Continuum mixture model

Bennon and Incropera [8-10] and Voller and coworkers [11-14] developed a continuum mixture model for the prediction of macroscopic transport behavior for a 

10 Numerical Modeling of Multiphase Flows in Materials Processing 

Integrating the generalized governing equation over this control volume and applying an upwind-differencing scheme, the discretized form of (10.1) becomes  

In (10.3)-(10.6) D and F represent, respectively, the diffusion and convection coefficients defined typically thus. 

The other diffusion and convection coefficients are similarly defined. 

---

Moholkar and Pandit (2001b) have also extended the nonlinear continuum mixture model to orifice-type reactors. Comparison of the bubble-dynamics profiles indicated that in the case of a venturi tube, a stable oscillatory radial bubble motion is obtained due to a linear pressure recovery (with low turbulence) gradient, whereas due to an additional oscillating pressure gradient due to turbulent velocity fluctuation, the radial bubble motion in the case of an orifice flow results in a combination of both stable and oscillatory type. Thus, the intensity of cavitation... 

The continuum mixture model and the control volume-based FDM are the predominant approaches employed by most researchers with only slight modifications. Those include the works of Christenson et al. [21], Voller et al. [13], Prescott and Incropera [22], Yoo and Viskanta [23], Shahani et al. [18], Chiang and Tsai [24], Chen and Tsai [25], Prescott et al. [26], Diao and Tsai [27], Schneider and... 

Prakash C, Voller V (1989) On the numerical solution of continuum mixture model equations describing binary solid-liquid phase change. Numer Heat Transf Part B 15 171-189... 

Mixture models differ from continuum models by virtue of the assertion that the separate contributions to any property from the several species are, in principle, measurable. When this assertion is abandoned, the differences between models often boil down to semantic niceties, and conflicting estimates of the adequacy of the starting points. 

Abstract In this contribution, the coupled flow of liquids and gases in capillary thermoelastic porous materials is investigated by using a continuum mechanical model based on the Theory of Porous Media. The movement of the phases is influenced by the capillarity forces, the relative permeability, the temperature and the given boundary conditions. In the examined porous body, the capillary effect is caused by the intermolecular forces of cohesion and adhesion of the constituents involved. The treatment of the capillary problem, based on thermomechanical investigations, yields the result that the capillarity force is a volume interaction force. Moreover, the friction interaction forces caused by the motion of the constituents are included in the mechanical model. The relative permeability depends on the saturation of the porous body which is considered in the mechanical model. In order to describe the thermo-elastic behaviour, the balance equation of energy for the mixture must be taken into account. The aim of this investigation is to provide with a numerical simulation of the behavior of liquid and gas phases in a thermo-elastic porous body. 

While the thermodynamic evidence may favor the mixture models, the diffraction studies from the static crystalline state tend to support the continuum model. The water molecules in the ices and high hydrates are always four-coordinated. 

The number of models that describe the structure and properties of liquid water is enormous. They can be subdivided into two groups the uniform continuum models and the cluster or mixture models. The main difference between these two classes of models is their treatment of the H-bond network in liquid water whereas the former assumes that a full network of H-bonds exists in liquid water, in the latter the network is considered broken at melting and that the liquid water is a mixture of various aggregates or clusters. The uniform continuum models stemmed from the classical publications of Bernal and Fowler, Pople, and Bernal.Among the cluster or mixture models, reviewed in refs 2—6 and 12, one should mention the models of Samoilov, Pauling, Frank and Quist, and Nemethy and Scheraga. ... 

The case to be considered here is that of a continuum mixture of chemical species in which chemical reactions occur. (A lumped-parameter model is given in (2).) The model for each different chemical compound is called a constituent. Were the constituents inert, then the amount of each would be an external constraint. However, since compounds can be produced or destroyed by chemical reactions in the mixtures being considered, the amount of each constituent is a non-conserved property. Then, if a mixture contains N constituents, and if R independent reactions are allowed to take place, N - R of the constituents can be selected as components and the amount of each component is a conserved property, i.e. its value can change only by being transported — it cannot be produced. Thus, every component is a constituent, but not every constituent is a component. However, the component amount, and the constituent amount of the same species are different. The amount of a constituent in a mixture at any instant represents the... 

The basic idea that the phase tfansition in category A-I SMPs can be phenomenologically modeled with a continuum mixture of a glassy and rubbery phase, each characterized by a volume fraction, was also picked up by Qi et al. [95]. A homogenization scheme is used to formulate the stress response of the SMP from the stress response of the phases. Similarly to the other models, constitutive relations are proposed for the temperature evolution of the volume fractions. 

Finally, we mention that very recently three other integral equation approaches to treating polymer systems have been proposed. Chiew [104] has used the particle-particle perspective to develop theories of the intermolecular structure and thermodynamics of short chain fluids and mixtures. Lipson [105] has employed the Born-Green-Yvon (BGY) integral equation approach with the Kirkwood superposition approximation to treat compressible fluids and blends. Initial work with the BGY-based theory has considered lattice models and only thermodynamics, but in principle this approach can be applied to compute structural properties and treat continuum fluid models. Most recently, Gan and Eu employed a Kirkwood hierarchy approximation to construct a self-consistent integral equation theory of intramolecular and intermolecular correlations [106]. There are many differences between these integral equation approaches and PRISM theory which will be discussed in a future review [107]. 

Fig. 3.10. Two different models of liquid water. (A) Continuum model (B) mixture model...

---