G. Cristofoli

A viscosity equation of state for R134a through a multi-layer feedforward neural network technique

A multi-layer feedforward neural network (MLFN) technique is adopted for developing a viscosity equation η = η(ρ, T) for R134a. The results obtained are very promising, with an average absolute deviation (AAD) of 0.63% for the currently available 571 primary data points, and are a significant improvement over those of a corresponding conventional equation in the literature. The method requires a high accuracy equation of state for the fluid in order to convert the experimental P, T into the independent variables p, T, but such an equation may not be available for the target fluid. Aiming at overcoming this difficulty, a viscosity implicit equation of state in the form T = T(η, P), avoiding the density variable, is developed for the liquid surface. The attained accuracy level is equivalent to that of the former equation. The proposed technique, being completely correlative and non theoretically founded, is also a powerful tool for experimental data screening.

A predictive density model in a corresponding states format. Application to pure and mixed refrigerants

A three parameters density model based on Corresponding States (CS) technique is proposed as a means of predicting the density of pure fluids and their mixtures on the entire PρT (PρTx) surface. The studied fluids belong to two conformal families of the new refrigerant fluids generation: the halogenated alkanes (HA) and the hydrofluoroethers (HFE). The new model is based on an original scaling factor parameter that is determined only on a saturated liquid density experimental value. Using two accurate dedicated equations of state (EoS) as references, the same structure of the Teja CS model is maintained, substituting the classical acentric factor with the new defined scaling parameter. Through this model, the density of the refrigerant fluids considered can be calculated on the whole surface with an accuracy level similar to that of the dedicated equations. The model is validated against experimental data for HFC refrigerants including fluoropropanes, fluorobutanes and fluoroethers. A comparison is also proposed with available density models regarded of high accuracy level.

Viscosity equations of pure fluids in an innovative extended corresponding states framework. II. Application to four fluids

Abstract This paper represents the second part of a work devoted to an innovative version of the historical extended corresponding states (ECS) technique for the development of viscosity equations on the whole ηTρ surface for individual fluids. In the first part [1] , the theoretical aspects of modelling the ECS transport properties modelling have been discussed, and the fundamental characteristics of the proposed method have been tested, using viscosity values generated from conventional dedicated viscosity equations. The very promising results suggest a move from generated to experimental data correlation to determine dedicated viscosity equations for a number of fluids for which conventional equations are available in literature. This is done in this second part, where the fluids studied are the alkane ethane and the haloalkane refrigerants R123, R134a and R152a. The absolute average deviations (AADs) obtained with primary data are, respectively, 1.04, 1.13, 0.92 and 0.71% with a significant improvement with respect to the conventional equations. Considering that the expected experimental accuracy for good viscosity data is generally in the range 1–2%, the obtained results seem to be very promising.

A corresponding states predictive model for the saturated liquid density of halogenated alkanes and of fluorinated propanes and ethers

Abstract A three-parameter corresponding states (CS) model is proposed here for the prediction of the saturated liquid density of pure fluids pertaining to the two conformal families of halogenated alkanes and hydrofluoroethers (HFE), most of which are either already used or proposed for use as refrigerants. Two fluids from each family were chosen for their acentric factor value and the availability of dedicated saturated liquid density equations and at first, on the basis of the three-parameter CS model by Teja et al., the saturated liquid density of a given fluid was obtained in reduced variables. Assuming experimental saturated liquid density data for several components of each of the two families of fluids, an improvement was then introduced, substituting the acentric factor with a new constant scaling factor. As a final result, the proposed liquid model has a predictive nature. The prediction accuracy reached by this new method is similar to that of the dedicated equations, where available, for all the fluids in a family. The result is particularly satisfactory for application requirements in refrigeration.

The viscosity surfaces of propane in the form of multilayer feed forward neural networks

The present work focuses on the development of a viscosity equation η=η(ρ,T) for propane through a multilayer feedforward neural network (MLFN) technique. Having been successfully applied to a variety of fluids so far, the proposed technique can be regarded as a general approach to viscosity modeling. The MLFN viscosity equation has been based on the available experimental data for propane: validation on the 969 primary data shows an average absolute deviation (AAD) of 0.29% in the temperature, pressure, and density range of applicability, i.e., 90 to 630 K, 0 to 60 MPa, and 0 to 730 kg⋅m−3. This result is very promising, especially when compared with experimental data uncertainty. The minimum amount of required data for setting up the MLFN has been investigated, to explore the minimum cost of the model. Comparisons with other viscosity models are presented regarding amount of input data, claimed accuracy, and range of applicability, with the aim of providing a guideline when viscosity has to be calculated for engineering purposes. A high accuracy equation of state for the conversion of variables from experimental P,T to operative ρ,T has to be provided. To overcome this requirement, two viscosity explicit equations in the form η=η(P,T) are also developed, for the liquid and for the vapor phases. The respective AADs are 0.58 and 0.22%, comparable with those of the former η=η(ρ,T) equation. Finally, the trend of the experimental viscosity second virial coefficient is reproduced and compared with that obtained from the MLFN.

Viscosity equations of pure fluids in an innovative extended corresponding states framework I. Modelling techniques

Abstract Application of the extended corresponding states (ECS) technique to thermodynamic and transport properties has demonstrated that there are different requirements for conformality of data for these two sets of properties. In addition to the thermodynamic shape factors, derived from accurate equations of state for both the target and reference fluids, there is a need for an additional shape factor, derived from transport property data of the target fluid, in order to fit the transport properties. As a result, a new ECS model is proposed here for viscosity, which uses a single, new viscosity shape factor, which is generated just from viscosity data over the available PρT surface. In this way, there is no need to determine shape factors from thermodynamics. By application to ethane and refrigerant R134a, it is shown that the scale function is a smooth function of temperature and pressure. The scale function is then represented through a neural network, because of the flexibility and the high data fitting capability of that technique. The accuracy of viscosity data representation on the basis of this model is similar to that obtained with the conventional approach, by summing the dilute gas, the excess and the critical enhancement contributions. The non-theoretical and completely heuristic nature of the model also allows its application to the statistical screening of the experimental data. Furthermore, the variables conversion T,P→T,ρ does not necessarily require an equation of state, as is the case for the historic ECS transport properties model, and this can be carried out by a density model like that recently presented by Cristofoli and co-workers [1] , [2] for the family of refrigerant fluids.

A corresponding states thermodynamic model for pure and mixed refrigerants in terms of the Helmholtz energy

Abstract For the alkane and halogenated alkane (HA) fluid families, several high accuracy dedicated equations of state (DEoS) have been recently published and models based on corresponding states (CS) format have been often used. These models aim at predicting the thermodynamic behavior of the fluids following different parameter contribution techniques, heedless of the “conformality” concept, which is nevertheless fundamental for a CS method application. Taking advantage of an extensive study on the thermodynamic “conformality”, and of the precise available DEoS, exact “scaling” parameters for the most significant state functions have been considered in a CS framework for HAs. A predictive volumetric model for pure fluids and mixtures, has been recently proposed and assumed as part of the present model. From a couple of standard liquid density and of vapor pressure values, as two inputs for each fluid of interest, a three parameters CS model for the residual Helmholtz free energy is reported, through which any thermodynamic residual property can be obtained in a semi-predictive mode. The model is then extended to mixtures, through both pseudocritical constants and scaling factors mixing rules. For pure fluids, the proposed model is validated on the main thermodynamic functions against DEoS of several HAs. For mixtures, the validation is developed on density, enthalpy and entropy against mixture DEoS and for VLE against data of non-azeotropic and azeotropic mixtures. In spite of the semi-predictive nature of this model, significant results are obtained for all the examined functions, assuring a very satisfactory representation of the thermodynamic properties of pure and blended HAs. The proposed models can be useful in view of the applications as refrigerants of the fluids studied.

Modelling enthalpy and entropy of pure and mixed refrigerants with an innovative corresponding states method

In this work an original improvement of the Corresponding States technique is developed and a new model, based on a three parameters CS format, is proposed to predict the enthalpy and the entropy of the new generation halogenated alkanes fluids together with some alkanes. Limiting the analysis of the selected fluids to a specific thermodynamic property behaviour, an appropriate conformality approach can be deduced, which allows to set up a predictive model of high accuracy level on a wide range of the enthalpy and entropy surfaces. The fundamentals of the model are innovative scaling parameters deduced from the enthalpy of vaporization and from two dedicated equations, belonging to the selected family of fluids. This allows to set up innovative models following a CS format. Through the introduction of advanced mixing rules, the models can be simply extended to calculate the corresponding properties for mixtures. The proposed models allow also the calculation of VLE for systems of rather regular behaviour. The required inputs for a pure target fluid are an ideal gas isobaric heat capacity correlation, a single value of saturated liquid density and of vaporization enthalpy; if the last one is lacking, a single value of vapor pressure can be alternatively supplied. For non azeotropic mixtures the enthalpy and entropy models are predictive, whereas in case of azeotropy VLE calculations are possibly only applying regressed interaction coefficients. Due to the lack of accurate experimental enthalpy data and to the particular nature of the entropy function, the validation of the models is proposed against fundamental dedicated EoS available, both for pure and mixtures, for a significant number of the studied family of fluids. The predictive character of the proposed approach as well as the high performances reached, make these models particularly suitable for the new families of fluids regarding advanced technological applications.

A new three-parameter corresponding states model for pure halocarbons viscosity prediction

Predictive and semipredictive models for viscosity calculation are currently needed and highly appreciated. Models developed for halogenated refrigerants (HR) and based on Corresponding States (CS) are leading to a prediction accuracy comparable to that of specifically developed models. In the present work, using recently published, highly accurate viscosity dedicated equations, it has been verified that viscosity conforms to a two-parameter CS model is then developed, based on Teja and coworkers three-parameter CS structure. Two fluids of the same family are taken for reference, and the reduced viscosity of a third fluid is obtained in reduced P,T variables. At first the Pitzer acentric factor is proposed as a third parameter, then it is substituted with a temperature-dependent function fitted on saturated viscosity data. The prediction accuracy of the model is comparable to that of the reference fluid equations and, considering its predictive nature, it is a satisfactory tool for the needs of technical applications.

Prediction of halocarbon mixture thermodynamics using an innovative gE-EoS mixing rule in a three-parameter CS framework

Abstract Historical models with g E -EoS mixing rules, which combine a cubic equation of state (EoS) requiring only the three parameters T c , P c and ω as individual inputs with a g E model for the liquid phase, are only applied to the saturation surfaces. A limitation of this rule is basically the number of parameters of the one-fluid EoS considered, which cannot exceed 2 or 3. By integrating an improved Teja corresponding states EoS (T), which requires the same individual parameters as a cubic equation, in a g E -EoS mixing rule framework, a new technique is obtained with two modes, one correlative and one predictive. In the correlative mode, the liquid phase γ is generated from input VLE data using the improved T model and the historical Wong–Sandler–Teja mixing rules for the pseudocritical functions, except for T cmix . In the predictive mode, the same general procedure is followed, but with the liquid phase γ coming from a UNIFAC g L E model, which makes the mixing rule predictive. The T cmix values are locally generated from these new rules and are correlated as an individual function, which in both cases presents a very smooth trend for the systems studied. The two proposed rules are applied to systems of halogenated alkanes, which are known to be polar and deviating, and are compared with the dedicated EoS available for these mixtures. The results for both modes show an interesting level of accuracy in representing the whole thermodynamic behaviour of a mixture.