Dimensionless property

The system modelled was described by Nguyen et al. (2). The parameters for the model are taken from independently conducted experiments (2-6). Computer simulations were conducted with the model to mimic the conditions of the experimental situation. The simulation results are presented for a disc device whose dimensions are given in Table I. The molecular weight of each monomeric unit of the polymer is about 350 with two sites of hydrolysis. The concentration of device components and the dimensionless property ratios corresponding to each disc are given in Table n. 

To allow an easy interpretation of this equation, we introduce the dimensionless properties... 

Accordingly, to convert from the dimensionless properties supplied, the conversion factor for the dimensionless flux value V". As previously derived,... 

Then, in 1991, Knox and Grant [5], working carefully with 3- and 5-/xm particles, showed that it was practical to achieve dimensionless property of less than 1 in the CEC mode. These results confirmed their strongly optimistic view of the future of this technique, and a few years later, interest in CEC rapidly accelerated. 

The two summations are dimensionless properties of the lattice structure, and accurate values can be obtained by summing over the first few sets of nearest neighbors (Table 21.3). The resulting total energy for the fee lattice is... 

It is important to realize that the diameters needed for thermodynamic calculations do not necessarily represent a true minimum attainable separation distance between molecules. The objective is rather to determine optimal or effective diameters which give best results when used with a particular method of dealing with the contributions of molecular attraction. In this chapter the effective diameters sought are to be used specifically with the hard-sphere expansion (HSE) conformal solution theory of Mansoori and Leland (3). This theory generates the proper pseudo parameters for a pure reference fluid to be used in predicting the excess of any dimensionless property of a mixture over the calculated value of this property for a hard-sphere mixture. The value of this excess is obtained from a known value of this type of excess for a pure reference fluid evaluated at temperature and density conditions made dimensionless with the pseudo parameters. For example, if Xm represents any dimensionless property for a mixture of n nonpolar constituents at mole fractions xu x2,. . . x -i at temperature T and density p, then ... 

The width of the range is selected ideally to determine at a given temperature and density, T and p, the first and second derivatives of the dimensionless property with respect to inverse temperatures and to predict the property at each temperature in the range with an accuracy within its experimental error by a quadratic function. For example, if the compressibility factor is being evaluated, the values of z at p at each point in the range about T are fit by least squares to ... 

Although only compressibility factor calculations are used as an example in the explanation of the method, other properties can be predicted equally well. Because of the temperature and density dependence of the diameters and shape factors needed to relate them to critical constants it is best to determine separate values of them for each component. Three basic dimensionless properties should be determined. These are the ones best suited to the use of the HSE method with an equation of state in terms of temperature and density. These are the compressibility factor, z the internal energy deviation (U — V)/RT and a dimensionless fugacity ratio, ln(f/pRT). All other desired properties can be obtained from them. The ln(f/pRT) and z are calculated similarly. The computation scheme is outlined as shown in Table III. 

To perform calculations of phase equilibrium we must obtain expressions for the fugacity of component in solution. The key excess property here is the Gibbs free energy of solution. We begin by defining a new dimensionless property, the activity coefficient (yi) of component i in solution ... 

Using an Augmented Property index (AUP) for each stream s, defined as the summation of the dimensionless property operators, the property cluster for property j of stream s is defined ... 

D is the relative permittivity or dielectrie eonstant of the medium it is a dimensionless property whose value is aroimd 80, for example, in the case of water at ambient temperature, is the electrical permittivity of a vacuum. Its value is 8.85 x lO " SI, which equates to arormd 10 / 36 r. 

We will give the results of this calculation in dimensionless properties as was done in the preceding text ... 

To solve this, we consider the following dimensionless properties ... 

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