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Equation of state modeling for the vapor pressure of biodiesel fuels
Accepted Manuscript Title: EQUATION OF STATE MODELING FOR THE VAPOUR PRESSURE OF BIODIESEL FUELS Author: O.Castellanos D´ıaz F. Schoeggl H.W. Yarranton M.A. Satyro PII: DOI: Reference:
S0378-3812(15)00004-7 http://dx.doi.org/doi:10.1016/j.fluid.2014.12.050 FLUID 10410
To appear in:
Fluid Phase Equilibria
Received date: Revised date: Accepted date:
14-5-2014 11-12-2014 31-12-2014
Please cite this article as: O.Castellanos D´iaz, F.Schoeggl, H.W.Yarranton, M.A.Satyro, EQUATION OF STATE MODELING FOR THE VAPOUR PRESSURE OF BIODIESEL FUELS, Fluid Phase Equilibria http://dx.doi.org/10.1016/j.fluid.2014.12.050 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.
EQUATION OF STATE MODELING FOR THE VAPOUR PRESSURE OF BIODIESEL FUELS O. Castellanos Díaz, F. Schoeggl, and H. W. Yarranton Department of Chemical and Petroleum Engineering, University of Calgary, AB, Canada M. A. Satyro Virtual Materials Group, Calgary, AB, Canada Abstract The vapour pressure and liquid heat capacity of seven biodiesel fuels and its components (fatty acid methyl esters - FAMEs) were modeled using the Advance Peng Robinson
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equation of state. The dataset used for the modeling was obtained from the literature and
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included FAME properties and the composition, vapour pressure, and liquid heat capacity of the biodiesel fuels from different sources at temperatures from 50 to 130ºC and -30 to
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75ºC, respectively. New values for the critical properties and acentric factor of FAMEs are introduced as well as new models for the ideal gas heat capacity for the FAMEs. The
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average AARD is 12% for vapour pressure and 3% for liquid heat capacity.
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Keywords: biodiesel fuels, fatty acid methyl ester, vapour pressure, heat capacity, cubic
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INTRODUCTION
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equation of state modeling
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Biodiesel fuels constitute one of the most promising alternatives to supplement or reduce petroleum diesel (petro-diesel) usage. They are renewable, non-mutagenic, non-
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carcinogenic, biodegradable fuels that can be domestically produced. Biodiesel fuels are refined mixtures of esters produced by the transesterification of fatty acids from
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vegetable oil and animal fat (fatty acid methyl esters or FAMEs for short). These fuels can be used directly or blended with petro-diesel to improve lubricity without adding any
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sulfur. These fuels may also improve engine firing due to their oxygen content [1-6].
The measurement and prediction of bio-diesel properties is required for their effective commercial use. One important property for the quality control of biodiesel fuels and their blends is volatility, which is directly related to the vapour pressure of their
individual constituent compounds [2]. For instance, vapour pressure is used to calculate the latent heat in order to compare rates of vapourization and injection characteristics with other fuels. Heat capacity is another important thermo-physical property used to characterize biodiesel for fuel blending with petro-diesel and is directly related to the vapour pressure.
Yuan et al. [7] modeled the vapour pressure data of three different biodiesel fuels at temperatures above 215°C [1,2]. They assumed that biodiesel fuels were ideal mixtures
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of their constituent FAMEs. They fitted the vapour pressure of the individual FAMEs with the Antoine equation and predicted the vapour pressures of the biodiesel fuels using
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Raoult's law. However, the prediction of vapour pressures at lower temperatures requires extrapolation using the fitted Antoine equations and the accuracy of the extrapolation is
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unknown.
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Castellanos-Diaz et al. [8] modeled the vapour pressure of several biodiesel fuels at
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temperatures from 50 to 130°C using Raoult’s law. They modeled the FAMEs vapour
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pressure using the Cox equation and constrained the equation parameters to also match
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the liquid heat capacities. They demonstrated that more accurate vapour pressure fits were obtained when calorimetric data are included in the optimization function used to
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determine the vapour pressure equation parameters. They also proposed new equations
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for calculating the vapour pressure and liquid heat capacities of FAMEs.
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Experimental data for the liquid heat capacity of biodiesel fuels are also scarce and, in most cases, are available only for specific biodiesel fuel sources. Conceicao et al. [3]
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presented liquid heat capacity data from thermo-gravimetric methods for castor oil-based biodiesel fuel at temperatures between 55 and 125°C. The same methodology was
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applied by Narvaez et al. [9] for palm oil-based biodiesel fuel at temperatures ranging from 30 to 100°C. Dzida and Prusakiewicz [10] measured the liquid heat capacity of rapeseed oil-based biodiesel fuel using differential scanning calorimetry at temperatures between 20 and 45°C. No modeling was included in any of these cases.
The model developed by Castellanos et al. [8] based on vapour pressure fitting using relatively simple equations is sufficient for calculations where only vapour pressure, enthalpy of vapourization and liquid heat capacity values are required. However, a more comprehensive approach is to model vapour pressure of FAMEs and biodiesel fuels using an equation of state (EoS), since an EoS can also be used for phase equilibrium calculations in processes with additional components and/or processing stages and also provides a consistent structure to introduce pressure effects on phase equilibrium calculations. An equation of state model is also more convenient for most commercial
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process simulation software.
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The purpose of this study is to develop an equation of state (EoS) model for the representation of vapour pressures and liquid heat capacities of fatty acid methyl esters
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and biodiesel fuels. The biodiesels were characterized as mixtures of FAMEs with a known composition. The first step was to characterize the FAMEs. Their critical
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properties were determined from literature data or correlations and then tuned to fit the
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equation of state model to measured FAME vapour pressures and enthalpies. Next, the
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ideal heat capacities of the FAMEs were calculated and the liquid heat capacity of
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FAMEs predicted. Finally, the vapour pressure and liquid heat capacity of biodiesels was determined. The vapour pressure and heat capacity datasets used to develop the model
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were taken from the literature and are summarized below. The equation of state model
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and optimization methodology are then presented followed by a discussion of the results.
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DATASETS
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2,1 FAMEs Dataset Eighteen FAMEs ranging in carbon number from 6 to 22 were examined, as presented in Table 1. The vapour pressure dataset [11] ranges in temperature from 25 to 300°C
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whereas liquid heat capacity data range from the FAME freezing point to 50 °C. Data that were not available from the database were predicted using the models developed by Castellanos-Diaz et al [8].
2 . 2 B i o d i e s e l F u e ls D a t a s e t The composition, liquid heat capacity, and vapour pressure data of seven biodiesel fuels were reported previously by Castellanos-Diaz et al. [8]. Briefly, the compositions were determined using GC and mass spectrometry [12-15]. The accuracy of the compositions based on standards is approximately ±2 wt%. Liquid heat capacities were measured using a differential scanning calorimeter (DSC) TA Q2000 V24.9 calibrated against indium with an average accuracy of ±2% [16]. Vapour pressures were measured using an inhouse static flash test apparatus at pressures ranging from 0.1 to 200 Pa. The average
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95% confidence interval for the measured vapour pressure data is ±13% [17]. Table 2 lists the seven biodiesels and the temperature range of their vapour pressure and heat
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capacity data. Table 3 summarizes the compositions of the biodiesels. FAMEs ranging from C6:0 to C20:1 were identified; with the exception of sample S070717, the majority
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of each fuel was composed of C18:0, C18:1, and C18:2.
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EQUATION OF STATE MODELING
The Advanced Peng-Robinson Equation of State (APR-EoS) [18-19] was used to
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calculate the vapour pressure and the heat capacity of the biodiesel fuels listed in Table 2.
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First, the FAMEs were modeled as pure components and then the biodiesel fuels were modeled as a mixture of the FAMEs. The APR-EoS is a modified version of the Peng-
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Robinson EoS that accounts for more recent experimental data in the -function and
is given by:
RT a (TR , ) v b v (v b) b(v b)
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P
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includes volume translation to improve on liquid density predictions [19]. The APR-EoS
[1]
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where P is the pressure, T is the absolute temperature, v is the molar volume, R is the universal gas constant, a and b are constants related to the attractive and repulsive forces,
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TR is the reduced temperature (T/TC), TC is the critical temperature, ω is the acentric factor, and α is a function that adjusts the EoS to vapor pressure experimental data. To facilitate the discussion below, recall that the APR-EoS can be expressed in terms of the compressibility factor as follows [20]:
Z 3 Z 2 1 B Z A 2B 3B 2 B A B B 2 0, [2]
where Z is the compressibility factor. A and B are defined by:
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a , TR P
RT 2
, [3]
bP , RT
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B
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[4]
where T is temperature, P is pressure, R is the universal gas constant, is the acentric
0.0777969 RTC ,i PC ,i
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,
,
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bi
PC ,i
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ai
0.457265 R 2TC2,i
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factor, TR is the reduced temperature, and a and b for any component i are given by:
[5]
[6]
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where TC and PC are the critical temperature and pressure, respectively. The α-function
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for the Peng-Robinson EoS is given by [21]:
2
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, TR 1 fW 1 TR0.5 ,
where fw is given by:
[7]
fW 0.37464 1.54226 0.26992 2 , 0.5
[8]
fW 0.3796 1.485 0.1644 2 0.01667 3 , 0.5
[9]
Generally, an EoS is tuned to fit vapour pressure data of a pure substance by adjusting the α-function, which depends on the acentric factor. For components where the critical properties are not known, these properties and/or the acentric factor may also be adjusted. Once the vapour pressure of the pure components is fitted to the EoS, the vapour pressure
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work, the classic Van der Waals mixing rules were used, given by:
bm i xi bi ,
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[10]
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am j i xi x j ai i a j j 1 kij ,
[11]
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of a mixture of these components can be modeled using appropriate mixing rules. In this
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Tuning of the interaction parameters used in the mixing rule may be required if the
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mixture exhibits non-ideal mixing behavior.
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3 . 1 F A M E s V a p o u r P re s s u r e
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Since vapour pressure and enthalpy are related through the Clapeyron equation and both can be determined directly from an equation of state, the tuning of the acentric factor and
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critical properties can be constrained through the simultaneous use of vapour pressure and enthalpy of vapourization data where available. The vapour pressure is an
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equilibrium property and, for a pure component, is determined through the equality of the fugacities of the component in the liquid and vapour phases [21]. The fugacity
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coefficients for each phase are calculated from the Peng-Robinson equation of state as follows [20]:
Z f Phase A 2.414 B ln Z Phase 1 ln( Z Phase B) ln Phase 2.828 B Z Phase 0.414 B P
[12]
The enthalpy of vapourization, HV, is simply the difference between the enthalpy of the liquid and vapour phases where, for the PR EoS, the enthalpy of each phase is given by:
2 Z j B (2 8) 1 da ln a T RT ( Z j 1), dT 8 B 2 Z j B (2 8)
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hj
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[13]
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where j stands for the phase at equilibrium.
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The optimization function used in this study is given by:
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OF i (ln Pi exp ln Pi calc ) 2 K H i ( H iexp H icalc ) 2 ,
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[14]
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where OF is the magnitude of the optimization function, KH is a weight factor used to
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scale the contribution of each variable to a similar magnitude; a value of 0.001 was found reasonable to balance the contribution of vapour pressure and enthalpy of vapourization
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data when calculating the OF. The enthalpy of vapourization in Equation 14 is in kJ/kg. Two strategies were used to minimize OF: 1) adjust only the acentric factor; 2) adjust the
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acentric factor and the critical properties. The procedure for both cases is the same and is shown in Figure 1. The vapour pressure, acentric factor, and critical properties were
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adjusted with the generalized reduced gradient algorithm in Excel Solver; in general, convergence was achieved within a few iterations. 3.2 FAMEs Heat Capa cit y The liquid heat capacity for each FAME was calculated as follows:
CP , L C PRe s C PV C PRe s C Po
[15]
where CP,L is the liquid heat capacity, CPV and CP° are the vapour and ideal gas heat capacity, respectively, and CPres is the residual heat capacity which can be determined from the PR EoS as follows:
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CPRe s
2 R da 1 1 d 2 a v b 1 2 v b dT v 2 2bv b 2 R, T ln 2 2a v b RT v b 1 2 b 8 dT 2 2 2 2 v b v 2bv b
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[16]
In order to complete the heat capacity calculations ideal gas heat capacities are required.
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The development of an ideal gas heat capacity correlation for FAMES is discussed later.
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3 . 3 B i o d i e s e l F u e ls
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Biodiesel fuels are simply a mixture of FAMEs and the EoS parameters for the mixture were determined from the van der Waals mixing rules. The vapour pressure was then
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determined by equating the fugacity of each phase as described for the FAMEs. The
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liquid heat capacity was determined from Equation 15. In this case, the residual heat capacity of the mixture was calculated from Equation 16 with the EOS mixture
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parameters. The ideal gas heat capacity of the mixture was calculated as follows:
[17]
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CP0 , Biodiesel i xi CP0 , FAME i ,
The binary interaction parameters in the van der Waals mixing rule, Equation 11, were initially set to zero and the model is predictive. If necessary, the model can be tuned by adjusting the binary interaction parameters based on experimental data. In this work, the
Gao et al. equation [22] was used to correlate the interaction parameters to critical temperatures as follows:
n
2 T T 0.5 ci cj , 1 kij Tci Tcj
[18]
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The exponent n in Equation 18 is the only adjustable parameter. Equation 14 was again
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used as the optimization function.
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RESULTS AND DISCUSSION
Recall that the following steps were required to complete the equation of state model: 1)
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determine FAME critical properties; 2) tune the predicted critical properties to fit the
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equation of state model to FAME vapour pressures and enthalpies; 3) determine FAME
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ideal heat capacities; 4) predict liquid heat capacity of FAMEs; 5) predict vapour
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pressure and liquid heat capacity of biodiesels. Each step is discussed below.
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4 . 1 F A M E s C r it ic a l P r o p e rt i e s Experimental critical properties for the majority of fatty acid methyl esters do not exist
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because the hypothetical critical temperature for most FAMEs is above the thermal cracking temperature. Therefore, both acentric factor and critical property tuning are
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considered for the FAMEs.
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As recommended by NIST [11], the hypothetical critical pressure, Pc, of the FAMEs was estimated using the Wilson-Jasperson and the Ambrose-Walton vapour pressure based
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methods. The hypothetical critical volume, Vc, and critical temperature, Tc were estimated using Joback’s group contribution method [20]. A group contribution method was considered appropriate because FAMEs consist of well known molecular groups including CH3-, -CH2-, -CH=, and COO-, and this method was also recommended and used by NIST’s TDE [11]. Initial estimates for the FAMEs acentric factors were
calculated from the vapour pressure curve (when available), using Pitzer’s definition [20] given by:
P Pitzer log10 v 1, PC AtT 0.7T C
[19]
where ωPitzer is the Pitzer’s acentric factor. The estimated acentric factors and critical
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properties for the FAMEs are shown in Table 4.
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4 . 2 F A M E s V a p o u r P re s s u r e
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Once the critical properties and acentric factors were estimated, the vapour pressure and enthalpy of vapourization of FAMEs were calculated using the APR-EoS. Table 5 shows
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deviation values for this approach ("No Tuning") and Figure 2 shows the results for
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methyl caprate, C10:0, as an example. The performance of the APR-EoS with initial critical values for saturated FAMEs is satisfactory, with an AARD of 36% over all
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FAMEs. The relatively high deviations for FAMEs are within the uncertainty of the
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experimental values at low temperatures. For instance, Genderen at al. [23] and NIST [11] report an average overall deviation of 5% for the vapor pressures of methyl caprilate;
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however, the error is only 2% near the normal boiling point but increases monotonically
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as the temperature decreases reaching values of 23% at the lowest temperatures (from -10 to 20 ºC, depending on the FAME). On the other hand, the performance of the APR-EoS
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with initial critical values for unsaturated FAMEs is poor, with an average AARD of 350% over all FAMEs. The errors may be higher for unsaturated FAMEs because their
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vapour pressures are lower than the saturated FAMEs at the same temperature. Another, more likely, explanation is that the critical property correlations shown in Table 4 are not
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appropriate for unsaturated FAMEs.
The performance of the EoS model was improved by adjusting the critical properties and acentric factor. The adjusted values are provided in Table 4. Table 5 shows deviations for vapour pressure and enthalpy of vapourization when only the acentric factor was
modified (“Tuning ω”) and when the acentric factor and the critical properties were tuned simultaneously (“Tuning Tc, Pc, ω"). Figure 2 shows the results for methyl caprylate (C10:0). The adjusted values of the critical properties and acentric factor do not differ significantly from the initial estimates, except for the unsaturated FAMEs. Note how the acentric factor, calculated from the available vapour pressure data and an unbounded extrapolation of the vapour pressure curve, increases considerably as the number of unsaturated carbon molecules is higher, Table 4. This increase may not reflect differences in the molecular structure but rather may be artifacts resulting from insufficient vapour
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pressure data at higher pressures, closer to a reduced temperature of 0.7 where the acentric factor is calculated, Equation 19. In any case, the changes in , Tc and/or Pc
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significantly improve the predicted values for the vapour pressure and the enthalpy of vapourization. The AARD for the vapour pressures of the FAMEs, including
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undersaturated FAMEs, are all lower than 30% when the tuned parameters are used., with
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values as low as 2%.
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4 . 3 F A M E s I d e a l G a s H e a t C a p a c i t y a n d L i q u i d H e a t C a p a c it y
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After the critical properties and acentric factor of the FAME were adjusted based on vapour pressure data, the heat capacity residual was calculated from the equation of state.
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As noted previously, the ideal gas heat capacity, CP0, is required to determine the liquid heat capacity. Since no experimental data for CP0 are available, correlations were
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developed for saturated and unsaturated FAMEs. The new correlations were compared with two well known predictive methods: the Joback method [20] and a modification of
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the Benson method [24].
Saturated FAMEs For each saturated FAME with available liquid heat capacity data, the ideal gas heat
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capacity was calculated as follows:
s CP0 , FAME C P , L , FAME C PRe, FAME ,
[20]
The ideal gas heat capacity of each of these FAMEs was regressed with a second degree polynomial:
CP0 (0) aCp 0 bCp 0 T cCp0 T 2 ,
[21]
where a Cp°, bCp°, and cCp° are constants. For example, Figure 3 shows the calculated and
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regressed ideal gas heat capacity values for methyl caprylate, C10:0.
To generalize Equation 21 for all saturated FAMEs, the constants for the set of FAMEs
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were plotted as a function of the molecular weight, Figure 4, and fitted as with the
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2.1079 x10 4 230.72 0.62516 MW 344.1759 MW 344.1759
,
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aCp 0
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following expressions:
[22]
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125.93 1.4529 5.5554 x103 MW 344.1759 MW 344.1759
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bCp 0
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0.19127 8.8626 x10 4 5.9999 x106 MW 344.1759 MW 344.1759
,
[24]
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cCp 0
[23]
Note that in developing Equations 22 to 24, outliers were not considered as can be
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observed in Figure 4.
Unsaturated FAMEs
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No data were available for C p0 or C L of unsaturated FAMEs and hence the modeling approach for the saturated FAMEs was not possible. Instead, the ideal gas heat capacities for the unsaturated FAMEs were determined from departure functions where the corresponding saturated FAME of the same carbon number is the reference fluid. The
departure function was calculated using Joback’s method [20]. After inserting the appropriate constants for the biodiesel fuel compositions, the Joback method can be written as follows:
CP0 NUC
1 NUC 6.1327 x102 1.5493 x10 4 T 1.842 x10 7 T 2 , C (0) 0 P
[25]
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where NUC is the number of unsaturated carbon molecules. The generalized liquid heat capacity model for FAMEs (Equations 21, and 25) was compared with the original
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Joback method and a modification of the Benson method [24]. Figure 5 shows the
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predicted liquid heat capacity of methyl caprilate, C10:0, using the APR-EoS and Equation 6. Table 6 shows the results for all of the FAMEs considered. The Joback and
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modified Benson methods provided ideal heat capacities that in conjunction with the residual heat capacity calculated with the APR EoS systematically under-predicted the
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APR-EoS, correct the under-prediction.
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liquid heat capacity of the FAMEs. The methods presented in this work, applied to the
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4 . 4 B i o d i e s e l F u e l V a p o u r P re s s u re
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The APR-EoS using the tuned critical properties, acentric factor, and the proposed method to estimate ideal gas heat capacities presented in the previous section for FAMEs
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was used to calculate both the vapour pressure and the heat capacity of the biodiesel fuels listed in Table 1. The classic van der Walls mixing rules were used; initially the
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interaction parameters, kij, were set to zero. Figures 6 and 7 show the measured and modeled vapour pressure and heat capacity, respectively, for all of the biodiesel fuels
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studied. The AARDs for the model are given in Table 7. The average AARD is 12% for vapour pressure and 3% for liquid heat capacity. The consistent positive deviation in the bubble pressure suggests that the model needs a small correction using interaction parameters. Over-predictions in the saturation pressure
of mixtures can be corrected for by decreasing the value of the interaction parameter or, in this case, adding a negative interaction parameter, using the Gao et al. correlation, Equation 18. A decrease in the n-value decreases the value of kij. Figure 8 shows the predicted and experimental values for coconut biodiesel fuel vapour pressure with n = 0 and n = -5. The AARD was improved by this reduction from 14 to 10%. Note, however, that the same change in n-value did not produce any significant difference in the prediction of vapour pressure for the rest of the biodiesel fuels (AARD values remained the same). The difference in sensitivity can be explained in terms of relative volatility
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(related to the ratio of vapour pressure) of the different FAMEs that comprise the biodiesel fuels. In the case of the coconut biodiesel fuel, as shown in Table 2, the
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composition distribution varies from C6 up to C18, which gives a wide range in relative volatility. On the other hand, the majority of the biodiesel fuels range from only C16 to
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C18 and there is little range in the volatility and therefore little sensitivity to the
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interaction parameters.
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Liquid heat capacity predictions, in general, were within 3% of the data; however, the
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predicted slopes versus temperature were slightly off. This deviation may be explained by
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non-idealities in the liquid phase corresponding to excess heat capacities different than those predicted by the APR EoS. Tuning with binary interaction parameters was again
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evaluated; however, it was found that the liquid heat capacity data had low sensitivity to
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the kij values. The liquid heat capacity is calculated using the residual heat capacity by the equation of state; the residual heat capacity depends only weakly on the attractive
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parameter (which contains the interaction parameter) and strongly on the first and second derivative of the attractive parameter (which do not contain the interaction parameter).
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Figure 9 shows the predicted and experimental values for coconut with n = 0 and n = -5.
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The AARD value remained the same within 3 significant figures.
One option to account for the apparent non-idealities in the liquid phase of the biodiesel fuels is to make the interaction parameters temperature dependent; for example, the measured Cp data could be used to estimate the variation of the several binary interaction parameters with temperature. Another option is to use a different approach for the kij
values, for instance, zero interaction parameters between saturated FAMEs pairs as well as between un-saturated FAMEs pairs, but tuned kij values between saturated/unsaturated pairs. Note that the APR EoS will generate non-ideal behavior even though the interaction parameters are zero. Although the positive deviations in vapour pressure and heat capacity suggest some non-ideality, it is possible that the solution is actually ideal and the interaction parameters in APR have to be added to make the predictions more ideal. However, since the APR EoS model presented here predicts both the vapour pressures and heat capacities with sufficient accuracy for most practical engineering
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applications, we elected to retain the straightforward application rather than to add more
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parameters to the model.
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CONCLUSIONS
The Advanced Peng-Robinson equation of state was applied to model the vapour pressure
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and enthalpy of vapourization of fatty acid methyl esters common to biodiesel fuels. New
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values for the critical properties and acentric factor of the FAMEs were proposed to
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reduce the absolute relative errors between predicted and experimental data from 5% for
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enthalpy of vapourization, 36% for vapour pressure of saturated FAMEs, and 350% for unsaturated vapour pressure of FAMEs down to 1%, 30%, and 30% respectively. New
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correlations for ideal heat capacity were proposed and the new set of equations used with the APR-EoS modeled liquid heat capacity data for the FAMEs with an AARD of 2%.
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The proposed heat capacity model showed significant improvement compared with two well known predictive methods for ideal heat capacity when used with the APR-EoS,
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decreasing the AARD from 7% to 2%.
The APR-EoS model for the fatty acid methyl esters was used to predict the vapour pressure and liquid heat capacity of different biodiesel fuels using ideal mixing rules. The
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AARD was 12% for vapour pressure and 3% for liquid heat capacity. The model proposed in this work predicted vapour pressure and heat capacities with enough accuracy for most practical engineering calculations and provides a more general model than analytical vapour pressure equations since it can be used to predict the phase
behavior, volumetric, and caloric properties of compounds making up biodiesel fuels, biodiesel fuel, and their blends
ACKNOWLEDGEMENTS The authors are grateful to the sponsors of the NSERC Industrial Research Chair in Heavy Oil Properties and Processing including the Natural Sciences and Engineering Research Council (NSERC), Petrobras, Schlumberger, and Shell. We thank Virtual
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Materials Group for the use of VMGSimTM software. We also thank Drs. Lovestead and Bruno of the National Institute of Standards and Technology for their previous
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contribution to the experimental data used in this study.
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NOMENCLATURE Definition
aPV,i
Adjustable parameter for Equation 5
ΔCP
Phase transition heat capacity
kJ/kmol·K
CP,L
Liquid heat capacity
kJ/kmol.K
CP 0
Ideal gas heat capacity
kJ/kmol.K
Residual heat capacity
kJ/kmol.K
Optimization objective function
-
Molecular mass
Kg/k-mol
N
A
M
CP
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Symbol
Res
D
J
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MW NUC
Units -
Number of unsaturated carbon
-
Pressure, Vapour pressure
kPa
Temperature
ºC
Reference temperature
-
xi
Mole fraction
-
α
Similarity function, Equation 13
-
ν
Stoichiometric value of an element in a compound
T
A
CC
TRef
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P, PV
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[8] Castellanos Díaz, O, Schoeggl, F., Yarranton, H.W., Satyro, M.A., Lovestead, T/M., Bruno, T.J. Modeling the vapour pressure of biodiesel fuels; World Academy of Science, Engineering and Technology. 65, (2012), 876-886
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[9] Narvaez, P. C., Rincon S. M., Castaneda, L. Z. and Sanchez, F. J. Determination of Some Physical and Transport Properties of Palm Oil and of its Methyl Esters, Latin American Applied Research, 38, 2008, 1-6
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[10] Dzida, M., and Prusakiewicz, P. The Effect of Temperature and pressure on the Physicochemical Properties of Petroleum Diesel Oil and Biodiesel Fuel, Fuel, 87, 2008, 1941-1948
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[11] National Institute of Standards and Technology (NIST). Thermo Data Engine (TDE) Version 6.0, Pure compounds, Equations of state, Binary mixtures, and Chemical Reactions. NIST Standard reference Database #103b. Thermodynamic Research Center. USA. 2011
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[12] Bruno, T. J., and Svoronos, P. D. N., CRC Handbook of Fundamental Spectroscopic Correlation Charts, Taylor and Francis Group, 2006
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[13] Bruno, T. J., and Svoronos, P. D. N., CRC Handbook of Basic Tables for Chemical Analysis, Third Edition, CRC Press, Boca Raton, 2011 [14] National Institute of Standards and Technology, NIST, NIST/EPA/NIH Mass Spectral Library Version 1.0, USA, 1995
[15] Weir, R. D., and de Loos, Th. W. Measurement of the Thermodynamic Properties of Multiple Phases, IUPAC, Physical Chemistry Division, Commission on Thermodynamics, Elsevier, The Netherlands, 2005 [16] Haines, P. J., Thermal Methods of Analysis: Principles, Applications and Problems, Springer, 2002 [17] Castellanos-Diaz, O., Schoeggl, F., Yarranton, H., and Satyro, M., Measurement of Heavy Oil and Bitumen Vapour Pressure for Fluid Characterization, Industrial Engineering Chemistry Research Journal, 52(8), 2013, 3027-3035
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[18] Peng, D.Y., Robinson, D.B. A new two-constant equation of state, Ind. Eng. Chem. Fund., 15(1), (1976), 59-64
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[19] Virtual Materials Group Inc (VMG). VMGSim Version 5.0.5, VMGSim User’s Manual. Calgary Alberta, 2010
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[20] Poling, B.E., Prausnitz, J.M., O’Connell, J.P., The Properties of Gases and Liquids, 5th edition, McGraw-Hill, 2001
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[21] Walas, S. M. Phase Equilibria in Chemical Engineering, Butterworth Publishers, U.S.A., 1985
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A
[22] Gao, G., Dadiron, J.L., Saint-Guirons, H., Xans, P., Montel, F. A simple correlation to evaluate binary interaction parameters of the Peng-Robinson equation of state: binary light hydrocarbon systems. Fluid Phase Equilibria, (1992), 74, 85-93
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[23] van Genderen, A. C. G., van Miltenburg, J. C.; Block, J. G.; Van Bommel, M. J.; van Ekeren, P. J.; van den Berg, G. J. K.; Oonk, H. A. J.; Liquid-Vapor Equilibria of the Methyl Esters of Alkanoic Acids: Vapor Pressures as a Function of Temperature and Standard Thermodynamic Function Changes, Fluid Phase Equilibria, 202, 2002, 109-120
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[24] Bureš, M.; Majer, V.; and Zábranský, M. Modification of Benson Method for Estimation of Ideal-Gas Heat Capacities, Chem. Eng. Sci., 36, 1981, 529-537
Table 1. Data available for selected FAMEs and temperature range in °C [11]
Points 65 53 70 112 90 29 110 27 101 29 12 4 33 -
C20:1(11)
8 18 12
26.85 26.85 26.85
176.85 214.95 185.7
Points 12 10 8 7 5 5 5 4 3 -
Tmax 76.85 76.85 76.85 76.85 76.85 76.85 76.85 76.85 76.85 -
-
-
1 1 1
N
C22:1 C18:2 C18:3
Tmax 146.52 145.70 188.20 226.85 237.8 226.85 321.95 226.85 346.95 226.85 258.95 176.85 218.50 -
∆Hv Points Tmin 2 25.0 3 25.0 2 25.0 4 25.0 4 25.0 4 25.0 2 25.0 2 25.0 2 25.0 2 25.0 2 25.0 1 25.0 2 25.0 1 25.0
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C6:0 C8:0 C10:0 C12:0 C14:0 C15:0 C16:0 C17:0 C18:0 C20:0 C22:0 C24:0 C16:1 C17:1 C18:1(11) C18:1(9)
CPL Tmin -33.15 -3.15 6.85 25 26.85 36.85 36.85 46.85 56.85 -
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Methyl hexanoate Methyl caprylate Methyl caprate Methyl laurate Methyl myristate Methyl pentadecanoate Methyl palmitate Methyl heptadecanoate Methyl stearate Methyl arachidate Methyl behenate Methyl lignocerate Methyl palmitoleate Methyl heptadecenoate Methyl oleate Methyl vaccenate Methyl cis-11eicosenoate Methyl erucate Methyl linoleate Methyl linolenate
PV Tmin 7.55 33.69 -12.74 -11.00 0.00 21.85 18.00 21.85 21.85 38.00 21.85 26.85 26.85 -
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Formula *
U
FAME
25.0 25.0 25.0
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* In the CN:S nomenclature for biodiesels, N stands for the carbon number of the carboxylic acid from which the biodiesel was generated, and S stands for the number of unsaturated carbon bonds. For instance, C18:1(11), methyl oleate, is a biodiesel formed from oleic acid (with 18 carbon atoms) and one double bond between the eleventh carbon and twelfth carbon atoms
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Table 2.Temperature range of vapour pressure and heat capacity data for selected biodiesel fuels [8]
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Biodiesel fuels Source Canola (South Alberta) Canola (Saskatchewan) Soy (Sunrise, US) Soy (Mountain Gold, US) Rapeseed (Europe) Palm (Europe) Coconut (Europe)
Code CB-01 I-25 SB100 MGB100 S102550 S090824 S070717
Vapour Pressure 60-196 °C 140 °C 80-110 °C 70-100 °C 95-125 °C
Liquid Heat Capacity 13-55 °C 12-55 °C 14-55 °C 10-55 °C 13-55 °C 23-55 °C 10-55 °C
Tmax 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0
Table 3.Composition in mole percentage of FAMEs in selected biodiesel fuels SB100 0 0 0 0 0 0 0 11.4 0 0 3.2 21.3 1.5 54.9 7.4 0.3 0 291.8
MGB100 0 0 0 0 0.6 0 0.5 12.5 0 0 4.9 27.0 1.6 46.6 6.1 0.3 0 291.3
S090824 0 0 0 0 1.5 0 0 45.1 0 0 3.6 39.5 0 9.8 0.2 0.3 0 283.7
S070717 1.0 12.6 7.7 48.3 16.6 0 0 6.7 0 0 1.5 4.4 0 1.1 0 0 0 218.2
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* For the CN:S nomenclature for biodiesels, refer to Table 1
S102550 0 0 0 0 0 0 0 4.8 0 0 1.28 59.91 3.68 19.44 9.08 1.26 0.55 294.6
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I-25 0 0 0 0 0 0 0.9 9.3 0.3 0 4.4 57.4 2.8 16.0 7.5 0.4 1.00 293.2
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CB-01 0 0 0 0 0 0 12.7 12.7 0 0 4.1 23.5 1.5 49.9 8.1 0.2 0 291.5
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FAMEs C6:0 C8:0 C10:0 C12:0 C14:0 C15:0 C16:1 C16:0 C17:0 C17:1 C18:0 C18:1(9) C18:1(11) C18:2 C18:3 C20:0 C20:1 MWavg
CC A
Not Tuned 2857 2064 1752 1755 1147 1384 1313 1225 1157 1031 999 1321 1231 1173 1153 1002 1204 1127
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Tc [ºC] Tune Tc, Pc, ω 329.6 374.8 409.8 439.0 453.8 462.8 476.6 486.4 498.8 514.8 530.8 476.8 496.8 503.8 514.8 524.8 509.8 516.8
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C6:0 C8:0 C10:0 C12:0* C14:0 C15:0 C16:0 C17:0 C18:0 C20:0 C22:0 C16:1 C17:1 C18:1(9) C20:1 C22:1 C18:2 C18:3
Not Tuned 328.8 362.4 398.0 439.0 447.8 462.8 481.8 486.8 498.8 516.8 542.8 477.8 488.8 494.8 523.8 543.8 499.8 500.8
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Formula
A
N
Table 4. Critical properties and acentric factor of selected FAMEs; comparison between contribution method predictions (not tuned) and tuned values for the APR EoS. Pc [kPa] Tuned Tc, Pc, ω 2950 1984 1570 1507 1175 1384 1250 1224 1158 1031 985 1830 1780 1710 1650 1600 3150 4850
*Critical temperature experimentally measured
Not Tuned 0.462 0.447 0.511 0.575 0.804 0.902 0.853 0.946 0.970 1.015 1.180 1.125 1.466 3.071
Tuned ω 0.584 0.660 0.715 0.829 0.876 0.906 0.937 0.969 1.032 0.966 0.942 0.965
Tuned Tc, Pc, ω 0.490 0.509 0.596 0.696 0.803 0.873 0.906 0.937 0.969 1.032 1.088 0.977 0.979 0.991 1.116 1.195 1.052 1.105
Table 5. Performance (AARD) of the APR EoS for the prediction of vapour pressure and enthalpy of vapourization for various FAMEs
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Case 3: “Tuning Tc, Pc, ω” PV [%] ∆HV [%] 5 11 1 12 1 12 1 8 0 4 1 6 1 2 4 1 11 1 27 13 6 1 1 10 12 -
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Case 2: “Tuning ω” PV [%] ∆HV [%] 8 5 4 4 6 2 5 1 5 1 7 1 2 4 1 11 1 4 14 14 -
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N
C6:0 C8:0 C10:0 C12:0 C14:0 C15:0 C16:0 C17:0 C18:0 C20:0 C22:0 C16:1 C17:1 C18:1(9) C20:1(11) C22:1 C18:2 C18:3
Case 1: “No Tuning” PV [%] ∆HV [%] 127 7 77 9 74 9 7 1 8 3 12 2 2 9 4 11 5 159 235 648 -
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Formula
This work 0.6 0.4 0.4 0.2 0.3 0.3 0.3 0.6 0.1
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Formula
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Table 6. AARD values for the APR-EoS for the prediction of liquid heat capacity for various FAMEs with three different methods for the ideal gas heat capacity prediction.
A
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C8:0 C10:0 C12:0 C14:0 C15:0 C16:0 C17:0 C18:0 C20:0
∆CPL AARD [%] Modified Benson 11.4 9.9 8.9 7.7 7.1 6.9 6.2 6.8 7.1
Joback 11.8 10.2 9.1 7.8 7.3 7.0 6.3 6.8 7.1
Table 7. Performance of APR-EoS prediction of biodiesel fuel vapour pressure and liquid heat capacity AARD [%] PV CPL Canola (South Alberta) CB-01 21.7 1.9 Canola (Saskatchewan) I-25 11.7 3.3 Soy (Sunrise, US) SB100 8.8 1.4 Soy (Mountain Gold, US) MGB100 15.7 2.7 Rapeseed (Europe) S102550 9.9 6.6 Palm (Europe) S090824 4.4 2.0 Coconut (Europe) S070717 14.5 1.3 Total 12.3 2.7 Code
Figure 1. Schematic of optimization procedure.
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Biodiesel Fuel Source
A
N
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Figure 2. Experimental and predicted vapour pressure and heat of vapourization data for methyl caprate, C10:0 using adjusted critical properties and acentric factor (Data from NIST [11]).
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Figure 3. Calculated and regressed ideal gas heat capacity for methyl caprate, C10:0.
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Figure 4. Parameters of Equation 21 as a function of molecular weight.
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Figure 5. Experimental and predicted liquid heat capacity data for methyl caprate, C10:0 (Data from NIST [11]).
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Figure 6. Experimental and APR-EoS predicted vapour pressure of biodiesel fuels listed in Table 7.
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Figure 7. Experimental and APR EoS predicted liquid heat capacity of biodiesel fuels listed in Table 7.
Figure 8. Experimental and predicted vapour pressure of coconut biodiesel fuel with the n values ranging from Zc to -5. Figure 9 Experimental and predicted liquid heat capacity of coconut biodiesel fuel with the n values ranging from Zc to -5.
Calculate from Eq. 19 Look up Tc, Pc or calculate from Eqs. 20-24
Calculate A and B from Eqs. 3 and 4 Calculate HV from Eq. 13
Choose vapour pressure (=P)
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Calculate flliquid and flvapour from Eq. 12
adjust P flliquid = flvapour ?
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no
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yes
adjust or adjust Tc, Pc
OF = minimum?
End
yes
A
N
no
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Calculate OF from Eq. 14
1.E+01
D TE
1.E-01 1.E-02 1.E-03
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1.E-04
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Pressure [kPa]
1.E+00
Experimental PR EoS with Adjusted Tc, Pc PR EoS with Predicted Tc, Pc
1.E-05
A
0
50
100 Temperature [C]
150
69 Heat Capacity Difference [kJ/mol]
M
1.E+02
200
Experimental Data Source 1 Experimental Data Source 2 PR EoS with Adjusted Tc, Pc PR EoS with Predicted Tc, Pc
68 67 66 65 64 63 -20
0
20 40 60 Temperature [C]
80
100
Calculated
340
Regressed
330 320 310 300 290 280
PT
1600
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5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 Temperature [C] 1 Parameters values
1200
Model
0
SC
1400
-1
1000
N
-3
600
A
400
0 150
200
250
300
A
CC
EP
TE
D
MW
M
200
350
1.E-02 8.E-03
U
bCP,0
aCP,0
-2
800
1.E-02
-4
cCP,0
Ideal Gas heat Capacity [kJ/kmol.K]
350
6.E-03
Parameter values Model Regressed
4.E-03
-5
2.E-03
-6
0.E+00 150
200
250 MW
300
350
Liquid Heat Capacity [kJ/kmol.K]
420 400 380 360 340 320
Experimental Data APR EoS with New Cp0 Correlation APR EoS with Jobak Method APR EoS with Benson Method
300
0
20
40 60 Temperature [C]
100
1.E+00
1.E+00 1.E-01
N
1.E-01
Pressure [kPa]
U
SC
1.E+01
A
1.E-02
Rapeseed Soy MG Canola I25 Coconut APR EoS Model kij = 0
M
Pressure [kPa]
1.E+01
1.E-03
25 50 75 100 125 150 175 200 225 Temperature [C]
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TE
0
D
1.E-04
A
80
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-20
PT
280
1.E-02 Palm
1.E-03
Soy S
1.E-04
Canola CB APR EoS Model kij = 0
1.E-05 0
25 50 75 100 125 150 175 200 225 Temperature [C]
750
Liquid Heat Capacity [kJ/kmol.K]
700 650 600
Rapeseed Soy MG Canola I25 Coconut APR EoS Model kij = 0
550 500 450 400
700 650 600 550 Palm Soy S Canola CB APR EoS Model kij = 0
500 450 400
5 10 15 20 25 30 35 40 45 50 55 60 Temperature [C]
0
1.E+01
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A
N
U
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Experimental Gao et al. Gao et al. n=0 Gao et al. n=-5
]a1.E+00 P k[ e r u ss e r P1.E-01
D
1.E-02
50 100 Temperature [C]
A
CC
EP
TE
0
5 10 15 20 25 30 35 40 45 50 55 60 Temperature [C]
RI
0
PT
Liquid Heat Capacity [kJ/kmol.K]
750
150
480 ] 470 K l. o 460 m k/ Jk [ 450 yt ic a440 p a C ta430 e H d i 420 u q iL 410
Experimental Gaoet al. Gaoet al. n=0 Gaoet al. n=-5 5 10 15 20 25 30 35 40 45 50 55 60 Temperature [C]
A
CC
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TE
D
M
A
N
U
SC
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0
PT
400
S0378-3812(15)00004-7 http://dx.doi.org/doi:10.1016/j.fluid.2014.12.050 FLUID 10410
To appear in:
Fluid Phase Equilibria
Received date: Revised date: Accepted date:
14-5-2014 11-12-2014 31-12-2014
Please cite this article as: O.Castellanos D´iaz, F.Schoeggl, H.W.Yarranton, M.A.Satyro, EQUATION OF STATE MODELING FOR THE VAPOUR PRESSURE OF BIODIESEL FUELS, Fluid Phase Equilibria http://dx.doi.org/10.1016/j.fluid.2014.12.050 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.
EQUATION OF STATE MODELING FOR THE VAPOUR PRESSURE OF BIODIESEL FUELS O. Castellanos Díaz, F. Schoeggl, and H. W. Yarranton Department of Chemical and Petroleum Engineering, University of Calgary, AB, Canada M. A. Satyro Virtual Materials Group, Calgary, AB, Canada Abstract The vapour pressure and liquid heat capacity of seven biodiesel fuels and its components (fatty acid methyl esters - FAMEs) were modeled using the Advance Peng Robinson
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equation of state. The dataset used for the modeling was obtained from the literature and
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included FAME properties and the composition, vapour pressure, and liquid heat capacity of the biodiesel fuels from different sources at temperatures from 50 to 130ºC and -30 to
SC
75ºC, respectively. New values for the critical properties and acentric factor of FAMEs are introduced as well as new models for the ideal gas heat capacity for the FAMEs. The
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average AARD is 12% for vapour pressure and 3% for liquid heat capacity.
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Keywords: biodiesel fuels, fatty acid methyl ester, vapour pressure, heat capacity, cubic
D
INTRODUCTION
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equation of state modeling
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Biodiesel fuels constitute one of the most promising alternatives to supplement or reduce petroleum diesel (petro-diesel) usage. They are renewable, non-mutagenic, non-
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carcinogenic, biodegradable fuels that can be domestically produced. Biodiesel fuels are refined mixtures of esters produced by the transesterification of fatty acids from
CC
vegetable oil and animal fat (fatty acid methyl esters or FAMEs for short). These fuels can be used directly or blended with petro-diesel to improve lubricity without adding any
A
sulfur. These fuels may also improve engine firing due to their oxygen content [1-6].
The measurement and prediction of bio-diesel properties is required for their effective commercial use. One important property for the quality control of biodiesel fuels and their blends is volatility, which is directly related to the vapour pressure of their
individual constituent compounds [2]. For instance, vapour pressure is used to calculate the latent heat in order to compare rates of vapourization and injection characteristics with other fuels. Heat capacity is another important thermo-physical property used to characterize biodiesel for fuel blending with petro-diesel and is directly related to the vapour pressure.
Yuan et al. [7] modeled the vapour pressure data of three different biodiesel fuels at temperatures above 215°C [1,2]. They assumed that biodiesel fuels were ideal mixtures
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of their constituent FAMEs. They fitted the vapour pressure of the individual FAMEs with the Antoine equation and predicted the vapour pressures of the biodiesel fuels using
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Raoult's law. However, the prediction of vapour pressures at lower temperatures requires extrapolation using the fitted Antoine equations and the accuracy of the extrapolation is
SC
unknown.
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Castellanos-Diaz et al. [8] modeled the vapour pressure of several biodiesel fuels at
N
temperatures from 50 to 130°C using Raoult’s law. They modeled the FAMEs vapour
A
pressure using the Cox equation and constrained the equation parameters to also match
M
the liquid heat capacities. They demonstrated that more accurate vapour pressure fits were obtained when calorimetric data are included in the optimization function used to
D
determine the vapour pressure equation parameters. They also proposed new equations
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for calculating the vapour pressure and liquid heat capacities of FAMEs.
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Experimental data for the liquid heat capacity of biodiesel fuels are also scarce and, in most cases, are available only for specific biodiesel fuel sources. Conceicao et al. [3]
CC
presented liquid heat capacity data from thermo-gravimetric methods for castor oil-based biodiesel fuel at temperatures between 55 and 125°C. The same methodology was
A
applied by Narvaez et al. [9] for palm oil-based biodiesel fuel at temperatures ranging from 30 to 100°C. Dzida and Prusakiewicz [10] measured the liquid heat capacity of rapeseed oil-based biodiesel fuel using differential scanning calorimetry at temperatures between 20 and 45°C. No modeling was included in any of these cases.
The model developed by Castellanos et al. [8] based on vapour pressure fitting using relatively simple equations is sufficient for calculations where only vapour pressure, enthalpy of vapourization and liquid heat capacity values are required. However, a more comprehensive approach is to model vapour pressure of FAMEs and biodiesel fuels using an equation of state (EoS), since an EoS can also be used for phase equilibrium calculations in processes with additional components and/or processing stages and also provides a consistent structure to introduce pressure effects on phase equilibrium calculations. An equation of state model is also more convenient for most commercial
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process simulation software.
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The purpose of this study is to develop an equation of state (EoS) model for the representation of vapour pressures and liquid heat capacities of fatty acid methyl esters
SC
and biodiesel fuels. The biodiesels were characterized as mixtures of FAMEs with a known composition. The first step was to characterize the FAMEs. Their critical
U
properties were determined from literature data or correlations and then tuned to fit the
N
equation of state model to measured FAME vapour pressures and enthalpies. Next, the
A
ideal heat capacities of the FAMEs were calculated and the liquid heat capacity of
M
FAMEs predicted. Finally, the vapour pressure and liquid heat capacity of biodiesels was determined. The vapour pressure and heat capacity datasets used to develop the model
D
were taken from the literature and are summarized below. The equation of state model
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and optimization methodology are then presented followed by a discussion of the results.
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DATASETS
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2,1 FAMEs Dataset Eighteen FAMEs ranging in carbon number from 6 to 22 were examined, as presented in Table 1. The vapour pressure dataset [11] ranges in temperature from 25 to 300°C
A
whereas liquid heat capacity data range from the FAME freezing point to 50 °C. Data that were not available from the database were predicted using the models developed by Castellanos-Diaz et al [8].
2 . 2 B i o d i e s e l F u e ls D a t a s e t The composition, liquid heat capacity, and vapour pressure data of seven biodiesel fuels were reported previously by Castellanos-Diaz et al. [8]. Briefly, the compositions were determined using GC and mass spectrometry [12-15]. The accuracy of the compositions based on standards is approximately ±2 wt%. Liquid heat capacities were measured using a differential scanning calorimeter (DSC) TA Q2000 V24.9 calibrated against indium with an average accuracy of ±2% [16]. Vapour pressures were measured using an inhouse static flash test apparatus at pressures ranging from 0.1 to 200 Pa. The average
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95% confidence interval for the measured vapour pressure data is ±13% [17]. Table 2 lists the seven biodiesels and the temperature range of their vapour pressure and heat
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capacity data. Table 3 summarizes the compositions of the biodiesels. FAMEs ranging from C6:0 to C20:1 were identified; with the exception of sample S070717, the majority
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of each fuel was composed of C18:0, C18:1, and C18:2.
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EQUATION OF STATE MODELING
The Advanced Peng-Robinson Equation of State (APR-EoS) [18-19] was used to
A
calculate the vapour pressure and the heat capacity of the biodiesel fuels listed in Table 2.
M
First, the FAMEs were modeled as pure components and then the biodiesel fuels were modeled as a mixture of the FAMEs. The APR-EoS is a modified version of the Peng-
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Robinson EoS that accounts for more recent experimental data in the -function and
is given by:
RT a (TR , ) v b v (v b) b(v b)
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P
TE
includes volume translation to improve on liquid density predictions [19]. The APR-EoS
[1]
CC
where P is the pressure, T is the absolute temperature, v is the molar volume, R is the universal gas constant, a and b are constants related to the attractive and repulsive forces,
A
TR is the reduced temperature (T/TC), TC is the critical temperature, ω is the acentric factor, and α is a function that adjusts the EoS to vapor pressure experimental data. To facilitate the discussion below, recall that the APR-EoS can be expressed in terms of the compressibility factor as follows [20]:
Z 3 Z 2 1 B Z A 2B 3B 2 B A B B 2 0, [2]
where Z is the compressibility factor. A and B are defined by:
A
a , TR P
RT 2
, [3]
bP , RT
PT
B
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[4]
where T is temperature, P is pressure, R is the universal gas constant, is the acentric
0.0777969 RTC ,i PC ,i
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A
,
,
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bi
PC ,i
D
ai
0.457265 R 2TC2,i
U
factor, TR is the reduced temperature, and a and b for any component i are given by:
[5]
[6]
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where TC and PC are the critical temperature and pressure, respectively. The α-function
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for the Peng-Robinson EoS is given by [21]:
2
A
, TR 1 fW 1 TR0.5 ,
where fw is given by:
[7]
fW 0.37464 1.54226 0.26992 2 , 0.5
[8]
fW 0.3796 1.485 0.1644 2 0.01667 3 , 0.5
[9]
Generally, an EoS is tuned to fit vapour pressure data of a pure substance by adjusting the α-function, which depends on the acentric factor. For components where the critical properties are not known, these properties and/or the acentric factor may also be adjusted. Once the vapour pressure of the pure components is fitted to the EoS, the vapour pressure
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work, the classic Van der Waals mixing rules were used, given by:
bm i xi bi ,
SC
[10]
N
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am j i xi x j ai i a j j 1 kij ,
[11]
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of a mixture of these components can be modeled using appropriate mixing rules. In this
A
Tuning of the interaction parameters used in the mixing rule may be required if the
M
mixture exhibits non-ideal mixing behavior.
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3 . 1 F A M E s V a p o u r P re s s u r e
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Since vapour pressure and enthalpy are related through the Clapeyron equation and both can be determined directly from an equation of state, the tuning of the acentric factor and
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critical properties can be constrained through the simultaneous use of vapour pressure and enthalpy of vapourization data where available. The vapour pressure is an
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equilibrium property and, for a pure component, is determined through the equality of the fugacities of the component in the liquid and vapour phases [21]. The fugacity
A
coefficients for each phase are calculated from the Peng-Robinson equation of state as follows [20]:
Z f Phase A 2.414 B ln Z Phase 1 ln( Z Phase B) ln Phase 2.828 B Z Phase 0.414 B P
[12]
The enthalpy of vapourization, HV, is simply the difference between the enthalpy of the liquid and vapour phases where, for the PR EoS, the enthalpy of each phase is given by:
2 Z j B (2 8) 1 da ln a T RT ( Z j 1), dT 8 B 2 Z j B (2 8)
PT
hj
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[13]
SC
where j stands for the phase at equilibrium.
N
U
The optimization function used in this study is given by:
A
OF i (ln Pi exp ln Pi calc ) 2 K H i ( H iexp H icalc ) 2 ,
M
[14]
D
where OF is the magnitude of the optimization function, KH is a weight factor used to
TE
scale the contribution of each variable to a similar magnitude; a value of 0.001 was found reasonable to balance the contribution of vapour pressure and enthalpy of vapourization
EP
data when calculating the OF. The enthalpy of vapourization in Equation 14 is in kJ/kg. Two strategies were used to minimize OF: 1) adjust only the acentric factor; 2) adjust the
CC
acentric factor and the critical properties. The procedure for both cases is the same and is shown in Figure 1. The vapour pressure, acentric factor, and critical properties were
A
adjusted with the generalized reduced gradient algorithm in Excel Solver; in general, convergence was achieved within a few iterations. 3.2 FAMEs Heat Capa cit y The liquid heat capacity for each FAME was calculated as follows:
CP , L C PRe s C PV C PRe s C Po
[15]
where CP,L is the liquid heat capacity, CPV and CP° are the vapour and ideal gas heat capacity, respectively, and CPres is the residual heat capacity which can be determined from the PR EoS as follows:
PT
RI
CPRe s
2 R da 1 1 d 2 a v b 1 2 v b dT v 2 2bv b 2 R, T ln 2 2a v b RT v b 1 2 b 8 dT 2 2 2 2 v b v 2bv b
SC
[16]
In order to complete the heat capacity calculations ideal gas heat capacities are required.
N
U
The development of an ideal gas heat capacity correlation for FAMES is discussed later.
A
3 . 3 B i o d i e s e l F u e ls
M
Biodiesel fuels are simply a mixture of FAMEs and the EoS parameters for the mixture were determined from the van der Waals mixing rules. The vapour pressure was then
D
determined by equating the fugacity of each phase as described for the FAMEs. The
TE
liquid heat capacity was determined from Equation 15. In this case, the residual heat capacity of the mixture was calculated from Equation 16 with the EOS mixture
EP
parameters. The ideal gas heat capacity of the mixture was calculated as follows:
[17]
A
CC
CP0 , Biodiesel i xi CP0 , FAME i ,
The binary interaction parameters in the van der Waals mixing rule, Equation 11, were initially set to zero and the model is predictive. If necessary, the model can be tuned by adjusting the binary interaction parameters based on experimental data. In this work, the
Gao et al. equation [22] was used to correlate the interaction parameters to critical temperatures as follows:
n
2 T T 0.5 ci cj , 1 kij Tci Tcj
[18]
PT
The exponent n in Equation 18 is the only adjustable parameter. Equation 14 was again
RI
used as the optimization function.
SC
RESULTS AND DISCUSSION
Recall that the following steps were required to complete the equation of state model: 1)
U
determine FAME critical properties; 2) tune the predicted critical properties to fit the
N
equation of state model to FAME vapour pressures and enthalpies; 3) determine FAME
A
ideal heat capacities; 4) predict liquid heat capacity of FAMEs; 5) predict vapour
M
pressure and liquid heat capacity of biodiesels. Each step is discussed below.
D
4 . 1 F A M E s C r it ic a l P r o p e rt i e s Experimental critical properties for the majority of fatty acid methyl esters do not exist
TE
because the hypothetical critical temperature for most FAMEs is above the thermal cracking temperature. Therefore, both acentric factor and critical property tuning are
EP
considered for the FAMEs.
CC
As recommended by NIST [11], the hypothetical critical pressure, Pc, of the FAMEs was estimated using the Wilson-Jasperson and the Ambrose-Walton vapour pressure based
A
methods. The hypothetical critical volume, Vc, and critical temperature, Tc were estimated using Joback’s group contribution method [20]. A group contribution method was considered appropriate because FAMEs consist of well known molecular groups including CH3-, -CH2-, -CH=, and COO-, and this method was also recommended and used by NIST’s TDE [11]. Initial estimates for the FAMEs acentric factors were
calculated from the vapour pressure curve (when available), using Pitzer’s definition [20] given by:
P Pitzer log10 v 1, PC AtT 0.7T C
[19]
where ωPitzer is the Pitzer’s acentric factor. The estimated acentric factors and critical
PT
properties for the FAMEs are shown in Table 4.
RI
4 . 2 F A M E s V a p o u r P re s s u r e
SC
Once the critical properties and acentric factors were estimated, the vapour pressure and enthalpy of vapourization of FAMEs were calculated using the APR-EoS. Table 5 shows
U
deviation values for this approach ("No Tuning") and Figure 2 shows the results for
N
methyl caprate, C10:0, as an example. The performance of the APR-EoS with initial critical values for saturated FAMEs is satisfactory, with an AARD of 36% over all
A
FAMEs. The relatively high deviations for FAMEs are within the uncertainty of the
M
experimental values at low temperatures. For instance, Genderen at al. [23] and NIST [11] report an average overall deviation of 5% for the vapor pressures of methyl caprilate;
D
however, the error is only 2% near the normal boiling point but increases monotonically
TE
as the temperature decreases reaching values of 23% at the lowest temperatures (from -10 to 20 ºC, depending on the FAME). On the other hand, the performance of the APR-EoS
EP
with initial critical values for unsaturated FAMEs is poor, with an average AARD of 350% over all FAMEs. The errors may be higher for unsaturated FAMEs because their
CC
vapour pressures are lower than the saturated FAMEs at the same temperature. Another, more likely, explanation is that the critical property correlations shown in Table 4 are not
A
appropriate for unsaturated FAMEs.
The performance of the EoS model was improved by adjusting the critical properties and acentric factor. The adjusted values are provided in Table 4. Table 5 shows deviations for vapour pressure and enthalpy of vapourization when only the acentric factor was
modified (“Tuning ω”) and when the acentric factor and the critical properties were tuned simultaneously (“Tuning Tc, Pc, ω"). Figure 2 shows the results for methyl caprylate (C10:0). The adjusted values of the critical properties and acentric factor do not differ significantly from the initial estimates, except for the unsaturated FAMEs. Note how the acentric factor, calculated from the available vapour pressure data and an unbounded extrapolation of the vapour pressure curve, increases considerably as the number of unsaturated carbon molecules is higher, Table 4. This increase may not reflect differences in the molecular structure but rather may be artifacts resulting from insufficient vapour
PT
pressure data at higher pressures, closer to a reduced temperature of 0.7 where the acentric factor is calculated, Equation 19. In any case, the changes in , Tc and/or Pc
RI
significantly improve the predicted values for the vapour pressure and the enthalpy of vapourization. The AARD for the vapour pressures of the FAMEs, including
SC
undersaturated FAMEs, are all lower than 30% when the tuned parameters are used., with
N
U
values as low as 2%.
A
4 . 3 F A M E s I d e a l G a s H e a t C a p a c i t y a n d L i q u i d H e a t C a p a c it y
M
After the critical properties and acentric factor of the FAME were adjusted based on vapour pressure data, the heat capacity residual was calculated from the equation of state.
D
As noted previously, the ideal gas heat capacity, CP0, is required to determine the liquid heat capacity. Since no experimental data for CP0 are available, correlations were
TE
developed for saturated and unsaturated FAMEs. The new correlations were compared with two well known predictive methods: the Joback method [20] and a modification of
CC
EP
the Benson method [24].
Saturated FAMEs For each saturated FAME with available liquid heat capacity data, the ideal gas heat
A
capacity was calculated as follows:
s CP0 , FAME C P , L , FAME C PRe, FAME ,
[20]
The ideal gas heat capacity of each of these FAMEs was regressed with a second degree polynomial:
CP0 (0) aCp 0 bCp 0 T cCp0 T 2 ,
[21]
where a Cp°, bCp°, and cCp° are constants. For example, Figure 3 shows the calculated and
PT
regressed ideal gas heat capacity values for methyl caprylate, C10:0.
To generalize Equation 21 for all saturated FAMEs, the constants for the set of FAMEs
RI
were plotted as a function of the molecular weight, Figure 4, and fitted as with the
U
2.1079 x10 4 230.72 0.62516 MW 344.1759 MW 344.1759
,
N
aCp 0
SC
following expressions:
[22]
A
125.93 1.4529 5.5554 x103 MW 344.1759 MW 344.1759
M
bCp 0
D
0.19127 8.8626 x10 4 5.9999 x106 MW 344.1759 MW 344.1759
,
[24]
EP
TE
cCp 0
[23]
Note that in developing Equations 22 to 24, outliers were not considered as can be
CC
observed in Figure 4.
Unsaturated FAMEs
A
No data were available for C p0 or C L of unsaturated FAMEs and hence the modeling approach for the saturated FAMEs was not possible. Instead, the ideal gas heat capacities for the unsaturated FAMEs were determined from departure functions where the corresponding saturated FAME of the same carbon number is the reference fluid. The
departure function was calculated using Joback’s method [20]. After inserting the appropriate constants for the biodiesel fuel compositions, the Joback method can be written as follows:
CP0 NUC
1 NUC 6.1327 x102 1.5493 x10 4 T 1.842 x10 7 T 2 , C (0) 0 P
[25]
PT
where NUC is the number of unsaturated carbon molecules. The generalized liquid heat capacity model for FAMEs (Equations 21, and 25) was compared with the original
RI
Joback method and a modification of the Benson method [24]. Figure 5 shows the
SC
predicted liquid heat capacity of methyl caprilate, C10:0, using the APR-EoS and Equation 6. Table 6 shows the results for all of the FAMEs considered. The Joback and
U
modified Benson methods provided ideal heat capacities that in conjunction with the residual heat capacity calculated with the APR EoS systematically under-predicted the
M
A
APR-EoS, correct the under-prediction.
N
liquid heat capacity of the FAMEs. The methods presented in this work, applied to the
D
4 . 4 B i o d i e s e l F u e l V a p o u r P re s s u re
TE
The APR-EoS using the tuned critical properties, acentric factor, and the proposed method to estimate ideal gas heat capacities presented in the previous section for FAMEs
EP
was used to calculate both the vapour pressure and the heat capacity of the biodiesel fuels listed in Table 1. The classic van der Walls mixing rules were used; initially the
CC
interaction parameters, kij, were set to zero. Figures 6 and 7 show the measured and modeled vapour pressure and heat capacity, respectively, for all of the biodiesel fuels
A
studied. The AARDs for the model are given in Table 7. The average AARD is 12% for vapour pressure and 3% for liquid heat capacity. The consistent positive deviation in the bubble pressure suggests that the model needs a small correction using interaction parameters. Over-predictions in the saturation pressure
of mixtures can be corrected for by decreasing the value of the interaction parameter or, in this case, adding a negative interaction parameter, using the Gao et al. correlation, Equation 18. A decrease in the n-value decreases the value of kij. Figure 8 shows the predicted and experimental values for coconut biodiesel fuel vapour pressure with n = 0 and n = -5. The AARD was improved by this reduction from 14 to 10%. Note, however, that the same change in n-value did not produce any significant difference in the prediction of vapour pressure for the rest of the biodiesel fuels (AARD values remained the same). The difference in sensitivity can be explained in terms of relative volatility
PT
(related to the ratio of vapour pressure) of the different FAMEs that comprise the biodiesel fuels. In the case of the coconut biodiesel fuel, as shown in Table 2, the
RI
composition distribution varies from C6 up to C18, which gives a wide range in relative volatility. On the other hand, the majority of the biodiesel fuels range from only C16 to
SC
C18 and there is little range in the volatility and therefore little sensitivity to the
U
interaction parameters.
N
Liquid heat capacity predictions, in general, were within 3% of the data; however, the
A
predicted slopes versus temperature were slightly off. This deviation may be explained by
M
non-idealities in the liquid phase corresponding to excess heat capacities different than those predicted by the APR EoS. Tuning with binary interaction parameters was again
D
evaluated; however, it was found that the liquid heat capacity data had low sensitivity to
TE
the kij values. The liquid heat capacity is calculated using the residual heat capacity by the equation of state; the residual heat capacity depends only weakly on the attractive
EP
parameter (which contains the interaction parameter) and strongly on the first and second derivative of the attractive parameter (which do not contain the interaction parameter).
CC
Figure 9 shows the predicted and experimental values for coconut with n = 0 and n = -5.
A
The AARD value remained the same within 3 significant figures.
One option to account for the apparent non-idealities in the liquid phase of the biodiesel fuels is to make the interaction parameters temperature dependent; for example, the measured Cp data could be used to estimate the variation of the several binary interaction parameters with temperature. Another option is to use a different approach for the kij
values, for instance, zero interaction parameters between saturated FAMEs pairs as well as between un-saturated FAMEs pairs, but tuned kij values between saturated/unsaturated pairs. Note that the APR EoS will generate non-ideal behavior even though the interaction parameters are zero. Although the positive deviations in vapour pressure and heat capacity suggest some non-ideality, it is possible that the solution is actually ideal and the interaction parameters in APR have to be added to make the predictions more ideal. However, since the APR EoS model presented here predicts both the vapour pressures and heat capacities with sufficient accuracy for most practical engineering
PT
applications, we elected to retain the straightforward application rather than to add more
RI
parameters to the model.
SC
CONCLUSIONS
The Advanced Peng-Robinson equation of state was applied to model the vapour pressure
U
and enthalpy of vapourization of fatty acid methyl esters common to biodiesel fuels. New
N
values for the critical properties and acentric factor of the FAMEs were proposed to
A
reduce the absolute relative errors between predicted and experimental data from 5% for
M
enthalpy of vapourization, 36% for vapour pressure of saturated FAMEs, and 350% for unsaturated vapour pressure of FAMEs down to 1%, 30%, and 30% respectively. New
D
correlations for ideal heat capacity were proposed and the new set of equations used with the APR-EoS modeled liquid heat capacity data for the FAMEs with an AARD of 2%.
TE
The proposed heat capacity model showed significant improvement compared with two well known predictive methods for ideal heat capacity when used with the APR-EoS,
CC
EP
decreasing the AARD from 7% to 2%.
The APR-EoS model for the fatty acid methyl esters was used to predict the vapour pressure and liquid heat capacity of different biodiesel fuels using ideal mixing rules. The
A
AARD was 12% for vapour pressure and 3% for liquid heat capacity. The model proposed in this work predicted vapour pressure and heat capacities with enough accuracy for most practical engineering calculations and provides a more general model than analytical vapour pressure equations since it can be used to predict the phase
behavior, volumetric, and caloric properties of compounds making up biodiesel fuels, biodiesel fuel, and their blends
ACKNOWLEDGEMENTS The authors are grateful to the sponsors of the NSERC Industrial Research Chair in Heavy Oil Properties and Processing including the Natural Sciences and Engineering Research Council (NSERC), Petrobras, Schlumberger, and Shell. We thank Virtual
PT
Materials Group for the use of VMGSimTM software. We also thank Drs. Lovestead and Bruno of the National Institute of Standards and Technology for their previous
RI
contribution to the experimental data used in this study.
SC
NOMENCLATURE Definition
aPV,i
Adjustable parameter for Equation 5
ΔCP
Phase transition heat capacity
kJ/kmol·K
CP,L
Liquid heat capacity
kJ/kmol.K
CP 0
Ideal gas heat capacity
kJ/kmol.K
Residual heat capacity
kJ/kmol.K
Optimization objective function
-
Molecular mass
Kg/k-mol
N
A
M
CP
U
Symbol
Res
D
J
TE
MW NUC
Units -
Number of unsaturated carbon
-
Pressure, Vapour pressure
kPa
Temperature
ºC
Reference temperature
-
xi
Mole fraction
-
α
Similarity function, Equation 13
-
ν
Stoichiometric value of an element in a compound
T
A
CC
TRef
EP
P, PV
References
[1] Allen, C.A.W., Watts, K.C., Ackman, R.G., Peg, M. J. Predicting the viscosity of biodiesel fuels from their fatty acid ester composition, Fuel, 78, (1999), 1319-1326
[2] Goodrum, J. W. Volatility and boiling points of biodiesel from vegetable oils and tallow, Biomass and Bioenergy, 22, (2002), 205-211 [3] Conceiçao, M.M., Roberlúcia, A.C., Silva, F.C., Bezerra, A.F., Fernandes Jr., V.J., Souza, A.G. Thermoanalytical characterization of castor oil biodiesel, Renewable & Sustainable Energy Reviews, 11, (2007), 964-975 [4] Ott, L., Bruno, T. Variability of biodiesel fuel and comparison to petroleum-derived diesel fuel: application of a composition and enthalpy explicit distillation curve method, Energy & Fuel, 22, (2008), 2861-2868
PT
[5] American Oil Chemist’s Society (AOCS), The Lipid Library, USA, 2001
RI
[6] Knothe, G.; van Gerpen, J., and Krahl, J., The Biodiesel Handbook, AOCS Press, 2005
SC
[7] Yuan, W., Hansen, A.C. Zhang, Q. Vapour pressure and normal boiling point predictions for pure methyl esters and biodiesel fuels, Fuel, 84, (2005), 943-95
N
U
[8] Castellanos Díaz, O, Schoeggl, F., Yarranton, H.W., Satyro, M.A., Lovestead, T/M., Bruno, T.J. Modeling the vapour pressure of biodiesel fuels; World Academy of Science, Engineering and Technology. 65, (2012), 876-886
M
A
[9] Narvaez, P. C., Rincon S. M., Castaneda, L. Z. and Sanchez, F. J. Determination of Some Physical and Transport Properties of Palm Oil and of its Methyl Esters, Latin American Applied Research, 38, 2008, 1-6
D
[10] Dzida, M., and Prusakiewicz, P. The Effect of Temperature and pressure on the Physicochemical Properties of Petroleum Diesel Oil and Biodiesel Fuel, Fuel, 87, 2008, 1941-1948
EP
TE
[11] National Institute of Standards and Technology (NIST). Thermo Data Engine (TDE) Version 6.0, Pure compounds, Equations of state, Binary mixtures, and Chemical Reactions. NIST Standard reference Database #103b. Thermodynamic Research Center. USA. 2011
CC
[12] Bruno, T. J., and Svoronos, P. D. N., CRC Handbook of Fundamental Spectroscopic Correlation Charts, Taylor and Francis Group, 2006
A
[13] Bruno, T. J., and Svoronos, P. D. N., CRC Handbook of Basic Tables for Chemical Analysis, Third Edition, CRC Press, Boca Raton, 2011 [14] National Institute of Standards and Technology, NIST, NIST/EPA/NIH Mass Spectral Library Version 1.0, USA, 1995
[15] Weir, R. D., and de Loos, Th. W. Measurement of the Thermodynamic Properties of Multiple Phases, IUPAC, Physical Chemistry Division, Commission on Thermodynamics, Elsevier, The Netherlands, 2005 [16] Haines, P. J., Thermal Methods of Analysis: Principles, Applications and Problems, Springer, 2002 [17] Castellanos-Diaz, O., Schoeggl, F., Yarranton, H., and Satyro, M., Measurement of Heavy Oil and Bitumen Vapour Pressure for Fluid Characterization, Industrial Engineering Chemistry Research Journal, 52(8), 2013, 3027-3035
PT
[18] Peng, D.Y., Robinson, D.B. A new two-constant equation of state, Ind. Eng. Chem. Fund., 15(1), (1976), 59-64
RI
[19] Virtual Materials Group Inc (VMG). VMGSim Version 5.0.5, VMGSim User’s Manual. Calgary Alberta, 2010
SC
[20] Poling, B.E., Prausnitz, J.M., O’Connell, J.P., The Properties of Gases and Liquids, 5th edition, McGraw-Hill, 2001
N
U
[21] Walas, S. M. Phase Equilibria in Chemical Engineering, Butterworth Publishers, U.S.A., 1985
M
A
[22] Gao, G., Dadiron, J.L., Saint-Guirons, H., Xans, P., Montel, F. A simple correlation to evaluate binary interaction parameters of the Peng-Robinson equation of state: binary light hydrocarbon systems. Fluid Phase Equilibria, (1992), 74, 85-93
TE
D
[23] van Genderen, A. C. G., van Miltenburg, J. C.; Block, J. G.; Van Bommel, M. J.; van Ekeren, P. J.; van den Berg, G. J. K.; Oonk, H. A. J.; Liquid-Vapor Equilibria of the Methyl Esters of Alkanoic Acids: Vapor Pressures as a Function of Temperature and Standard Thermodynamic Function Changes, Fluid Phase Equilibria, 202, 2002, 109-120
A
CC
EP
[24] Bureš, M.; Majer, V.; and Zábranský, M. Modification of Benson Method for Estimation of Ideal-Gas Heat Capacities, Chem. Eng. Sci., 36, 1981, 529-537
Table 1. Data available for selected FAMEs and temperature range in °C [11]
Points 65 53 70 112 90 29 110 27 101 29 12 4 33 -
C20:1(11)
8 18 12
26.85 26.85 26.85
176.85 214.95 185.7
Points 12 10 8 7 5 5 5 4 3 -
Tmax 76.85 76.85 76.85 76.85 76.85 76.85 76.85 76.85 76.85 -
-
-
1 1 1
N
C22:1 C18:2 C18:3
Tmax 146.52 145.70 188.20 226.85 237.8 226.85 321.95 226.85 346.95 226.85 258.95 176.85 218.50 -
∆Hv Points Tmin 2 25.0 3 25.0 2 25.0 4 25.0 4 25.0 4 25.0 2 25.0 2 25.0 2 25.0 2 25.0 2 25.0 1 25.0 2 25.0 1 25.0
PT
C6:0 C8:0 C10:0 C12:0 C14:0 C15:0 C16:0 C17:0 C18:0 C20:0 C22:0 C24:0 C16:1 C17:1 C18:1(11) C18:1(9)
CPL Tmin -33.15 -3.15 6.85 25 26.85 36.85 36.85 46.85 56.85 -
RI
Methyl hexanoate Methyl caprylate Methyl caprate Methyl laurate Methyl myristate Methyl pentadecanoate Methyl palmitate Methyl heptadecanoate Methyl stearate Methyl arachidate Methyl behenate Methyl lignocerate Methyl palmitoleate Methyl heptadecenoate Methyl oleate Methyl vaccenate Methyl cis-11eicosenoate Methyl erucate Methyl linoleate Methyl linolenate
PV Tmin 7.55 33.69 -12.74 -11.00 0.00 21.85 18.00 21.85 21.85 38.00 21.85 26.85 26.85 -
SC
Formula *
U
FAME
25.0 25.0 25.0
D
M
A
* In the CN:S nomenclature for biodiesels, N stands for the carbon number of the carboxylic acid from which the biodiesel was generated, and S stands for the number of unsaturated carbon bonds. For instance, C18:1(11), methyl oleate, is a biodiesel formed from oleic acid (with 18 carbon atoms) and one double bond between the eleventh carbon and twelfth carbon atoms
TE
Table 2.Temperature range of vapour pressure and heat capacity data for selected biodiesel fuels [8]
A
CC
EP
Biodiesel fuels Source Canola (South Alberta) Canola (Saskatchewan) Soy (Sunrise, US) Soy (Mountain Gold, US) Rapeseed (Europe) Palm (Europe) Coconut (Europe)
Code CB-01 I-25 SB100 MGB100 S102550 S090824 S070717
Vapour Pressure 60-196 °C 140 °C 80-110 °C 70-100 °C 95-125 °C
Liquid Heat Capacity 13-55 °C 12-55 °C 14-55 °C 10-55 °C 13-55 °C 23-55 °C 10-55 °C
Tmax 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0
Table 3.Composition in mole percentage of FAMEs in selected biodiesel fuels SB100 0 0 0 0 0 0 0 11.4 0 0 3.2 21.3 1.5 54.9 7.4 0.3 0 291.8
MGB100 0 0 0 0 0.6 0 0.5 12.5 0 0 4.9 27.0 1.6 46.6 6.1 0.3 0 291.3
S090824 0 0 0 0 1.5 0 0 45.1 0 0 3.6 39.5 0 9.8 0.2 0.3 0 283.7
S070717 1.0 12.6 7.7 48.3 16.6 0 0 6.7 0 0 1.5 4.4 0 1.1 0 0 0 218.2
U
* For the CN:S nomenclature for biodiesels, refer to Table 1
S102550 0 0 0 0 0 0 0 4.8 0 0 1.28 59.91 3.68 19.44 9.08 1.26 0.55 294.6
PT
I-25 0 0 0 0 0 0 0.9 9.3 0.3 0 4.4 57.4 2.8 16.0 7.5 0.4 1.00 293.2
RI
CB-01 0 0 0 0 0 0 12.7 12.7 0 0 4.1 23.5 1.5 49.9 8.1 0.2 0 291.5
SC
FAMEs C6:0 C8:0 C10:0 C12:0 C14:0 C15:0 C16:1 C16:0 C17:0 C17:1 C18:0 C18:1(9) C18:1(11) C18:2 C18:3 C20:0 C20:1 MWavg
CC A
Not Tuned 2857 2064 1752 1755 1147 1384 1313 1225 1157 1031 999 1321 1231 1173 1153 1002 1204 1127
M
D
Tc [ºC] Tune Tc, Pc, ω 329.6 374.8 409.8 439.0 453.8 462.8 476.6 486.4 498.8 514.8 530.8 476.8 496.8 503.8 514.8 524.8 509.8 516.8
EP
C6:0 C8:0 C10:0 C12:0* C14:0 C15:0 C16:0 C17:0 C18:0 C20:0 C22:0 C16:1 C17:1 C18:1(9) C20:1 C22:1 C18:2 C18:3
Not Tuned 328.8 362.4 398.0 439.0 447.8 462.8 481.8 486.8 498.8 516.8 542.8 477.8 488.8 494.8 523.8 543.8 499.8 500.8
TE
Formula
A
N
Table 4. Critical properties and acentric factor of selected FAMEs; comparison between contribution method predictions (not tuned) and tuned values for the APR EoS. Pc [kPa] Tuned Tc, Pc, ω 2950 1984 1570 1507 1175 1384 1250 1224 1158 1031 985 1830 1780 1710 1650 1600 3150 4850
*Critical temperature experimentally measured
Not Tuned 0.462 0.447 0.511 0.575 0.804 0.902 0.853 0.946 0.970 1.015 1.180 1.125 1.466 3.071
Tuned ω 0.584 0.660 0.715 0.829 0.876 0.906 0.937 0.969 1.032 0.966 0.942 0.965
Tuned Tc, Pc, ω 0.490 0.509 0.596 0.696 0.803 0.873 0.906 0.937 0.969 1.032 1.088 0.977 0.979 0.991 1.116 1.195 1.052 1.105
Table 5. Performance (AARD) of the APR EoS for the prediction of vapour pressure and enthalpy of vapourization for various FAMEs
RI
PT
Case 3: “Tuning Tc, Pc, ω” PV [%] ∆HV [%] 5 11 1 12 1 12 1 8 0 4 1 6 1 2 4 1 11 1 27 13 6 1 1 10 12 -
SC
Case 2: “Tuning ω” PV [%] ∆HV [%] 8 5 4 4 6 2 5 1 5 1 7 1 2 4 1 11 1 4 14 14 -
A
N
C6:0 C8:0 C10:0 C12:0 C14:0 C15:0 C16:0 C17:0 C18:0 C20:0 C22:0 C16:1 C17:1 C18:1(9) C20:1(11) C22:1 C18:2 C18:3
Case 1: “No Tuning” PV [%] ∆HV [%] 127 7 77 9 74 9 7 1 8 3 12 2 2 9 4 11 5 159 235 648 -
U
Formula
This work 0.6 0.4 0.4 0.2 0.3 0.3 0.3 0.6 0.1
D
Formula
M
Table 6. AARD values for the APR-EoS for the prediction of liquid heat capacity for various FAMEs with three different methods for the ideal gas heat capacity prediction.
A
CC
EP
TE
C8:0 C10:0 C12:0 C14:0 C15:0 C16:0 C17:0 C18:0 C20:0
∆CPL AARD [%] Modified Benson 11.4 9.9 8.9 7.7 7.1 6.9 6.2 6.8 7.1
Joback 11.8 10.2 9.1 7.8 7.3 7.0 6.3 6.8 7.1
Table 7. Performance of APR-EoS prediction of biodiesel fuel vapour pressure and liquid heat capacity AARD [%] PV CPL Canola (South Alberta) CB-01 21.7 1.9 Canola (Saskatchewan) I-25 11.7 3.3 Soy (Sunrise, US) SB100 8.8 1.4 Soy (Mountain Gold, US) MGB100 15.7 2.7 Rapeseed (Europe) S102550 9.9 6.6 Palm (Europe) S090824 4.4 2.0 Coconut (Europe) S070717 14.5 1.3 Total 12.3 2.7 Code
Figure 1. Schematic of optimization procedure.
SC
RI
PT
Biodiesel Fuel Source
A
N
U
Figure 2. Experimental and predicted vapour pressure and heat of vapourization data for methyl caprate, C10:0 using adjusted critical properties and acentric factor (Data from NIST [11]).
M
Figure 3. Calculated and regressed ideal gas heat capacity for methyl caprate, C10:0.
D
Figure 4. Parameters of Equation 21 as a function of molecular weight.
TE
Figure 5. Experimental and predicted liquid heat capacity data for methyl caprate, C10:0 (Data from NIST [11]).
EP
Figure 6. Experimental and APR-EoS predicted vapour pressure of biodiesel fuels listed in Table 7.
A
CC
Figure 7. Experimental and APR EoS predicted liquid heat capacity of biodiesel fuels listed in Table 7.
Figure 8. Experimental and predicted vapour pressure of coconut biodiesel fuel with the n values ranging from Zc to -5. Figure 9 Experimental and predicted liquid heat capacity of coconut biodiesel fuel with the n values ranging from Zc to -5.
Calculate from Eq. 19 Look up Tc, Pc or calculate from Eqs. 20-24
Calculate A and B from Eqs. 3 and 4 Calculate HV from Eq. 13
Choose vapour pressure (=P)
PT
Calculate flliquid and flvapour from Eq. 12
adjust P flliquid = flvapour ?
RI
no
SC
yes
adjust or adjust Tc, Pc
OF = minimum?
End
yes
A
N
no
U
Calculate OF from Eq. 14
1.E+01
D TE
1.E-01 1.E-02 1.E-03
CC
1.E-04
EP
Pressure [kPa]
1.E+00
Experimental PR EoS with Adjusted Tc, Pc PR EoS with Predicted Tc, Pc
1.E-05
A
0
50
100 Temperature [C]
150
69 Heat Capacity Difference [kJ/mol]
M
1.E+02
200
Experimental Data Source 1 Experimental Data Source 2 PR EoS with Adjusted Tc, Pc PR EoS with Predicted Tc, Pc
68 67 66 65 64 63 -20
0
20 40 60 Temperature [C]
80
100
Calculated
340
Regressed
330 320 310 300 290 280
PT
1600
RI
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 Temperature [C] 1 Parameters values
1200
Model
0
SC
1400
-1
1000
N
-3
600
A
400
0 150
200
250
300
A
CC
EP
TE
D
MW
M
200
350
1.E-02 8.E-03
U
bCP,0
aCP,0
-2
800
1.E-02
-4
cCP,0
Ideal Gas heat Capacity [kJ/kmol.K]
350
6.E-03
Parameter values Model Regressed
4.E-03
-5
2.E-03
-6
0.E+00 150
200
250 MW
300
350
Liquid Heat Capacity [kJ/kmol.K]
420 400 380 360 340 320
Experimental Data APR EoS with New Cp0 Correlation APR EoS with Jobak Method APR EoS with Benson Method
300
0
20
40 60 Temperature [C]
100
1.E+00
1.E+00 1.E-01
N
1.E-01
Pressure [kPa]
U
SC
1.E+01
A
1.E-02
Rapeseed Soy MG Canola I25 Coconut APR EoS Model kij = 0
M
Pressure [kPa]
1.E+01
1.E-03
25 50 75 100 125 150 175 200 225 Temperature [C]
CC
EP
TE
0
D
1.E-04
A
80
RI
-20
PT
280
1.E-02 Palm
1.E-03
Soy S
1.E-04
Canola CB APR EoS Model kij = 0
1.E-05 0
25 50 75 100 125 150 175 200 225 Temperature [C]
750
Liquid Heat Capacity [kJ/kmol.K]
700 650 600
Rapeseed Soy MG Canola I25 Coconut APR EoS Model kij = 0
550 500 450 400
700 650 600 550 Palm Soy S Canola CB APR EoS Model kij = 0
500 450 400
5 10 15 20 25 30 35 40 45 50 55 60 Temperature [C]
0
1.E+01
M
A
N
U
SC
Experimental Gao et al. Gao et al. n=0 Gao et al. n=-5
]a1.E+00 P k[ e r u ss e r P1.E-01
D
1.E-02
50 100 Temperature [C]
A
CC
EP
TE
0
5 10 15 20 25 30 35 40 45 50 55 60 Temperature [C]
RI
0
PT
Liquid Heat Capacity [kJ/kmol.K]
750
150
480 ] 470 K l. o 460 m k/ Jk [ 450 yt ic a440 p a C ta430 e H d i 420 u q iL 410
Experimental Gaoet al. Gaoet al. n=0 Gaoet al. n=-5 5 10 15 20 25 30 35 40 45 50 55 60 Temperature [C]
A
CC
EP
TE
D
M
A
N
U
SC
RI
0
PT
400