Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

The rheological model of biodiesels at elevated pressures and temperatures

The rheological model of biodiesels at elevated pressures and temperatures Abstract In the present study, an approximation is used to study viscosity as a function of pressure at different temperatures. The correlation so obtained is applied to study the viscosity of biodiesels extracted from soybean, Vistive soybean, canola, used canola, coconut and rapeseed. The computed values of viscosity from the proposed model were found to be in good agreement with experimental data throughout the range of pressure and temperature studied. The maximum average absolute relative deviation (AARD%) and mean AARD% are found to be 0.52 and 0.20, respectively, over the entire range of pressure (0.1–140 MPa) and temperature (283.15–373.15 K) for all biodiesels except rapeseed biodiesel, for which the values are 1.1 and 0.62, respectively. Furthermore, this work includes the very first investigation conducted so far on the variation of the pressure–viscosity coefficient (PVC) with pressure at different temperatures for biodiesels. The variation in PVC with the temperature is more sensitive at elevated pressures as compared to atmospheric pressure whereas the variation in PVC with pressure is more sensitive at elevated temperatures as compared to room temperature. Open in new tabDownload slide renewable fuel, biodiesels, viscosity, pressure–viscosity coefficient, pressure, temperature Introduction Biodiesel is a biodegradable and non-toxic fuel, which is a renewable source of energy. Due to the high cetane number and non-aromaticity of the fuel, it has many advantages over diesel fuel. Biodiesel, being free of sulphur, contains ~10–12% oxygen by weight and is used neat or as a blend with conventional diesel fuel. Biodiesel, when blended with conventional diesel fuel, produces a significantly lower amount of greenhouse gases (GHG), carbon monoxide (CO), hydrocarbons (HC) and particulate emissions [1, 2], though the output is lower. Moreover, being more viscous than diesel fuel, biodiesel improves lubricity, which results in a longer life for engine components [3]. Conventionally, biodiesel is produced through the transesterification of fats and oils in the presence of alcohol, such as methanol, ethanol, etc. The typical profile of the fatty acid methyl ester, which is the main constituent of biodiesel, depends on the composition of the oil feedstock or the fat and is responsible for the chemical and physical properties of the biodiesel yield. For example, the biodiesel produced as a result of transesterification of soybean oil typically contains oleate (C18:1) and linoleate (C18:2), while biodiesel from coconut oil contains laurate (C12:0) and myristate (C14:0). The difference in the chain length and absence of branching are responsible for different physico-chemical properties, such as cloud point, viscosity, density, oxidative stability, etc. [4, 5]. Viscosity is considered to have a great impact on fuel quality. The prediction of viscosity under extreme conditions of high pressure and high temperature would be a useful tool not only in the optimization of biodiesel production, but also in managing the blending of raw materials and refined products. Moreover, the understanding of the effect of pressure and temperature on the viscosity of biodiesel is desirable for the optimization of engine performance under high-pressure and high-temperature conditions. The lack of high-pressure viscosity data at elevated temperatures motivated us to investigate a simple model, which can be used to extrapolate the viscosity of biodiesels under extreme conditions (real conditions in a diesel engine). The majority of the experimental and theoretical work reported in the literature is limited to predicting viscosity variation with temperature, while viscosity variation with pressure is scarce, particularly in biodiesels. There are few methods for predicting the variation of viscosity with pressure [6–17]. The well-known relationship describing the effect of pressure on viscosity was introduced by Barus [9] and tested successfully for moderate pressure in many liquids including biodiesels [10–12]. The Barus model is based on the free-volume theory, which reveals that a chain of molecules needs a larger free-volume void to move within the liquid [13]. Freitas et al. [14] introduced a method to study viscosity as a function of pressure for the high-pressure range, which was valid at high temperatures only since a significant deviation from experimental data was observed at low temperatures. Paluch et al. [17] used the analogy of the Vogel–Tammann–Fulcher (VTF) equation [15, 16] (initially proposed to study the temperature dependence of viscosity) to study the pressure dependence of viscosity as well. However, this equation also failed to represent the viscosity at low pressures. The Munro, Block and Piermarini [18] model was useful for liquids exhibiting non-exponential behaviour at low pressures. Besides these, other methods are also reported in the literature but they are more complex owing to the involvement of a large number of unknown parameters. In addition to the methods stated above, the following soft computing methods for predicting the viscosity of biodiesels and their blends have been developed: multilayer perceptron neural network (MLP-NN), radial basis function (RBF) optimized by particle swarm optimization (PSO-RBF) and adaptive neuro-fuzzy inference system optimized by the hybrid approach (Hybrid-ANFIS) [19, 20]. However, these methods necessitated a significant amount of experimental data to train the system, as well as a relatively long computational time. The pressure–viscosity coefficient (PVC) is an important parameter of lubricants that is rarely investigated for biodiesels. The fluid film thickness that protects the mechanical device from high friction and premature wear is determined by the high value of the PVC. Biodiesel with a higher PVC forms thicker lubricant films and increases the expected life of rolling element bearings, gears, etc. An ideal lubricant forms a thick separating layer between the rubbing surfaces over a wide temperature and pressure range. There are broadly two techniques to estimate the PVC based on (i) viscosity-versus-pressure curves [21, 22] and (ii) elastohydrodynamic lubrication (EHL) film thickness measurement [23]. Using the free-volume theory for calculating the PVC of several liquids, Wu et al. [24] derived a non-linear equation with four adjustable parameters that provided good accuracy when tested over a wide viscosity range. Another empirical relation for the PVC proposed by So and Klaus [25] was successfully applied to mineral oils, resin–polymer blends, pure hydrocarbons and non-hydrocarbons. However, the Wu and So–Klaus models involve a large number of fitting parameters and are complex. Thus, the purpose of this work is to investigate the simple model, which can predict the viscosity of biodiesel as a function of pressure at different temperatures. An approximation is made to deduce the equation, which enables the prediction of the viscosity of biodiesels at high pressures and temperatures. Biodiesels extracted from soybean (S), Vistive soybean, canola, used canola, coconut and rapeseed (R) are chosen to validate the proposed model. The proposed model is also applied to predict the PVC as a function of pressure at different temperatures, which is an important tribological parameter, investigated for the first time for biodiesels in this work. 1 Methodology The Barus equation [9] assumed that the ratio of the first pressure derivative of viscosity to the viscosity itself is independent of pressure but it does not match the experimental viscosity data adequately, particularly at high pressures. However, Kamal and Dass [26] assumed that the ratio of the second pressure derivative of viscosity to the first pressure derivative of viscosity is independent of pressure, as represented by the following equation: (d2η(p,T)╱dp2)T(dη(p,T)╱dp)T=−z(1) where z is a constant. Integrating Equation (1) with respect to pressure, within the limit of p0 to p, yields Equation (2): ∫pp0ddp((dη(p,T)╱dp)T)(dη(p,T)╱dp)Tdp=∫pp0−zpln((dη(p,T)╱dp)T(dη(p0,T)╱dp)T)=−z(p−p0)(dη(p,T)╱dp)T= (dη(p0,T)╱dp)Te−z(p−p0)or η′(p,T)=η′(p0,T)exp(z(p−p0))(2) One more integration of Equation (2) with respect to pressure yields Equation (3): ∫pp0(dη(p,T)╱dp)Tdp=η′(p0,T) ∫pp0e−z(p−p0)dporη(p,T)−η(p0,T)=η′(p0,T)[exp(z(p−p0))−1z]orη(p,T)=η(p0,T)+ η′(p0,T)z[exp(z(p−p0))−1](3) where η′(p0,T) is the first pressure derivative of viscosity η(p,T) at pressure p0 and temperature T. Equation (3) shows that the viscosity will increase with the increase in pressure. Another advantage of this model is that the second and higher pressure derivatives of the viscosity can also be computed with the help of the general relation: (∂Nη∂pN)T=(±z)N−1(∂η∂p)T(4) The VTF equation has been successfully applied in the literature to study the temperature dependence of viscosity in liquids [15, 16]. Paluch et al. [17] used the VTF equation analogy to study the pressure dependence of viscosity in a low-molecular-weight glass-forming liquid, which was later extended to other liquids [27]. The VTF equation has been written by Paluch et al. as: η(p,T)=η (p0, T)exp[C (p−p0)p∞+ (p−p0)](5) The parameter p∞ is referred to as the pressure at which the viscosity diverges and η(p0,T) is the viscosity at pressure p0 and temperature T. C is an empirical constant. In order to find out the PVC, Equation (2) can be rewritten to define the pressure–viscosity coefficient as: α=∂lnη∂p=z η′(p0,T)exp(z(p−p0))η(p0,T)z+η′(p0,T)[exp(z(p−p0))−1](6) However, the PVC derived from Equation (5) is given as: α=∂lnη∂p=bc(c−p)2(7) Equation (3) was successfully applied to predict the viscosity as a function of pressure and temperature in liquids [26]. Thus in the present work, Equation (3) is applied to investigate the viscosity of biodiesels as a function of pressure at different temperatures, whereas Equation (6) is used to study the PVC of biodiesels as a function of pressure at different temperatures. 2 Experimental data The experimental high-pressure viscosity data at different temperatures are taken from Duncan et al. [28] for biodiesel samples extracted from soybean, Vistive soybean, canola, used canola and coconut while the data for the biodiesel sample extracted from rapeseed are taken from Samuel et al. [29]. For comparison, conventional highly paraffinic (HPF) diesel fuel is also considered and high-pressure viscosity data for conventional diesel fuel are taken from Rowane et al. [30]. Duncan et al. have used the ViscoPro2000 system 4-SPL viscometer for the high-pressure viscosity measurement at different temperatures. The equipment is a temperature-controlled oven (0.1 K) housing the high-pressure viscometer sensor as well as a precision pressure transducer and a resistance temperature detector (RTD) (±0.05 K). To pressurize the diesel samples, the viscometer is linked to a manual high-pressure syringe pump acquired from High Pressure Equipment Company (HIP) (model 50-575-30; 30 000 psi, capacity of 10 cm3, with PolyPak). The viscometer uses annular flow principles around an axially oscillating piston and was recently certified by the American Society for Testing and Materials (ASTM). Their viscometer is capable of measuring viscosity in the temperature range of 233.15–463.15 K and at pressures of ≤137.9 MPa. The maximum deviations in the measurement of temperature and viscosity are observed as 0.1 K and 0.036 mPa-s, respectively. Samuel et al. have used the vibrating wire viscometer, which is capable of measuring the viscosity in the temperature range of 273.15–423.15 K and at pressures of ≤140 MPa. Toluene has been used as a standard fluid for calibration purposes and the maximum uncertainty observed in the measurement of viscosity is 1%. In terms of viscosity, the predicted error is 1%. Their vibrating wire viscometer is capable to measure viscosities ranging from 0.3 to 30 mPa-s. The basic component of the apparatus is a sensor with a tungsten wire (length 50 mm and nominal radius 75 lm) anchored at both ends, placed inside a pressure vessel and an external magnetic field oriented perpendicular to the wire length using a magnet set around the vessel. Rowane et al. have used the Rolling-Ball Viscometer/Densimeter for high-pressure viscosity measurement at different temperatures ranging from 298 to 528 K at pressures of ≤300 MPa. The universal calibration method is used for calibration purposes. Standard uncertainty in the measurement of pressure is 0.07 MPa at pressures of <63.9 MPa and 0.41 MPa at pressures of >68.9 MPa, while the standard uncertainty in the measurement of viscosity is 0.025η (mPa-s) with a conversion factor of 2. 3 Results and discussion Equation (3) represents the viscosity as a function of pressure at different temperatures for all biodiesels. Initially, the adjustable parameters η(p0,T), η′(p0,T) and z in Equation (3) were determined using a non-linear fitting toolbox in MATLAB®. The experimental data used to determine the fitting parameters were taken from literature [28–30]. The values of these parameters are given in Table 1 along with root mean square deviation (RMSD) for all biodiesels. Since Equation (5) has been used in the literature to represent the viscosity as a function of pressure, the results of Equation (3) were compared with those of Equation (5). The fitted parameters of Equation (5) are reported in Table 1 in small parentheses. Rheological behaviour of conventional diesel fuel is also compared with chosen biodiesels in this study. The fitted parameters for conventional diesel fuel are also given in Table 1. Table 1: Fitted parameters of Equations (3) and (5) at different temperatures along with RMSD, AARD% and pressure range for all biodiesels and HPF diesel. The fitted parameters (η(p0,T), C and p∞) of Equation (5) and corresponding AARD% are reported in parentheses Liquid . T . η(p0,T) (mPa-s) . η′(p0,T) (mPa-s/MPa) . Z (× 10–3) (MPa–1) . RMSD . AARD% . Pressure range (MPa) . Soybeana (p0 = 0.1 MPa) 283.15 7.452 (7.458) 0.09383 (7.524) 0.007925 (607.7) 0.02234 (0.02629) 0.158 (0.183) 0.1–131 298.15 5.037 (5.061) 0.05823 (9.169) 0.007178 (829.6) 0.01792 (0.02561) 0.119 (0.214) 313.15 3.533 (3.548) 0.03841 (12.38) 0.007704 (1181) 0.02179 (0.02899) 0.281 (0.405) 373.15 1.379 (1.386) 0.01527 (4.598) 0.003971 (437.6) 0.004213 (0.00644) 0.124 (0.214) Vistive soybeana (p0 = 0.1 MPa) 283.15 7.76 (7.762) 0.0984 (6.951) 0.00759 (558.5) 0.01085 (0.009111) 0.075 (0.075) 0.1–131 298.15 5.084 (5.112) 0.06033 (8.411) 0.006992 (744.7) 0.008696 (0.02199) 0.0652 (0.187) 313.15 3.613 (3.629) 0.04008 (11.72) 0.007694 (1097) 0.02726 (0.03236) 0.3 (0.383) 373.15 1.37 (1.377) 0.01539 (4.722) 0.004139 (443.9) 0.004231 (0.00762) 0.0417 (0.2) Canolaa (p0 = 0.1 MPa) 283.15 8.595 (8.609) 0.1069 (13.54) 0.009496 (1110) 0.08151 (0.08658) 0.328 (0.379) 0.1–131 298.15 5.605 (5.644) 0.06502 (9.914) 0.007675 (886.3) 0.08905 (0.04607) 0.462 (0.557) 313.15 3.907 (3.926) 0.04398 (7.958) 0.006508 (740.8) 0.01675 (0.02525) 0.205 (0.324) 373.15 1.426 (1.433) 0.01639 (4.713) 0.004211 (433) 0.006893 (0.004) 0.209 (0.124) Used canolaa (p0 = 0.1 MPa) 283.15 8.642 (8.652) 0.1055 (13.4) 0.009379 (1111.5) 0.03745 (0.0414) 0.171 (0.214) 0.1–131 298.15 5.541 (5.572) 0.06712 (8.462) 0.007125 (734.9) 0.01766 (0.02786) 0.095 (0.190) 313.15 3.914 (3.933) 0.04468 (7.594) 0.006415 (697.5) 0.009975 (0.01392) 0.114 (0.171) 373.15 1.437 (1.444) 0.01616 (5.021) 0.004479 (470.8) 0.004109 (0.004751) 0.124 (0.157) Coconuta (p0 = 0.1 MPa) 283.15 4.848 (4.859) 0.05969 (7.228) 0.007143 (605.3) 0.008689 (0.01067) 0.087 (0.086) 0.1–131 298.15 3.279 (3.296) 0.03747 (8.019) 0.006603 (736.4) 0.02481 (0.03013) 0.257 (0.391) 313.15 2.196 (2.208) 0.02615 (6.631) 0.006019 (587.5) 0.01331 (0.01381) 0.290 (0.286) 373.15 0.9258 (0.9297) 0.01001 (3.395) 0.002147 (330.8) 0.01053 (0.01169) 0.519 (0.552) Rapeseedb (p0 = 0.1 MPa) 293.15 6.93 (6.709) 0.07982 (3392) 0.01604 (245 700) 0.1812 (0.3266) 1.07 (2.13) 0.1–140 313.15 4.22 (4.264) 0.04939 (17.3) 0.008759 (1556) 0.08324 (0.08973) 0.653 (0.800) 333.15 2.86 (2.892) 0.03343 (8.225) 0.006516 (753.4) 0.03016 (0.03451) 0.542 (0.593) 353.15 2.11 (2.139) 0.02514 (5.036) 0.004281 (460.3) 0.02811 (0.03055) 0.550 (0.529) 373.15 1.68 (1.681) 0.01851 (4.333) 0.003669 (410.8) 0.01785 (0.021) 0.486 (0.536) 393.15 1.33 (1.338) 0.0157 (3.304) 0.001921 (298) 0.01054 (0.009716) 0.421 (0.371) Highly paraffinic dieselc (HFPD) (p0 = 0 MPa) 298.3 3.086 (3.483) 0.05339 (29.22) 0.0103 (2229) 0.4225 (4530) 1.17 (1.74) 3.8–300 323.2 1.716 (1.879) 0.02611 (18.02) 0.008327 (1522) 0.1398 (0.1367) 1.68 (1.44) 350.4 1.138 (1.198) 0.01521 (13.10) 0.006922 (1203) 0.03742 (0.05768) 0.828 (1.60) 433.1 0.4564 (0.4762) 0.005952 (6.061) 0.004255 (560.8) 0.01762 (0.01834) 1.13 (1.24) 528.7 0.2244 (0.2341) 0.003339 (4.404) 0.002766 (362.7) 0.009343 (0.0114) 1.45 (1.90) Liquid . T . η(p0,T) (mPa-s) . η′(p0,T) (mPa-s/MPa) . Z (× 10–3) (MPa–1) . RMSD . AARD% . Pressure range (MPa) . Soybeana (p0 = 0.1 MPa) 283.15 7.452 (7.458) 0.09383 (7.524) 0.007925 (607.7) 0.02234 (0.02629) 0.158 (0.183) 0.1–131 298.15 5.037 (5.061) 0.05823 (9.169) 0.007178 (829.6) 0.01792 (0.02561) 0.119 (0.214) 313.15 3.533 (3.548) 0.03841 (12.38) 0.007704 (1181) 0.02179 (0.02899) 0.281 (0.405) 373.15 1.379 (1.386) 0.01527 (4.598) 0.003971 (437.6) 0.004213 (0.00644) 0.124 (0.214) Vistive soybeana (p0 = 0.1 MPa) 283.15 7.76 (7.762) 0.0984 (6.951) 0.00759 (558.5) 0.01085 (0.009111) 0.075 (0.075) 0.1–131 298.15 5.084 (5.112) 0.06033 (8.411) 0.006992 (744.7) 0.008696 (0.02199) 0.0652 (0.187) 313.15 3.613 (3.629) 0.04008 (11.72) 0.007694 (1097) 0.02726 (0.03236) 0.3 (0.383) 373.15 1.37 (1.377) 0.01539 (4.722) 0.004139 (443.9) 0.004231 (0.00762) 0.0417 (0.2) Canolaa (p0 = 0.1 MPa) 283.15 8.595 (8.609) 0.1069 (13.54) 0.009496 (1110) 0.08151 (0.08658) 0.328 (0.379) 0.1–131 298.15 5.605 (5.644) 0.06502 (9.914) 0.007675 (886.3) 0.08905 (0.04607) 0.462 (0.557) 313.15 3.907 (3.926) 0.04398 (7.958) 0.006508 (740.8) 0.01675 (0.02525) 0.205 (0.324) 373.15 1.426 (1.433) 0.01639 (4.713) 0.004211 (433) 0.006893 (0.004) 0.209 (0.124) Used canolaa (p0 = 0.1 MPa) 283.15 8.642 (8.652) 0.1055 (13.4) 0.009379 (1111.5) 0.03745 (0.0414) 0.171 (0.214) 0.1–131 298.15 5.541 (5.572) 0.06712 (8.462) 0.007125 (734.9) 0.01766 (0.02786) 0.095 (0.190) 313.15 3.914 (3.933) 0.04468 (7.594) 0.006415 (697.5) 0.009975 (0.01392) 0.114 (0.171) 373.15 1.437 (1.444) 0.01616 (5.021) 0.004479 (470.8) 0.004109 (0.004751) 0.124 (0.157) Coconuta (p0 = 0.1 MPa) 283.15 4.848 (4.859) 0.05969 (7.228) 0.007143 (605.3) 0.008689 (0.01067) 0.087 (0.086) 0.1–131 298.15 3.279 (3.296) 0.03747 (8.019) 0.006603 (736.4) 0.02481 (0.03013) 0.257 (0.391) 313.15 2.196 (2.208) 0.02615 (6.631) 0.006019 (587.5) 0.01331 (0.01381) 0.290 (0.286) 373.15 0.9258 (0.9297) 0.01001 (3.395) 0.002147 (330.8) 0.01053 (0.01169) 0.519 (0.552) Rapeseedb (p0 = 0.1 MPa) 293.15 6.93 (6.709) 0.07982 (3392) 0.01604 (245 700) 0.1812 (0.3266) 1.07 (2.13) 0.1–140 313.15 4.22 (4.264) 0.04939 (17.3) 0.008759 (1556) 0.08324 (0.08973) 0.653 (0.800) 333.15 2.86 (2.892) 0.03343 (8.225) 0.006516 (753.4) 0.03016 (0.03451) 0.542 (0.593) 353.15 2.11 (2.139) 0.02514 (5.036) 0.004281 (460.3) 0.02811 (0.03055) 0.550 (0.529) 373.15 1.68 (1.681) 0.01851 (4.333) 0.003669 (410.8) 0.01785 (0.021) 0.486 (0.536) 393.15 1.33 (1.338) 0.0157 (3.304) 0.001921 (298) 0.01054 (0.009716) 0.421 (0.371) Highly paraffinic dieselc (HFPD) (p0 = 0 MPa) 298.3 3.086 (3.483) 0.05339 (29.22) 0.0103 (2229) 0.4225 (4530) 1.17 (1.74) 3.8–300 323.2 1.716 (1.879) 0.02611 (18.02) 0.008327 (1522) 0.1398 (0.1367) 1.68 (1.44) 350.4 1.138 (1.198) 0.01521 (13.10) 0.006922 (1203) 0.03742 (0.05768) 0.828 (1.60) 433.1 0.4564 (0.4762) 0.005952 (6.061) 0.004255 (560.8) 0.01762 (0.01834) 1.13 (1.24) 528.7 0.2244 (0.2341) 0.003339 (4.404) 0.002766 (362.7) 0.009343 (0.0114) 1.45 (1.90) Experimental data taken from a[28], b[29] and c[30]. Open in new tab Table 1: Fitted parameters of Equations (3) and (5) at different temperatures along with RMSD, AARD% and pressure range for all biodiesels and HPF diesel. The fitted parameters (η(p0,T), C and p∞) of Equation (5) and corresponding AARD% are reported in parentheses Liquid . T . η(p0,T) (mPa-s) . η′(p0,T) (mPa-s/MPa) . Z (× 10–3) (MPa–1) . RMSD . AARD% . Pressure range (MPa) . Soybeana (p0 = 0.1 MPa) 283.15 7.452 (7.458) 0.09383 (7.524) 0.007925 (607.7) 0.02234 (0.02629) 0.158 (0.183) 0.1–131 298.15 5.037 (5.061) 0.05823 (9.169) 0.007178 (829.6) 0.01792 (0.02561) 0.119 (0.214) 313.15 3.533 (3.548) 0.03841 (12.38) 0.007704 (1181) 0.02179 (0.02899) 0.281 (0.405) 373.15 1.379 (1.386) 0.01527 (4.598) 0.003971 (437.6) 0.004213 (0.00644) 0.124 (0.214) Vistive soybeana (p0 = 0.1 MPa) 283.15 7.76 (7.762) 0.0984 (6.951) 0.00759 (558.5) 0.01085 (0.009111) 0.075 (0.075) 0.1–131 298.15 5.084 (5.112) 0.06033 (8.411) 0.006992 (744.7) 0.008696 (0.02199) 0.0652 (0.187) 313.15 3.613 (3.629) 0.04008 (11.72) 0.007694 (1097) 0.02726 (0.03236) 0.3 (0.383) 373.15 1.37 (1.377) 0.01539 (4.722) 0.004139 (443.9) 0.004231 (0.00762) 0.0417 (0.2) Canolaa (p0 = 0.1 MPa) 283.15 8.595 (8.609) 0.1069 (13.54) 0.009496 (1110) 0.08151 (0.08658) 0.328 (0.379) 0.1–131 298.15 5.605 (5.644) 0.06502 (9.914) 0.007675 (886.3) 0.08905 (0.04607) 0.462 (0.557) 313.15 3.907 (3.926) 0.04398 (7.958) 0.006508 (740.8) 0.01675 (0.02525) 0.205 (0.324) 373.15 1.426 (1.433) 0.01639 (4.713) 0.004211 (433) 0.006893 (0.004) 0.209 (0.124) Used canolaa (p0 = 0.1 MPa) 283.15 8.642 (8.652) 0.1055 (13.4) 0.009379 (1111.5) 0.03745 (0.0414) 0.171 (0.214) 0.1–131 298.15 5.541 (5.572) 0.06712 (8.462) 0.007125 (734.9) 0.01766 (0.02786) 0.095 (0.190) 313.15 3.914 (3.933) 0.04468 (7.594) 0.006415 (697.5) 0.009975 (0.01392) 0.114 (0.171) 373.15 1.437 (1.444) 0.01616 (5.021) 0.004479 (470.8) 0.004109 (0.004751) 0.124 (0.157) Coconuta (p0 = 0.1 MPa) 283.15 4.848 (4.859) 0.05969 (7.228) 0.007143 (605.3) 0.008689 (0.01067) 0.087 (0.086) 0.1–131 298.15 3.279 (3.296) 0.03747 (8.019) 0.006603 (736.4) 0.02481 (0.03013) 0.257 (0.391) 313.15 2.196 (2.208) 0.02615 (6.631) 0.006019 (587.5) 0.01331 (0.01381) 0.290 (0.286) 373.15 0.9258 (0.9297) 0.01001 (3.395) 0.002147 (330.8) 0.01053 (0.01169) 0.519 (0.552) Rapeseedb (p0 = 0.1 MPa) 293.15 6.93 (6.709) 0.07982 (3392) 0.01604 (245 700) 0.1812 (0.3266) 1.07 (2.13) 0.1–140 313.15 4.22 (4.264) 0.04939 (17.3) 0.008759 (1556) 0.08324 (0.08973) 0.653 (0.800) 333.15 2.86 (2.892) 0.03343 (8.225) 0.006516 (753.4) 0.03016 (0.03451) 0.542 (0.593) 353.15 2.11 (2.139) 0.02514 (5.036) 0.004281 (460.3) 0.02811 (0.03055) 0.550 (0.529) 373.15 1.68 (1.681) 0.01851 (4.333) 0.003669 (410.8) 0.01785 (0.021) 0.486 (0.536) 393.15 1.33 (1.338) 0.0157 (3.304) 0.001921 (298) 0.01054 (0.009716) 0.421 (0.371) Highly paraffinic dieselc (HFPD) (p0 = 0 MPa) 298.3 3.086 (3.483) 0.05339 (29.22) 0.0103 (2229) 0.4225 (4530) 1.17 (1.74) 3.8–300 323.2 1.716 (1.879) 0.02611 (18.02) 0.008327 (1522) 0.1398 (0.1367) 1.68 (1.44) 350.4 1.138 (1.198) 0.01521 (13.10) 0.006922 (1203) 0.03742 (0.05768) 0.828 (1.60) 433.1 0.4564 (0.4762) 0.005952 (6.061) 0.004255 (560.8) 0.01762 (0.01834) 1.13 (1.24) 528.7 0.2244 (0.2341) 0.003339 (4.404) 0.002766 (362.7) 0.009343 (0.0114) 1.45 (1.90) Liquid . T . η(p0,T) (mPa-s) . η′(p0,T) (mPa-s/MPa) . Z (× 10–3) (MPa–1) . RMSD . AARD% . Pressure range (MPa) . Soybeana (p0 = 0.1 MPa) 283.15 7.452 (7.458) 0.09383 (7.524) 0.007925 (607.7) 0.02234 (0.02629) 0.158 (0.183) 0.1–131 298.15 5.037 (5.061) 0.05823 (9.169) 0.007178 (829.6) 0.01792 (0.02561) 0.119 (0.214) 313.15 3.533 (3.548) 0.03841 (12.38) 0.007704 (1181) 0.02179 (0.02899) 0.281 (0.405) 373.15 1.379 (1.386) 0.01527 (4.598) 0.003971 (437.6) 0.004213 (0.00644) 0.124 (0.214) Vistive soybeana (p0 = 0.1 MPa) 283.15 7.76 (7.762) 0.0984 (6.951) 0.00759 (558.5) 0.01085 (0.009111) 0.075 (0.075) 0.1–131 298.15 5.084 (5.112) 0.06033 (8.411) 0.006992 (744.7) 0.008696 (0.02199) 0.0652 (0.187) 313.15 3.613 (3.629) 0.04008 (11.72) 0.007694 (1097) 0.02726 (0.03236) 0.3 (0.383) 373.15 1.37 (1.377) 0.01539 (4.722) 0.004139 (443.9) 0.004231 (0.00762) 0.0417 (0.2) Canolaa (p0 = 0.1 MPa) 283.15 8.595 (8.609) 0.1069 (13.54) 0.009496 (1110) 0.08151 (0.08658) 0.328 (0.379) 0.1–131 298.15 5.605 (5.644) 0.06502 (9.914) 0.007675 (886.3) 0.08905 (0.04607) 0.462 (0.557) 313.15 3.907 (3.926) 0.04398 (7.958) 0.006508 (740.8) 0.01675 (0.02525) 0.205 (0.324) 373.15 1.426 (1.433) 0.01639 (4.713) 0.004211 (433) 0.006893 (0.004) 0.209 (0.124) Used canolaa (p0 = 0.1 MPa) 283.15 8.642 (8.652) 0.1055 (13.4) 0.009379 (1111.5) 0.03745 (0.0414) 0.171 (0.214) 0.1–131 298.15 5.541 (5.572) 0.06712 (8.462) 0.007125 (734.9) 0.01766 (0.02786) 0.095 (0.190) 313.15 3.914 (3.933) 0.04468 (7.594) 0.006415 (697.5) 0.009975 (0.01392) 0.114 (0.171) 373.15 1.437 (1.444) 0.01616 (5.021) 0.004479 (470.8) 0.004109 (0.004751) 0.124 (0.157) Coconuta (p0 = 0.1 MPa) 283.15 4.848 (4.859) 0.05969 (7.228) 0.007143 (605.3) 0.008689 (0.01067) 0.087 (0.086) 0.1–131 298.15 3.279 (3.296) 0.03747 (8.019) 0.006603 (736.4) 0.02481 (0.03013) 0.257 (0.391) 313.15 2.196 (2.208) 0.02615 (6.631) 0.006019 (587.5) 0.01331 (0.01381) 0.290 (0.286) 373.15 0.9258 (0.9297) 0.01001 (3.395) 0.002147 (330.8) 0.01053 (0.01169) 0.519 (0.552) Rapeseedb (p0 = 0.1 MPa) 293.15 6.93 (6.709) 0.07982 (3392) 0.01604 (245 700) 0.1812 (0.3266) 1.07 (2.13) 0.1–140 313.15 4.22 (4.264) 0.04939 (17.3) 0.008759 (1556) 0.08324 (0.08973) 0.653 (0.800) 333.15 2.86 (2.892) 0.03343 (8.225) 0.006516 (753.4) 0.03016 (0.03451) 0.542 (0.593) 353.15 2.11 (2.139) 0.02514 (5.036) 0.004281 (460.3) 0.02811 (0.03055) 0.550 (0.529) 373.15 1.68 (1.681) 0.01851 (4.333) 0.003669 (410.8) 0.01785 (0.021) 0.486 (0.536) 393.15 1.33 (1.338) 0.0157 (3.304) 0.001921 (298) 0.01054 (0.009716) 0.421 (0.371) Highly paraffinic dieselc (HFPD) (p0 = 0 MPa) 298.3 3.086 (3.483) 0.05339 (29.22) 0.0103 (2229) 0.4225 (4530) 1.17 (1.74) 3.8–300 323.2 1.716 (1.879) 0.02611 (18.02) 0.008327 (1522) 0.1398 (0.1367) 1.68 (1.44) 350.4 1.138 (1.198) 0.01521 (13.10) 0.006922 (1203) 0.03742 (0.05768) 0.828 (1.60) 433.1 0.4564 (0.4762) 0.005952 (6.061) 0.004255 (560.8) 0.01762 (0.01834) 1.13 (1.24) 528.7 0.2244 (0.2341) 0.003339 (4.404) 0.002766 (362.7) 0.009343 (0.0114) 1.45 (1.90) Experimental data taken from a[28], b[29] and c[30]. Open in new tab The RMSD values given in Table 1 indicate the applicability of Equation (3) to describe the viscosity as a function of the pressure and temperature for all biodiesels and conventional diesel fuel. The maximum AARD% and mean AARD% are found to be 0.52 and 0.20, respectively, in the entire range of pressure and temperature for all biodiesels except rapeseed oil. For rapeseed oil, the maximum AARD% and mean AARD% values are 1.1 and 0.62, respectively. The applicability of Equation (5) is also similar but deviates slightly in the present study, particularly at low pressure as depicted in terms of the RMSD and AARD% values given in Table 1. The maximum AARD% and mean AARD% values determined using Equation (5) are found to be 0.57 and 0.26, respectively, in the entire range of pressures and temperatures for all biodiesels except rapeseed oil, for which these values are 2.1 and 0.83, respectively. The calculated viscosities as a function of pressure at different temperatures from Equations (3) and (5) are compared to experimentally determined the viscosities for soybean, canola, used canola and coconut biodiesels in Fig. 1. Fig. 1: Open in new tabDownload slide (a)–(d) Variation in viscosity as a function of pressure at different temperatures in biodiesels derived from soybean, Vistive soybean, canola and used canola. As shown in Fig. 1, the maximum error in Equation (3) is observed at low pressures and at room temperature. Based on the comparison above, it is clear that Equation (3) is superior to Equation (5) and other models [27] for studying viscosity as a function of pressure, especially in the high-temperature regime. Similar comparison for coconut, rapeseed biodiesels and conventional diesel fuel is shown in Fig. 2. Fig. 2: Open in new tabDownload slide (a) and (b) Variation in viscosity as a function of pressure at different temperatures in biodiesels derived from coconut and rapeseed, and (c) HPF diesel. The PVC values were calculated as a function of pressure at different temperatures using Equations (6) and (7) for each biodiesel and conventional diesel fuel with the help of data given in Table 1. PVC values are compared for soybean and conventional diesel as shown in Fig. 3. Fig. 3: Open in new tabDownload slide (a) and (b) Variation in pressure–viscosity coefficient as a function of pressure at different temperatures in biodiesels derived from HPF diesel and soybean. The understanding of variation of viscosity with pressure and temperature is desirable to optimize the performance of an engine. For example, a common automotive diesel engine may have operating injector pressure and temperature as high as 300 MPa and 363 K, respectively, whereas the temperature in the combustion chamber will be much higher than this [12, 30]. The high value of viscosity increases the problem of atomization and hence may damage the fuel injectors, thereby resulting in incomplete combustion and the deposition of solid unburned particles. On the other hand, biodiesel with lower viscosity may be lacking in providing lubrication to the injectors and increase the problem of leakage [12]. Therefore, biodiesel must be optimized to have a viscosity that lies within the range prescribed by ASTM and European Norm (EN) standards. It is clear from Fig. 1 that the viscosity increases with an increase in pressure and decreases with an increase in temperature; however, the effect of temperature is more dominating than pressure. The effect of pressure on viscosity is lower at higher temperatures, particularly in HPF diesel, as the curvature of the viscosity pressure curve is decreasing with temperature in Fig. 2. The viscosity of biodiesel is greater than that of conventional diesel; the behaviour of viscosity with pressure at elevated temperatures can help to optimize the viscosity as per ASTM and EN standards by controlling the proportions of the biodiesel and conventional diesel in the blend. This may lead to better performance of diesel engines in terms of lower emission of greenhouse gases and other emissions. Fig. 3 shows the variation in PVC with pressure as calculated from Equations (6) and (7) at different temperatures for soybean biodiesel and conventional diesel fuel. The PVC is found to decrease with pressure at different temperatures for all biodiesels except rapeseed oil, for which it is increasing with pressure at 293.15 K. The physical reason for this unusual behaviour is not clear but the curvature of the viscosity pressure curve at 293.15 K is more than that at other temperatures, i.e. the first pressure derivative of viscosity is increasing faster than the viscosity itself. It is further noted that the PVC is decreasing with temperature, but the variation is very small (almost temperature-independent) at atmospheric pressure as compared to the variation at elevated pressures. In the literature, variation of PVC with temperature is reported for biodiesels but the variation with pressure is very scarce. However, the reported values of PVC at different temperatures are significantly different from each other and largely depend on the method employed for determination [27]. In the present study, the PVC is changing from 13 to 5 GPa–1 in the entire range of investigation for pressure and temperature. Similar trends are observed for PVC calculated from Equation (7). The high PVC values suggest that biodiesels have flexible hydrocarbon structures with little or no branching or layering at atmospheric pressure. But at elevated pressures, biodiesels with PVC values of <10 GPa–1 may behave like layered molecular structures [31]. Moreover, Fig. 3 shows that the PVC of HPF diesel also decreases with pressure at a faster rate in low-pressure regions whereas variation is sluggish at high pressures. 3 Conclusions Because the viscosity and PVCs are key parameters to evaluate the quality of biodiesel, models to predict the dependence of these parameters on pressure and temperature were established. The proposed models were validated using biodiesels extracted from various feedstocks. It is evident from Table 1 and Figs 1 and 2 that the present model is quite appropriate to accurately predict the pressure dependence of viscosity of biodiesels at different temperatures as the AARD% is quite small in the entire temperature range (283–528 K) and pressure range (0.1–300 MPa). The highest departure from experimental data is recorded in the low-pressure and high-temperature regions. With respect to experimental data, Equation (5) exhibits much higher divergence at low pressure and high temperature. The PVC of biodiesels was also calculated using a novel correlation developed as Equation (6). The calculated PVC values of biodiesels were found to be lower than those of conventional diesel. Therefore, EHL must be used at a relatively high pressure. To summarize, the current model successfully represents viscosity as a function of pressure and temperature. The current model can interpolate and extrapolate viscosity at different pressures and temperatures. The findings of this study can be used for simulation and planning by researchers and industries without the need for energy-intensive experiments. The results obtained here can also be used to prepare feedstocks with the appropriate properties in order to generate biodiesel fuels for the efficient operation of diesel engines with lower emissions. Acknowledgements We are thankful to the University of Petroleum and Energy Studies, Dehradun for providing the infrastructure for this work. Funding This research received no external funding. Conflict of interest statement None declared. References [1] Graboski MS , McCormik RL. Combustion of fat and vegetable oil derived fuels in diesel engines . Progress in Energy and Combustion Science , 1998 , 24 : 125 – 164 . Google Scholar Crossref Search ADS WorldCat [2] Tomasevic AV , Marinkovic SS. Methanolysis of used frying oil . Fuel Processing Technology , 2003 , 81 : 1 – 6 . Google Scholar Crossref Search ADS WorldCat [3] Kinast JA. Production of Biodiesels from Multiple Feedstocks and Properties of Biodiesels and Biodiesel-Diesel Blends, NREL/SR-510-31460 . Des Plaines, IL : National Renewable Energy Laboratory , 2001 . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC [4] Rodrigues J , Cardoso F, Lachter E, et al. Correlating chemical structure and physical properties of vegetable oil esters . Journal of the American Oil Chemists' Society , 2006 , 83 : 353 – 357 . Google Scholar Crossref Search ADS WorldCat [5] Ramos M , Fernandez C, Casas A, et al. Influence of fatty acid composition of raw materials on biodiesel properties . Bioresource Technology , 2009 , 100 : 261 – 268 . Google Scholar Crossref Search ADS PubMed WorldCat [6] Taghizadehfard M , Hosseinim SM, Alavianmehr MM. Viscosity modeling of fatty acid esters and biodiesels based on friction theory and perturbed hard-dimer-chain equation of state . Journal of Molecular Liquids , 2021 , 325 : 115048 . Google Scholar Crossref Search ADS WorldCat [7] Chum-in T , Sudaprasert K, Phankosol S, et al. Gibbs energy additivity approaches to QSPR in modeling of high pressure density and kinematic viscosity of FAME and biodiesel . Fuel Processing Technology , 2017 , 156 : 385 – 393 . Google Scholar Crossref Search ADS WorldCat [8] Chum-in T , Sudaprasert K, Phankosol S, et al. Gibbs energy additivity approaches to QSPR in modeling of high pressure dynamic viscosity of FAME and biodiesel . Journal of Molecular Liquids , 2016 , 223 : 1006 – 1012 . Google Scholar Crossref Search ADS WorldCat [9] Barus C . Isothermals, isopiestics and isometrics relative to viscosity . American Journal of Science , 1983 , 45 : 87 – 96 . Google Scholar OpenURL Placeholder Text WorldCat [10] Paton JM , Schaschke CJ. Viscosity measurement of biodiesel at high pressure with a falling sinker viscometer . Chemical Engineering Research and Design , 2009 , 87 : 1520 – 1526 . Google Scholar Crossref Search ADS WorldCat [11] Robertson LX , Schaschke CJ. Combined high pressure and low temperature viscosity measurement of biodiesel . Energy Fuels , 2010 , 24 : 1293 – 1297 . Google Scholar Crossref Search ADS WorldCat [12] Schaschke CJ . Experimental viscosity measurements of biodiesels at high pressure . Chemical Industry and Chemical Engineering Quarterly , 2016 , 22 : 453 – 460 . Google Scholar Crossref Search ADS WorldCat [13] Kioupis LI , Maginn EJ. Impact of molecular architecture on the high-pressure rheology of hydrocarbon fluids . Journal of Physical Chemistry B , 2000 , 104 : 7774 – 7783 . Google Scholar Crossref Search ADS WorldCat [14] Freitas Samuel VD , Segovia Jose J, Carmen Martín M, et al. Measurement and prediction of high-pressure viscosities of biodiesel fuels . Fuel , 2014 , 122 : 223 – 228 . Google Scholar Crossref Search ADS WorldCat [15] Angell CA . Formation of glasses from liquids and biopolymers . Science , 1995 , 276 : 1924 – 1935 . Google Scholar OpenURL Placeholder Text WorldCat [16] Ivaniš GR , Radovic IR, Valada B, et al. Thermodynamic properties of biodiesel and petro-diesel blends at high pressures and temperatures. Experimental and modeling . Fuel , 2016 , 184 : 277 – 288 . Google Scholar Crossref Search ADS WorldCat [17] Paluch M , Denzik Z, Rzoska SJ. Scaling of high-pressure viscosity data in low-molecular-weight glass-forming liquids . Physical Review B , 1999 , 60 : 2979 – 2982 . Google Scholar Crossref Search ADS WorldCat [18] Lima , TA , Ribeiro, et al. . Low-frequency Raman spectra of a glass-forming ionic liquid at low temperature and high pressure . Journal of Chemical Physics , 2019 , 150 : 164502 . Google Scholar Crossref Search ADS PubMed WorldCat [19] Hosseini SM , Pierantozzi M, Moghadasi J. Viscosities of some fatty acid esters and biodiesel fuels from a rough hard-sphere-chain model and artificial neural network . Fuel , 1083 , 2019 : 1091 . Google Scholar OpenURL Placeholder Text WorldCat [20] Masoud M , Hadi NF, Zand N, et al. . Accurate prediction of kinematic viscosity of biodiesels and their blends with diesel fuels . Journal of the American Oil Chemists Society , 2020 , 97 : 1083 – 1094 . Google Scholar OpenURL Placeholder Text WorldCat [21] Harris KR , Bair S. Temperature and pressure dependence of the viscosity of diisodecyl phthalate at temperatures between 0 and 100 C and at pressures to 1 GPa . Journal of Chemical & Engineering Data , 2007 , 52 : 272 – 278 . Google Scholar Crossref Search ADS WorldCat [22] Dai W , Xiao H, George E, et al. Chapter 46 | lubrication fundamentals . In: Totten GE, Shah RJ, Westbrook SR (eds). Fuels and Lubricants Handbook: Technology, Properties, Performance, and Testing . 2nd edn. West Conshohocken, PA : ASTM International , 2019 , 1565 – 1630 . Google Scholar Crossref Search ADS Google Preview WorldCat COPAC [23] Biresaw G . Elastohydrodynamic properties of seed oils . Journal of the American Oil Chemists' Society , 2006 , 83 : 559 – 566 . Google Scholar Crossref Search ADS WorldCat [24] Wu CS , Klaus EE, Duda JL. Development of a method for the prediction of pressure–viscosity coefficients of lubricating oils based on free-volume theory . Transactions of the ASME, Journal of Tribology , 1989 , 111 : 121 – 128 . Google Scholar Crossref Search ADS WorldCat [25] Ramasamy US , Bair , S, Martini A. Predicting pressure–viscosity behavior from ambient viscosity and compressibility: challenges and opportunities . Tribology Letters , 2015 , 57 : 11 . Google Scholar Crossref Search ADS WorldCat [26] Kapoor K , Dass N. A model for the pressure dependence of viscosity in liquids . Journal of Applied Physics , 2005 , 98 : 066105 . Google Scholar Crossref Search ADS WorldCat [27] Ferreira AG , Talvera-Prieto NM, Portugal AA, et al. Models for predicting viscosities of biodiesel fuels over extended ranges of temperature and pressure . Fuel , 2021 , 287 : 119544 . Google Scholar Crossref Search ADS WorldCat [28] Duncan AM , Azita A, McHenry R, et al. High-pressure viscosity of biodiesel from soybean, canola, and coconut oils . Energy Fuels , 2010 , 24 : 5708 – 5716 . Google Scholar Crossref Search ADS WorldCat [29] Samuel V , Freitas D, Segovia JJ, et al. Measurement and prediction of high-pressure viscosities of biodiesel fuels . Fuel , 2014 , 122 : 223 – 228 . Google Scholar OpenURL Placeholder Text WorldCat [30] Rowane AJ , Babu VM, Rokni HB, et al. Effect of composition, temperature, and pressure on the viscosities and densities of three diesel fuels . Journal of Chemical & Engineering Data , 2019 , 64 : 5529 – 5547 . Google Scholar Crossref Search ADS WorldCat [31] Biresaw G , Bantchev GB. Pressure viscosity coefficient of vegetable oils . Tribology Letters , 2013 , 49 : 501 – 512 . Google Scholar Crossref Search ADS WorldCat © The Author(s) 2022. Published by Oxford University Press on behalf of National Institute of Clean-and-Low-Carbon Energy This is an Open Access article distributed under the terms of the Creative Commons Attribution-NonCommercial License (https://creativecommons.org/licenses/by-nc/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the original work is properly cited. For commercial re-use, please contact journals.permissions@oup.com © The Author(s) 2022. Published by Oxford University Press on behalf of National Institute of Clean-and-Low-Carbon Energy http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Clean Energy Oxford University Press

The rheological model of biodiesels at elevated pressures and temperatures

Clean Energy , Volume 6 (3): 8 – Jun 1, 2022
8 pages

Loading next page...
 
/lp/oxford-university-press/the-rheological-model-of-biodiesels-at-elevated-pressures-and-4Z8kAvTu70
Publisher
Oxford University Press
Copyright
Copyright © 2022 National Institute of Clean-and-Low-Carbon Energy
ISSN
2515-4230
eISSN
2515-396X
DOI
10.1093/ce/zkac024
Publisher site
See Article on Publisher Site

Abstract

Abstract In the present study, an approximation is used to study viscosity as a function of pressure at different temperatures. The correlation so obtained is applied to study the viscosity of biodiesels extracted from soybean, Vistive soybean, canola, used canola, coconut and rapeseed. The computed values of viscosity from the proposed model were found to be in good agreement with experimental data throughout the range of pressure and temperature studied. The maximum average absolute relative deviation (AARD%) and mean AARD% are found to be 0.52 and 0.20, respectively, over the entire range of pressure (0.1–140 MPa) and temperature (283.15–373.15 K) for all biodiesels except rapeseed biodiesel, for which the values are 1.1 and 0.62, respectively. Furthermore, this work includes the very first investigation conducted so far on the variation of the pressure–viscosity coefficient (PVC) with pressure at different temperatures for biodiesels. The variation in PVC with the temperature is more sensitive at elevated pressures as compared to atmospheric pressure whereas the variation in PVC with pressure is more sensitive at elevated temperatures as compared to room temperature. Open in new tabDownload slide renewable fuel, biodiesels, viscosity, pressure–viscosity coefficient, pressure, temperature Introduction Biodiesel is a biodegradable and non-toxic fuel, which is a renewable source of energy. Due to the high cetane number and non-aromaticity of the fuel, it has many advantages over diesel fuel. Biodiesel, being free of sulphur, contains ~10–12% oxygen by weight and is used neat or as a blend with conventional diesel fuel. Biodiesel, when blended with conventional diesel fuel, produces a significantly lower amount of greenhouse gases (GHG), carbon monoxide (CO), hydrocarbons (HC) and particulate emissions [1, 2], though the output is lower. Moreover, being more viscous than diesel fuel, biodiesel improves lubricity, which results in a longer life for engine components [3]. Conventionally, biodiesel is produced through the transesterification of fats and oils in the presence of alcohol, such as methanol, ethanol, etc. The typical profile of the fatty acid methyl ester, which is the main constituent of biodiesel, depends on the composition of the oil feedstock or the fat and is responsible for the chemical and physical properties of the biodiesel yield. For example, the biodiesel produced as a result of transesterification of soybean oil typically contains oleate (C18:1) and linoleate (C18:2), while biodiesel from coconut oil contains laurate (C12:0) and myristate (C14:0). The difference in the chain length and absence of branching are responsible for different physico-chemical properties, such as cloud point, viscosity, density, oxidative stability, etc. [4, 5]. Viscosity is considered to have a great impact on fuel quality. The prediction of viscosity under extreme conditions of high pressure and high temperature would be a useful tool not only in the optimization of biodiesel production, but also in managing the blending of raw materials and refined products. Moreover, the understanding of the effect of pressure and temperature on the viscosity of biodiesel is desirable for the optimization of engine performance under high-pressure and high-temperature conditions. The lack of high-pressure viscosity data at elevated temperatures motivated us to investigate a simple model, which can be used to extrapolate the viscosity of biodiesels under extreme conditions (real conditions in a diesel engine). The majority of the experimental and theoretical work reported in the literature is limited to predicting viscosity variation with temperature, while viscosity variation with pressure is scarce, particularly in biodiesels. There are few methods for predicting the variation of viscosity with pressure [6–17]. The well-known relationship describing the effect of pressure on viscosity was introduced by Barus [9] and tested successfully for moderate pressure in many liquids including biodiesels [10–12]. The Barus model is based on the free-volume theory, which reveals that a chain of molecules needs a larger free-volume void to move within the liquid [13]. Freitas et al. [14] introduced a method to study viscosity as a function of pressure for the high-pressure range, which was valid at high temperatures only since a significant deviation from experimental data was observed at low temperatures. Paluch et al. [17] used the analogy of the Vogel–Tammann–Fulcher (VTF) equation [15, 16] (initially proposed to study the temperature dependence of viscosity) to study the pressure dependence of viscosity as well. However, this equation also failed to represent the viscosity at low pressures. The Munro, Block and Piermarini [18] model was useful for liquids exhibiting non-exponential behaviour at low pressures. Besides these, other methods are also reported in the literature but they are more complex owing to the involvement of a large number of unknown parameters. In addition to the methods stated above, the following soft computing methods for predicting the viscosity of biodiesels and their blends have been developed: multilayer perceptron neural network (MLP-NN), radial basis function (RBF) optimized by particle swarm optimization (PSO-RBF) and adaptive neuro-fuzzy inference system optimized by the hybrid approach (Hybrid-ANFIS) [19, 20]. However, these methods necessitated a significant amount of experimental data to train the system, as well as a relatively long computational time. The pressure–viscosity coefficient (PVC) is an important parameter of lubricants that is rarely investigated for biodiesels. The fluid film thickness that protects the mechanical device from high friction and premature wear is determined by the high value of the PVC. Biodiesel with a higher PVC forms thicker lubricant films and increases the expected life of rolling element bearings, gears, etc. An ideal lubricant forms a thick separating layer between the rubbing surfaces over a wide temperature and pressure range. There are broadly two techniques to estimate the PVC based on (i) viscosity-versus-pressure curves [21, 22] and (ii) elastohydrodynamic lubrication (EHL) film thickness measurement [23]. Using the free-volume theory for calculating the PVC of several liquids, Wu et al. [24] derived a non-linear equation with four adjustable parameters that provided good accuracy when tested over a wide viscosity range. Another empirical relation for the PVC proposed by So and Klaus [25] was successfully applied to mineral oils, resin–polymer blends, pure hydrocarbons and non-hydrocarbons. However, the Wu and So–Klaus models involve a large number of fitting parameters and are complex. Thus, the purpose of this work is to investigate the simple model, which can predict the viscosity of biodiesel as a function of pressure at different temperatures. An approximation is made to deduce the equation, which enables the prediction of the viscosity of biodiesels at high pressures and temperatures. Biodiesels extracted from soybean (S), Vistive soybean, canola, used canola, coconut and rapeseed (R) are chosen to validate the proposed model. The proposed model is also applied to predict the PVC as a function of pressure at different temperatures, which is an important tribological parameter, investigated for the first time for biodiesels in this work. 1 Methodology The Barus equation [9] assumed that the ratio of the first pressure derivative of viscosity to the viscosity itself is independent of pressure but it does not match the experimental viscosity data adequately, particularly at high pressures. However, Kamal and Dass [26] assumed that the ratio of the second pressure derivative of viscosity to the first pressure derivative of viscosity is independent of pressure, as represented by the following equation: (d2η(p,T)╱dp2)T(dη(p,T)╱dp)T=−z(1) where z is a constant. Integrating Equation (1) with respect to pressure, within the limit of p0 to p, yields Equation (2): ∫pp0ddp((dη(p,T)╱dp)T)(dη(p,T)╱dp)Tdp=∫pp0−zpln((dη(p,T)╱dp)T(dη(p0,T)╱dp)T)=−z(p−p0)(dη(p,T)╱dp)T= (dη(p0,T)╱dp)Te−z(p−p0)or η′(p,T)=η′(p0,T)exp(z(p−p0))(2) One more integration of Equation (2) with respect to pressure yields Equation (3): ∫pp0(dη(p,T)╱dp)Tdp=η′(p0,T) ∫pp0e−z(p−p0)dporη(p,T)−η(p0,T)=η′(p0,T)[exp(z(p−p0))−1z]orη(p,T)=η(p0,T)+ η′(p0,T)z[exp(z(p−p0))−1](3) where η′(p0,T) is the first pressure derivative of viscosity η(p,T) at pressure p0 and temperature T. Equation (3) shows that the viscosity will increase with the increase in pressure. Another advantage of this model is that the second and higher pressure derivatives of the viscosity can also be computed with the help of the general relation: (∂Nη∂pN)T=(±z)N−1(∂η∂p)T(4) The VTF equation has been successfully applied in the literature to study the temperature dependence of viscosity in liquids [15, 16]. Paluch et al. [17] used the VTF equation analogy to study the pressure dependence of viscosity in a low-molecular-weight glass-forming liquid, which was later extended to other liquids [27]. The VTF equation has been written by Paluch et al. as: η(p,T)=η (p0, T)exp[C (p−p0)p∞+ (p−p0)](5) The parameter p∞ is referred to as the pressure at which the viscosity diverges and η(p0,T) is the viscosity at pressure p0 and temperature T. C is an empirical constant. In order to find out the PVC, Equation (2) can be rewritten to define the pressure–viscosity coefficient as: α=∂lnη∂p=z η′(p0,T)exp(z(p−p0))η(p0,T)z+η′(p0,T)[exp(z(p−p0))−1](6) However, the PVC derived from Equation (5) is given as: α=∂lnη∂p=bc(c−p)2(7) Equation (3) was successfully applied to predict the viscosity as a function of pressure and temperature in liquids [26]. Thus in the present work, Equation (3) is applied to investigate the viscosity of biodiesels as a function of pressure at different temperatures, whereas Equation (6) is used to study the PVC of biodiesels as a function of pressure at different temperatures. 2 Experimental data The experimental high-pressure viscosity data at different temperatures are taken from Duncan et al. [28] for biodiesel samples extracted from soybean, Vistive soybean, canola, used canola and coconut while the data for the biodiesel sample extracted from rapeseed are taken from Samuel et al. [29]. For comparison, conventional highly paraffinic (HPF) diesel fuel is also considered and high-pressure viscosity data for conventional diesel fuel are taken from Rowane et al. [30]. Duncan et al. have used the ViscoPro2000 system 4-SPL viscometer for the high-pressure viscosity measurement at different temperatures. The equipment is a temperature-controlled oven (0.1 K) housing the high-pressure viscometer sensor as well as a precision pressure transducer and a resistance temperature detector (RTD) (±0.05 K). To pressurize the diesel samples, the viscometer is linked to a manual high-pressure syringe pump acquired from High Pressure Equipment Company (HIP) (model 50-575-30; 30 000 psi, capacity of 10 cm3, with PolyPak). The viscometer uses annular flow principles around an axially oscillating piston and was recently certified by the American Society for Testing and Materials (ASTM). Their viscometer is capable of measuring viscosity in the temperature range of 233.15–463.15 K and at pressures of ≤137.9 MPa. The maximum deviations in the measurement of temperature and viscosity are observed as 0.1 K and 0.036 mPa-s, respectively. Samuel et al. have used the vibrating wire viscometer, which is capable of measuring the viscosity in the temperature range of 273.15–423.15 K and at pressures of ≤140 MPa. Toluene has been used as a standard fluid for calibration purposes and the maximum uncertainty observed in the measurement of viscosity is 1%. In terms of viscosity, the predicted error is 1%. Their vibrating wire viscometer is capable to measure viscosities ranging from 0.3 to 30 mPa-s. The basic component of the apparatus is a sensor with a tungsten wire (length 50 mm and nominal radius 75 lm) anchored at both ends, placed inside a pressure vessel and an external magnetic field oriented perpendicular to the wire length using a magnet set around the vessel. Rowane et al. have used the Rolling-Ball Viscometer/Densimeter for high-pressure viscosity measurement at different temperatures ranging from 298 to 528 K at pressures of ≤300 MPa. The universal calibration method is used for calibration purposes. Standard uncertainty in the measurement of pressure is 0.07 MPa at pressures of <63.9 MPa and 0.41 MPa at pressures of >68.9 MPa, while the standard uncertainty in the measurement of viscosity is 0.025η (mPa-s) with a conversion factor of 2. 3 Results and discussion Equation (3) represents the viscosity as a function of pressure at different temperatures for all biodiesels. Initially, the adjustable parameters η(p0,T), η′(p0,T) and z in Equation (3) were determined using a non-linear fitting toolbox in MATLAB®. The experimental data used to determine the fitting parameters were taken from literature [28–30]. The values of these parameters are given in Table 1 along with root mean square deviation (RMSD) for all biodiesels. Since Equation (5) has been used in the literature to represent the viscosity as a function of pressure, the results of Equation (3) were compared with those of Equation (5). The fitted parameters of Equation (5) are reported in Table 1 in small parentheses. Rheological behaviour of conventional diesel fuel is also compared with chosen biodiesels in this study. The fitted parameters for conventional diesel fuel are also given in Table 1. Table 1: Fitted parameters of Equations (3) and (5) at different temperatures along with RMSD, AARD% and pressure range for all biodiesels and HPF diesel. The fitted parameters (η(p0,T), C and p∞) of Equation (5) and corresponding AARD% are reported in parentheses Liquid . T . η(p0,T) (mPa-s) . η′(p0,T) (mPa-s/MPa) . Z (× 10–3) (MPa–1) . RMSD . AARD% . Pressure range (MPa) . Soybeana (p0 = 0.1 MPa) 283.15 7.452 (7.458) 0.09383 (7.524) 0.007925 (607.7) 0.02234 (0.02629) 0.158 (0.183) 0.1–131 298.15 5.037 (5.061) 0.05823 (9.169) 0.007178 (829.6) 0.01792 (0.02561) 0.119 (0.214) 313.15 3.533 (3.548) 0.03841 (12.38) 0.007704 (1181) 0.02179 (0.02899) 0.281 (0.405) 373.15 1.379 (1.386) 0.01527 (4.598) 0.003971 (437.6) 0.004213 (0.00644) 0.124 (0.214) Vistive soybeana (p0 = 0.1 MPa) 283.15 7.76 (7.762) 0.0984 (6.951) 0.00759 (558.5) 0.01085 (0.009111) 0.075 (0.075) 0.1–131 298.15 5.084 (5.112) 0.06033 (8.411) 0.006992 (744.7) 0.008696 (0.02199) 0.0652 (0.187) 313.15 3.613 (3.629) 0.04008 (11.72) 0.007694 (1097) 0.02726 (0.03236) 0.3 (0.383) 373.15 1.37 (1.377) 0.01539 (4.722) 0.004139 (443.9) 0.004231 (0.00762) 0.0417 (0.2) Canolaa (p0 = 0.1 MPa) 283.15 8.595 (8.609) 0.1069 (13.54) 0.009496 (1110) 0.08151 (0.08658) 0.328 (0.379) 0.1–131 298.15 5.605 (5.644) 0.06502 (9.914) 0.007675 (886.3) 0.08905 (0.04607) 0.462 (0.557) 313.15 3.907 (3.926) 0.04398 (7.958) 0.006508 (740.8) 0.01675 (0.02525) 0.205 (0.324) 373.15 1.426 (1.433) 0.01639 (4.713) 0.004211 (433) 0.006893 (0.004) 0.209 (0.124) Used canolaa (p0 = 0.1 MPa) 283.15 8.642 (8.652) 0.1055 (13.4) 0.009379 (1111.5) 0.03745 (0.0414) 0.171 (0.214) 0.1–131 298.15 5.541 (5.572) 0.06712 (8.462) 0.007125 (734.9) 0.01766 (0.02786) 0.095 (0.190) 313.15 3.914 (3.933) 0.04468 (7.594) 0.006415 (697.5) 0.009975 (0.01392) 0.114 (0.171) 373.15 1.437 (1.444) 0.01616 (5.021) 0.004479 (470.8) 0.004109 (0.004751) 0.124 (0.157) Coconuta (p0 = 0.1 MPa) 283.15 4.848 (4.859) 0.05969 (7.228) 0.007143 (605.3) 0.008689 (0.01067) 0.087 (0.086) 0.1–131 298.15 3.279 (3.296) 0.03747 (8.019) 0.006603 (736.4) 0.02481 (0.03013) 0.257 (0.391) 313.15 2.196 (2.208) 0.02615 (6.631) 0.006019 (587.5) 0.01331 (0.01381) 0.290 (0.286) 373.15 0.9258 (0.9297) 0.01001 (3.395) 0.002147 (330.8) 0.01053 (0.01169) 0.519 (0.552) Rapeseedb (p0 = 0.1 MPa) 293.15 6.93 (6.709) 0.07982 (3392) 0.01604 (245 700) 0.1812 (0.3266) 1.07 (2.13) 0.1–140 313.15 4.22 (4.264) 0.04939 (17.3) 0.008759 (1556) 0.08324 (0.08973) 0.653 (0.800) 333.15 2.86 (2.892) 0.03343 (8.225) 0.006516 (753.4) 0.03016 (0.03451) 0.542 (0.593) 353.15 2.11 (2.139) 0.02514 (5.036) 0.004281 (460.3) 0.02811 (0.03055) 0.550 (0.529) 373.15 1.68 (1.681) 0.01851 (4.333) 0.003669 (410.8) 0.01785 (0.021) 0.486 (0.536) 393.15 1.33 (1.338) 0.0157 (3.304) 0.001921 (298) 0.01054 (0.009716) 0.421 (0.371) Highly paraffinic dieselc (HFPD) (p0 = 0 MPa) 298.3 3.086 (3.483) 0.05339 (29.22) 0.0103 (2229) 0.4225 (4530) 1.17 (1.74) 3.8–300 323.2 1.716 (1.879) 0.02611 (18.02) 0.008327 (1522) 0.1398 (0.1367) 1.68 (1.44) 350.4 1.138 (1.198) 0.01521 (13.10) 0.006922 (1203) 0.03742 (0.05768) 0.828 (1.60) 433.1 0.4564 (0.4762) 0.005952 (6.061) 0.004255 (560.8) 0.01762 (0.01834) 1.13 (1.24) 528.7 0.2244 (0.2341) 0.003339 (4.404) 0.002766 (362.7) 0.009343 (0.0114) 1.45 (1.90) Liquid . T . η(p0,T) (mPa-s) . η′(p0,T) (mPa-s/MPa) . Z (× 10–3) (MPa–1) . RMSD . AARD% . Pressure range (MPa) . Soybeana (p0 = 0.1 MPa) 283.15 7.452 (7.458) 0.09383 (7.524) 0.007925 (607.7) 0.02234 (0.02629) 0.158 (0.183) 0.1–131 298.15 5.037 (5.061) 0.05823 (9.169) 0.007178 (829.6) 0.01792 (0.02561) 0.119 (0.214) 313.15 3.533 (3.548) 0.03841 (12.38) 0.007704 (1181) 0.02179 (0.02899) 0.281 (0.405) 373.15 1.379 (1.386) 0.01527 (4.598) 0.003971 (437.6) 0.004213 (0.00644) 0.124 (0.214) Vistive soybeana (p0 = 0.1 MPa) 283.15 7.76 (7.762) 0.0984 (6.951) 0.00759 (558.5) 0.01085 (0.009111) 0.075 (0.075) 0.1–131 298.15 5.084 (5.112) 0.06033 (8.411) 0.006992 (744.7) 0.008696 (0.02199) 0.0652 (0.187) 313.15 3.613 (3.629) 0.04008 (11.72) 0.007694 (1097) 0.02726 (0.03236) 0.3 (0.383) 373.15 1.37 (1.377) 0.01539 (4.722) 0.004139 (443.9) 0.004231 (0.00762) 0.0417 (0.2) Canolaa (p0 = 0.1 MPa) 283.15 8.595 (8.609) 0.1069 (13.54) 0.009496 (1110) 0.08151 (0.08658) 0.328 (0.379) 0.1–131 298.15 5.605 (5.644) 0.06502 (9.914) 0.007675 (886.3) 0.08905 (0.04607) 0.462 (0.557) 313.15 3.907 (3.926) 0.04398 (7.958) 0.006508 (740.8) 0.01675 (0.02525) 0.205 (0.324) 373.15 1.426 (1.433) 0.01639 (4.713) 0.004211 (433) 0.006893 (0.004) 0.209 (0.124) Used canolaa (p0 = 0.1 MPa) 283.15 8.642 (8.652) 0.1055 (13.4) 0.009379 (1111.5) 0.03745 (0.0414) 0.171 (0.214) 0.1–131 298.15 5.541 (5.572) 0.06712 (8.462) 0.007125 (734.9) 0.01766 (0.02786) 0.095 (0.190) 313.15 3.914 (3.933) 0.04468 (7.594) 0.006415 (697.5) 0.009975 (0.01392) 0.114 (0.171) 373.15 1.437 (1.444) 0.01616 (5.021) 0.004479 (470.8) 0.004109 (0.004751) 0.124 (0.157) Coconuta (p0 = 0.1 MPa) 283.15 4.848 (4.859) 0.05969 (7.228) 0.007143 (605.3) 0.008689 (0.01067) 0.087 (0.086) 0.1–131 298.15 3.279 (3.296) 0.03747 (8.019) 0.006603 (736.4) 0.02481 (0.03013) 0.257 (0.391) 313.15 2.196 (2.208) 0.02615 (6.631) 0.006019 (587.5) 0.01331 (0.01381) 0.290 (0.286) 373.15 0.9258 (0.9297) 0.01001 (3.395) 0.002147 (330.8) 0.01053 (0.01169) 0.519 (0.552) Rapeseedb (p0 = 0.1 MPa) 293.15 6.93 (6.709) 0.07982 (3392) 0.01604 (245 700) 0.1812 (0.3266) 1.07 (2.13) 0.1–140 313.15 4.22 (4.264) 0.04939 (17.3) 0.008759 (1556) 0.08324 (0.08973) 0.653 (0.800) 333.15 2.86 (2.892) 0.03343 (8.225) 0.006516 (753.4) 0.03016 (0.03451) 0.542 (0.593) 353.15 2.11 (2.139) 0.02514 (5.036) 0.004281 (460.3) 0.02811 (0.03055) 0.550 (0.529) 373.15 1.68 (1.681) 0.01851 (4.333) 0.003669 (410.8) 0.01785 (0.021) 0.486 (0.536) 393.15 1.33 (1.338) 0.0157 (3.304) 0.001921 (298) 0.01054 (0.009716) 0.421 (0.371) Highly paraffinic dieselc (HFPD) (p0 = 0 MPa) 298.3 3.086 (3.483) 0.05339 (29.22) 0.0103 (2229) 0.4225 (4530) 1.17 (1.74) 3.8–300 323.2 1.716 (1.879) 0.02611 (18.02) 0.008327 (1522) 0.1398 (0.1367) 1.68 (1.44) 350.4 1.138 (1.198) 0.01521 (13.10) 0.006922 (1203) 0.03742 (0.05768) 0.828 (1.60) 433.1 0.4564 (0.4762) 0.005952 (6.061) 0.004255 (560.8) 0.01762 (0.01834) 1.13 (1.24) 528.7 0.2244 (0.2341) 0.003339 (4.404) 0.002766 (362.7) 0.009343 (0.0114) 1.45 (1.90) Experimental data taken from a[28], b[29] and c[30]. Open in new tab Table 1: Fitted parameters of Equations (3) and (5) at different temperatures along with RMSD, AARD% and pressure range for all biodiesels and HPF diesel. The fitted parameters (η(p0,T), C and p∞) of Equation (5) and corresponding AARD% are reported in parentheses Liquid . T . η(p0,T) (mPa-s) . η′(p0,T) (mPa-s/MPa) . Z (× 10–3) (MPa–1) . RMSD . AARD% . Pressure range (MPa) . Soybeana (p0 = 0.1 MPa) 283.15 7.452 (7.458) 0.09383 (7.524) 0.007925 (607.7) 0.02234 (0.02629) 0.158 (0.183) 0.1–131 298.15 5.037 (5.061) 0.05823 (9.169) 0.007178 (829.6) 0.01792 (0.02561) 0.119 (0.214) 313.15 3.533 (3.548) 0.03841 (12.38) 0.007704 (1181) 0.02179 (0.02899) 0.281 (0.405) 373.15 1.379 (1.386) 0.01527 (4.598) 0.003971 (437.6) 0.004213 (0.00644) 0.124 (0.214) Vistive soybeana (p0 = 0.1 MPa) 283.15 7.76 (7.762) 0.0984 (6.951) 0.00759 (558.5) 0.01085 (0.009111) 0.075 (0.075) 0.1–131 298.15 5.084 (5.112) 0.06033 (8.411) 0.006992 (744.7) 0.008696 (0.02199) 0.0652 (0.187) 313.15 3.613 (3.629) 0.04008 (11.72) 0.007694 (1097) 0.02726 (0.03236) 0.3 (0.383) 373.15 1.37 (1.377) 0.01539 (4.722) 0.004139 (443.9) 0.004231 (0.00762) 0.0417 (0.2) Canolaa (p0 = 0.1 MPa) 283.15 8.595 (8.609) 0.1069 (13.54) 0.009496 (1110) 0.08151 (0.08658) 0.328 (0.379) 0.1–131 298.15 5.605 (5.644) 0.06502 (9.914) 0.007675 (886.3) 0.08905 (0.04607) 0.462 (0.557) 313.15 3.907 (3.926) 0.04398 (7.958) 0.006508 (740.8) 0.01675 (0.02525) 0.205 (0.324) 373.15 1.426 (1.433) 0.01639 (4.713) 0.004211 (433) 0.006893 (0.004) 0.209 (0.124) Used canolaa (p0 = 0.1 MPa) 283.15 8.642 (8.652) 0.1055 (13.4) 0.009379 (1111.5) 0.03745 (0.0414) 0.171 (0.214) 0.1–131 298.15 5.541 (5.572) 0.06712 (8.462) 0.007125 (734.9) 0.01766 (0.02786) 0.095 (0.190) 313.15 3.914 (3.933) 0.04468 (7.594) 0.006415 (697.5) 0.009975 (0.01392) 0.114 (0.171) 373.15 1.437 (1.444) 0.01616 (5.021) 0.004479 (470.8) 0.004109 (0.004751) 0.124 (0.157) Coconuta (p0 = 0.1 MPa) 283.15 4.848 (4.859) 0.05969 (7.228) 0.007143 (605.3) 0.008689 (0.01067) 0.087 (0.086) 0.1–131 298.15 3.279 (3.296) 0.03747 (8.019) 0.006603 (736.4) 0.02481 (0.03013) 0.257 (0.391) 313.15 2.196 (2.208) 0.02615 (6.631) 0.006019 (587.5) 0.01331 (0.01381) 0.290 (0.286) 373.15 0.9258 (0.9297) 0.01001 (3.395) 0.002147 (330.8) 0.01053 (0.01169) 0.519 (0.552) Rapeseedb (p0 = 0.1 MPa) 293.15 6.93 (6.709) 0.07982 (3392) 0.01604 (245 700) 0.1812 (0.3266) 1.07 (2.13) 0.1–140 313.15 4.22 (4.264) 0.04939 (17.3) 0.008759 (1556) 0.08324 (0.08973) 0.653 (0.800) 333.15 2.86 (2.892) 0.03343 (8.225) 0.006516 (753.4) 0.03016 (0.03451) 0.542 (0.593) 353.15 2.11 (2.139) 0.02514 (5.036) 0.004281 (460.3) 0.02811 (0.03055) 0.550 (0.529) 373.15 1.68 (1.681) 0.01851 (4.333) 0.003669 (410.8) 0.01785 (0.021) 0.486 (0.536) 393.15 1.33 (1.338) 0.0157 (3.304) 0.001921 (298) 0.01054 (0.009716) 0.421 (0.371) Highly paraffinic dieselc (HFPD) (p0 = 0 MPa) 298.3 3.086 (3.483) 0.05339 (29.22) 0.0103 (2229) 0.4225 (4530) 1.17 (1.74) 3.8–300 323.2 1.716 (1.879) 0.02611 (18.02) 0.008327 (1522) 0.1398 (0.1367) 1.68 (1.44) 350.4 1.138 (1.198) 0.01521 (13.10) 0.006922 (1203) 0.03742 (0.05768) 0.828 (1.60) 433.1 0.4564 (0.4762) 0.005952 (6.061) 0.004255 (560.8) 0.01762 (0.01834) 1.13 (1.24) 528.7 0.2244 (0.2341) 0.003339 (4.404) 0.002766 (362.7) 0.009343 (0.0114) 1.45 (1.90) Liquid . T . η(p0,T) (mPa-s) . η′(p0,T) (mPa-s/MPa) . Z (× 10–3) (MPa–1) . RMSD . AARD% . Pressure range (MPa) . Soybeana (p0 = 0.1 MPa) 283.15 7.452 (7.458) 0.09383 (7.524) 0.007925 (607.7) 0.02234 (0.02629) 0.158 (0.183) 0.1–131 298.15 5.037 (5.061) 0.05823 (9.169) 0.007178 (829.6) 0.01792 (0.02561) 0.119 (0.214) 313.15 3.533 (3.548) 0.03841 (12.38) 0.007704 (1181) 0.02179 (0.02899) 0.281 (0.405) 373.15 1.379 (1.386) 0.01527 (4.598) 0.003971 (437.6) 0.004213 (0.00644) 0.124 (0.214) Vistive soybeana (p0 = 0.1 MPa) 283.15 7.76 (7.762) 0.0984 (6.951) 0.00759 (558.5) 0.01085 (0.009111) 0.075 (0.075) 0.1–131 298.15 5.084 (5.112) 0.06033 (8.411) 0.006992 (744.7) 0.008696 (0.02199) 0.0652 (0.187) 313.15 3.613 (3.629) 0.04008 (11.72) 0.007694 (1097) 0.02726 (0.03236) 0.3 (0.383) 373.15 1.37 (1.377) 0.01539 (4.722) 0.004139 (443.9) 0.004231 (0.00762) 0.0417 (0.2) Canolaa (p0 = 0.1 MPa) 283.15 8.595 (8.609) 0.1069 (13.54) 0.009496 (1110) 0.08151 (0.08658) 0.328 (0.379) 0.1–131 298.15 5.605 (5.644) 0.06502 (9.914) 0.007675 (886.3) 0.08905 (0.04607) 0.462 (0.557) 313.15 3.907 (3.926) 0.04398 (7.958) 0.006508 (740.8) 0.01675 (0.02525) 0.205 (0.324) 373.15 1.426 (1.433) 0.01639 (4.713) 0.004211 (433) 0.006893 (0.004) 0.209 (0.124) Used canolaa (p0 = 0.1 MPa) 283.15 8.642 (8.652) 0.1055 (13.4) 0.009379 (1111.5) 0.03745 (0.0414) 0.171 (0.214) 0.1–131 298.15 5.541 (5.572) 0.06712 (8.462) 0.007125 (734.9) 0.01766 (0.02786) 0.095 (0.190) 313.15 3.914 (3.933) 0.04468 (7.594) 0.006415 (697.5) 0.009975 (0.01392) 0.114 (0.171) 373.15 1.437 (1.444) 0.01616 (5.021) 0.004479 (470.8) 0.004109 (0.004751) 0.124 (0.157) Coconuta (p0 = 0.1 MPa) 283.15 4.848 (4.859) 0.05969 (7.228) 0.007143 (605.3) 0.008689 (0.01067) 0.087 (0.086) 0.1–131 298.15 3.279 (3.296) 0.03747 (8.019) 0.006603 (736.4) 0.02481 (0.03013) 0.257 (0.391) 313.15 2.196 (2.208) 0.02615 (6.631) 0.006019 (587.5) 0.01331 (0.01381) 0.290 (0.286) 373.15 0.9258 (0.9297) 0.01001 (3.395) 0.002147 (330.8) 0.01053 (0.01169) 0.519 (0.552) Rapeseedb (p0 = 0.1 MPa) 293.15 6.93 (6.709) 0.07982 (3392) 0.01604 (245 700) 0.1812 (0.3266) 1.07 (2.13) 0.1–140 313.15 4.22 (4.264) 0.04939 (17.3) 0.008759 (1556) 0.08324 (0.08973) 0.653 (0.800) 333.15 2.86 (2.892) 0.03343 (8.225) 0.006516 (753.4) 0.03016 (0.03451) 0.542 (0.593) 353.15 2.11 (2.139) 0.02514 (5.036) 0.004281 (460.3) 0.02811 (0.03055) 0.550 (0.529) 373.15 1.68 (1.681) 0.01851 (4.333) 0.003669 (410.8) 0.01785 (0.021) 0.486 (0.536) 393.15 1.33 (1.338) 0.0157 (3.304) 0.001921 (298) 0.01054 (0.009716) 0.421 (0.371) Highly paraffinic dieselc (HFPD) (p0 = 0 MPa) 298.3 3.086 (3.483) 0.05339 (29.22) 0.0103 (2229) 0.4225 (4530) 1.17 (1.74) 3.8–300 323.2 1.716 (1.879) 0.02611 (18.02) 0.008327 (1522) 0.1398 (0.1367) 1.68 (1.44) 350.4 1.138 (1.198) 0.01521 (13.10) 0.006922 (1203) 0.03742 (0.05768) 0.828 (1.60) 433.1 0.4564 (0.4762) 0.005952 (6.061) 0.004255 (560.8) 0.01762 (0.01834) 1.13 (1.24) 528.7 0.2244 (0.2341) 0.003339 (4.404) 0.002766 (362.7) 0.009343 (0.0114) 1.45 (1.90) Experimental data taken from a[28], b[29] and c[30]. Open in new tab The RMSD values given in Table 1 indicate the applicability of Equation (3) to describe the viscosity as a function of the pressure and temperature for all biodiesels and conventional diesel fuel. The maximum AARD% and mean AARD% are found to be 0.52 and 0.20, respectively, in the entire range of pressure and temperature for all biodiesels except rapeseed oil. For rapeseed oil, the maximum AARD% and mean AARD% values are 1.1 and 0.62, respectively. The applicability of Equation (5) is also similar but deviates slightly in the present study, particularly at low pressure as depicted in terms of the RMSD and AARD% values given in Table 1. The maximum AARD% and mean AARD% values determined using Equation (5) are found to be 0.57 and 0.26, respectively, in the entire range of pressures and temperatures for all biodiesels except rapeseed oil, for which these values are 2.1 and 0.83, respectively. The calculated viscosities as a function of pressure at different temperatures from Equations (3) and (5) are compared to experimentally determined the viscosities for soybean, canola, used canola and coconut biodiesels in Fig. 1. Fig. 1: Open in new tabDownload slide (a)–(d) Variation in viscosity as a function of pressure at different temperatures in biodiesels derived from soybean, Vistive soybean, canola and used canola. As shown in Fig. 1, the maximum error in Equation (3) is observed at low pressures and at room temperature. Based on the comparison above, it is clear that Equation (3) is superior to Equation (5) and other models [27] for studying viscosity as a function of pressure, especially in the high-temperature regime. Similar comparison for coconut, rapeseed biodiesels and conventional diesel fuel is shown in Fig. 2. Fig. 2: Open in new tabDownload slide (a) and (b) Variation in viscosity as a function of pressure at different temperatures in biodiesels derived from coconut and rapeseed, and (c) HPF diesel. The PVC values were calculated as a function of pressure at different temperatures using Equations (6) and (7) for each biodiesel and conventional diesel fuel with the help of data given in Table 1. PVC values are compared for soybean and conventional diesel as shown in Fig. 3. Fig. 3: Open in new tabDownload slide (a) and (b) Variation in pressure–viscosity coefficient as a function of pressure at different temperatures in biodiesels derived from HPF diesel and soybean. The understanding of variation of viscosity with pressure and temperature is desirable to optimize the performance of an engine. For example, a common automotive diesel engine may have operating injector pressure and temperature as high as 300 MPa and 363 K, respectively, whereas the temperature in the combustion chamber will be much higher than this [12, 30]. The high value of viscosity increases the problem of atomization and hence may damage the fuel injectors, thereby resulting in incomplete combustion and the deposition of solid unburned particles. On the other hand, biodiesel with lower viscosity may be lacking in providing lubrication to the injectors and increase the problem of leakage [12]. Therefore, biodiesel must be optimized to have a viscosity that lies within the range prescribed by ASTM and European Norm (EN) standards. It is clear from Fig. 1 that the viscosity increases with an increase in pressure and decreases with an increase in temperature; however, the effect of temperature is more dominating than pressure. The effect of pressure on viscosity is lower at higher temperatures, particularly in HPF diesel, as the curvature of the viscosity pressure curve is decreasing with temperature in Fig. 2. The viscosity of biodiesel is greater than that of conventional diesel; the behaviour of viscosity with pressure at elevated temperatures can help to optimize the viscosity as per ASTM and EN standards by controlling the proportions of the biodiesel and conventional diesel in the blend. This may lead to better performance of diesel engines in terms of lower emission of greenhouse gases and other emissions. Fig. 3 shows the variation in PVC with pressure as calculated from Equations (6) and (7) at different temperatures for soybean biodiesel and conventional diesel fuel. The PVC is found to decrease with pressure at different temperatures for all biodiesels except rapeseed oil, for which it is increasing with pressure at 293.15 K. The physical reason for this unusual behaviour is not clear but the curvature of the viscosity pressure curve at 293.15 K is more than that at other temperatures, i.e. the first pressure derivative of viscosity is increasing faster than the viscosity itself. It is further noted that the PVC is decreasing with temperature, but the variation is very small (almost temperature-independent) at atmospheric pressure as compared to the variation at elevated pressures. In the literature, variation of PVC with temperature is reported for biodiesels but the variation with pressure is very scarce. However, the reported values of PVC at different temperatures are significantly different from each other and largely depend on the method employed for determination [27]. In the present study, the PVC is changing from 13 to 5 GPa–1 in the entire range of investigation for pressure and temperature. Similar trends are observed for PVC calculated from Equation (7). The high PVC values suggest that biodiesels have flexible hydrocarbon structures with little or no branching or layering at atmospheric pressure. But at elevated pressures, biodiesels with PVC values of <10 GPa–1 may behave like layered molecular structures [31]. Moreover, Fig. 3 shows that the PVC of HPF diesel also decreases with pressure at a faster rate in low-pressure regions whereas variation is sluggish at high pressures. 3 Conclusions Because the viscosity and PVCs are key parameters to evaluate the quality of biodiesel, models to predict the dependence of these parameters on pressure and temperature were established. The proposed models were validated using biodiesels extracted from various feedstocks. It is evident from Table 1 and Figs 1 and 2 that the present model is quite appropriate to accurately predict the pressure dependence of viscosity of biodiesels at different temperatures as the AARD% is quite small in the entire temperature range (283–528 K) and pressure range (0.1–300 MPa). The highest departure from experimental data is recorded in the low-pressure and high-temperature regions. With respect to experimental data, Equation (5) exhibits much higher divergence at low pressure and high temperature. The PVC of biodiesels was also calculated using a novel correlation developed as Equation (6). The calculated PVC values of biodiesels were found to be lower than those of conventional diesel. Therefore, EHL must be used at a relatively high pressure. To summarize, the current model successfully represents viscosity as a function of pressure and temperature. The current model can interpolate and extrapolate viscosity at different pressures and temperatures. The findings of this study can be used for simulation and planning by researchers and industries without the need for energy-intensive experiments. The results obtained here can also be used to prepare feedstocks with the appropriate properties in order to generate biodiesel fuels for the efficient operation of diesel engines with lower emissions. Acknowledgements We are thankful to the University of Petroleum and Energy Studies, Dehradun for providing the infrastructure for this work. Funding This research received no external funding. Conflict of interest statement None declared. References [1] Graboski MS , McCormik RL. Combustion of fat and vegetable oil derived fuels in diesel engines . Progress in Energy and Combustion Science , 1998 , 24 : 125 – 164 . Google Scholar Crossref Search ADS WorldCat [2] Tomasevic AV , Marinkovic SS. Methanolysis of used frying oil . Fuel Processing Technology , 2003 , 81 : 1 – 6 . Google Scholar Crossref Search ADS WorldCat [3] Kinast JA. Production of Biodiesels from Multiple Feedstocks and Properties of Biodiesels and Biodiesel-Diesel Blends, NREL/SR-510-31460 . Des Plaines, IL : National Renewable Energy Laboratory , 2001 . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC [4] Rodrigues J , Cardoso F, Lachter E, et al. Correlating chemical structure and physical properties of vegetable oil esters . Journal of the American Oil Chemists' Society , 2006 , 83 : 353 – 357 . Google Scholar Crossref Search ADS WorldCat [5] Ramos M , Fernandez C, Casas A, et al. Influence of fatty acid composition of raw materials on biodiesel properties . Bioresource Technology , 2009 , 100 : 261 – 268 . Google Scholar Crossref Search ADS PubMed WorldCat [6] Taghizadehfard M , Hosseinim SM, Alavianmehr MM. Viscosity modeling of fatty acid esters and biodiesels based on friction theory and perturbed hard-dimer-chain equation of state . Journal of Molecular Liquids , 2021 , 325 : 115048 . Google Scholar Crossref Search ADS WorldCat [7] Chum-in T , Sudaprasert K, Phankosol S, et al. Gibbs energy additivity approaches to QSPR in modeling of high pressure density and kinematic viscosity of FAME and biodiesel . Fuel Processing Technology , 2017 , 156 : 385 – 393 . Google Scholar Crossref Search ADS WorldCat [8] Chum-in T , Sudaprasert K, Phankosol S, et al. Gibbs energy additivity approaches to QSPR in modeling of high pressure dynamic viscosity of FAME and biodiesel . Journal of Molecular Liquids , 2016 , 223 : 1006 – 1012 . Google Scholar Crossref Search ADS WorldCat [9] Barus C . Isothermals, isopiestics and isometrics relative to viscosity . American Journal of Science , 1983 , 45 : 87 – 96 . Google Scholar OpenURL Placeholder Text WorldCat [10] Paton JM , Schaschke CJ. Viscosity measurement of biodiesel at high pressure with a falling sinker viscometer . Chemical Engineering Research and Design , 2009 , 87 : 1520 – 1526 . Google Scholar Crossref Search ADS WorldCat [11] Robertson LX , Schaschke CJ. Combined high pressure and low temperature viscosity measurement of biodiesel . Energy Fuels , 2010 , 24 : 1293 – 1297 . Google Scholar Crossref Search ADS WorldCat [12] Schaschke CJ . Experimental viscosity measurements of biodiesels at high pressure . Chemical Industry and Chemical Engineering Quarterly , 2016 , 22 : 453 – 460 . Google Scholar Crossref Search ADS WorldCat [13] Kioupis LI , Maginn EJ. Impact of molecular architecture on the high-pressure rheology of hydrocarbon fluids . Journal of Physical Chemistry B , 2000 , 104 : 7774 – 7783 . Google Scholar Crossref Search ADS WorldCat [14] Freitas Samuel VD , Segovia Jose J, Carmen Martín M, et al. Measurement and prediction of high-pressure viscosities of biodiesel fuels . Fuel , 2014 , 122 : 223 – 228 . Google Scholar Crossref Search ADS WorldCat [15] Angell CA . Formation of glasses from liquids and biopolymers . Science , 1995 , 276 : 1924 – 1935 . Google Scholar OpenURL Placeholder Text WorldCat [16] Ivaniš GR , Radovic IR, Valada B, et al. Thermodynamic properties of biodiesel and petro-diesel blends at high pressures and temperatures. Experimental and modeling . Fuel , 2016 , 184 : 277 – 288 . Google Scholar Crossref Search ADS WorldCat [17] Paluch M , Denzik Z, Rzoska SJ. Scaling of high-pressure viscosity data in low-molecular-weight glass-forming liquids . Physical Review B , 1999 , 60 : 2979 – 2982 . Google Scholar Crossref Search ADS WorldCat [18] Lima , TA , Ribeiro, et al. . Low-frequency Raman spectra of a glass-forming ionic liquid at low temperature and high pressure . Journal of Chemical Physics , 2019 , 150 : 164502 . Google Scholar Crossref Search ADS PubMed WorldCat [19] Hosseini SM , Pierantozzi M, Moghadasi J. Viscosities of some fatty acid esters and biodiesel fuels from a rough hard-sphere-chain model and artificial neural network . Fuel , 1083 , 2019 : 1091 . Google Scholar OpenURL Placeholder Text WorldCat [20] Masoud M , Hadi NF, Zand N, et al. . Accurate prediction of kinematic viscosity of biodiesels and their blends with diesel fuels . Journal of the American Oil Chemists Society , 2020 , 97 : 1083 – 1094 . Google Scholar OpenURL Placeholder Text WorldCat [21] Harris KR , Bair S. Temperature and pressure dependence of the viscosity of diisodecyl phthalate at temperatures between 0 and 100 C and at pressures to 1 GPa . Journal of Chemical & Engineering Data , 2007 , 52 : 272 – 278 . Google Scholar Crossref Search ADS WorldCat [22] Dai W , Xiao H, George E, et al. Chapter 46 | lubrication fundamentals . In: Totten GE, Shah RJ, Westbrook SR (eds). Fuels and Lubricants Handbook: Technology, Properties, Performance, and Testing . 2nd edn. West Conshohocken, PA : ASTM International , 2019 , 1565 – 1630 . Google Scholar Crossref Search ADS Google Preview WorldCat COPAC [23] Biresaw G . Elastohydrodynamic properties of seed oils . Journal of the American Oil Chemists' Society , 2006 , 83 : 559 – 566 . Google Scholar Crossref Search ADS WorldCat [24] Wu CS , Klaus EE, Duda JL. Development of a method for the prediction of pressure–viscosity coefficients of lubricating oils based on free-volume theory . Transactions of the ASME, Journal of Tribology , 1989 , 111 : 121 – 128 . Google Scholar Crossref Search ADS WorldCat [25] Ramasamy US , Bair , S, Martini A. Predicting pressure–viscosity behavior from ambient viscosity and compressibility: challenges and opportunities . Tribology Letters , 2015 , 57 : 11 . Google Scholar Crossref Search ADS WorldCat [26] Kapoor K , Dass N. A model for the pressure dependence of viscosity in liquids . Journal of Applied Physics , 2005 , 98 : 066105 . Google Scholar Crossref Search ADS WorldCat [27] Ferreira AG , Talvera-Prieto NM, Portugal AA, et al. Models for predicting viscosities of biodiesel fuels over extended ranges of temperature and pressure . Fuel , 2021 , 287 : 119544 . Google Scholar Crossref Search ADS WorldCat [28] Duncan AM , Azita A, McHenry R, et al. High-pressure viscosity of biodiesel from soybean, canola, and coconut oils . Energy Fuels , 2010 , 24 : 5708 – 5716 . Google Scholar Crossref Search ADS WorldCat [29] Samuel V , Freitas D, Segovia JJ, et al. Measurement and prediction of high-pressure viscosities of biodiesel fuels . Fuel , 2014 , 122 : 223 – 228 . Google Scholar OpenURL Placeholder Text WorldCat [30] Rowane AJ , Babu VM, Rokni HB, et al. Effect of composition, temperature, and pressure on the viscosities and densities of three diesel fuels . Journal of Chemical & Engineering Data , 2019 , 64 : 5529 – 5547 . Google Scholar Crossref Search ADS WorldCat [31] Biresaw G , Bantchev GB. Pressure viscosity coefficient of vegetable oils . Tribology Letters , 2013 , 49 : 501 – 512 . Google Scholar Crossref Search ADS WorldCat © The Author(s) 2022. Published by Oxford University Press on behalf of National Institute of Clean-and-Low-Carbon Energy This is an Open Access article distributed under the terms of the Creative Commons Attribution-NonCommercial License (https://creativecommons.org/licenses/by-nc/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the original work is properly cited. For commercial re-use, please contact journals.permissions@oup.com © The Author(s) 2022. Published by Oxford University Press on behalf of National Institute of Clean-and-Low-Carbon Energy

Journal

Clean EnergyOxford University Press

Published: Jun 1, 2022

References