Journal of Chemistry

Volume 2015, Article ID 163262, 12 pages

http://dx.doi.org/10.1155/2015/163262

## A New Equation Relating the Viscosity Arrhenius Temperature and the Activation Energy for Some Newtonian Classical Solvents

^{1}Laboratoire de Valorisation des Matériaux Utiles, Centre National des Recherche en Sciences des Matériaux, B.P. 95, Borj Cedria, 2050 Hammam Lif, Tunisia^{2}Laboratoire d’Ingénierie Mathématique, Ecole Polytechnique de Tunisie, Université de Carthage, Rue El Khawarizmi, B.P. 743, 2078 La Marsa, Tunisia^{3}Department of Mathematics, Faculty of Basic Education, PAAET, 92400 Al-Ardiya, Kuwait^{4}Department of Mathematics, Faculty of Science, University of Alexandria, Alexandria 21511, Egypt^{5}Laboratoire Biophysique et de Technologies Médicales LR13ES04, Institut Supérieur des Technologies Médicales de Tunis, Université de Tunis El Manar, 9 Avenue Dr. Zouhaier Essafi, 1006 Tunis, Tunisia

Received 15 December 2014; Accepted 15 April 2015

Academic Editor: Demeter Tzeli

Copyright © 2015 Aymen Messaâdi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In transport phenomena, precise knowledge or estimation of fluids properties is necessary, for mass flow and heat transfer computations. Viscosity is one of the important properties which are affected by pressure and temperature. In the present work, based on statistical techniques for nonlinear regression analysis and correlation tests, we propose a novel equation modeling the relationship between the two parameters of viscosity Arrhenius-type equation, such as the energy () and the preexponential factor (). Then, we introduce a third parameter, the Arrhenius temperature (), to enrich the model and the discussion. Empirical validations using 75 data sets of viscosity of pure solvents studied at different temperature ranges are provided from previous works in the literature and give excellent statistical correlations, thus allowing us to rewrite the Arrhenius equation using a single parameter instead of two. In addition, the suggested model is very beneficial for engineering data since it would permit estimating the missing parameter value, if a well-established estimate of the other parameter is readily available.

#### 1. Introduction

Among the physicochemical properties of pure liquids and their mixtures that are constantly in demand for optimizing and designing industrial processes is viscosity. Being one of the most important factors in its own right in transport equations, nutrition, and chemical, cosmetic, and pharmaceutical industries, liquids viscosity parameters are essential for energy transference calculations and for hydraulic calculations of fluid transport [1–16]. Most cases found in industrial settings involve the difficulty posed by the nonlinear behavior of mixtures, against the mole fraction of one of the pure components constituting the corresponding binary liquid mixtures. As a result, rigorous and reliable data must be available with models that can provide a reliable estimation of the viscous behavior of mixtures [16].

The viscosity of fluids is determined both by collision among particles and by the force fields which determines interactions among molecules. The theoretical description of viscosity is therefore quite complex [16]. This is why several models have been proposed in the literature essentially based on Eyring theory or empirical or semiempirical equations that are not always applicable to a large number of mixtures [1, 2, 7–10]. On the other hand, excess thermodynamic functions (like enthalpy of hydration) and deviations of analogous nonthermodynamic functions (like viscosity) of binary liquid mixtures are fundamental for understanding different types of intermolecular interactions in these mixtures.

Many empirical and semiempirical models have been developed to describe the viscosity of pure liquids and binary liquid mixtures. This paper aims to contribute to describing the viscosity of pure liquids. For that, we will use statistical correlation analysis techniques for determining a relationship between the two viscosity Arrhenius parameters, allowing the reduction of the parameters number and facilitating thus calculations in engineering of fluid transport. Also, this will open a field for new theoretical concept and treatment. In addition, the suggested practical equation is useful when one of the two Arrhenius parameters data is absent. Indeed, it can be used to estimate the nonavailable value of one parameter using the information provided by the other one. In fact, the viscosity Arrhenius energy () can be related to the enthalpy of vaporization () at the same pressure [11]. Also, for the second parameter, the preexponential factor () can be closely related to the viscosity of the pure system in vapor state at the same studied pressure [12–15].

Recently, Ben Haj-Kacem et al. [16] proposed an empirical power law-type equation for modeling the relationship between the two parameters of viscosity Arrhenius-type equation for some pure classical solvents, such as the Arrhenius energy () or the preexponential factor (). We note that this is called HajKacem-Ouerfelli equation [16] which presents good concordance only for the low and moderate viscous fluids which have no very high values of activation energy ( kJ·mol^{−1}) and no very low values of preexperimental factor (−17 < ln(/Pa·s) < −10). Then, taking some mathematical considerations, we try in the present work to suggest an empirical exponential law-type equation valid on more extended intervals, that is, for the very viscous fluids and also for the very fluid liquids like the liquefied gas ( kJ·mol^{−1}) and (−25 < ln(/Pa·s) < −9). In addition, we have tried to give some physical meaning of the proposed equation parameters. We add that the suggested equation is important since it allows rewriting the viscosity Arrhenius-type equation by using a single parameter instead of two and thus it is very useful for engineering data which can permit estimating one nonavailable parameter when the second is available or can be precisely evaluated by some theories suggested in the literature.

#### 2. Viscosity-Temperature Dependence

Available data of transport properties of liquids are essential for mass and heat flow. As it is one of the important properties of fluids, liquid viscosity needs to be measured or estimated given that it influences the cases of design, handling, operation of mixing, transport, injection, combustion efficiency, pumping, pipeline, atomization and transportation, and so forth. The characteristics of liquid flow depend on viscosity which is affected principally by temperature and pressure.

##### 2.1. Theoretical Background

Due to the complex aspect of fluids, several theoretical methods for estimating viscosity are suggested in the literature [16]. Among these theories, we can cite the distribution function theory proposed by Kirkwood et al. [17], the molecular dynamic approach reported by Cummings and Evans [18], and the reaction rate theory of Eyring [19–21]. Generally, empirical and semiempirical methods provide reasonable results but they lack generality of approach, especially near or above the boiling temperature [11]. Hence, experimental data available in literature show that the liquid viscosity decreases with absolute temperature in nonlinear and concave fashion, and it is slightly dependent on low pressure.

##### 2.2. Empirical Equations

Numerous expressions have been suggested for representing the variation of liquid viscosity, () upon temperature () through available experimental data for interpolation purpose [19–43].

We can summarize the most different forms of temperature dependence of viscosity proposed under correlation methods by the following equation [16]:

We can classify the viscosity-temperature dependence on two great families such as liquid systems with linear or nonlinear behavior, when we plot the logarithm of dynamic viscosity () against the reciprocal of absolute temperature (). Furthermore, some supplement multiconstant equations (1) are proposed for numerous fluids deviating strongly to the Arrhenius behavior. We can cite the case of melting salts, glasses and metals, ionic liquids, heavy and vegetable oils, fuels and biofuels, and so forth [11, 16]. Moreover, for the nonlinear behavior, it is found that the temperature dependence of dynamic viscosity can be fitted frequently with the Vogel-Fulcher-Tammann-type equation [16, 22–24], given for constants, , , and, , by

In addition, for a long time, a favored theoretical base for the interpretation of viscosity was provided by the application of the transition-state theory of Arrhenius chemical kinetics to transport phenomena [6, 44]. The exponential form is a common expression of the variation against the reciprocal absolute temperature, of some physicochemical quantities related to the transport properties such as chemical rate, diffusion, electrical conductivity, gas kinetics, and viscosity. In fact, the preexponential factor is correlated to motion, rate, disorder, speed, and entropy. Also, some theories such as kinetic theory, Maxwell-Boltzmann statistics, thermal agitation, activation, and free volume theory lead to a similar expression [6, 11]. The second parameter () expresses the activation energy or a gap where the studied phenomenon or property may proceed through an intermediate “transition state.”

#### 3. Methodology

In addition, for the linear Arrhenius behavior, it is found that the temperature dependence of dynamic viscosity can be fitted frequently with the Arrhenius-type equation for numerous Newtonian classic solvents, which can be rewritten in the logarithmic form:where , , and are the gas constant, the Arrhenius activation energy, and the preexponential (entropic) factor of the Arrhenius equation for the liquid system, respectively.

The plot of the logarithm of shear viscosity, , against the reciprocal of absolute temperature () for numerous liquid systems is practically linear and the Arrhenius parameters, which are the activation energy () and the preexponential factor (), are thus independent of temperature over different studied temperature ranges (from 278.15 to 328.15) K approximately around the room temperature at constant atmospheric pressure. Using both graphical and linear least-squares fitting methods, the slope of the straight line is equal to () and the intercept on the ordinate is equal to (). In addition to these two main parameters, we added a third parameter () called the Arrhenius temperature deduced from the intercept with the abscissa axis: which can simplify the viscosity-temperature dependence following the Eyring [5, 11, 19–21] form as

Figure 1 shows how to determine graphically and how to proceed by extrapolation to reach the two parameters and .