Pseudo parameter

A general method of predicting the effective molecular diameters and the thermodynamic properties for fluid mix-tures based on the hard-sphere expansion conformal solution theory is developed. The method of Verlet and Weis produces effective hard-sphere diameters for use with this method for those fluids whose intermolecular potentials are known. For fluids with unknown potentials, a new method has been developed for obtaining the effective diameters from isochoric behavior of pure fluids. These methods have been extended to polar fluids by adding a new polar excess function, to account for polar contributions in a mixture. A new set of pseudo parameters has been developed for this purpose. The calculation of thermodynamic properties for several fluid mixtures including CH —C02 has been carried out successfully. 

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 ... 

In Equation 1 XnSM is the value of X for a hard-sphere mixture of diameters dn, d22. . . etc., and XIIS is the value of X for a pure hard-sphere fluid with diameter d. The value of XHs is calculated from the Camahan-Starling (CS) equation (4). XREf represents the value of X as obtained from a reduced equation of state for the pure reference fluid evaluated at T and p made dimensionless by the pseudo parameters c and d3. 

The pseudo parameters of the HSE theory are derived from an equation of state expanded in powers of 1/kT about a hard-sphere fluid, as is developed by the perturbation theory. Consequently, it is reasonable to expect that procedures for defining optimal diameters for the perturbation theory should work well with the HSE procedure. The first portion of this chapter shows that this is indeed correct. The Verlet-Weis (VW) (5) modification of the Weeks, Chandler, and Anderson (WCA) (6) procedure was used here to determine diameters in a mixture of Len-nard-Jones (LJ) (12-6) fluids. These diameters then were used in the HSE procedure to predict the mixture properties. 

The column headed HSE uses an approximation made originally by Mansoori and Leland (3) that the diameter used in the hard sphere equations of state is c0o-, the LJ a parameter for each molecule multiplied by a universal constant for conformal fluids. This approximation then requires that be replaced by equations defining the HSE pseudo parameters, Equations 10 and 11. The results in the HSE column use c0 = 0.98, the value for LJ fluids obtained empirically by Mansoori and Leland. This procedure is correct only for a Kihara-type potential and it is not consistent with the LJ fluids in Table I. Furthermore, this causes only the high temperature limit of the repulsion effects to be included in the hard-sphere calculation. Soft repulsions are predicted by the reference fluid. 

Table I shows that the HSE theory is much better than either of the VDW theories, which justifies the direct computations of the hard-sphere contributions as opposed to their prediction by the reference fluid as is the case with the VDW theories. Furthermore, derivation of the HSE pseudo parameters considers (1/kT)2 terms in the expansion about hard-sphere properties, whereas in the VDW theories the expansion is truncated after the first order 1/kT term and all of the repulsion properties of the fluid are predicted by the reference fluid. The importance of including the second order (1/kT)2 terms is indicated by the excellent results of the HSE-VW method for the enthalpy, in that this property is much less dependent on the choice of the diameter and the hard-sphere contribution.

Substitution of Equations 18 and 19 into Equations 10 and 11 converts the pseudo parameters to pseudo criticals. The result is ... 

For (17 — V)/RT no hard-sphere property calculations are made and the a0 term of the quadratic fit along the compressibility factor isochores can be equated to ZHs(pd3). This is then solved for the diameter used in the pseudo parameter computations. 

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. 

In this figure the next definitions are used A - projection operator, B - pseudo-inverse operator for the image parameters a,( ), C - empirical posterior restoration of the FDD function w(a, ), E - optimal estimator. The projection operator A is non-observable due to the Kalman criteria [10] which is the main singularity for this problem. This leads to use the two step estimation procedure. First, the pseudo-inverse operator B has to be found among the regularization techniques in the class of linear filters. In the second step the optimal estimation d (n) for the pseudo-inverse image parameters d,(n) has to be done in the presence of transformed noise j(n). 

The adaptive estimation of the pseudo-inverse parameters a n) consists of the blocks C and E (Fig. 1) if the transformed noise ( ) has unknown properties. Bloek C performes the restoration of the posterior PDD function w a,n) from the data a (n) + (n). It includes methods and algorithms for the PDD function restoration from empirical data [8] which are based on empirical averaging. Beeause the noise is assumed to be a stationary process with zero mean value and the image parameters are constant, the PDD function w(a,n) converges, at least, to the real distribution. The posterior PDD funetion is used to built a back loop to block B and as a direct input for the estimator E. For the given estimation criteria f(a,d) an optimal estimation a (n) can be found from the expression... 

The pseudo-wavefunction within this frame work is guaranteed to be nodeless. The parameters (a, p, y, 8) are... 

Step 4 - it is initially assumed that the flow field in the entire domain is incompressible and using the initial and boundary conditions the corresponding flow equations are solved to obtain the velocity and pressure distributions. Values of the material parameters at different regions of the domain are found via Equation (3.70) using the pseudo-density method described in Chapter 3, Section 5.1. 

Use a pseudo reaction coordinate with one parameter constrained followed by a quasi-Newton optimization. 

The numerical solution of these equations is shown in Fig. 23-28. This is a plot of the enhancement fac tor E against the Hatta number, with several other parameters. The factor E represents an enhancement of the rate of transfer of A caused by the reaction compared with physical absorption with zero concentration of A in the liquid. The uppermost line on the upper right represents the pseudo-first-order reaction, for which E = P coth p. 

From this equation it is seen that parameters have been introduced into the QM-MM method, with K, L, M, and corresponding to the pseudo s orbital on the classical atom. These parameters can be optimized to reproduce experimental or high level theoretical data. Field et al. [9] perfonned extensive investigations of the values of these extra parameters and suggested that the parameters K, L, and M (i = 1,. . . , 4) can be set to zero and that should take a value of 5.0. These are generally the values used in most current QM-MM implementations that employ semiempirical methods in the quantum region. 

A parameter such as a rate constant is usually obtained as a consequence of various arithmetic manipulations, and in order to estimate the uncertainly (error) in the parameter we must know how this error is related to the uncertainties in the quantities that contribute to the parameter. For example, Eq. (2-33) for a pseudo-first-order reaction defines k, which can be determined by a semilogarithmic plot according to Eq. (2-6). By a method to be described later in this section the uncertainty in itobs (expressed as its variance associated with cb. Thus, we need to know how the errors in fcobs and cb are propagated into the rate constant k. 

Recently Thiel and Voityuk have constructed a workable NDDO model which also includes d-orbitals for use in connection with MNDO, called MNDO/d. With reference to the above description for MNDO/AM1/PM3, it is clear that there are immediately three new parameters Cd, Ud and (dd (eqs. (3.82) and (3.83)). Of the 12 new one-centre two-electron integrals only one (Gjd) is taken as a freely varied parameter. The other 11 are calculated analytically based on pseudo-orbital exponents, which are assigned so that the analytical formulas regenerate Gss, Gpp and Gdd. 

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