Viscosities of Fatty Acid Methyl and Ethyl Esters under High Pressure

Article pubs.acs.org/jced

Viscosities of Fatty Acid Methyl and Ethyl Esters under High Pressure: Methyl Caprate and Ethyl Caprate Matthieu Habrioux, Jean-Patrick Bazile, Guillaume Galliero, and Jean Luc Daridon* Laboratoire des Fluides Complexes et leurs Réservoirs, UMR 5150, Université de Pau, BP 1155, 64013 Pau Cedex, France ABSTRACT: The paper reports high pressure data of the viscosity of methyl caprate (C11H22O2) and ethyl caprate (C12H24O2). The measurements cover the temperature range from (293.15 to 353.15) K, from atmospheric pressure up to 200 MPa and were carried out by two different methods. One is based on a falling-body viscometer, whereas the other rests on a quartz crystal resonator technique. Based on the two sets of data, high pressure viscosity correlations are proposed to correlate within the experimental uncertainty the viscosity values as a function of temperature and pressure. Finally, a thermodynamic scaling method was used to describe the viscosity in terms of density and temperature.

fatty acid alkyl esters by using simple mixing rules10−12 as biodiesels are mainly composed of nonassociated components. However, the application of such predictive techniques required reliable experimental data on viscosity pure components. To this end, we have initiated a program of measurement of viscosity of pure fatty acid methyl (or ethyl) esters in an extended range of pressure. In the present work, viscosity measurements have been carried out in liquid methyl caprate (C11H22O2) and ethyl caprate (C12H24O2) by two different apparatus and techniques to check the consistency of the data. The first one is a falling body viscometer in which viscosity is determined from the measurement of the time taken for a sinker to fall freely inside a vertical cylinder containing the liquid sample. The second viscometer adopted is based on Thickness Shear Mode Quartz Crystal Resonator technique developed by our group. In this viscometer, the viscosity is evaluated from the measurement of the conductance spectra near the resonance of several overtones of a quartz crystal fully immersed in the liquid under pressure. The data cover the pressures from atmospheric to 200 MPa over the temperature range from (293.15 to 353.15) K. Considering these measurements, high pressure viscosity correlations are proposed to represent both sets of experimental data.

1. INTRODUCTION Biodiesels consist of mixtures of alkyl esters of fatty acid obtained by a transesterification reaction where a vegetable oil or an animal fat is combined with a short chain alcohol such as methanol or ethanol. The distribution of fatty acid alkyl esters may change a lot from one biodiesel to another according to which type of feedstock is used for the synthesis.1 Changes in fatty compounds profile affect the thermophysical properties2,3 of biodiesels and, therefore, influence engine performances as well as the composition of the exhaust gases. In the case of compression ignition engine, one of the most important factor that influences the quality of the combustion and, therefore, the engine performance, fuel consumption, and harmful emission is the fuel injection.4,5 During this operation, fuel flows at high velocity and under high pressure through one or more nozzle holes. This results in an instantaneous breakup of the liquid fuel in a small droplets spray. During its penetration in the combustion chamber, the atomized fuel heats, vaporizes, and mixes with the high temperature−high pressure surrounding air, causing its autoignition. The time delay between the beginnings of the injection and the injection depends broadly on the spray pattern and the droplet distribution, which is itself related to hole geometry, injection pressure air temperature, and fuel viscosity.6,7 An increase in viscosity lowers the breakup and increases the size of the droplet leading to a worsening of the combustion quality and consequently to a rising of harmful emissions. On the other hand, a decrease in viscosity may produce excessive leakage and may not provide sufficient lubrication of injectors and pumps. Therefore, having a good knowledge of fuel viscosity over an extended range of pressure and temperature is essential to optimize injection process in diesel engine. In the case of biodiesel, viscosity can either be measured under pressure8,9 for each biodiesel fuel or determined from the viscosity of pure © XXXX American Chemical Society

2. EXPERIMENTAL SECTION 2.1. Materials. Table 1 shows the sample descriptions of methyl caprate (decanoic acid methyl ester, CAS Number: 11042-9, molar mass: 186.2912 g·mol−1) and ethyl caprate (decanoic acid ethyl ester, CAS Number: 110-38-3, molar Received: October 24, 2014 Accepted: January 8, 2015

A

DOI: 10.1021/je500980a J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Journal of Chemical & Engineering Data

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resonant behavior of a piezoelectric quartz working in the thickness shear mode. The apparatus is mainly composed of a high pressure vessel initially designed to measure speed of sound.15 It is made up of a stainless steel autoclave cylinder closed at one end by a plug in which two high-pressure electric connections were located. These electrical connections are used to hold and plug the sensor into the cell. The microacoustic sensor supplied by International Crystal Manufacturing Co. (Oklahoma City, Oklahoma, U.S.A.) consists in a highly polished AT-cut quartz disk having a fundamental frequency of 2 MHz and a blank diameter of 25.4 mm. It is electrically excited by means of two electrodes (6.4 mm in diameter) deposited on both its lateral sides by vacuum evaporation of an adhesive layer of titanium of 10 nm thickness followed by a 100 nm thick layer of gold. It is connected to a network analyzer working in the frequency range 0.1 MHz to 5 GHz (Agilent E5071C) that monitors its admittance spectra. The pressure is produced in the cell by the liquid itself by means of a volumetric pump, and it is measured by using two pressure gauges (HBM) fixed between the pump and the cell. The former is calibrated in the full pressure scale (0.1 to 200) MPa, whereas the later is only calibrated between (0.1 and 100) MPa in order to reduce the uncertainty in this range.The temperature of the investigated liquid is controlled by entirely immersing the cell in a thermoregulated bath and it is measured by a Pt100 probe housed in a metal finger used to isolate it from the pressure When a radiofrequency (rf) voltage is applied between the electrodes, the quartz disk exhibits a shear deformation that generates a transversal wave propagating out through the contacting media. In a liquid environment, the shear acoustic wave is strongly absorbed in the first liquid layers and the quartz resonator is damped due to viscous friction. The mechanical coupling between the resonator and the surrounding media influences the resonant behavior of the quartz crystal. In the case of full immersion of the quartz crystal in a Newtonian liquid, the surrounding media causes a decrease of the resonance frequency and an increase of the dissipation compared to the unloaded quartz crystal. Both effects are related to the square root of the density−viscosity product. Therefore, measurement of these changes permits the use of a quartz resonator to determine the density−viscosity product by two independent methods, and from knowledge of the density, the viscosity can be derived. Although the change in resonance frequency provides an absolute method for measurement of viscosity, it is less accurate than dissipation technique under pressure as resonance frequency is strongly affected by pressure changes.16 Consequently, the dissipation method was only considered in this work to evaluate the viscosity. In this technique, the shift of the half-band-half width ΔΓn of the resonant peak of an overtone n higher than 1 is considered to quantify the dissipation and the liquid viscosity is deduced from its measurement by using the working equation17,18

Table 1. Details of the Chemicals Used in This Study chemical name

source

initial mole fraction purity

purification method

methyl caprate ethyl caprate decane

Sigma-Aldrich Sigma-Aldrich Sigma-Aldrich

0.99 0.99 0.994

none none none

mass: 200.3178 g·mol−1) used in the present work. Decane (CAS Number: 124-18-5) has also been used as reference fluid for calibration of the falling body viscometer and his purity is also given in Table 1. 2.2. Falling Body Viscometer. The first method deployed to carry out measurements of viscosity in compressed liquids is based on a falling body viscometer. In this technique, the dynamic viscosity is determined by measuring the time taken for a sinker to fall freely due to gravity inside a vertical tube containing the investigated liquid. The apparatus, which has been described previously in detail,13 consists mainly of a highpressure cylindrical cell containing an open cylindrical tube with a perfect cylinder shape in which the sinker moves through the liquid to be measured. The internal tube is placed vertically and concentrically inside the vessel in such a way that both its internal and external faces are surrounded by the pressurized fluid. The main aim of this configuration is to minimize the geometrical deformations of the tube due to pressure in order to maintain constant the narrow gap between the tube internal wall and sinker. In this case, the sinker falls in a self-centering concentric way with a constant terminal velocity and the liquid viscosity ηL can be evaluated from the time t taken for the sinker to fall between two points of known position according to the following working relationship: η = Κ(ρS − ρL )t

(1)

where ρS − ρL is the difference in density between the sinker, ρS, and liquid ρL, whereas K is a parameter specific of the instrument. It accounts for the geometrical dimensions of both the tube and the sinker. Its value is obtained at each working P, T condition by calibration with a liquid of known viscosity and density. For the present work, decane was considered for calibration using the viscosity data reported by Huber et al.14 The fall time is measured by observing the change in the inductance of two coils wrapped around the outer face of the cell and separated from each other by a distance of 150 mm. The experiments are repeated several times for each P, T condition and the falling time taken in the working relationship is the average value of six measurements having a reproducibility better than 0.5 %. The temperature of the viscometer is regulated by an external circulating fluid and is measured with a Pt100 with an uncertainty of ± 0.05 K in the temperature range investigated. The pressure is transmitted to the cell by the liquid itself using an external pump and measured thanks to a HBM-P3M pressure gauge (with an uncertainty of 0.2 MPa) fixed on the circuit linking the pump to the measurement cell. Taking into account the error in temperature, pressure, and fall time measurements as well as the uncertainty in the reference values, the overall experimental uncertainty in the viscosity measured by this first technique is estimated to be ± 2 % up to 100 MPa and ± 4 % between 100 MPa and 200 MPa. 2.3. Quartz Crystal Resonator Viscometer. The second viscometer employed to conduct viscosity measurements under high pressure is based on microacoustic sensors that use the

ηL =

⎞2 πZq2 ⎛ cexp ⎜ ⎟ 4ρL f03 ⎝ 1 + R interface ⎠

(2)

where Zq is the acoustic impedance of AT-cut quartz, ρL is the density of the investigated liquid, and f0 stands for the fundamental resonance frequency of the unloaded quartz crystal. cexp is the ratio of the shift of the half-band-half and the square root of the overtone number B



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Table 2. Experimental Values of Viscosity ηFB Measured with Falling Body Viscometer at Temperatures T and Pressures p for the Liquid Methyl Caprate and Ethyl Capratea p

T

η

T

η

T

η

T

η

MPa

K

mPa·s

K

mPa·s

K

mPa·s

K

mPa·s

1.07 1.32 1.58 1.86 2.17 2.50 2.86 3.26 3.70 4.18 4.70

353.15 353.15 353.15 353.15 353.15 353.15 353.15 353.15 353.15 353.15 353.15

0.825 1.01 1.21 1.42 1.64 1.88 2.13 2.41 2.71 3.03 3.38

1.16 1.44 1.74 2.06 2.42 2.81 3.23 3.70 4.22 4.78 5.41

353.15 353.15 353.15 353.15 353.15 353.15 353.15 353.15 353.15 353.15 353.15

0.888 1.10 1.32 1.56 1.82 2.10 2.40 2.73 3.08 3.46 3.88

0.1013 20 40 60 80 100 120 140 160 180 200

293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15

2.15 2.66 3.24 3.91 4.67 5.53 6.52 7.66 8.96

313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15

0.1013 20 40 60 80 100 120 140 160 180 200

293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15

2.34 2.94 3.62 4.40 5.29 6.30 7.45 8.78 10.30 12.06 14.10

313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15

Methyl Caprate 1.46 333.15 1.83 333.15 2.21 333.15 2.65 333.15 3.12 333.15 3.63 333.15 4.21 333.15 4.85 333.15 5.56 333.15 6.35 333.15 7.24 333.15 Ethyl Caprate 1.59 333.15 1.96 333.15 2.38 333.15 2.85 333.15 3.37 333.15 3.96 333.15 4.62 333.15 5.35 333.15 6.18 333.15 7.11 333.15 8.16 333.15

a

Standard uncertainties u are u(T) = 0.05 K, u(p) = 0.2 MPa, and the combined expanded uncertainties Uc (level of confidence = 0.95) is Uc(η) = 0.035 η.

cexp =

ΔΓn n

technique using 10 MPa steps up to 100 MPa and 20 MPa steps beyond. In this latter table, measurements undertaken at atmospheric conditions used for calibration of quartz crystal correspond to Ubbelohde tube experiments. The viscosity data at atmospheric pressure were correlated to temperature by using a Vogel−Fulcher−Tammann19−21 like equation

(3)

It is independent of the overtone number beyond the fundamental and any overtone can be considered to obtain its value. In this work, only the third harmonic was used to measure Cexp as this overtone presents the highest signal-tonoise ratio and it appears relatively unaffected by spurious modes. Finally, Rinterface is a coefficient related to the roughness of the quartz surface. It must be calibrated for each quartz crystal before use. As Rinterface is pressure-independent, the calibration was done by self-reference considering measurements performed at atmospheric pressure in the same liquid. For this purpose, a Ubbelohde tube connected to an automatic AVS350 Schott Gerate Analyzer was used. Taking into account the uncertainty of the temperature, the pressure, the calibration procedure as well as the error in the measurements of the shift of the half-band-half width, the overall experimental uncertainty in the viscosity measured by this second technique is estimated to be ± 2 % up to 100 MPa and ± 4 % between 100 MPa and 200 MPa.

⎡ B ⎤ η0 = A exp⎢ ⎣ T − C ⎥⎦

(4)

Table 4 list the parameters for each set of atmospheric data. The change in viscosity with respect to pressure were fitted to the following equation: ⎛η⎞ ⎛ F + Δp ⎞ ⎟ ln⎜⎜ ⎟⎟ = DΔp + E ln⎜ ⎝ F ⎠ ⎝ η0 ⎠

(5)

where Δp = p − patm. In this equation, D is temperature independent, whereas E and F are expressed as a function of temperature by considering polynomial forms

3. RESULTS AND DISCUSSION Measurements were performed along isotherms spaced at 20 K intervals from 293.15 K to 353.15 K in the pressure range from atmospheric pressure to 200 MPa. The density values ρL needed in eqs 1 and 2 have been taken from measurements reported by Ndiaye et al.15 for both methyl caprate and ethyl caprate. The viscosities measured every 20 MPa steps with the falling body viscometer are reported in Table 2, whereas Table 3 lists the viscosities measured with the quartz crystal resonator

E(T ) = E0 + E1T

(6)

2 F(T ) = F0 + FT 1 + F2T

(7)

Keeping the values of A, B, and C determined from atmospheric viscosity data, the six coefficients of eqs 5 to 7 have been estimated by minimizing the following objective function: C



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Table 3. Experimental Values of Viscosity ηQCR Measured with Quartz Crystal Resonator at Temperatures T and Pressures p for the Liquid Methyl Caprate and Ethyl Capratea p

T

η

T

η

T

η

T

η

MPa

K

mPa·s

K

mPa·s

K

mPa·s

K

mPa·s

1.08 1.19 1.30 1.43 1.55 1.68 1.81 1.97 2.11 2.26 2.43 2.76 3.19 3.62 4.07 4.54

353.15 353.15 353.15 353.15 353.15 353.15 353.15 353.15 353.15 353.15 353.15 353.15 353.15 353.15 353.15 353.15

0.837 0.927 1.02 1.12 1.22 1.31 1.42 1.53 1.63 1.75 1.87 2.13 2.41 2.72 3.05 3.41

1.16 1.28 1.43 1.59 1.74 1.87 2.07 2.20 2.37 2.59 2.82 3.27 3.72 4.25 4.70 5.36

353.15 353.15 353.15 353.15 353.15 353.15 353.15 353.15 353.15 353.15 353.15 353.15 353.15 353.15 353.15 353.15

0.893 1.00 1.10 1.21 1.32 1.45 1.58 1.72 1.84 1.98 2.13 2.42 2.74 3.14 3.53 3.98

0.1013 10 20 30 40 50 60 70 80 90 100 120 140 160 180 200

293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15

2.13 2.36 2.62 2.92 3.21 3.53 3.87 4.22 4.61 5.02 5.47 6.47 7.82 9.17

313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15

0.1013 10 20 30 40 50 60 70 80 90 100 120 140 160 180 200

293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15

2.33 2.59 2.91 3.22 3.57 3.95 4.35 4.80 5.23 5.74 6.25 7.46 8.83 10.7 12.4 14.5

313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15

Methyl Caprate 1.50 333.15 1.66 333.15 1.84 333.15 2.00 333.15 2.20 333.15 2.39 333.15 2.59 333.15 2.82 333.15 3.06 333.15 3.29 333.15 3.56 333.15 4.10 333.15 4.75 333.15 5.43 333.15 6.29 333.15 7.13 333.15 Ethyl Caprate 1.59 333.15 1.77 333.15 1.97 333.15 2.18 333.15 2.39 333.15 2.62 333.15 2.86 333.15 3.14 333.15 3.44 333.15 3.72 333.15 4.04 333.15 4.69 333.15 5.52 333.15 6.27 333.15 7.31 333.15 8.36 333.15

a Standard uncertainties u are u(T) = 0.1 K, u(p) = 0.01 MPa up to 100 MPa, u(p) =0.1 MPa between (100 and 200) MPa, and the combined expanded uncertainties Uc (level of confidence = 0.95) are Uc(η) = 0.02 η up to 100 MPa, Uc(η) = 0.04 η between (100 and 200) MPa.

⎛ ⎛ ⎞2 ⎞ ⎜ ⎜ ⎟ ⎛ ⎞ ⎟ + Δ F p ηi i OF = ∑ ⎜ln⎜ ⎟⎟ ⎟ − DΔpi − E ln⎜ ⎡ B ⎤⎟ ⎝ F ⎠⎟ i ⎜ ⎜ A exp⎢ ⎣ Ti − C ⎥⎦ ⎠ ⎝ ⎝ ⎠

Notice also that for each viscosity set, the average deviation is much smaller than the absolute average deviations and, therefore, the equation does not introduce any systematic error. Finally, it can be noted from Figures 1 and 2, showing the deviation as a function of pressure at a fixed temperature (333.15 K), that the deviations do not increase systematically with pressure. All these comparisons show that the function provides a very good representation of viscosity in the pressure−temperature range investigated. This good agreement also emphasizes the overall consistency between the viscosity measurements carried out with the two different techniques. For the components investigated in this work, no value of viscosity under pressure is available in the open literature, but several authors reported viscosity measurements carried out at different temperature under at atmospheric pressure. The measurements reported by Shigley et al.,22 Liew et al.,23 Pratas et al.24 represent the majority of the available data as a function of temperature. The other references25−28 provide only one or two temperatures conditions. Comparison between the measurements of Pratas et al.24 and eqs 4 to 7 shows a good agreement (Table 5) with an absolute average deviation of 0.5

Nexp

(8)

that takes into account experimental data obtained by both methods at pressures higher than atmospheric pressure. The values of the six coefficients determined in this way are given in Table 4 along with the average deviation, the average absolute deviation, and the maximum deviation with experimental data. In order to compare the deviation with the estimated uncertainties, the pressure ranges used to calculate the average and the maximum deviation were splitted in two domains: 0.1 MPa to 100 MPa and 120 MPa to 200 MPa. The former corresponds to an experimental uncertainty of 2 %, whereas the uncertainty associate to the in the later domain is twice greater. From this table it can be observed that the absolute average deviation observed between the experimental and calculated viscosity is much lower than the experimental uncertainty. In the worst case, the maximum deviation does not exceed 3.3 %. D



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Table 4. Parameters of Eqs 4 to 7 for Methyl Caprate and Ethyl Caprate from 293.15 K to 353.15 K and for a Pressure Range from 0.1 MPa to 200 MPa and Relative Deviations Δη/η in Averagea, Absolute Averageb, and Maximum Deviationc parameters

methyl caprate

ethyl caprate

4.779037 × 10−2 1.780816 × 10−2 3 1.178821 × 10 7.137804 × 102 4.682152 × 101 1.094214 × 102 −3 2.339591 × 10 1.779129 × 10−3 8.401486 9.273287 −2.127898 × 10−2 −2.307281 × 10−2 −1.908791 × 103 −1.124078 × 103 1 1.545424 × 10 1.054466 × 101 −2 −2.758378 × 10 −1.989928 × 10−2 Deviations with Falling Body Values (0.1 MPa to 100 MPa) AD % −0.5 −0.04 AAD % 1 0.7 MD % 2.5 1.4 Deviations with Falling Body Values (120 to 200 MPa) AD % −0.5 0.8 AAD % 0.7 0.9 MD % 1.9 1.9 Deviations with QCR Values (0.1 to 100 MPa) AD % 0.3 −0.1 AAD % 0.8 0.6 MD % 2.0 1.9 Deviations with QCR Values (120 to 200 MPa) AD % 0.5 −0.6 AAD % 1.2 1.2 MD % 3.0 3.3 A B C D E0 E1 F0 F1 F2

a

Figure 2. Relative differences Δη/η = {η(cal) − η(exp)}/η(exp) of the values η(cal) obtained from eqs 4 through 7 for ethyl caprate at T = 333.15 K from experimental viscosities η(exp) as a function of pressure: △, falling body viscometer at 0 MPa to 100 MPa ; ▲, falling body viscometer at 120 MPa to 200 MPa; ○, quartz crystal resonator viscometer at 0 MPa to 100 MPa ; ●, quartz crystal resonator viscometer at 120 MPa to 200 MPa.

Table 5. Deviations with Previous Reported Measurements at Atmospheric Pressure T range

reference Bonhorst et al.25 Gros and Feuge26 Gouw et al.27 Liew et al.23 Knothe and Steidley28 Pratas et al.24 Shigley et al.22 Gros and Feuge26 Knothe and Steidley28 Pratas et al.24

AD %. bAAD %. cMD %.

Methyl Caprate 293−371 348 293−343 293−353 313 293−353 Ethyl Caprate 308−368 348 313 283−353

AAD %

MD %

1.4 10 1.7 2.5 1.15 0.5

2 10 2.9 3.9 1.15 0.8

0.4 5.1 0.16 0.4

0.7 5.1 0.16 0.9

of T and V and, consequently, to the steepness of the intermolecular potential. Therefore, one can correlate viscosity measurements over a wide range of temperatures and pressures by using a function of a single variable ⎛T ⎞ η = f⎜ γ⎟ ⎝ρ ⎠

(9)

This viscosity scaling scheme, named thermodynamic scaling, has been applied here to correlate the reduced viscosity31 rather than viscosity itself. This dimensionless viscosity η* is defined as

Figure 1. Relative differences Δη/η = {η(cal) − η(exp)}/η(exp) of the values η(cal) obtained from eqs 4 through 7 for methyl caprate at T = 333.15 K from experimental viscosities η(exp) as a function of pressure: △, falling body viscometer at 0 MPa to 100 MPa ; ▲, falling body viscometer at 120 MPa to 200 MPa; ○, quartz crystal resonator viscometer at 0 MPa to 100 MPa ; ●, quartz crystal resonator viscometer at 120 MPa to 200 MPa.

⎛ MN 2 ⎞1/6 η ∗ = η⎜ 4 A 3 ⎟ ⎝ ρ (RT ) ⎠ −1

(10) −1

where NA (mol ) is Avogadro’s constant, M (kg·mol ) the molar mass, R (J·mol−1·K−1) the gas constant, and ρ and η are the experimental density and viscosity, respectively. The exponents γ of both components are first determined by plotting η* as a function of ργ/T and by estimating the value for which all isotherms overlap in narrow master curve. The optimal γ value is obtained by minimizing the distance between the experimental points and the master curve fitted for each γ value by a polynomial of order 6. With optimized γ values, reduced viscosity data can be gathered in a compact master curve when expressed as a function ργ/T as can be seen in

% for methyl caprate and 0.4 % for ethyl caprate. A good match (0.4 % in AAD %) is also observed with the data reported by Shigley et al. for ethyl caprate, whereas data of Liew et al. for methyl caprate present a much higher deviation (2.5 % in AAD %). It was previously observed29,30 that transport properties in liquids can be related to a single thermodynamic quantity defined by the product of temperature and volume raised to the power of γ, an exponent correlated to the relative contribution E



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Figure 3. Thus, the thermodynamic scaling approach yields a consistent scaling for transport properties of alkyl ester in T, ρ

Table 6. Parameters of Eq 11 Methyl Caprate and Ethyl Caprate from 293.15 K to 353.15 K and for Pressure Range from 0.1 MPa to 200 MPa and Relative Deviations Δη/η in Averagea, absolute averageb, and maximum deviationc parameters γ a0 a1 a2 AD % AAD % MD % a

methyl caprate

ethyl caprate

4.35 4.38 −1.219933 × 10−1 −1.178580 × 10−1 2.607660 × 109 2.878340 × 109 1.447590 × 10−12 1.231450 × 10−12 Deviations with Experimental Values 0.05 −0.09 1.3 1.6 3.8 4.5

AD %. bAAD %. cMD %.

Figure 3. Reduced viscosity η* versus ργ/T: △, methyl caprate measured with falling body viscometer; ○, methyl caprate measured with quartz crystal resonator viscometer ; ▲, ethyl caprate measured with falling body viscometer; ●, ethyl caprate measured with quartz crystal resonator viscometer.

range of investigation. By plotting the inverse of the reduce viscosity as a function T/ργ in Figure 4, it can also be observed

Figure 5. Relative differences Δη/η = {η(cal) − η(exp)}/η(exp) of the values η(cal) obtained from the thermodynamic scaling (eq 11) for methyl caprate at different temperatures from experimental viscosities η(exp) as a function of pressure: ▲, falling body viscometer; ○, quartz crystal resonator viscometer.

Figure 4. Inverse of reduced viscosity 1/η* versus T/ργ: △, methyl caprate measured with falling body viscometer; ○, methyl caprate measured with quartz crystal resonator viscometer ; ▲, ethyl caprate measured with falling body viscometer; ●, ethyl caprate measured with quartz crystal resonator viscometer.

1/η* appears as a linear function of T/ργ in the low densities region. Consequently, the following function was chosen to correlate reduce viscosity: ργ 1 T = a0 + a1 γ + a 2 η∗ ρ T

Figure 6. Relative differences Δη/η = {η(cal) − η(exp)}/η(exp) of the values η(cal) obtained from the thermodynamic scaling (eq 11) for ethyl caprate at different temperatures from experimental viscosities η(exp) as a function of pressure: ▲, falling body viscometer; ○, quartz crystal resonator viscometer.

(11)

Table 6 lists the coefficients γ and the parameters a0, a1, a2 as well as the global deviations (AD, AAD, MD). The fitting procedure yields an average deviation of 1.3 % for methyl decanoate and 1.6 % for ethyl decanoate. Moreover, it can be observed from Figures 5 and 6 that the deviations do not increase systematically with pressure. The maximum deviation is not higher than 4.5 %. Notwithstanding the low number of fitting parameters (only three), the correlation provides a good estimation of the reduced viscosity as a function volume and temperature. This confirms that the thermodynamic scaling method yields a consistent scaling for the full range of temperature density conditions investigate in the present work.

4. CONCLUSIONS Viscosity was measured in methyl caprate and ethyl caprate over a temperature range between 293.15 K to 353.15 K with pressure ranging from atmospheric to 200 MPa using falling body and quartz resonator viscometers. A correlation was proposed to correlate within the experimental uncertainty the viscosity values as a function of temperature and pressure. Finally, a thermodynamic scaling method was used to describe the viscosity in terms of density and temperature. This scaling technique provides a description of viscosity with a three parameter function that leads to an absolute average deviation F



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of 1.3 % and 1.6 % for methyl caprate and ethyl caprate, respectively.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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REFERENCES

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Viscosities of Fatty Acid Methyl and Ethyl Esters under High Pressure

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