Three-parameter

Three parameters are estimated from binary VLE data and correspond to ... 

The following example will help to clarify the distinctions between these three parameters. 

The estimation of the three parameters —pseudo-critical temperature, pseudo-critical pressure, and the acentric factor— should be done using the same method because these constants should be coherent. 

This method utilizes essentially the concept developed by Fitzer in 1955. According to the principle of three-parameter corresponding states, the compressibility factor z, for a fluid of acentric factor w, is obtained by interpolating between the compressibilities Zj and Z2 for the two fluids having acentric factors w, and (p -... 

Introduction and Commercial Application The objective of reservoir geology is the description and quantification of geologically controlled reservoir parameters and the prediction of their lateral variation. Three parameters broadly define the reservoir geology of a field ... 

We will explore the effect of three parameters 2 -and < )> that is, the time delay between the pulses, the tuning or detuning of the carrier frequency from resonance with an excited-state vibrational transition and the relative phase of the two pulses. We follow closely the development of [22]. Using equation (Al.6.73). 

For each pair of interacting atoms (/r is their reduced mass), three parameters are needed D, (depth of the potential energy minimum, k (force constant of the par-tictilar bond), and l(, (reference bond length). The Morse ftinction will correctly allow the bond to dissociate, but has the disadvantage that it is computationally very expensive. Moreover, force fields arc normally not parameterized to handle bond dissociation. To circumvent these disadvantages, the Morse function is replaced by a simple harmonic potential, which describes bond stretching by Hooke s law (Eq. (20)). 

The simplest and most commonly used flux model is provided by the dusty gas equations (3.17)-(3.19), All the conditions (i)-(iv) above are satisfied by these equations, and the three parameters K, , and intro-... 

The presence of the q B term with its implied distance dependency means that the charges depend upon the molecular geometry. Thus, should the conformation of a molecule change the atomic charges will also change. Just three parameters are required for each atom in the system (the electronegativity, the idempotential and the covalent radius). 

The kinetic data are essentially always treated using the pseudophase model, regarding the micellar solution as consisting of two separate phases. The simplest case of micellar catalysis applies to unimolecTilar reactions where the catalytic effect depends on the efficiency of bindirg of the reactant to the micelle (quantified by the partition coefficient, P) and the rate constant of the reaction in the micellar pseudophase (k ) and in the aqueous phase (k ). Menger and Portnoy have developed a model, treating micelles as enzyme-like particles, that allows the evaluation of all three parameters from the dependence of the observed rate constant on the concentration of surfactant". ... 

To meet the point that the amount of resonance interaction in the transition state will be dependent upon the nature of the electrophile, Yukawa and Tsuno have put forward a modified equation with three parameters. The physical interpretation of such an equation is interesting, but it is not surprising that it correlates experimental data better than does the equation with two parameters. ... 

The Price-Alfrey approach begins by defining three parameters-P, Q, and e-for each of the comonomers in a reaction system. We shall see presently that the parameter P is rapidly eliminated from the theory. As a result, the Price-Alfrey system is also called the Q-e scheme for copolymerization. 

Haeany Solution Model The initial model (37) considered the adsorbed phase to be a mixture of adsorbed molecules and vacancies (a vacancy solution) and assumed that nonideaUties of the solution can be described by the two-parameter Wilson activity coefficient equation. Subsequendy, it was found that the use of the three-parameter Flory-Huggins activity coefficient equation provided improved prediction of binary isotherms (38). 

In plotting on WeibuU paper, a downward concave plot implies a non2ero minimum life. Values for S < can be selected by trial and error. When they are subtracted from each /, a relatively straight line is produced. This essentially translates the three-parameter WeibuU distribution back to a two-parameter distribution. 

The capacity of any specific tank configuration, in terms of metric ton equivalents, is deterrnined by one of three parameters. (/) The solubiHty of waste salts. Precipitates can settle and cause thermal hot spots, which in turn can result in accelerated corrosion rates. Thus it is important to maintain the... 

The characteristic separation curve can be deterrnined for any size separation device by sampling the feed, and coarse and fine streams during steady-state operation. A protocol for determining such selectivity functions has been pubHshed (4). This type of testing, when properly conducted, provides the relationships among d K, and a at operating conditions. These three parameters completely describe a size separation device and can be used to predict the size distribution of the fine and coarse streams. 

AH three parameters, the cut size, sharpness index, and apparent bypass, are used to evaluate a size separation device because these are assumed to be independent of the feed size distribution. Other measures, usually termed efficiencies, are also used to evaluate the separation achieved by a size separation device. Because these measures are dependent on the feed size distribution, they are only usefiil when making comparisons for similar feeds. AH measures reduce to either recovery efficiency, classification efficiency, or quantitative efficiency. Recovery efficiency is the ratio of the amount of material less than the cut size in the fine stream to the amount of material less than the cut size in the feed stream. Classification efficiency is defined as a corrected recovery efficiency, ie, the recovery efficiency minus the ratio of the amount of material greater than the cut size in the fine stream to the amount of material greater than the cut size in the feed stream. Quantitative efficiency is the ratio of the sum of the amount of material less than the cut size in the fine stream plus the amount of material greater than the cut size in the coarse stream, to the sum of the amount of material less than the cut size in the feed stream plus the amount of material greater than the cut size in the feed stream. Thus, if the feed stream analyzes 50% less than the cut size and the fine stream analyzes 95% less than the cut size and the fine stream flow rate is one-half the feed stream flow rate, then the recovery efficiency is 95%, the classification efficiency is 90%, and the quantitative efficiency is 95%. 

Sodium bromide has a very high water solubiUty. At 25°C a saturated solution contains 48.6% sodium bromide by weight. Values for the solubiUty at several temperatures are known (1). Three parameter equations for calculating the solubiUty iu terms of mole fraction of both the anhydrous and dihydrate salts are available (2). Convenient equations for calculating the solubiUty iu weight percent of sodium bromide iu water at various temperatures, t iu °C, are as follows ... 

Computerized optimization using the three-parameter description of solvent interaction can facihtate the solvent blend formulation process because numerous possibihties can be examined quickly and easily and other properties can also be considered. This approach is based on the premise that solvent blends with the same solvency and other properties have the same performance characteristics. Eor many solutes, the lowest cost-effective solvent blends have solvency that is at the border between adequate and inadequate solvency. In practice, this usually means that a solvent blend should contain the maximum amount of hydrocarbon the solute can tolerate while still remaining soluble. 

The volumetric properties of fluids are represented not only by equations of state but also by generalized correlations. The most popular generalized correlations are based on a three-parameter theorem of corresponding states which asserts that the compressibiHty factor is a universal function of reduced temperature, reduced pressure, and a parameter CO, called the acentric factor ... 

Whereas the Wilson model has been found to represent a wide variety of nonideal VLE, it caimot handle the case of partial immiscihility of the Hquid phase for this purpose a three-parameter relationship, the nonrandom, two-Hquid (NRTL) model was developed (20). 

This gives two choices ia interpreting calculated NRR values, ie, a direct comparison of NRR values for different options or a comparison of the NRR value of each option with a previously defined NRR cutoff level for acceptabiUty. The NPV, DTC, and NRR can be iaterpreted as discounted measures of the return, iavestment, and return rate, analogous to the parameters of the earher example. These three parameters characterize a venture over its entire life. Additional parameters can be developed to characterize the cash flow pattern duting the early venture years. Eor example, the net payout time (NPT) is the number of operating years for the cumulative discounted cash flow to sum to zero. This characterizes the early cash flow pattern it can be viewed as a discounted measure of the expected operating time that the investment is at risk. 

The exchange current density, depends on temperature, the composition of the electrolyte adjacent to the electrode, and the electrode material. The exchange current density is a measure of the kinetic resistance. High values of correspond to fast or reversible kinetics. The three parameters, a, a. ... 

Three Parameter Models. Most fluids deviate from the predicted corresponding states values. Thus the acentric factor, CO, was introduced to account for asymmetry in molecular stmcture (79). The acentric factor is defined as the deviation of reduced vapor pressure from 0.1, measured at a reduced temperature of 0.7. In equation form this becomes ... 

Four Parameter Models. Two- and three-parameter theories are only accurate for simple, normal, and some slightly polar fluids. In order to accurately predict polar fluid behavior a fourth parameter is needed (80). The Stiel polarity factor, is one such fourth parameter and follows from the... 

One of the most versatile and accurate generalized correlations for the prediction of the fugacity coefficient (3) involves a three-parameter generalized correlation which takes advantage of the acentric factor. The correlation breaks the fugacity coefficient into two parts (j) and ( ). ... 

Pitzer s Corresponding-States Correlation A three-parameter corresponding-states correlation of the type developed by Pitzer, K.S. Thennodynamic.s, 3ded., App. 3, McGraw-HiU, New York, 1995) is described in Sec. 2. It has as its basis an equation for the compressibility factor ... 

The NRTL equation contains three parameters for a binary system and is written ... 

This three-parameter equation behaves linearly in the Henry s law region and reduces to the Langmuir isotherm for m = 1. Other well-known isotherms include the Radke-Prausnitz isotherm [Radke and Prausnitz, Ind. Eng. Chem. Fundam., 11, 445 (1972) AIChE J., 18, 761 (1972)]... 

Another three-parameter equation that often fits data well and is linear in the Henry s law region is the UNILAN equation [Honig and... 

Three parameters define the performance of a classifier. These are cut size, shaipness of cut, and eapaeity. Cut size, X50, is the size at which 50 percent of the material goes into the coarse product and 50 percent into the fine. (This should not be confused with the cutoff size, a name sometimes given to the top size of the fine product .)... 

Although two-parameter models are rather restrictive, three-parameter models of the intermolecular potential have been developed which provide reasonable descriptions of the thermodynamic behavior of solids. Examples include the Morse potential, the exponential-six potential, and, more recently, a form proposed by Rose et al. (1984) for metals. 

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