Fluid phase equilibria .259 surface

Because of the close relationship between the MNM transition and the vapor-liquid transition, it is to be expected that immiscibility in the mercury-helium system reaches up to the critical point, or even into the supercritical region. This expectation is confirmed by measurements of the phase diagram at very low helium concentrations and at pressures close to the critical pressure of pure mercury. The experiments extend up to 1610 °C and to pressures up to 3325 bar (Marceca et al., 1996). The p — T — X phase equilibrium surface obtained is qualitatively like the one shown schematically in Fig. 6.4 for a binary fiuid-fluid system of the first kind. The critical line starts at the critical point of pure mercury (Tc(l) = 1478 °C, Pc(l) = 1673 bar) and runs to higher temperatures and pressures as the helium composition X2 increases. 

Adsorption is a dynamic process in which some adsorbate molecules are transferring from the fluid phase onto the solid surface, while others are releasing from the surface back into the fluid. When the rate of these two processes becomes equal, adsorption equilibrium has been established. The equilibrium relationship between a speeific adsorbate and adsorbent is usually defined in terms of an adsorption isotherm, which expresses the amount of adsorbate adsorbed as a fimetion of the gas phase coneentration, at a eonstant temperature. 

Hougen- Watson Models for Cases where Adsorption and Desorption Processes are the Rate Limiting Steps. When surface reaction processes are very rapid, the overall conversion rate may be limited by the rate at which adsorption of reactants or desorption of products takes place. Usually only one of the many species in a reaction mixture will not be in adsorptive equilibrium. This generalization will be taken as a basis for developing the expressions for overall conversion rates that apply when adsorption or desorption processes are rate limiting. In this treatment we will assume that chemical reaction equilibrium exists between various adsorbed species on the catalyst surface, even though reaction equilibrium will not prevail in the fluid phase. 

When we derived the phase rule, we assumed that all phases are at the same pressure. In mineral systems, fluid phases can be at a pressure different from the solid phases if the rock column above them is permeable to the fluid. Under these circumstances, the system has an additional degree of freedom and the equilibrium at any depth depends on both the fluid pressure Pp and the pressure on the solid Ps at that level. Each pressure is determined by p, the density of the phase, and h, the height of the column between the surface and the level being studied. 

Some components in a gas or liquid interact with sites, termed adsorption sites, on a solid surface by virtue of van der Waals forces, electrostatic interactions, or chemical binding forces. The interaction may be selective to specific components in the fluids, depending on the characteristics of both the solid and the components, and thus the specific components are concentrated on the solid surface. It is assumed that adsorbates are reversibly adsorbed at adsorption sites with homogeneous adsorption energy, and that adsorption is under equilibrium at the fluid- adsorbent interface. Let (m" ) be the number of adsorption sites and (m 2) the number of molecules of A adsorbed at equilibrium, both per unit surface area of the adsorbent. Then, the rate of adsorption r (kmol m s ) should be proportional to the concentration of adsorbate A in the fluid phase and the number of unoccupied adsorption sites. Moreover, the rate of desorption should be proportional to the number of occupied sites per unit surface area. Here, we need not consider the effects of mass transfer, as we are discussing equilibrium conditions at the interface. At equilibrium, these two rates should balance. Thus,... 

Thus, the sorption of chemicals on the surface of the solid matrix may become important even for substances with medium or even small solid-fluid equilibrium distribution coefficients. For the case of strongly sorbing chemicals only a tiny fraction of the chemical actually remains in the fluid. As diffusion on solids is so small that it usually can be neglected, only the chemical in the fluid phase is available for diffusive transport. Thus, the diffusivity of the total (fluid and sorbed) chemical, the effective diffusivity DieS, may be several orders of magnitude smaller than diffusivity of a nonsorbing chemical. We expect that the fraction which is not directly available for diffusion increases with the chemical s affinity to the sorbed phase. Therefore, the effective diffusivity must be inversely related to the solid-fluid distribution coefficient of the chemical and to the concentration of surface sites per fluid volume. 

Initiation und Growth of Cells. The initiation or nucleation of cells is ihe formation of cells of such size that they are capable of growth under the given conditions of foam expansion. The growth of a hole or cell in a fluid medium at equilibrium is controlled hy the pressure difference (AP) between ihe inside and the outside of the cell, ihe surface tension of the fluid phase y, and Ihe radius t of flic cell ... 

There have been few studies reported in the literature in the area of multi-component adsorption and desorption rate modeling (1, 2,3., 4,5. These have generally employed simplified modeling approaches, and the model predictions have provided qualitative comparisons to the experimental data. The purpose of this study is to develop a comprehensive model for multi-component adsorption kinetics based on the following mechanistic process (1) film diffusion of each species from the fluid phase to the solid surface (2) adsorption on the surface from the solute mixture and (3) diffusion of the individual solute species into the interior of the particle. The model is general in that diffusion rates in both fluid and solid phases are considered, and no restrictions are made regarding adsorption equilibrium relationships. However, diffusional flows due to solute-solute interactions are assumed to be zero in both fluid and solid phases. 

The equilibrium surface concentrations on the solid phase qsl and qS2 are given by equation (3). To obtain the concentrations Cs and CS2 In the fluid phase, the two equations in equation (3) must be solved at each time step. This was done using the Newton-Raphson Method for solving nonlinear equations. 

Phenol and dodecyl benzene sulfonate are two solutes that have markedly different adsorption characteristics. The surface diffusion coefficient of phenol is about fourteen times greater than that for dodecyl benzene sulfonate. The equilibrium adsorption constants indicate that dodecyl benzene sulfonate has a much higher energy of adsorption than phenol (20,22). The adsorption rates from a mixture of these solutes can be predicted accurately, if (1) an adequate representation is obtained for the mixture equilibria, and (2) the diffusion rates in the solid and fluid phases are not affected by solute-solute interactions. 

In summary, we refer to Figure 5.5, which may be considered as the projection of the entire equilibrium surface on the entropy-volume plane. All of the equilibrium states of the system when it exists in the single-phase fluid state lie in the area above the curves alevd. All of the equilibrium states of the system when it exists in the single-phase solid state lie in the area bounded by the lines bs and sc. These areas are the projections of the primary surfaces. The two-phase systems are represented by the shaded areas alsb, lev, and csvd. These areas are the projections of the derived surfaces for these states. Finally, the triangular area slv represents the projection of the tangent plane at the triple point, and represents all possible states of the system at the triple point. This area also is a projection of a derived surface. 

The fluid phase that fills the voids between particles can be multiphase, such as oil-and-water or water-and-air. Molecules at the interface between the two fluids experience asymmetric time-average van der Waals forces. This results in a curved interface that tends to decrease in surface area of the interface. The pressure difference between the two fluids A/j = v, — 11,2 depends on the curvature of the interface characterized by radii r and r-2, and the surface tension, If (Table 2). In fluid-air interfaces, the vapor pressure is affected by the curvature of the air-water interface as expressed in Kelvin s equation. Curvature affects solubility in liquid-liquid interfaces. Unique force equilibrium conditions also develop near the tripartite point where the interface between the two fluids approaches the solid surface of a particle. The resulting contact angle 0 captures this interaction. 

Consider a physical system shown schematically in Figure 1. A fluid stream containing reactant A is moving upwards in plug flow with a constant velocity U. The reactant is adsorbed by a stream of solid catalytic particles falling downwards with a constant velocity V and occupying the void fraction of 1 - e. On the surface of catalyst an irreversible chemical reaction A - B is occurring and the product B is then rapidly desorbed back into the fluid phase. Instantaneous adsorption equilibrium for the species A is assumed. 

Ice point. In the Dewar flask, prepare a slushy mixture of finely shaved clean ice and distilled water. This should be fluid enough to permit the gas thermometer bulb to be lowered into place and to allow a metal ring stirrer to be operated but should have sufficient ice to maintain two-phase equilibrium over the entire surface of the bulb. Mount the bulb in place, and commence stirring the mixture, moving the stirrer slowly up and down from the very top to the very bottom of the Dewar flask. Do not force the stirrer, since the bulb and its capillary stem are fragile. After equilibrium is achieved, take the ice-point pressure readings and record p. ... 

To obtain the friction coefficient X as the transport coefficient in LRT, we would like to take a nonequilibrium steady state average of the frictional force. Thus we identify F with the phase variable B. The frictional force in our system is the force exerted by the surface on the fluid. At equilibrium, the average of this force will be zero, because there is equal likelihood of fluid particles flowing in any given direction. Thus, the first term in Eq. [210] will be zero. Under shear, the surface will exert on average a nonzero force on the fluid due to the directionality of the flow. The frictional force is given by... 

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