Rackett Equation Research Articles

Predicting the temperature dependent density of biodiesel–diesel–bioethanol blends

Density is a very important fuel property because it influences production, transportation, and distribution processes as well as all processes that take place in the internal combustion engine. Elaborating models to describe the density of biodiesel–diesel–bioethanol blends with a high content of biofuel facilitates the production of blends that comply with the standard quality requirements for density and the modeling and simulation of injection and combustion processes. For predicting density of rapeseed oil biodiesel–diesel–bioethanol and used cooking oil biodiesel–diesel–bioethanol blends, 2×15 ternary mixtures were prepared, having a maximum amount of biofuel of 30%v/v. The concentration of biodiesel and bioethanol varies between 5% and 25%, in increments of 5%. Their density values were determined in 15 steps in the temperature range of 273.15–343.15K. Based on experimental density values of components and blends, common linear mixing rules were evaluated, obtaining an average relative deviation under 0.125% and a regression coefficient above 0.996. In order to improve accuracy, four new mixing rules with different complexities were elaborated. The temperature dependent density of components was modeled based on experimental density values and constituents composition by linear and polynomial regression, by free version of Rackett equation and by group contribution methods. The elaborated mixing rules were tested for various combinations of component densities, yielding a very good accuracy, with an average deviation of under 0.053%. Taking into account the complexity and accuracy of the elaborated models, recommendations were made regarding their uses according to their concrete applications.

Density and Speed of Sound Measurements of Four Bioderived Aviation Fuels

Compressed-liquid densities and ambient-pressure densities and speeds of sound of four biomass-derived fuels have been measured. The compressed-liquid measurements were made from (270 to 470) K and (0.5 to 50) MPa. The ambient-pressure measurements were made from (278 to 343) K. Compressed-liquid density data at 10 MPa and below were extrapolated to 0.083 MPa and, in combination with the ambient-pressure data, fitted to a Rackett equation, to allow comparison of the two sources of data. Additionally, the compressed-liquid density data have been correlated to a Tait equation, and parameters are given for each fuel.

Developing models for correlating ionic liquids density: Part 1 – Density at 0.1 MPa

In this work, the experimental data of ionic liquids density at 0.1MPa were correlated with three models. Rackett equation which is a well known equation for calculating liquid density, has been considered as the first approach to calculate ionic liquids density. As Rackett equation could not reproduce well experimental data, the first model was developed to give a simple modification to Rackett equation. The basis of the first model was similar to Rackett equation which is an observed linear relation between logarithm of reduced volume and reduced temperature. The second model is obtained by expanding ionic liquid volume as a function of reduced temperature in the form of Taylor series. The third model was similar to Bhirud's model and logarithm of reduced density was expressed as function of logarithm of reduced temperature. All proposed models involve adjustable parameters in terms of ionic liquids thermodynamic properties which are obtained by correlating experimental data. As results show, these correlations are accurate enough to calculate ionic liquids density and the average absolute percent deviation was 0.83%, 1.10% and 2.04% for first, second and third model. The calculation of ionic liquids density at high pressures by aid of developed models is the subject of upcoming paper.

Multiplicative algorithm for computing D-optimum designs for pVT measurements

Multifactor models are present in industry and in biological experiments; therefore optimum designs are a valuable tool for experimenters, leading to estimators of the parameters with minimum variance. A generalization of the Multiplicative algorithm to find locally D-optimum designs for multifactor models as Tait-like equations is here proposed. The method consists of transforming a conceived set of design points over a finite interval into proportions of the design interval defined by the sub-intervals between successive points. Examples for different modifications of the Tait equation are here presented: a three parameter model under isothermal conditions with pressure as the independent variable and two different modifications of an eight parameter model (Rackett equation) with pressure and temperature as design variables. Optimum design for the first example is equally supported at three points while the two-factor modifications of the Tait equation show designs with more support points than unknown parameters and different weights for each point. Efficiencies of real experimental designs are computed and suitable more efficient experimental designs, satisfying experimental demands are also proposed.

Excess Volumes and Deviations of Viscosities of Binary Blends of Sunflower Biodiesel + Diesel and Fish Oil Biodiesel + Diesel at Various Temperatures

This paper presents the values of viscosity (η) and density (ρ) of binary blends of sunflower biodiesel + diesel and of fish oil biodiesel + diesel over the entire range of proportion at several temperatures. Temperatures of (293.15, 313.15, 333.15, 353.15, and 373.15) K at atmospheric pressure were studied. Excess volumes (VE) and deviations of viscosities (Δη) have been calculated from the experimental data. These thermodynamic properties are essential in the understanding of intermolecular interactions between different molecules of biodiesel, and it is important in the design of several equipments used in the biodiesel industry. The excess viscosity has been fitted using the Redlich–Kister polynomial equation. A modified Rackett's equation was used to estimate the density. The viscosity for the biodiesels was estimated using three models: Thomas, Orrick–Erbar, and modified Sastri–Rao. The critical properties were estimated by the Rackett's equation, together with the Marrero–Gani method of group contribution.

Prediction of deep eutectic solvents densities at different temperatures

Predicting densities of nonconventional solvents like deep eutectic solvents (DESs) as a function of temperature is of considerable importance in the development and design of new processes utilizing these solvents. Because of the nature of bonding existing between the salt and the hydrogen bond donor, conventional methods result in very large deviations. In this study, the density of DESs based on three different salts was estimated using empirical method. Nine different salts:hydrogen bond donor combinations were selected to test this method. The densities of all DESs were measured at a temperature range (298.15–368.1 K). The critical properties of salt and hydrogen bond donor were estimated using the Modified Lydersen–Joback–Reid method, while that of the mixture were calculated using Lee–Kesler equation. The Rackett equation modified by Spencer and Danner was employed to predict the DES density. The values of measured and predicted densities were compared and the average of absolute relative error percentage (ARPE) for all DESs was found to be 1.9%. The effect of salt to HBD molar ratio on ARPE in predicted DESs densities was also investigated.

Compressibilities and viscosities of reference and vegetable oils for their use as hydraulic fluids and lubricants

The use of biodegradable lubricants based on vegetable oils is receiving increasing attention, being an important area of research, in order to promote products with a reduced environmental impact during their entire life cycle. Compressibility is one of the key properties that should be taken into account to develop efficient hydraulic fluids and gear oils. The isothermal compressibility of four reference fluids, a sunflower base oil and a biodegradable oil up to 50 MPa and from 298.15 K to 373.15 K has been determined. For this aim an experimental device based on a vibrating-tube densimeter was used to obtain the density values. The density values at atmospheric pressure were correlated within an absolute average deviation of 0.07% with the Rackett equation whereas a modified Tait equation was used to correlate the experimental data over all temperatures and at higher pressures. The sunflower base oil analyzed in this work has a slightly lower compressibility than those of the reference oils for hydraulic and two stroke engines applications. Moreover, viscosities from 278.15 K to 373.15 K and the viscosity index (VI) were determined for all the analyzed oils using a SVM 3000 Anton Paar rotational Stabinger viscometer.

Sorbate densities on 5A zeolite above and below the critical conditions: n alkane data evaluation and modeling

The sorbate densities of n-alkanes C1–C20 in 5A zeolite are modeled. For validation of this model, the n-alkane adsorption data for gases and liquids on 5A zeolite is critically evaluated using the concept of sorbate densities. The adsorbed phase critical temperatures appear to occur at a reduced temperature of 0.975 different from the vapor-liquid critical reduced temperatures of 1.0 for C3 to C20n-alkanes. For methane and ethane species, the critical adsorbate reduced temperatures TCAR occur earlier at reduced temperatures of 0.83 and 0.96 respectively. The modified Rackett equation of Spencer and Danner (1972) is satisfactorily used to calculate the adsorbate loading for qmax below adsorbed phase critical reduced temperature, TCAR.

Density of Jatropha curcas Seed Oil and its Methyl Esters: Measurement and Estimations

Density data as a function of temperature have been measured for Jatropha curcas seed oil, as well as biodiesel jatropha methyl esters at temperatures from above their melting points to 90 °C. The data obtained were used to validate the method proposed by Spencer and Danner using a modified Rackett equation. The experimental and estimated density values using the modified Rackett equation gave almost identical values with average absolute percent deviations less than 0.03% for the jatropha oil and 0.04% for the jatropha methyl esters. The Janarthanan empirical equation was also employed to predict jatropha biodiesel densities. This equation performed equally well with average absolute percent deviations within 0.05%. Two simple linear equations for densities of jatropha oil and its methyl esters are also proposed in this study.

Densities of Ethyl Esters Produced from Different Vegetable Oils

Biodiesel density data as a function of temperature is needed to model the combustion process. In this work, the densities of ethyl ester biodiesel obtained from various vegetable oils were measured at various temperatures from (15 to 90) °C. The data obtained were used to validate the method proposed by Spencer and Danner using a modified Rackett equation. The experimental and estimated density values using the modified Rackett equation gave almost identical values with deviations less than (0.21, 0.35, 0.22, 0.15, and 0.24) % for the ethyl esters of palm, soybean, canola, corn, and ricebran oil, respectively. Simple linear equations for density of various vegetable oil ethyl ester biodiesels are also proposed in this work.

Density of Palm Oil-Based Methyl Ester

Biodiesel density data as a function of temperature is needed to model the combustion process. In this work, the density and specific gravity of pure palm oil-based methyl ester biodiesel was measured at various temperatures from (15 to 90) °C. The data obtained was validated with Janarthanan empirical equation and the method proposed by Spencer and Danner using a modified Rackett equation. The experimental and estimated density values using the modified Rackett equation gave almost identical values with differences less than 0.03 %. The Janarthanan empirical equation performs equally with diffrences within 0.03 %. A simple linear equation for density is also proposed in this work.

A new method for predicting saturated liquid viscosity at temperatures above the normal boiling point

A new temperature viscosity relationship similar to the Rackett equation for saturated liquid volumes is proposed for predicting saturated liquid viscosity at temperatures above the normal boiling point. The first order group contribution values proposed earlier by the authors for predicting the viscosity at the normal boiling point are revised. Combining these two, the viscosities of organic liquids can be predicted at any temperature above the normal boiling point using the normal boiling point, critical temperature and chemical structure only as inputs. The accuracy of the correlation compares favourably with those used for the low temperature region. It is observed that the three properties, volume, thermal conductivity and viscosity, of saturated liquids can be predicted by using the same generalised empirical approach starting from a form of the Rackett equation.

A Modified Rackett Equation Applied to Water and Aqueous NaCl and KCl Solutions

The Rackett equation is modified for the calculations of the liquid volumes of H2O, H2O + NaCl (up to 25.0 NaCl mass %), and H2O + KCl (up to 4.0 KCl mass %) solutions under vapor-saturated conditions. Two terms are added to the index part of the Rackett equation for describing the volumetric property of liquid water. The average of the absolute deviations from the NBS/NRC Steam Tables at 293.15 K to 643.15 K in steps of 10 K becomes 0.39%. The effect of the dissolved salt on the volumetric property is included in the index part. The equations for molar volumes of aqueous NaCl solutions at 373.15 K to 773.15 K and aqueous KCl solutions at 373.15 K to 643.15 K are obtained on the basis of the previous experimental results. The average of the absolute deviations for aqueous NaCl solutions is 0.77%, and that for the aqueous KCl solutions is 0.63%.

Influencia de los parámetros críticos en la estimación de la densidad de algunos ácidos grasos por el método de Rackett

In this work a comparative study is carried out of the critical properties of temperature and pressure applied for the palmitic, stearic and oleic fatty acids, provided by different bibliographical sources. These properties are used in the Rackett's equation to estimate the density of the pure substances. The most commonly employed methods are described and applied in these cases to estimate these critical properties. A group of values of critical properties is selected to estimate the density using the Rackett's equation. The values of the critical properties that give a better estimation of the density, according to Rackett's equation, when compared with the experimental results are proposed. It is demonstrated that the selection of the critical properties influences in a decisive way in the estimation of physical properties by means of the Rackett's equation.

Estudio de la densidad y de la viscosidad de algunos ácidos grasos puros

An experimental study is carried out according to ASTM analysis methods of two physical properties, density and viscosity, of palmitic, stearic and oleic fatty acids; compounds which are present in the deodorization effluents of edible oils, with a view to their physics chemical characterisation needed for the design of surface condensers and freezers that could be employed in the recovery of these effluents. The measures are carried out every 5 °C, from temperatures close to those of solidification in each compound, of great interest in our study, to 100 °C. The values of these properties are contrasted with those that appear in the literature, and the experimental measures are correlated. Values of the density are also obtained applying Rackett's equation and those of dynamic viscosity using Andrade's equation, and then compared with experimental measures. Experimental data of the dynamic viscosity for oleic acid at low temperatures (between 0 and 24 °C) are also presented, because of their interest in the design of the previously mentioned equipment.

A new temperature–thermal conductivity relationship for predicting saturated liquid thermal conductivity

A new temperature–thermal conductivity relationship similar to the Rackett equation for saturated liquid densities is proposed. The first-order group contribution values proposed earlier for predicting the thermal conductivity at the normal boiling point [S.R.S. Sastri, K.K. Rao, Chem. Eng. 100(8) (1993) 106–107] are revised. Combining these two, the thermal conductivities of organic liquids are predicted over the entire saturated liquid region using the normal boiling point, critical temperature and chemical structure as inputs. Below the normal boiling point, the average absolute deviation and the maximum absolute deviation between the predicted data and the literature data are 5.6% and 28.9%, respectively. These figures compare favourably with those of the Nagvekar method [M. Nagvekar, T.E. Daubert, Ind. Eng. Chem. Res. 26 (1987) 1562–1565], claimed to be the best available one applicable to all types of liquids. The deviations between the predicted and literature values are of the same order as the variations in the experimental data, even at temperatures upto 0.98 T r, where T r is the reduced temperature. The proposed correlation is in agreement with the observation of B.C. Sakiadis and J. Coates [AIChE J. 1 (1955) 275–288] regarding the near constancy of the pseudo critical thermal conductivity for members of the homologous series, but it does not support their other expectations regarding the regularity in the variation in the thermal conductivity and temperature coefficient of thermal conductivity for these liquids. The periodic variation in the temperature coefficient with the number of carbon atoms observed by J.D. Raal and R.L. Rijisdijk [J. Chem. Eng. Data 26 (1981) 351–359] for n-alcohols is also not supported.

Temperature dependence of excess molar volumes in ( n-alkane (C 6–C 9) or alcohol (C 2–C 4))+olive oil mixtures

In the scope of investigating physical properties related to equipment design for edible oil industries we present the temperature dependence of excess molar volumes of n-alkane ( n-hexane, n-heptane, n-octane and n-nonane) or alkanol (ethanol, 1-propanol, 2-propanol, 1-butanol) with natural olive oil being measured at the range 283.15–298.15 K and atmospheric pressure. The phase compositions on every partial miscible binary mixture were obtained by measurement of density and application of the fitting polynomials. Application of contribution methods (UNIFAC and UNIFAC-Lyngby) were applied in order to compare its capability in predicting the experimental values. Derived properties such as partial excess molar volumes, isobaric expansibility and isothermal coefficient of pressure excess molar enthalpy were computed due to its importance in the study of specific molecular interactions. Because of current processes, design is strongly computer oriented, consideration was also given to how accurate the corresponding states prediction works. The validity of predicting densities was tested by the Rackett equation, this model being selected attending to ease of use, accuracy and range of application.

Densities of ternary mixtures of benzene + cyclohexane + hexane between 298.15 and 473.15 K

The results of measurements of the densities of the ternary mixtures of benzene + cyclohexane + hexane using a high pressure stainless steel pycnometer system in the whole composition range over a wide variation of temperature from 298.15 to 473.15 K are presented. The experimental densities were compared with those predicted by the Spencer-Danner-modified Rackett equation (SDR) and the Hankinson-Brobst-Thompson correlation (HBT). The SDR showed an average absolute deviation of 0.33% while the HBT predicted the densities of the mixtures with an average absolute deviation of 0.75%.

Prediction of saturated liquid volumes of organic compounds

A method based on a generalised form of the Rackett equation is proposed for predicting the volumes of organic liquids in the entire saturated liquid region. The method requires data on the normal boiling point, critical temperature and the chemical structure only as inputs. Liquid volumes predicted by the proposed method gave 1.7% average absolute deviation and a maximum absolute deviation of 8.3% from the literature values for the 144 organic liquids (836 points) tested over temperatures ranging from the freezing point to the critical point.

Generalized method for prediction of saturated liquid volumes using Van der Waals volumes

The modified Rackett equation represents a simple and very widely used method to predict saturated liquid volumes. In addition to being fairly accurate, this equation is mathematically continuous and renders expressions for first and second derivatives of saturated liquid volume needed for computations of thermodynamic properties such as enthalpies and heat capacities from certain liquid state equations of state that use saturated liquid volume. The objective of this work is to further enhance the applicability of the modified Rackett equation by developing a correlation for the parameter Z{sub RA} with Van der Waals volume of the molecule and with a volume parameter calculated by summing the UNIFAC volume parameters (R) of individual groups making up the molecule. The data used in this work included 124 compounds classified into various families. These compounds included high molecular weight aromatics and heterocycles, phenols, and some amines.