Model fluids

Stell G, Patey G N and H0ye J S 1981 Dielectric constant of fluid models statistical mechanical theory and its quantitative implementation Adv. Chem. Phys. 48 183... 

Analytic teclmiques often use a time-dependent generalization of Landau-Ginzburg ffee-energy fiinctionals. The different universal dynamic behaviours have been classified by Hohenberg and Halperin [94]. In the simple example of a binary fluid (model B) the concentration difference can be used as an order parameter m.. A gradient in the local chemical potential p(r) = 8F/ m(r) gives rise to a current j... 

The simplest case of fluid modeling is the technique known as computational fluid dynamics. These calculations model the fluid as a continuum that has various properties of viscosity, Reynolds number, and so on. The flow of that fluid is then modeled by using numerical techniques, such as a finite element calculation, to determine the properties of the system as predicted by the Navier-Stokes equation. These techniques are generally the realm of the engineering community and will not be discussed further here. 

The Maxwell model is also called Maxwell fluid model. Briefly it is a mechanical model for simple linear viscoelastic behavior that consists of a spring of Young s modulus (E) in series with a dashpot of coefficient of viscosity (ji). It is an isostress model (with stress 5), the strain (f) being the sum of the individual strains in the spring and dashpot. This leads to a differential representation of linear viscoelasticity as d /dt = (l/E)d5/dt + (5/Jl)-This model is useful for the representation of stress relaxation and creep with Newtonian flow analysis. 

Das, N. C., Elastohydrodynamic Lubrication Theory of Line [39] Contacts Couple Stress Fluid Model, Tribal. Trans., Vol. 40, No.2,1997,pp.353-359. 

Lin, J. R., Squeeze Film Characteristics Between a Sphere and [40] a Flat Plate Couple Stress Fluid Model, Camputers Structures, Vol. 75,2000, pp. 73-80. 

Eulerian two-fluid model coupled with dispersed itequations was applied to predict gas-liquid two-phase flow in cyclohexane oxidation airlift loop reactor. Simulation results have presented typical hydrodynamic characteristics, distribution of liquid velocity and gas hold-up in the riser and downcomer were presented. The draft-tube geometry not only affects the magnitude of liquid superficial velocity and gas hold-up, but also the detailed liquid velocity and gas hold-up distribution in the reactor, the final construction of the reactor lies on the industrial technical requirement. The investigation indicates that CFD of airlift reactors can be used to model, design and scale up airlift loop reactors efficiently. 

In a silane-hydrogen discharge the feedstock gases SiHa and H2 take part in all the processes that occur. A large number of reactions have been proposed (see e.g. Kushner [190]). Nienhuis et al. [191] have performed a sensitivity analysis in their self-consistent fluid model, from which a minimum set of reactions have been extracted for a typical low-pressure RF discharge. Tables II and III list these reactions. They will be used in the plasma models described in subsequent sections. The review articles on silane chemistry by Perrin et al. [192] and on hydrogen by Phelps [193] and Tawara et al. [194] have been used. The electron collision data are compiled in Figure 13 [189]. 

Nienhuis et al. [189, 191] have developed a self-consistent fluid model that describes the electron kinetics, the silane-hydrogen chemistry, and the deposition... 

In the fluid model the momentum balance is replaced by the drift-diffusion approximation, where the particle flux F consists of a diffusion term (caused by density gradients) and a drift term (caused by the electric field ) ... 

In a fluid model the correct calculation of the source terms of electron impact collisions (e.g. ionization) is important. These source terms depend on the EEDF. In the 2D model described here, the source terms as well as the electron transport coefficients are related to the average electron energy and the composition of the gas by first calculating the EEDF for a number of values of the electric field (by solving the Boltzmann equation in the two-term approximation) and constructing a lookup table. 

In the ASTER reactor deposition experiments were performed in order to compare with the 2D model results. Normalized deposition rates are plotted in Figure 22 as a function of radial position for data taken at 25 and 18 Pa. The deposition takes place on a square glass plate. For each pressure two profile measurements were performed, each profile perpendicular to the other (a and b in Fig. 22). A clear discrepancy is present. The use of the simplified deposition model is an explanation for this. Another recent 2D fluid model also shows discrepancies between the measured and calculated deposition rate [257], which are attributed to the relative simplicity of the deposition model. 

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. 

Especially for the electrons, the fluid model has the advantage of a lower computational effort than the PIC/MC method. Their low mass (high values of the transport coefficients) and consequent high velocities give rise to small time steps in the numerical simulation (uAf < Aa) if a so-called explicit method is used. This restriction is easily eliminated within the fluid model by use of an implicit method. Also, the electron density is strongly coupled with the electric field, which results in numerical Instabilities. This requires a simultaneous implicit solution of the Poisson equation for the electric field and the transport equation for the electron density. This solution can be deployed within the fluid model and gives a considerable reduction of computational effort as compared to a nonsi-multaneous solution procedure [179]. Within the PIC method, only fully explicit methods can be applied. 

The disadvantage of the fluid model is that no kinetic information is obtained. Also, transport (diffusion, mobility) and rate coefficients (ionization, attachment) are needed, which can only be obtained from experiments or from kinetic calculations in simpler settings (e.g. Townsend discharges). Experimental data on... 

Subsequently, because the electron temperature is known as a function of the electric field E, the temperature Te can be used instead of as a parameter for coefficients and rates, by elimination of E. Thus, the coefficients are available both as a function of E and of Te. Both the loeal electrie field [225, 269] and the electron temperature [239,268] have been used as parameters in fluid modeling. 

In order to be able to explain the observed results plasma modeling was applied. A one-dimensional fluid model was used, which solves the particle balances for both the charged and neutral species, using the drift-diffusion approximation for the particle fluxes, the Poisson equation for the electric field, and the energy balance for the electrons [191] (see also Section 1.4.1). 

This map has been checked by many researchers, indicating that it is applicable to a wide range of conditions. Also shown in Figure 3.4 are correlations derived by Mishima and Ishii (1984), which used similar basic principles except for the slug-to-churn transition. These authors pointed out that, in view of the practical applications of the separate-fluid model to transient analysis, flow regime criteria based on the superficial velocities of the liquid and gas may not be consistent with the separate-flow model formulation. A direct geometric parameter such as the... 

If this model is further simplified by considering unidirectional flow, the number of equations is reduced to four (Wallis, 1969). Another example is Bankoff s variable-density, single-fluid model for two-phase flow (Bankoff, 1960). Since it is based on an intimate mixture, both mechanical equilibrium (i.e., same velocity) and thermal equilibrium (same temperature) between the two phases must logically be assumed (Boure, 1975). 

For other discussions of two-phase models and numerical solutions, the reader is referred to the following references thermofluid dynamic theory of two-phase flow (Ishii, 1975) formulation of the one-dimensional, six-equation, two-phase flow models (Le Coq et al., 1978) lumped-parameter modeling of one-dimensional, two-phase flow (Wulff, 1978) two-fluid models for two-phase flow and their numerical solutions (Agee et al., 1978) and numerical methods for solving two-phase flow equations (Latrobe, 1978 Agee, 1978 Patanakar, 1980). 

Bankoff, S. G., 1960, A Variable-Density, Single-Fluid Model for Two-Phase Flow with Particular Reference to Steam-Water Flow, Trans. ASME, J. Heat Transfer 52 265-272. (3)... 

Granziera, R., and M. S. Kazimi, 1980, A Two-Dimensional Two-Fluid Model for Sodium Boiling in LMFBR Fuel Assemblies, Energy Laboratory Rep. No. MIT-EL-80-011, Massachusetts Institute of Technology, Cambridge, MA. (4)... 

Staub, F. W., 1967, The Void Fraction in Subcooled Boiling—Prediction of the Initial Point of Net Vapor Generation, ASME Paper 67-HT-36, National Heat Transfer Conf., ASME, New York. (3) Staub, F. W., 1969, Two Phase Fluid Modeling, The Critical Heat Flux, Nuclear Sci. Eng. 3J.T 90-199. (5)... 

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