Bender Equation Research Articles

Numerical solution of particle size distribution equation: Rapid expansion of supercritical solution (RESS) process

The present study aimed to numerically simulate the rapid expansion of the supercritical solution (RESS) process including particle generation, hydrodynamics, and solving the population balance equation (PBE) to predict the particle-size distribution (PSD) of solid-supercritical carbon dioxide binary systems. Energy, momentum, and mass equations, in addition to the extended generalized Bender equation of state (EoS), were applied to predict the hydrodynamic behavior of a supercritical solution under several operating conditions using a nozzle and expansion vessel. The tetraphenylporphyrin (TBTPP) solubility in supercritical carbon dioxide was calculated using the Peng–Robinson EoS/Kwak–Mansoori as a mixing rule. Subsequently, TBTPP, aspirin, ibuprofen, and salicylic acid nucleation as well as the supersaturation rate were calculated. Finally, we solved the time dependence of the parameters of the size distribution numerically. The established models are compared over a wide parameter range using a reference model that refers to the method of moment log-normal size distribution functions through the RESS process to predict a solid PSD. The results obtained are presented with and without coagulation phenomena. The average absolute percent deviation of solubility of TBTPP was 3.98, and the hydrodynamic behavior of supercritical carbon dioxide showed a similar trend as the results presented in the published research work. Furthermore, a particle size distribution prediction using coagulation showed acceptable agreement with the experimental PSDs.

Smart Implementation of Bender Equation of State

In this paper matrix and vector products are exploited to reformulate Bender Equation of State. Finally, the new formulation is used to generate results which has been compared with experimental data sets available in literature and the analogous findings coming from different thermodynamic packages commonly used in Aspen Hysys for Air Separation Unit.

Investigation on hydrodynamic and formation of nano particle by RESS process: The numerical study

The particle formation by supercritical technology is an active research field. Rapid expansion of supercritical fluid (RESS) is a promising new technology for particle formation using supercritical fluids. In this study, numerical modeling of particle formation of TBTPP (5, 10, 15, 20-tetrakis (3, 5-bis (trifluoromethyl) phenyl) porphyrin) through RESS process is investigated. The RESS process contains two pre-expansion (nozzle part) and expansion vessel (supersonic free jet) parts. In the first section, mass, momentum and energy balances in addition to extended generalized Bender equation of state (egB-EoS) as accurate equation of state is used to calculate temperature, pressure, density and velocity of supercritical carbon dioxide (SC-CO2) based on mathematical modeling and computational fluid dynamics (CFD) simulation methods. Both models include friction in the nozzle and heat exchange in the expansion vessel and in the nozzle region. Using calculated pressure and temperature, the solubility of TBTPP in the supercritical CO2 is predicted by Peng-Robinson EoS with Kwak-Mansoori mixing rule and the results are compared with experimental data and a semi-empirical correlation. Supersaturation, homogenous nucleation and critical radius of TBTPP based on classical nucleation theory are calculated. The effect of pre-expansion temperature on supersaturation, nucleation and critical radius parameters is also studied. The results of hydrodynamic and particle formation modeling present the same trend as reported in the literature.

Numerical simulation of rapid expansion of supercritical carbon dioxide

Axisymmetric rapid expansion of supercritical carbon dioxide is investigated in this article. The extended generalized Bender equation of state is used to give a good description of the fluids over a wide range of pressure and temperature conditions. The locations of Mach disks are analyzed and compared with an experimental correlation for the case where there is no plate positioned in front of the nozzle exit. It is found that the disagreement between our numerical results and the experimental formula is very small when the pressure ratio is small, and increases as the pressure ratio increases. It is also found that with different equations of state, the predicted positions of Mach disks do not differ a lot, but the temperature profiles in the chamber differ a lot. The case where there is a plate positioned in front of the nozzle exit is also studied in this article. A universal similarity solution is obtained. © 2014 American Institute of Chemical Engineers AIChE J, 61: 317–332, 2015

Evaluation of the parameters of the Bender equation of state for low acentric factor fluids and carbon dioxide

Volumetric properties of several low acentric factor fluids (Ar, CH 4, C 2H 6, Kr, N 2, Ne, O 2, Xe) as well as CO 2 are modeled using the Bender equation of state. This equation is a linear function of 19 adjustable parameters, which are evaluated from properties data, using a linear numerical procedure. The validity of the EOS is tested by calculating the Joule–Thomson inversion curve. A simple model is in particular used to correlate the inversion properties predicted by the Bender equation, expressed in term of reduced pressure as a function of reduced temperatures ranging from 0.8 to 6. The simple correlation reproduces accurately the used data. We employ data on state behaviour ρ ( P , T ) of homogeneous fluid phases, vapour–liquid equilibrium, second virial coefficient and the coordinates of the critical point.

High-Pressure Adsorption of Supercritical Gases on Activated Carbons: An Improved Approach Based on the Density Functional Theory and the Bender Equation of State

Adsorption of nitrogen, argon, methane, and carbon dioxide on activated carbon Norit R1 over a wide range of pressure (up to 50 MPa) at temperatures from 298 to 343 K (supercritical conditions) is analyzed by means of the density functional theory modified by incorporating the Bender equation of state, which describes the bulk phase properties with very high accuracy. It has allowed us to precisely describe the experimental data of carbon dioxide adsorption slightly above and below its critical temperatures. The pore size distribution (PSD) obtained with supercritical gases at ambient temperatures compares reasonably well with the PSD obtained with subcritical nitrogen at 77 K. Our approach does not require the skeletal density of activated carbon from helium adsorption measurements to calculate excess adsorption. Instead, this density is treated as a fitting parameter, and in all cases its values are found to fall into a very narrow range close to 2000 kg/m3. It was shown that in the case of high-pressure adsorption of supercritical gases the PSD could be reliably obtained for the range of pore width between 0.6 and 3 nm. All wider pores can be reliably characterized only in terms of surface area as their corresponding excess local isotherms are the same over a practical range of pressure.

Modeling of High-Pressure Adsorption Using the Bender Equation of State

A model describing high-pressure adsorption based on the simplified local density model and the Bender equation of state is proposed. The Bender equation of state was used for the description of both bulk and adsorbed fluid. The model is capable of characterizing adsorbent material in terms of structural parameters such as the skeletal density, the micropore volume, and the energy of adsorption. The model was successfully applied to high-pressure adsorption isotherms of nitrogen and methane up to 50 MPa at 40 °C on a series of synthetic carbons and at 25−70 °C on the commercial carbon Norit R1. The structural parameters of the investigated carbons obtained by the proposed method are in good agreement with those obtained by independent methods based on standard and advanced characterization from nitrogen adsorption at 77 K.

Modeling of Gas Adsorption Equilibrium over a Wide Range of Pressure: A Thermodynamic Approach Based on Equation of State

A thermodynamic approach based on the Bender equation of state is suggested for the analysis of supercritical gas adsorption on activated carbons at high pressure. The approach accounts for the equality of the chemical potential in the adsorbed phase and that in the corresponding bulk phase and the distribution of elements of the adsorption volume (EAV) over the potential energy for gas–solid interaction. This scheme is extended to subcritical fluid adsorption and takes into account the phase transition in EAV. The method is adapted to gravimetric measurements of mass excess adsorption and has been applied to the adsorption of argon, nitrogen, methane, ethane, carbon dioxide, and helium on activated carbon Norit R1 in the temperature range from 25 to 70°C. The distribution function of adsorption volume elements over potentials exhibits overlapping peaks and is consistently reproduced for different gases. It was found that the distribution function changes weakly with temperature, which was confirmed by its comparison with the distribution function obtained by the same method using nitrogen adsorption isotherm at 77 K. It was shown that parameters such as pore volume and skeleton density can be determined directly from adsorption measurements, while the conventional approach of helium expansion at room temperature can lead to erroneous results due to the adsorption of helium in small pores of activated carbon. The approach is a convenient tool for analysis and correlation of excess adsorption isotherms over a wide range of pressure and temperature. This approach can be readily extended to the analysis of multicomponent adsorption systems.

A simple method for evaluation of parameters of the Bender equation of state from experimental data

A simple numerical method for evaluation of parameters (constants) of Bender equation of state for pure fluids is proposed. The minimisation of the objective function leads to a set of linear equations. The method employs experimental data on state behaviour ( p– ρ– T) of fluid phases, vapour–liquid equilibrium data (saturated vapour pressures and orthobaric densities), second virial coefficients, and the coordinates of the gas–liquid critical point. Results of the tests using data for two fluids (methane and n-pentane) are presented.

Simulation of particle formation during the rapid expansion of supercritical solutions

Experimental and theoretical results for the rapid expansion of supercritical solutions (RESS-process) are presented. Experiments were carried out with the solvents carbon dioxide (CO 2) and trifluoromethane (CHF 3) and the organic solutes cholesterol, benzoic acid and griseofulvin. It is shown that for each of these solutes it is possible to produce particles in the size range of 200 nm which leads to improved dissolution. Furthermore, to demonstrate the higher bioavailability of the micronized pharmaceutical griseofulvin as of the original material, results of a dissolution experiment are shown. Besides the experiments the flow and particle formation and growth in the expansion unit (inlet–capillary nozzle–supersonic free jet) of the RESS-process is modelled numerically. The model to calculate the flow field consists of mass, momentum and energy balances as well as the extended generalized Bender equation of state (egB-EoS). The flow is assumed to be steady, one-dimensional, viscid and non-adiabatic. The general dynamic equation (GDE) is included to model the particulate phase. The results of the modelling show that particle formation occurs in most of the cases mainly in the supersonic free jet and that the main mechanism for particle growth is coagulation. Furthermore, it can be inferred from the high particle number concentration at the Mach disk that particle growth by coagulation continues in the subsonic part of the free jet (expansion chamber). Experimental results with variation of additional air supply to the expansion chamber are presented to confirm the theoretical results.

Parameters of the Bender Equation of State for Chloro Derivatives of Methane and Chlorobenzene

Values of adjustable parameters of the Bender equation of state evaluated for chloromethane, dichloromethane, trichloromethane, tetrachloromethane, and chlorobenzene from published experimental data are presented. Experimental data employed in the evaluation included the data on state behaviour (p-ρ-T) of fluid phases, vapour-liquid equilibrium data (saturated vapour pressures and orthobaric densities), second virial coefficients, and the coordinates of the gas-liquid critical point. The description of second virial coefficient by the equation of state is examined.

High-pressure (vapour + liquid) equilibria in (carbon dioxide + acetone or 2-propanol) at temperatures from 293 K to 333 K

High-pressure (vapour + liquid) equilibria in (carbon dioxide + acetone, or 2-propanol) at temperatures from 293 K to 333 K and pressures between 1.1 MPa and 8.1 MPa are reported. The new experimental results are in good agreement with selected literature data. The new data are correlated by two equations of state: the Peng–Robinson equation of state as modified by Melhem et al. and the generalized Bender equation of state. Both equations give a good description of the pressure above such liquid mixtures. The correlations give good results for the vapour phase composition of 2-propanol, whereas the calculated vapour phase composition of acetone (for a given temperature and a liquid phase composition) is somewhat less than the experimental results.

The Bender Equation of State and Virial Coefficients

The second and third virial coefficients calculated from the Bender equation of state (BEOS) are tested against experimental virial coefficient data. It is shown that the temperature dependences of the second and third virial coefficients as predicted by the BEOS are sufficiently accurate. We conclude that experimental second virial coefficients should be used to determine independently five of twenty constants of the Bender equation. This would improve the performance of the equation in a region of low-density gas, and also suppress correlations among the BEOS constants, which is even more important. The third virial coefficients cannot be used for the same purpose because of large uncertainties in their experimental values.

Mixing behaviour of a mixture of equal amounts of substance of 1,1,1,2-tetrafluoroethane and 1,1-difluoroethane. II. Representation of thermal properties by equations of state

In a preceding investigation, the heat capacity at constant pressure and the Joule–Thomson coefficient of the refrigerants R134a (CF3CH2F), R152a (CHF2CH3) and of (0.5CF3CH2F+0.5CHF2CH3) have been measured in the gaseous phase for temperatures from 298 K to 423 K at pressures up to 2.5 MPa. In the present work, the Joule–Thomson coefficient and residual parts of the heat capacity and enthalpy have been calculated with a fundamental equation of state, developed previously by Tillner–Roth on the basis of a large number (3433) of experimental data points, and with a generalized Bender equation of state, the parameters of which are based only on experimental values of the vapour pressure and critical temperature. Both equations describe the dependence of the thermal properties on temperature and pressure rather well, although the deviations from the experimental data are somewhat larger than the experimental uncertainty. In spite of the much smaller experimental basis of the generalized Bender equation, the deviations from the measurements are, on average, only twice as large as with the fundamental equation. Also, the calculated values of the excess molar heat capacity and of the excess molar enthalpy, which have the opposite sign to most other gaseous mixtures, are in reasonable agreement with the experimental results.

Solubility and micronization of griseofulvin in supercritical CHF3

Griseofulvin (GF) is an antimicotic compound whose pharmaceutical activity can be improved by particle size reduction. Its micronization was obtained by rapid expansion from a supercritical solution (RESS). Trifluoromethane (CHF 3 ) was chosen as a supercritical solvent and the Bender equation of state was selected to represent its volumetric properties. GF solubility in supercritical CHF 3 was studied at temperatures of 40 and 60 °C and in the pressure range from 120 to 330 bar. The solubility data was fitted by using the Peng-Robinson (PR) equation of state in which the sublimation pressure of CHF 3 and the k 12 interaction parameter were considered as adjustable parameters. RESS experiments were performed at different preexpansion pressures (from 180 to 220 bar) and temperatures (from 60 to 150 °C) and at different expansion chamber temperatures (from 10 to 25 °C). Two different morphologies of GF crystals were observed : quasispherical particles and needles. The process parameter controlling the transition from one morphology to the other was the preexpansion chamber temperature. The orientation and the coalescence of GF crystals was also sometimes observed. Our hypothesis was that this could be due to the presence of electrostatic charges induced by the fast motion in the expansion chamber and located on the surface of the GF particles.

High-pressure vapor-liquid equilibria in binary mixtures of carbon dioxide and aromatic hydrocarbons: Experimental data and correlation for CO2 + acetophenone, CO2 + 1-chloronaphthalene, CO2 + methyl benzoate and CO2 + n-propylbenzene

Experimental data for the high-pressure vapor-liquid equilibrium in binary mixtures of carbon dioxide and benzene derivatives (acetophenone, 1-chloronaphthalene, methyl benzoate, and n-propylbenzene) are reported. Measurements were performed at temperatures between 313 and 393 K and pressures up to 18 MPa using a flow-type apparatus. The experimental phase compositions are compared with literature values and are successfully correlated applying a generalized Bender equation of state. Furthermore, the new results for the mixtures containing acetophenone and n-propylbenzene are compared with predictions from the group-contribution equation of state (GC-EOS) by Skjold-Jørgensen.

High-pressure vapour-liquid equilibria in binary mixtures of carbon dioxide and benzene compounds: experimental data for mixtures with ethylbenzene, isopropylbenzene, 1,2,4-trimethylbenzene, 1,3,5-trimethylbenzene, ethenylbenzene and isopropenylbenzene, and their correlation with the generalized Bender and Skjold-Jørgensen's group contribution equation of state

Bamberger, A., Schmelzer, J., Walther, D. and Maurer, G., 1994. High-pressure vapour—liquid Equilibria in binary mixtures of carbon dioxide and benzene compounds: experimental data for mixtures with ethylbenzene, isopropyl-benzene, 1,2,4-trimethylbenzene, 1,3,5-trimethylbenzene, ethenylbenzene and isopropenylbenzene, and their correlation with the generalized Bender and Skjold-Jørgensen's group contribution equation of state. Fluid Phase Equilibria, 97: 167-189. The binary vapour-liquid Equilibrium of six mixtures of carbon dioxide and a benzene derivative (ethylbenzene, isopropylbenzene, 1,2,4-trimethylbenzene, 1,3,5-trimethylbenzene, ethenylbenzene and isopropenylbenzene) was studied by a flow technique at temperatures between 313 and 393 K and pressures between about 6 and 18 MPa. The experimental results are correlated using a generalized Bender equation of state and the group contribution equation of state (GC-EOS) by Skjold-Jørgensen. The application of that equation of state was extended by estimating new parameters for interactions between carbon dioxide and aromatic groups, using some of the new experimental results as well as literature data.

Application of a generalized Bender equation of state to the description of vapour-liquid equilibria in binary systems

Abstract Platzer, B. and Maurer, G., 1993. Application of a generalized Bender equation of state to the description of vapour-liquid equilibria in binary systems. Fluid Phase Equilibria , 84: 79-110. A recently published four-parameter corresponding state generalization of the Bender equation of state is extended to describe fluid mixtures. Interactions between species of different kinds are considered by introducing three temperature-independent binary parameters which were fitted to experimental vapour-liquid equilibrium data. Altogether 71 binary mixtures are treated. Data correlated by the generalized Bender equation are compared with experimental results and with data correlated by the Lee-Kesler-Plocker (LKP) procedure. The generalized Bender equation needs less computing time than the LKP method and gives a better representation of the experimental data, especially when the components differ considerably in molecular size and/or structure, while there is no or only a minor improvement for simple systems such as argon/methane.

High pressure vapour-liquid equilibria in binary mixtures of carbon dioxide and benzaldehyde, bromobenzene, chlorobenzene, 1,2-dichlorobenzene and 2-chloro-1-methylbenzene at temperatures between 313 and 393 k and pressures up to 22 MPa

The high-pressure vapour-liquid equilibrium in binary mixtures of carbon dioxide and five aromatic compounds (benzaldehyde, bromobenzene, chlorobenzene, 1,2-dichlorobenzene and 2-chloro-1-methyl-benzene) has been investigated by a flow technique at temperatures between 313 and 393 K at pressures up to about 22 MPa. For none of these mixtures – with the exception of CO2 + benzaldehyde – experimental results have been reported before. The experimental arrangement is described. The results of the measurements are reported, discussed and correlated by a recently developed generalized Bender equation of state.

High-pressure (vapour + liquid) equilibria of (carbon dioxide + methylbenzene or 1,2-dimethylbenzene or 1,3-dimethylbenzene or 1,4-dimethylbenzene) at temperatures between 313 K and 393 K and pressures up to 17.3 MPa

A simple flow apparatus has been developed for studying high-pressure (vapour + liquid) equilibria at temperatures from 300 K to 420 K and pressures up to 30 MPa. Experimental results for [(1−x)CO2 + x{C6H5CH3 or o-C6H4(CH3)2 or m-C6H4(CH3)2 or p-C6H4(CH3)2}](1) = [(1−y)CO2 + y{C6H5CH3 or o-C6H4(CH3)2 or m-C6H4(CH3)2 or p-C6H4(CH3)2}](g) have been determined. A recently developed generalized Bender equation of state is successfully applied to correlating the results. As no temperature-dependent mixing rules are used, the correlation should be a valuable instrument for predictions.