Empirical modifications

Theory of Calcination. The reversible reaction involved in the calcination and recarbonation of lime—limestone is one of the simplest and most fundamental of all chemical reactions. In practice, lime burning can be quite complex, however, and many empirical modifications are often necessary for efficient performance. 

Wilke-Chang This correlation for D°b is one of the most widely used, and it is an empirical modification of the Stokes-Einstein equation. It is not very accurate, however, for water as the solute. Otherwise, it apphes to diffusion of very dilute A in B. The average absolute error for 251 different systems is about 10 percent. ( )b is an association factor of solvent B that accounts for hydrogen bonding. 

As the basis for predicting ionic activity coefficients we chose to adopt an. empirical modification of Bromley s ( 5) extension of the Debye-Huckel model. The mean activity coefficient of a pure salt in water is given by... 

Many other empirical modifications of the Einstein equation have been made to predict actual viscosities. Since the modulus (M) is related to viscosity, these empirical equations, such as the Einstein-Guth-Gold (EGG) equation (8.3), have been used to predict changes in modulus when spherical fillers are added. 

A large number of empirical modifications to this expression have been proposed which model the viscosity of a liquid containing moderate concentrations of spherical particles [5] These include Mooney [6], Maron-Pierce [7] and Krieger-Dougherty [8] expressions which take into account the maximum packing fraction of the particles, and where interaction effects are absent, and can be represented by the general form ... 

Diffusion of small solute particles (atoms, molecules) in a dense liquid of larger particles is an important but ill-understood problem of condensed matter physics and chemistry. In this case one does not expect the Stokes-Einstein (SE) relation between the diffusion coefficient D of the tagged particle of radius R and the viscosity r/s of the medium to be valid. Indeed, experiments [83, 112-115] have repeatedly shown that in this limit SE relation (with slip boundary condition) significantly underestimates the diffusion coefficient. The conventional SE relation is D = C keT/Rr]s, where k T is the Boltzmann constant times the absolute temperature and C is a numerical constant determined by the hydrodynamic boundary condition. To explain the enhanced diffusion, sometimes an empirical modification of the SE relation of the form... 

When accurate vapor pressure data are plotted as log 0 versus 1/T over an appreciable pressure range, it is usually found that there is some curvature to the plot. This defect in Eq. (3) is largely overcome by an empirical modification proposed by Antoine ... 

A PVT EOS for a single species, based on concepts of molecular interactions, rather than semi-empirical modifications of the Abel EOS, was developed by Jacobs (Ref 17). It was then broadened to obtain relationships for detonation products by inclusion of mixt rules for n moles of s species. The most recent modification is called the JCZ-3 EOS (Jacobs—Cowperthwaite— Zwlsler). For a general summary of JCZ-3 we quote from Ref 19 ... 

Solutions to the musical noise problem. Various empirical modifications of the basic approach have been proposed to overcome this problem. A first possibility consists of taking advantage of the musical noise characteristics more precisely, the short duration of the musical noise components (typically a few short-time frames) [Boll, 1979, Vaseghi and Fray ling-Cork, 1992) and the fact that the appearance of musical noise in one sub-band is independent of that in other sub-bands [Sondhi et al., 1981]. The main shortcoming of this type of approach is that, since they are based on average statistical properties, the musical noise is reduced (ie. its appearance is made less frequent) but not completely eliminated. 

An older, but still widely used, correlation for D°AB, the Wilke-Chang technique, is, in essence, an empirical modification of the Stokes-Einstein relation (Eq. 4). Several systems were studiedby these investigators, and an average error of about 10% was noted ... 

A convenient way to classify solvents of chromatographic interest in terms of their polarity and the specific chemical interactions is the empirical scheme proposed by Snyder [214,215]. This scheme is based on experimental (gas chromatographic) distribution coefficients for three test solutes ( probes ) on a large number of stationary phases, which were published by Rohrschneider [216]. The probe compounds are ethanol (e), 1,4-dioxane (d) and nitromethane (n). The experimental values for the distribution coefficients undergo several empirical modifications ... 

Both the slope and intercept point in Fig. 23 change with increasing extent of conversion, which may be due to chainging NJs and a changing interdipole correlation. To simplify the analysis, an empirical modification to the Onsager equation was used to analyze the data ... 

H — -H + H2 within the collinear collision model This pioneering work states the following For any system driven by a bent potential valley, the reaction proceeds through a gradual transformation of the collision energy into vibrational energy of the product molecule. The first 3-dimensional trajectories were for the same reac-tion Since then, much important work has been undertaken. For instance, the way in which an empirical modification of the potential modifies the reaction-probability, the intramolecular states of the products, the deflection angle, etc... all these were the subject of many studies " and also of a review article ... 

We have deduced a set of equations with such properties based on empirical modifications of the quasi-chemical theory. (7 8) For binary systems the excess free energy of mixing is given by the expression... 

The two-phase k — e model analyzed was based on the Favre averaged transport equation for turbulent kinetic energy developed by [73, 74]. The resulting transport equation for kinetic energy is similar to the one obtained from the single phase model (5.2), supporting the semi-empirical modification introduced in that model. 

---