One-fluid model

This equation bases on the Camahan-Starling-One-Fluid Model for the repulsion term and the Square-Well-Pade-approximant for the attraction term. 

The simplest case is called the one-fluid model, that is, k — 1, assuming that particles are distributed in the fluid discretely. It was used for modeling voidage distributions in fast fluidized beds (Li and Kwauk, 1980 Bai et al., 1988 Zhang et al., 1990). 

The first suceessful method of generalizing a pure fluid EOS to mixtures was the one-fluid model proposed by van der Waals. The underlying assumption of this model is that the same EOS used for pure fluids can be used for mixtures if a satisfactory way is found of obtaining the mixture EOS parameter. The common method for doing this is based on expanding the EOS in virial form, that is, in powers of (1/ V). For the Peng-Robinson equation one obtains... 

S, Nonquadracic Combining Rules for the van der Waals One-Fluid Model (2PVDW Model)... 

An empirical approach to overcome the shortcomings of the van der Waals one-fluid model for a cubic EOS has simply been to add an additional composition dependence and parameters to the combining rule for the a parameter, generally leaving the b parameter rule unchanged. Some examples are the combining rules of Panagiotopoulos and Reid (1986)... 

Note that with this modification to the van der Waals one-fluid model, the fugac-ity coefficient expression for species i given in eqn. (3.3.9) will change because an additional compositional dependence has been introduced to the a term of the EOS. For the PR EOS, with the van der Waals one-fluid model, a more general form of the fugacity coefficient expression of species i in a mixture is... 

Mathias 1986 Michel, Hooper, and Prausnitz 1989 Sandler el al. 1986) and retains an important feature of the one-fluid model that the EOS for the pure fluids and the mixture have the sarrie density dependence. 

We have studied one-fluid model of binary fluids with polyamorphic components and found that multicritical point scenario gives opportunity to consider the continuous critical lines as the pathways linking isolated critical points of components on the global equilibria surface of binary mixture. It enhances considerably the landscape of mixture phase behavior in a stable region at the account of hidden allocation of other critical points in metastable region. 

In general the two-fluid average potential model gives better results than the one-fluid model. The three-fluid model counts all binary inter actions and is, therefore, properly used at low densities. The van der Waals one-fluid... 

Mathematical formalism has been developed using semi-empirical considerations [36, 37]. Computer simulation smdies show that resulting equation predicts oscillations. Attempt has been made to provide justification on the bases of Navier-Stokes equation but it is open to question. Dimensional analysis has recently been employed for investigating the phenomena [31]. Flow dynamics and stability in a density oscillator have been examined by Steinbock and co-workers [38], They have related it to Rayleigh-Taylor instability of two different dense viscous liquids. A theoretical description has been presented which is based on a one-fluid model and a steady state approximation for a two-dimensional flow using Navier-Stokes equation. However, the treatment is quite complex and cannot explain the generation of electric potential oscillations. 

A completely different approach to mixing rules is the one fluid model, in which it is not a and b which are modified, but the intermolecular potentials. We have a look at this method in 13.7.1. 

There are two general approaches to the prediction of the viscosity of the mixtures by the methods considered here. The first approach involves estimating the pure component viscosity of each of the constituents by some method and then combining these values to obtain the viscosity of the mixture. We refer to this approach as the multi-fluid model. A second approach is the so-called one-fluid model, in which the mixture is treated as a pseudo-pure fluid, with mixing rules for obtaining the parameters of the mixture from those of the pure components. 

The one-fluid approach can be used with some of the models presented earlier. In the method of Mehrotra [II], the one-fluid model can be used to obtain the needed normal boiling temperature, T, from = [ZxjT y ], if the mixture is well defined. This proposal has not been tested in that method. However, Orbey and Sandler [ 13] found this boiling-point estimation method to be suitable for use in their viscosity model as discussed later. 

The pressure change calculated from the so-called van der Waals one-fluid model is also shown in Figure 8. In this modeF the polydisperse system is characterized by a single size parameter... 

Thus, we have at our disposal an infinite set of terms (coefficients of I ") from which we can choose two for the determination of the potential parameters of the hypothetical pure fluid. In the van der Waals one fluid model, the first two members of the series are chosen, giving... 

The Lennard-Jones parameters for this one-fluid model are estimated from the... 

John Prausnitz had mentioned the excellent agreement with experiment that Mollerup had obtained in the gas-liquid critical region of binary hydrocarbon mixtures. Mollerup s results were obtained with a good reference equation of state for methane (but one which is classical in form and so which does not describe accurately the known nonclassical singularities in the thermodynamic functions at the critical point), and with a one-fluid model based on a mole fraction average (or "mole fraction based mixing rules"). 

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