and Curve 3, glycerol.

temperature, but they are always much smaller than unity. This explains a substantial shift of the equilibrium toward an increased yield of the end target products.

methanol, the equilibrium of the reaction strongly shifts toward the end-target products. The reaction runs mainly via route 1, 3, 6. The reaction coordinates are as follows: X1 = 0.852 and X2 = 0.148. The yields of OFAME and PFAME are virtually equal to the maximum values, 25% and 50%, respectively. Only trace amounts of intermediate products are detected in the reaction mixture. The transformations of OLLFAG look quite different (see Figure 15). Similar to the previous case, here the reaction also runs mainly via route 1, 3, 6. The reaction coordinates are X1 = 0.855 and X2 = 0.145. The initial substance is converted completely. However, reaction 6 remains strongly reversible, and the concentration of LFAMonoG increases with temperature. Accordingly, the yield of LFAME target product decreases. To conclude, let us illustrate the effect of reaction mixture nonideality on the chemical equilibrium using the equilibrium constants KF(T, P, yeq). As noted above (formula 20b), these constants can significantly alter the KY values. At KF(T, P, yeq) < 1, which is typical of supercritical conditions, equilibrium is shifted toward the target products. Nontrivial behavior of the equilibrium constants KF(T, P, yeq) can be explained by the diversity of critical parameters and acentric factor of individual components, as well as by the dependence of these constants on the equilibrium composition of the mixture. Note that KF(T, P, yeq) values increase with

6. CONCLUSION Using the authors’-modified Joback group method, thermodynamic characteristics (critical parameters, acentricity factor, entropy, enthalpy, thermal capacity) were calculated for practically all fatty acid esters, monoglycerides and diglycerides, simple and mixed triglycerides, free fatty acids involved in transesterification reactions of vegetable oils; respective databases were accumulated. Based on the methods and algorithms developed, investigation of phase equilibrium of the mixtures containing triglycerides and lower alcohols, along with the reaction products of triglyceride transesterificationglycerol and esters of fatty acidsmade it possible to choose the reaction conditions for various lower alcohols, providing the supercritical state of thereaction mixture. Chemical equilibrium at transesterification with methanol of simple and mixed triglycerides containing radicals of palmitic, oleic, and linoleic acids in a wide range of temperatures, pressures, and initial compositions has been studied in detail. The calculations of chemical equilibrium allowed predicting reaction behavior both with ideal approximation and with an accounting for nonideality, particularly in the supercritical region of the reaction mixture. Calculations allowed us to find the reaction conditions that give a minimum yield of monoglycerides and diglycerides and a maximum yield of biodiesel as the target product. 4794



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ASSOCIATED CONTENT

S Supporting Information *

This information is available free of charge via the Internet at http://pubs.acs.org/.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS

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REFERENCES

The authors are grateful to the Russian Foundation for Basic Research for financial support, Project No. 10-08-00282.

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