Cell model of liquids

MSE.IO. I. Prigogine et L. Saraga, Sur la tension superficielle et le modele ceUulaire de I etat liquide (On the surface tension and the cell model of liquid state), J. Chim. Phys. 49, 399-407 (1952). 

Thus surface tensions or surface energies of some simple liquids have already been calculated in terms of intermolecular forces from theories which are based on some simplifying assumptions such as the cell model of liquid structure. The results are in good agreement with observed values. 

Given the character of the water-water interaction, particularly its strength, directionality and saturability, it is tempting to formulate a lattice model, or a cell model, of the liquid. In such models, local structure is the most important of the factors determining equilibrium properties. This structure appears when the molecular motion is defined relative to the vertices of a virtual lattice that spans the volume occupied by the liquid. In general, the translational motion of a molecule is either suppressed completely (static lattice model), or confined to the interior of a small region defined by repulsive interactions with surrounding molecules (cell model). Clearly, the nature of these models is such that they describe best those properties which are structure determined, and describe poorly those properties which, in some sense, depend on the breakdown of positional and orientational correlations between molecules. 

The theoretical description based on the lattice or cell models of the liquid uses the language contributing states of occupancy . Nevertheless, these states ot occupancy are not taken to be real, and the models are, fundamentally, of the continuum type. The contribution to the free energy function of different states of occupancy of the basic lattice section is analogous to the contribution to the energy of a quantum mechanical system of terms in a configuration interaction series. 

To summarize, to properly model liquid water transport and ensuing flooding effect on cell performance, one must consider four submodels (1) a model of catalytic surface coverage by liquid water inside the catalyst layer, (2) a model of liquid water transport through hydrophobic microporous layer and GDL, (3) an interfacial droplet model at the GDL surface, and last (4) a two-phase flow model in the gas channel. Both experimental and theoretical works, in academia and industry alike, are ongoing to build models for the four key steps of water generation, transport, and removal from a PEFC. 

Figure 4.26 shows a cell model of the three phases. Gas in the upper region has a very low density and the molecules are free to fly around. When the vapor condenses into a liquid (shown lower right), the density is greatly increased so that there is very little free volume space the molecules have limited ability to move around, and they have random orientation that is, they can rotate and point in random directions. When the liquid freezes into a solid (shown lower left), the density is slightly increased to eliminate the void space, the molecules have assigned positions and are not free to move around, and there is now an orientation order that is, they cannot rotate freely and they all point at the same direction. 

Figure 4.26 Cell model of gases, liquids, and solids...

Lamont, J. C., and D. S. Scott, An eddy cell model of mass transfer into the surface of a turbulent liquid , AlChE J., 16,4, 513-519 (1970). 

Indirectly related to the cell models of this section is the work of Davis and Brenner (1981) on the rheological and shear stability properties of three-phase systems, which consist of an emulsion formed from two immiscible liquid phases (one, a discrete phase wholly dispersed in the other continuous phase) together with a third, solid, particulate phase dispersed within the interior of the discontinuous liquid phase. An elementary analysis of droplet breakup modes that arise during the shear of such three-phase systems reveals that the destabilizing presence of the solid particles may allow the technological production of smaller size emulsion droplets than could otherwise be produced (at the same shear rate). 

Figure 6 The radial distribution function for a Lennard-Jones model of liquid argon at a temperature T = 300 K. A simulation cell of 35 A containing 864 atoms with periodic boundary conditions was used. The simulation was carried out by coupling each degree of freedom to an MTK thermostat, and the equation of motion was integrated using the methods discussed in Ref. 28.

FIGURE 9.28 (a) U nit-cell model of the equiaxed dendritic growth of a crystal. The liquid within the grain envelope and within the element are shown as well as the mass fraction distribution of the species A (b) the idealized phase diagram (c) the mass fraction distribution of the species A for two different elapsed times. (From Rappaz and Thevos, Ref. 152, reproduced by permission 1987 Pergamon.)... 

In view of the failure of the rigid sphere model to yield the correct isochoric temperature coefficient of the viscosity, the investigation of other less approximate models of the liquid state becomes desirable. In particular, a study making use of the Lennard-Jones and Devonshire cell theory of liquids would be of interest because it makes use of a realistic intermolecular potential function while retaining the essential simplicity of a single particle theory. The main task is to calculate the probability density of the molecule within its cell as perturbed by the steady-state transport process. 

M. E. Orazem and J. Newman, Mathematical modeling of liquid-junction photovoltaic cells I. Governing equations, J. Electrochem. Soc. 131 (1984) 2569-2574. 

Equations of state derived from statisticai thermodynamics arise from proper con-figurationai partition functions formuiated in the spirit of moiecuiar modeis. A comprehensive review of equations of state, inciuding the historicai aspects, is provided in Chapter 6. Therefore, we touch briefly in oniy a few points. Lennard-Jones and Devonshire [1937] developed the cell model of simple liquids, Prigogine et al. [1957] generalized it to polymer fluids, and Simha and Somcynsky [1969] modified Pri-gogine s cell model, allowing for more disorder in the system by lattice imperfections or holes. Their equations of state have been compared successfully with PVT data on polymers [Rodgers, 1993]. 

Marlow and Rowell (37) working with coal/water slurries and using the CVP technique have shown that, at the frequencies of their measurements (200 kHz), the effect of particle concentration can be adequately described by introducing a factor (1 — g ) into their equivalent of Eq. (1) where again, g was very close to unity. In their review article Marlow et al. (6) discuss the way the cell model of Levine and coworkers (38, 39) is introduced into the CVP theory and show that, for thin double layers, the result is that the hydrodynamic and electrostatic interactions essentially cancel one another and one is left with only the factor (1—d)) to take account of the backflow of liquid caused by the particle motion. 

Wang ZH, Wang CY (2003) Mathematical modeling of liquid-feed direct methanol fuel cells. J Electrochem Soc 150 A508-A519... 

Sinha, P.K. and Wang, C.-Y. (2007) Pore-network modeling of liquid water transport in gas diffusion layer of a polymer electrolyte fuel cell. Bectrochim. Acta, 52, 7936 7945. 

The basic lattice models of liquid state are the quasi lattice model, the cell model, the free volume model, the hole model, the cluster model, the tunnel model, etc. The use of models in thermodynamic treatment of solutions to express deviation from ideality, such as excess thermodynamic functions, offers the advantage of compensating for the approximation involved in models, affecting to an equal extent the functions of the mixture and the single components. 

Figure 3. Different phase diagrams of water generated by changing parameters in the cell model of Stokely et al. [32]. (a) Singularity free (SF). (b) Liquid-liquid critical point (LLCP). (c) Liquid-liquid critical point at negative pressure, (d) Critical point free with reentrant stability limit (CPF/SL). HDL and LDL refer to high- and low-density liquids, respectively. The L-L Widom line is the locus of maxima in the correlation length emanating from the LLCP. Reprinted with permission from Ref. [32].

Fig. 14 Back flow cell model of BCR with arbitrary liquid inlet [36]...

Yan, T. Z., and Jen, T.-C. 2008. Two-phase flow modeling of liquid-feed direct methanol fuel cell. lui J J QlMassTmns, 51, 1192-1204. 

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