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Vapor-Pressure Measurements and Modeling of Dipentaerythritol Ester Lubricants Josefa García,† Ramy Abou Naccoul,‡ Josefa Fernandez,*,§ Antonio Razzouk,‡ and Ilham Mokbel‡ †
Departamento Física Aplicada, Edificio Ciencias Experimentales, Univ. de Vigo, E-36310 Vigo, Spain Laboratoire des Sciences Analytiques, UMR 5180, Universite Claude Bernard, Lyon1, 69622 Villeurbanne, France § Laboratorio de Propiedades Termofísicas, Departamento de Física Aplicada, Universidade de Santiago de Compostela, E-15782 Santiago de Compostela, Spain ‡
ABSTRACT: The aim of this work is to present new vapor-pressure measurements and provide a good description of the volumetric and phase behavior of dipentaerythritol ester (DiPEs) lubricants, using the Statistical Associating Fluid Theory (SAFT) model (PC-SAFT and SAFT-VR versions). Characteristic parameters of these versions were optimized for dipentaerythritol hexapentanoate (DiPEC5), dipentaerythritol hexaheptanoate (DiPEC7), and dipentaerythritol isononanoate (DiPEiC9), using experimental vapor pressures and densities. With these parameters, compressed densities were predicted. Experimental vapor pressures, determined by a gas saturation apparatus, range between 2 10-5 Pa and 16 Pa, whereas the absolute deviations obtained in the correlations for both versions of the model (PC-SAFT and SAFT-VR) are between 10-6 Pa and 1 Pa. In the case of saturated densities, the average absolute deviation (AAD) values, for both models, range from 0.2% to 1.3%, except for DiPEC7, where the AAD is 4.4% when using the SAFT-VR model. The SAFT-VR version gives slightly better correlations, although we must note that this version has four parameters, whereas the PC-SAFT version has three.
1. INTRODUCTION Organic esters are one of the most used synthetic fluids since World War II. The two more-used synthetic ester lubricants are diesters and polyolesters (POE). For most applications, diesters have been substituted by polyolesters, because of the higher thermal stability and higher viscosity-grade range of POEs.1 There are four types of polyol esters, depending on the alcohol used for their synthesis: neopentyl glycol, trimetylolpentane, pentaerythritol, and dipentaerythritol, which have two, three, four, and six ester groups per molecule, respectively.2 Their viscosity increases with the number of ester groups. The primary use of POE was in engine jet oils. In addition, they are used mainly in air compressor oils and as refrigeration oils in combination with hydrofluorocarbons (HFCs). In the case where a carbon dioxide refrigerant is used, lubricant oils with higher viscosity should be used, because of the high solubility of CO2 in esters and the low viscosity of carbon dioxide lead to a marked viscosity reduction of the lubricant in the compressor. This must be compensated by an increase in viscosity of the lubricant.3-5 Hence, it is convenient to use dipentaerythritol ester (DiPE) alone or blended with pentaerythritol esters (PE) to get the correct viscosity grade, to prevent wear in the compressor. With high viscosity and excellent heat and oxidation stability, dipentaerythritol esters are also suitable base oils for aviation gas turbine lubricants, high-temperature chain oil, and high-temperature greases (for example, heat conduction greases). Vapor pressures are fundamental data that are required in several applications, among them, use in the design of several process and products in the petroleum, refrigeration, industrial fluid, and lubrication industries. In lubrication design, this property is very useful, because it provides information about the tendency r 2011 American Chemical Society
of vaporization of the different oils or other lubricants and, thus, the potential of contamination from evaporative loss: the lower the vapor pressure, the better the lubricant, especially in the case of many space applications and lubricants for the disk drive industry.6,7 In the case of engines, there is a direct association between oil volatility and oil consumption rates.1 Because of the great difficulty in measuring low vapor pressures, the usual techniques in the lubrication field are based on weight loss due to evaporation: the Knudsen effusion method, thermogravimetric analysis (TGA), and nonequilibrium weight loss measurements under vacuum.8 Karis9 recently developed an estimation method in which the evaporation rate is determined by isothermal thermogravimetric analysis, using a stagnant film diffusion model. Nevertheless, these methods are not precise and the uncertainty is unknown, because several assumptions are taken into account without verification. For these reasons, a gas saturation apparatus, described elsewhere,10,11 has been used to determine the vapor pressures of some lubricant bases. In a previous work,12 the vapor pressures of four pentaerythritol esters were measured. In the present work, we extended this study to lower volatility compounds: dipentaerythritol hexapentanoate (DiPEC5), dipentaerythritol hexaheptanoate (DiPEC7), and dipentaerythritol isononanoate (DiPEiC9). In addition, a description of the volumetric and phase behavior of these dipentaerythritol ester lubricants, using PC-SAFT and SAFT-VR versions of the SAFT model, is provided. Received: October 25, 2010 Accepted: February 24, 2011 Revised: February 16, 2011 Published: March 15, 2011 4231
dx.doi.org/10.1021/ie102166r | Ind. Eng. Chem. Res. 2011, 50, 4231–4237
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2. EXPERIMENTAL SECTION 2.1. Materials. Dipentaerythritol hexapentanoate (CAS No. 76185-96-1, C40H70O13), dipentaerythritol hexaheptanoate (CAS No. 76939-66-7, C52H94O13), and dipentaerythritol isononanoate (CAS No. 127304-08-9, C64H118O13) were provided by Croda-Uniqema (United Kingdom). The samples of DiPEC5 and DiPEC7 are aliquots of those used by Paredes et al.13 These last products were analyzed by 1H NMR, 13C NMR, and mass spectroscopy. The estimated mole fraction purity for both compounds was better than 0.95. 2.2. Apparatus and Operation. The apparatus for vaporpressure determination is based on a gas saturation method, which also is known as the transpiration method. The apparatus allows reliable measurements within a large pressure interval ranging from 10-6 Pa to 103 Pa. The description in details of the saturation apparatus can be found elsewhere.10,11 Hence, we only give the most significant information and modifications made to improve the apparatus. The experimental apparatus, presented in Figure 1, is composed of two parts. The sampling part consists of an equilibrium oven that contains two saturators, which consist of stainless steel columns, filled with a porous gas chromatography (GC) support, respectively impregnated with the sample and the standard compounds. The second part is a GC system equipped with a SGE BPX1 column and a FID detector. When thermal equilibrium is reached, the sample and the standard compounds are simultaneously swept by the inert gas N2 into the cold GC column, where they are trapped. To limit adsorption and desorption phenomena, we modified the connection between the saturators and the GC system. A silica capillary tube (T) is connected to the outlet tube of the saturators on one side, whereas the other end penetrates inside the GC column. Under these conditions, the carrier gas does not pass through the union tube of fused silica and desorbs only the compounds trapped in the analysis column. By heating the capillary column, the two
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compounds are eluted and detected by the flame ionization detector (FID). The present apparatus is totally automatic, because all the valves are controlled by the GC output. 2.2.1. Impregnation of the Support. The impregnation of the support by the sample or the standard compounds is done by batch, where ∼1.5 g of the compounds was dissolved in toluene. The chromatographic support was then added to the solution in a way to have a “compound weight/support weight” ratio equal to 20%. The dry impregnated support is then introduced into the saturation stainless steel column (length = 2 m, id = 31 mm). 2.2.2. Saturation Gas Flow Rate and Purge Time. A preheated nitrogen steam was passed through the saturators at constant temperature. The flow rates were measured with an uncertainty of 1%, using mass flow meters from Bronkhorst. The flow rates were optimized to reach the saturation equilibrium of the gas. Thus, experiments were carried out using flow rates in the range of 3-8 mL min-1, according to the equilibrium temperature and the volatility of the compounds. The same trap time was applied to both sample and reference compounds. It varies between 30 min to 20 h, depending on the saturation limit of the analytical column. 2.2.3. Vapor-Pressure Determination. The vapor pressures were calculated using the following equation, which supposes ideal behavior of the vapor phase: P1 A1 M2 F2 ¼k ð1Þ P2 A2 M1 F1 Here, the subscripts i refer, respectively, to the standard and the sample; Pi is the vapor pressure; Ai is the chromatographic peak area; Mi is the molar mass; Fi is the saturation gas flow rate; and k is the relative response factor of the FID. This factor was determined for each couple (sample/reference) by means of standard solutions.
3. EXPERIMENTAL RESULTS The relative uncertainty on the pressure ratio from eq 1 is ∼4%.10 In the present study, the reference compounds used were octacosane for DiPEC5 and DiPEC7 studies and tetracontane for DiPEiC9. The vapor pressure of octacosane has been measured in our laboratory using the same apparatus. The estimated uncertainty was ∼7%.10 By a quadratic combination with the pressure ratio, the relative uncertainty of the vapor pressures for DiPEC5 and DiPEC7 is 8%. As for tetracontane, the vapor pressure has been extracted from the literature.14 Applying the same method, the uncertainty of DiPEiC9 is 9%. The uncertainty of the temperature is (0.02 K. The experimental vapor-pressure values of the three esters are reported in Table 1 and are plotted in Figure 2. As expected, the vapor pressures decrease with the length of the chain of the molecule. The vapor pressures of the three esters were determined and fitted to the Clausius-Clapeyron equation: Figure 1. Schematic of the saturation apparatus. [Legend: A - Chemstation acquisition, from Agilent; C - analytical capillary column; F1, F2, F3 - mass flow meters from Bronkhorst, with a flow range of 0-10 mL min-1 and an uncertainty of 1%; H - heated zone; S1, S2 - saturation stainless steel columns (L = 2 m; inner diameter (id) = 2.1 mm) containing Gas Chrom P support (particle diameter of 147-175 μm); T - capillary silica tube (L = 25 cm ; id = 0.32); T1, T2, T3 - stainless steel tubing (L = 3 m; id = 0.50 mm) for saturation and carrier gas preheating; and V1, V2, V3 - electrovalves that control gas flow.]
ln P ðmmHgÞ ¼ B -
A TðKÞ
with
A¼
ΔH vap R
ð2Þ
The Clapeyron parameters, along with the standard deviation, are reported in Table 2. From the slopes of the straight lines of Figure 2, mean vaporization enthalpies were estimated assuming that (i) the vapor phase behaves as an ideal gas, (ii) the molar volume of the liquid is very small, with regard to the vapor phase, 4232
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and (iii) the inert gas is not soluble in the condensed phase. The estimated vaporization enthalpies for DiPEC5, DiPEC7, and DiPEiC9 are, respectively, 168.7, 266.7, and 185.9 kJ mol-1, respectively. These values are significantly higher than those obtained in a previous study12 for pentaerythritol tetrapentanoate (PEC5), pentaerythritol tetraheptanoate (PEC7), pentaerythritol tetranonanoate (PEC9), and pentaerythritol tetra 2-ethylhexanoate (PEBE8), for which the mean vaporization enthalpy ranges from 119 kJ mol-1 for PEC5 to 154 kJ mol-1 for PEC9. The vaporization enthalpies increase with the chain length and decrease with the branching degree of the DiPEs molecules. Table 1. Vapor Pressure of Dipentaerythritol Esters temperature, T/K
pressure, P/Pa DiPEC5
392.71 412.53
0.0000201 0.000220
432.41
0.00340
475.42
0.186
494.45
1.09
515.61
5.84
534.15
16.7 DiPEC7
473.15
0.000232
494.45 516.32
0.00704 0.129
534.15
0.515 DiPEiC9
473.15
0.00446
493.15
0.0342
513.15
0.186
543.15
1.91
553.15
4.57
4. THEORY A careful characterization of lubricants for a specific application can be obtained by experimental techniques, simulations, and/or theoretical approaches. A main advantage of a theory or an equation of state versus the other techniques is the speed and low cost in which these calculations are performed to reproduce the experimental data. Therefore, in this article, there is interest in a physically based model that might be more useful for process engineering computations, such as PC-SAFT and SAFT-VR equations of state. Based on Wertheim’s theory, Chapman et al.15,16 derived the Statistical Associating Fluid Theory (SAFT) equation of state for chain molecules. Subsequently, several modifications of the SAFT model were suggested, such as the version of Gil-Villegas et al.,17 which is known as SAFT Variable-Range (SAFT-VR), or the Perturbed-Chain SAFT (PC-SAFT) version reported by Gross and Sadowski.18 In the SAFT-VR approach, the nonassociating chain molecules are modeled as flexible chains formed from m tangent spherical segments. Each segment in a chain has the same diameter σ, but segments belonging to different species can have different diameters. The dispersive interactions between the segments can be modeled using any standard attractive pair potential of depth ε and variable range λ. We use SAFT-VR EoS with the square-well (SW) potential. The general form of the Helmholtz free energy, for nonassociating chain molecules, Agnrl, which interact only through van der Waals dispersive forces, is written as Agnrl Aideal Amono Achain ¼ þ þ NkT NkT NkT NkT
ð3Þ
where N is the number of molecules, T the temperature, and k the Boltzmann constant. Aideal is the ideal free energy. Amono is the monomer free energy, which is the contribution due to the segment-segment interactions obtained from the high-temperature perturbation theory of Barker and Henderson,19-21 with a hard-sphere fluid as the reference system. This is expressed as a sum of the free energy of the hard-sphere reference system (AHS)
Figure 2. Plot of the Clapeyron equation of the three dipentaerythritol esters: () DiPEC5, (0) DiPEC7, and (4) DiPEiC9. 4233
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Table 2. Clausius-Clapeyron Equation Parameters B and A,a Standard Deviations of the Parameters (sA and sB), Average Relative Deviation (AAD),b and Estimated Vaporization Enthalpy (ΔvapH, from eq 2)
a
A (sA)
B (sB)
100 AAD
ΔvapH/kJ mol-1(s)
compound
temp range T/K
DiPEC5
392.71-534.15
20276 (258)
36.07 (0.55)
12
168.7 (2.1)
DiPEC7 DiPEiC9
473.15-534.15 473.15-553.15
32077 (1903) 22366 (284)
54.68 (3.79) 37.01 (0.57)
26 5.8
266.7 (19.4) 185.9 (2.4)
A = ΔvapH/R. b AAD = (1/n) ∑[|Pexp-Pcal|/Pexp].
and a perturbation contribution: Amono AHS Apert ¼ þ NkT NkT NkT
ð4Þ
VR where the perturbation contribution has two terms: AVR 1 and A2 .
Apert VR ¼ AVR 1 þ A2 NkT
ð5Þ
The two perturbation terms are defined in terms of the pair distribution function for the reference hard spheres and the segment-segment pair potential. Finally, Achain is the contribution due to the formation of chains of segments. The reader can find full details in refs 17 and 22. In the version reported by Gross and Sadowski,18 a nonassociated pure fluid consists of equal spherical segments m, of diameter σ, interacting according to a modified square-well suggested by Chen and Kreglewski23 of depth ε. The PC-SAFT equation, when expressed in terms of the Helmholtz free energy for nonassociated pure fluid, consists of the ideal contribution, hard-chain reference contribution, and the perturbative contribution for the dispersive forces. Agnrl Aideal AHC Apert ¼ þ þ ð6Þ NkT NkT NkT NkT The difference between eq 6 and eqs 3 and 4, for SAFT-VR, is that the second term on the right-hand side of the equation now corresponds to the free energy of a reference hard-sphere-chain fluid, whereas the perturbation and chain terms in eqs 3 and 4 have been replaced here by a chain perturbation contribution, Apert. As with SAFT-VR, the perturbation contribution is that obtained from the high-temperature perturbation theory of Barker and Henderson,19,20 but, in this case, with a hardsphere-chain fluid as the reference system: Apert Adisp PC ¼ ¼ APC 1 þ A2 NkT NkT
ð7Þ
PC where the perturbation terms APC 1 and A2 give the attractive interactions, which are treated using an accurate fitted alkanechain-segment potential. For a better performance, the integral expressions for the first- and second-order terms were fitted to appropriate Taylor series expansions in density using square-well chain analytical data and the saturated liquid-density and vaporpressure data for the first eight pure n-alkanes. Full details are given in ref 18. Thus, the most important difference for these two versions is that the perturbation terms contain two integral terms with expressions for the radial distribution function for hard spheres in the case of SAFT-VR and for the site-site radial distribution function for hard-sphere chains in the case of PC-SAFT. In SAFT-VR and PC-SAFT, the molecules are characterized by m, σ, and ε. However, PC-SAFT lacks the extra parameter λ of
SAFT-VR. The influences of the size and energy parameters in SAFT-VR and PC-SAFT are subtly different. In SAFT-VR, σ and ε (together with the range parameter λ) describe the potential that appears in the EoS, whereas, in the PC-SAFT equation of state (EoS), the segment-segment pair potential no longer appears explicitly as a result of the procedure for fitting the form of the free energy. Thus, in SAFT-VR with the SW potential used in this study, σ is explicitly the segment diameter and ε is explicitly the depth of the segment-segment potential well, whereas this is not necessarily true for the PC-SAFT EoS, in which these are more loosely defined size and energy parameters.22
5. MODELING RESULTS 5.1. Characteristic Parameters. In a previous paper,5 we
calculated the SAFT and PC-SAFT characteristic parameters of four pentaerythritol ester lubricants: PEC5, PEC7, PEC9, and PEB8. Nevertheless, as commented previously, dipentaerythritol esters (DiPEs) seem to be lubricants that are more adequate for the natural refrigerant CO2 than pentaerythritol esters (PEs).13,24 For this reason, in this work, the characteristic parameters of SAFT- VR and PC-SAFT were optimized for three DiPEs lubricants—DiPEC5, DiPEC7, and DiPEiC9— using experimental vapor pressures of this work and the saturated liquid densities obtained from Tammann-Tait correlation densities.25,26 In the optimization, two algorithms were used: for SAFT-VR, a simplex method was used;27 and for PC-SAFT, a simplex Nelder-Mead algorithm from phase Equilibria (PE2000, version 2.9.9a) software was used.28 The values of the parameters and deviations are given in Table 3. For the three compounds, the values obtained for the number of segments m are higher for PC-SAFT than for SAFT-VR, whereas the contrary happens for the diameter (σ) and energy (ε/k) parameters. The same trends occur for alkanes.17,18 The values found for the energy parameters ε/k, using PC-SAFT EoS for the pentaerythritol esters,12 are ∼30 K higher than those found for dipentaerythritol esters, whereas the numbers of segments for the PEs are quite lower (12.1 for PEC7, compared to 24.6 for DiPEC7). Taking into account the vapor-pressure values and the estimated vaporization enthalpy, it can be concluded that, overall, the total interaction energies in the DiPE compound are larger than in PEs of similar chain size. In addition, we previously obtained12 that, for PEs, the increased branching decreases the number of segments m and a similar effect was found for the DiPEiC9 in this work. This means that PC-SAFT models the branched molecules with less segments, to take into account the decrease in the number of interactions due to steric hindrance caused by the side groups of the chains of the DiPEs. For saturated pressures, the average deviation (AD) was determined by the very small values of this property for the lubricants studied (experimental vapor pressures in the range of 0.00002-16 Pa, such as that which can be seen in Table 1), 4234
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Table 3. Lubricant, Molecular Weight, and Characteristic Parameters of SAFT-VR and PC-SAFT Versions and the Number of Fitted Experimental Data for Saturated Pressures and Densities (n), Absolute Deviations for Saturated Pressures (AD), and Average Percentage Absolute Deviations for Saturated Densities (AAD%) in the Temperature Range of 283-398 K compound
Mw/g mol-1
m
s (Å)
DiPEC5
758.98
12.8387
4.3252
DiPEC7
927.29
15.9686
4.4998
DiPEiC9
1095.61
8.6369
6.1209
DiPEC5 DiPEC7
758.98 927.29
20.2826 24.6438
3.6823 3.7699
DiPEiC9
1095.61
19.5071
4.3525
ε/k/K
λ
n
349.42
1.4697
7
403.07
1.4162
4
652.57
1.2894
n
AAD%Psat
0.55
7
0.5
0.037
7
4.4
5
0.016
7
0.2
230.85 240.38
7 4
0.74 0.2 10-6
7 7
1.3 0.6
250.50
5
0.32 10-6
7
1.0
ADPsat/Pa
SAFT-VR
PC-SAFT
Figure 3. Experimental vapor pressures [this work] and densities,25,26 together with correlated values for DiPEs: (a) ln P (Pa) vs 1/T (K) and (b) density data vs T. Data points represent experimental values ((]) DiPEC5, (4) DiPEC7, and (O) DiPEiC9); lines represent correlations ((- - -) SAFTVR and (—) PC-SAFT).
whereas for the saturated density, the average percentage absolute deviation (AAD%) has been calculated. The expressions of these deviations are given for the following equations: 1 ð8Þ AD ¼ Psat;exp - Psat;theo n
Table 4. Absolute Average Deviation (AAD%), Maximum Deviation (MD%), and Bias (Bias%) for the Compressed Density of DiPEs, Using SAFT-VR and PC-SAFT Versions
∑
1 AAD% ¼ 100 n
∑
F sat;exp - Fsat;theo Fsat;exp
compound
na
Trange/K
AAD%
MD%
Bias%
-1.3
SAFT-VR (from 1 MPa to 70 MPa)
ð9Þ
where Psat,exp and Psat,theo, and Fsat,exp and Fsat,theo, are the experimental and theoretical saturated pressures and densities, respectively, and n is the number of fitted experimental data. The SAFT-VR version gives slightly better correlations (see Figure 3), because this version has four parameters, whereas PC-SAFT has three. We should note that the two versions are difficult to describe well with the same parameters, for both saturated pressures and densities, especially the densities of DiPEC7 with SAFT-VR. This could be due to the fact that the experimental temperature ranges for both properties are different. For vapor pressures, the temperature range is in the range of 392-553 K, whereas, for the saturated densities, the range is 283-398 K. An additional difficulty is the very small values of the vapor pressures (
