Local fluid density, calculation

The simulations are repeated several times, starting from different matrix configurations. We have found that about 10 rephcas of the matrix usually assure good statistics for the determination of the local fluid density. However, the evaluation of the nonuniform pair distribution functions requires much longer runs at least 100 matrix replicas are needed to calculate the pair correlation functions for particles parallel to the pore walls. However, even as many as 500 replicas do not ensure the convergence of the simulation results for perpendicular configurations. 

We calculate the dependence of gm on concentration from the Peng-Robinson equation of state. The resulting source of energy 5 h then depends on the local gradient of concentration, local fluid velocity, and fluid density ... 

For laminar flow, the characteristic time of the fluid phase Tf can be deflned as the ratio between a characteristic velocity Uf and a characteristic dimension L. For example, in the case of channel flows confined within two parallel plates, L can be taken equal to the distance between the plates, whereas Uf can be the friction velocity. Another common choice is to base this calculation on the viscous scale, by dividing the kinematic viscosity of the fluid phase by the friction velocity squared. For turbulent flow, Tf is usually assumed to be the Kolmogorov time scale in the fluid phase. The dusty-gas model can be applied only when the particle relaxation time tends to zero (i.e. Stp 1). Under these conditions, Eq. (5.105) yields fluid flow. This typically happens when particles are very small and/or the continuous phase is highly viscous and/or the disperse-to-primary-phase density ratio is very small. The dusty-gas model assumes that there is only one particle velocity field, which is identical to that of the fluid. With this approach, preferential accumulation and segregation effects are clearly not predicted since particles are transported as scalars in the continuous phase. If the system is very dilute (one-way coupling), the properties of the continuous phase (i.e. density and viscosity) are assumed to be equal to those of the fluid. If the solid-particle concentration starts to have an influence on the fluid phase (two-way coupling), a modified density and viscosity for the continuous phase are generally introduced in Eq. (4.92). 

We presented a novel quenched solid non-local density functional (QSNLDFT) model, which provides a r istic description of adsorption on amorphous surfaces without resorting to computationally expensive two- or three-dimensional DFT formulations. The main idea is to consider solid as a quenched component of the solid-fluid mixture rather than a source of the external potential. The QSNLDFT extends the quenched-annealed DFT proposed recently by M. Schmidt and cowoikers [23,24] for systems with hard core interactions to porous solids with attractive interactions. We presented several examples of calculated adsorption isotherms on amorphous and microporous solids, which are in qualitative agreement with experimental measurements on typical polymer-templated silica materials like SBA-15, FDU-1 and oftiers. Introduction of the solid density distribution in QSNLDFT eliminates strong layering of the fluid near the walls that was a characteristic feature of NLDFT models with smoodi pore walls. As the result, QSNLDFT predicts smooth isotherms in the region of polymolecular adsorption. The main advantage of the proposed approach is that QSNLDFT retains one-dimensional solid and fluid density distributions, and thus, provides computational efficiency and accuracy similar to conventional NLDFT models. 

The simultaneous calculation of the amount adsorbed and the distribution of adsorbate within the pores of a reconstructed solid can be impl ented afier the i lication of a Draisity Functional Theory (DFT) mean field model, which is particularly suited for on-lattice simulations on di tised structures [31]. In a lattice model the spatial distribution of adsorbate can be described at each site by the local density fimction. The equilibrium density profile for a given matrix realization and chemical potential ft (xin then be determined by minimising the grand potential 0 p with respect to the fluid density on each lattice site, / (x), leading to ... 

The second essential part of the mean-field approximation is the prescription for calculating u. Here is the value of Umr that is felt by a test particle at any point at which its hard core fits among those of the molecules of the fluid. Ihe prescription of the mean-field approximation is now that be identified with die hypothetical that would be felt by the test particle if the molecules of the fluid were distributed around it with the local macroscopic density, but otherwise at random. If the fluid were inhomogeneous, with density p(t) at i, the ii f so calculated would be r-dependent, and, assuming pair forces alone, would be given by... 

The fluid model is a description of the RF discharge in terms of averaged quantities [268, 269]. Balance equations for particle, momentum, and/or energy density are solved consistently with the Poisson equation for the electric field. Fluxes described by drift and diffusion terms may replace the momentum balance. In most cases, for the electrons both the particle density and the energy are incorporated, whereas for the ions only the densities are calculated. If the balance equation for the averaged electron energy is incorporated, the electron transport coefficients and the ionization, attachment, and excitation rates can be handled as functions of the electron temperature instead of the local electric field. 

In the Lagrangian approach, individual parcels or blobs of (miscible) fluid added via some feed pipe or otherwise are tracked, while they may exhibit properties (density, viscosity, concentrations, color, temperature, but also vorti-city) that distinguish them from the ambient fluid. Their path through the turbulent-flow field in response to the local advection and further local forces if applicable) is calculated by means of Newton s law, usually under the assumption of one-way coupling that these parcels do not affect the flow field. On their way through the tank, these parcels or blobs may mix or exchange mass and/or temperature with the ambient fluid or may adapt shape or internal velocity distributions in response to events in the surrounding fluid. 

Fluorescence spectra and quantum yields of pyrene in supercritical CO2 have been determined systematically as functions of temperature, CO2 density, and pyrene concentration. Under near-critical conditions, contributions of the pyrene excimer emission in observed fluorescence spectra are abnormally large. The results cannot be explained in the context of the classical photophysical mechanism well established for pyrene in normal liquid solvents. The photophysical behavior of pyrene in a supercritical fluid is indeed unusual. The experimental results can be rationalized with a proposal that the local concentration of pyrene monomer in the vicinity of an excited pyrene molecule is higher than the bulk in a supercritical solvent environment. It is shown that the calculated ratios between the local and bulk concentrations deviate from unity more significantly under near-critical conditions (Sun and Bunker, 1995). 

Fig. 5. VET rate constants of benzene in scC02 as a function of reduced density (filled circles). The solid line represents calculations of the local density at the position of the first maximum of the radial distribution function around an attractive solute in a Lennard-Jones fluid (see Fig. 7 and text for details). Experimental conditions pred = 2.1 (500bar, 318K), prei = 1.6 (150 bar, 318K), pred= 1.2 (lOObar, 318K), pred= 0.7 (lOObar, 328K).

Once the free energy of an inhomogeneous system is given, one can calculate by standard methods the properties of the interface—for example, the interfacial tension or the density profile perpendicular the interface [285]. Weiss and Schroer compared the various approximations within square-gradient theory discussed earlier in Section IV.F for studying the interfacial properties for pure DH and FL theory [241, 242], In theories based on local density approximations the interfacial thickness and the interfacial tension were found to differ by up to a factor of four in the various approximations. This contrasts with nonionic fluids, where the density profiles and interfacial... 

We have now collected almost all the pieces required for a first version of COSMO-RS, which starts from the QM/COSMO calculations for the components and ends with thermodynamic properties in the fluid phase. Although some refinements and generalizations to the theory will be added later, it is worthwhile to consider such a basic version of COSMO-RS because it is simpler to describe and to understand than the more elaborate complete version covered in chapter 7. In this model we make an assumption that all relevant interactions of the perfectly screened COSMO molecules can be expressed as local contact energies, and quantified by the local COSMO polarization charge densities a and a of the contacting surfaces. These have electrostatic misfit and hydrogen bond contributions as described in Eqs. (4.31) and (4.32) by a function for the surface-interaction energy density... 

The local density of solvent about the solute may be determined by comparing the experimental and calculated curves. Consider points A and B in Figure 5 at a constant value of E, i.e., 55 kcal/mol. A hypothetical homogeneous fluid at point B gives the same "solvent strength" as he actual fluid at point A. The local density about the solute exceeds the bulk density due to compression, such that... 

The crudest approximation to the density matrix for the system is obtained by assuming that there are no statistical correlations between the elementary excitations (perfect fluid), so that can be written as a simple product of molecular density matrices A. A better approximation is obtained if one does a quantum field theory calculation of the local field effects in the system which in a certain approximation gives the Lorentz-Lorenz correction L(TT) in terms of the refractive index n53). One then writes,... 

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