Fluid-phase equations

The cell theory plus fluid phase equation of state has been extensively applied by Cottin and Monson [101,108] to all types of solid-fluid phase behavior in hard-sphere mixtures. This approach seems to give the best overall quantitative agreement with the computer simulation results. Cottin and Monson [225] have also used this approach to make an analysis of the relative importance of departures from ideal solution behavior in the solid and fluid phases of hard-sphere mixtures. They showed that for size ratios between 1.0 and 0.7 the solid phase nonideality is much more important and that using the ideal solution approximation in the fluid phase does not change the calculated phase diagrams significantly. 

Theoretical approaches have been applied to binary Lennard-Jones 12-6 mixtures by Rick and Haymet, who used DFT [222], and by Cottin and Monson [109], who used their cell theory plus fluid phase equation of state approach. The latter approach gave qualitatively good agreement with experimental results for binary mixtures involving argon, krypton, and methane. Quantitative agreement with experiment was found to be sensitive... 

In the two-medium treatment of the single-phase flow and heat transfer through porous media, no local thermal equilibrium is assumed between the fluid and solid phases, but it is assumed that each phase is continuous and represented with an appropriate effective total thermal conductivity. Then the thermal coupling between the phases is approached either by the examination of the microstructure (for simple geometries) or by empiricism. When empiricism is applied, simple two-equation (or two-medium) models that contain a modeling parameter hsf (called the interfacial convective heat transfer coefficient) are used. As is shown in the following sections, only those empirical treatments that contain not only As/but also the appropriate effective thermal conductivity tensors (for both phases) and the dispersion tensor (in the fluid-phase equation) are expected to give reasonably accurate predictions. 

However, usually catalysts are porous materials to strongly increase the active reaction area per unit of reactor volume. Therefore, the intra-particle gradients, i.e., the mass and heat transfer inside the catalyst particle, have to be considered as well. Fluid phase equations are the same as Eqs. 4.10, 4.11, while solid phase equations have to be modified as follows ... 

In other words, free energy is related to pressure by a logarithmic function. If there are several components, we have to use partial pressure. Furthermore, partial pressure is proportional to the concentration of the substance in the fluid phase. Equation 5.79 may therefore be used for the fluid phase if we replace the pressure with concentration in Equation 5.79. 

If the first two inequalities hold /8,9/, it can be expected that heat and mass transport outside the catalyst particle is not limiting and therefore there is no need for a heterogeneous model a pseudo-homogeneous model may be sufficient, in which the whole catalyst particle is regarded as a sink and/or source for heat and mass in the fluid phase. This results in a drastic reduction of the number of model equations, and the transfer terms in the fluid phase equations are lumped into effective reaction rate terms. If the second inequality holds /9/, pore diffusion is not limiting. The third inequality is valid if axial dispersion can be neclected /lO/ and the fourth if radial dispersion is of no importance /lO/. 

Predictions of the single-phase particle bed model were confronted with experimental observations of gas-fluidized beds in Chapter 9. The assumption of pp pf, which enabled the fluid-phase equations to be effectively removed from consideration in this case, would appear to render this approximation inappropriate for most cases of liquid fluidization. The above results, however, show that the single-phase approximation leads to stability predictions in reasonable harmony with the full two-phase model for liquid fluidization over a substantial range of particle density, down to perhaps three times that of the fluid. In this section we confront reported experimental observations relating to the stability of... 

The CS pressures are close to the machine calculations in the fluid phase, and are bracketed by the pressures from the virial and compressibility equations using the PY approximation. Computer simulations show a fluid-solid phase transition tiiat is not reproduced by any of these equations of state. The theory has been extended to mixtures of hard spheres with additive diameters by Lebowitz [35], Lebowitz and Rowlinson [35], and Baxter [36]. 

Figure A2.5.26. Molar heat capacity C y of a van der Waals fluid as a fimction of temperature from mean-field theory (dotted line) from crossover theory (frill curve). Reproduced from [29] Kostrowicka Wyczalkowska A, Anisimov M A and Sengers J V 1999 Global crossover equation of state of a van der Waals fluid Fluid Phase Equilibria 158-160 532, figure 4, by pennission of Elsevier Science.

Kostrowicka Wyczalkowska A, Anisimov M A and Sengers J V 1999 Global crossover equation of state of a van der Waals fluid Fluid Phase Equilibria 158-160 523-35... 

The fugacity coefficient of thesolid solute dissolved in the fluid phase (0 ) has been obtained using cubic equations of state (52) and statistical mechanical perturbation theory (53). The enhancement factor, E, shown as the quantity ia brackets ia equation 2, is defined as the real solubiUty divided by the solubihty ia an ideal gas. The solubiUty ia an ideal gas is simply the vapor pressure of the sohd over the pressure. Enhancement factors of 10 are common for supercritical systems. Notable exceptions such as the squalane—carbon dioxide system may have enhancement factors greater than 10. Solubihty data can be reduced to a simple form by plotting the logarithm of the enhancement factor vs density, resulting ia a fairly linear relationship (52). 

As a reactant molecule from the fluid phase surrounding the particle enters the pore stmcture, it can either react on the surface or continue diffusing toward the center of the particle. A quantitative model of the process is developed by writing a differential equation for the conservation of mass of the reactant diffusing into the particle. At steady state, the rate of diffusion of the reactant into a shell of infinitesimal thickness minus the rate of diffusion out of the shell is equal to the rate of consumption of the reactant in the shell by chemical reaction. Solving the equation leads to a result that shows how the rate of the catalytic reaction is influenced by the interplay of the transport, which is characterized by the effective diffusion coefficient of the reactant in the pores, and the reaction, which is characterized by the first-order reaction rate constant. 

Erinciple) represent both the vapor- and liqmd-phase volumetric ehavior of pure fluids are equations cubic in molar volume. All such expressions are encompasseci by the generic equation... 

Axial Dispersion Effects In adsorption bed calculations, axial dispersion effects are typically accounted for by the axial diffusionhke term in the bed conservation equations [Eqs. (16-51) and (16-52)]. For nearly linear isotherms (0.5 < R < 1.5), the combined effects of axial dispersion and mass-transfer resistances on the adsorption behavior of packed beds can be expressed approximately in terms of an apparent rate coefficient for use with a fluid-phase driving force (column 1, Table 16-12) ... 

In the SMB operation, the countercurrent motion of fluid and solid is simulated with a discrete jump of injection and collection points in the same direction of the fluid phase. The SMB system is then a set of identical fixed-bed columns, connected in series. The transient SMB model equations are summarized below, with initial and boundary conditions, and the necessary mass balances at the nodes between each column. 

In certain circumstances even the parabolic rate law may be observed under conditions in which the oxide is porous and permeated by the oxidising environment". In these cases it has been shown that it is diffusion of one or other of the reactants through the fluid phase which is rate controlling. More usually however the porous oxide is thought to grow on the surface of a lower oxide which is itself growing at a parabolic rate. The overall rate of growth is then said to be paralinear - and may be described by the sum of linear and parabolic relationships (see equations 1.197 and 1.198). 

The fundamental idea of this procedure is as follows For a system of two fluid phases containing N components, we are concerned with N — 1 independent mole fractions in each phase, as well as with two other intensive variables, temperature T and total pressure P. Let us suppose that the two phases (vapor and liquid) are at equilibrium, and that we are given the total pressure P and the mole fractions of the liquid phase x, x2,. .., xN. We wish to find the equilibrium temperature T and the mole fractions of the vapor phase yu y2,. .., yN-i- The total number of unknowns is N + 2 there are N — 1 unknown mole fractions, one unknown temperature, and two unknown densities corresponding to the two limits of integration in Eq. (6), one for the liquid phase and the other for the vapor phase. To solve for these N +2 unknowns, we require N + 2 equations of equilibrium. For each component i we have an equation of the form... 

Sublimation, dissolution, and ablative processes in general can be fit using H = 0 in Equation (11.48). The actual reaction, if there is one, occurs in the fluid phase with kinetics independent of Equation (11.48). 

Equations (11.54) and (11.55) apply to any distribution of particle residence times provided the linear consumption rate is constant. They do not require that the fluid phase is perfectly mixed, only that the consumption rate is strictly controlled by the surface reaction. For the special case of... 

The flow velocity, pressure and dynamic viscosity are denoted u, p and fj and the symbol (...) represents an average over the fluid phase. Kim et al. used an extended Darcy equation to model the flow distribution in a micro channel cooling device [118]. In general, the permeability K has to be regarded as a tensor quantity accounting for the anisotropy of the medium. Furthermore, the description can be generalized to include heat transfer effects in porous media. More details on transport processes in porous media will be presented in Section 2.9. 

The governing equation for the transport of heat contains convective and diffusive contributions. Inside the fluid phase, convective transport often dominates. Within... 

Among interface capturing methods, one of the most popular and most successful schemes is the volume-of-fluid (VOF) method dating back to the work of Hirt and Nichols [174]. The VOF method is based on a volume-fraction field c, assuming values between 0 and 1. A value of c = 1 indicates cells that are filled with phase 1, and phase 2 corresponds to c= 0. Intermediate values of c indicate the position of the interface between the phases however, the goal is to maintain a sharp interface in order to identify the different fluid phases uniquely. Volumes assigned to the different phases are moving with the local flow velocity W , and therefore the evolution of c is determined by a convection equation ... 

As will be shown later the equation above is identical to the mass balance equation for a continuous stirred-tank reactor. The recycle can be provided either by an external pump as shown in Fig. 5.4-18 or by an impeller installed within the reaction chamber. The latter design was proposed by Weychert and Trela (1968). A commercial and advantageously modified version of such a reactor has been developed by Berty (1974, 1979), see Fig. 5.4-19. In these reactors, the relative velocity between the catalyst particles and the fluid phases is incretised without increasing the overall feed and outlet flow rates. 

Equation (A9) accounts for the change in the amount of daughter within the solid while Equation (A 10) accounts for that in the fluid. The transfer of the nuclide from the solid to the fluid phase is governed by the first term on the right side of both equations. Note that in this formulation, as given in Spiegelman and Elliott (1993), the time of melt transport is accounted for and depends on the physically based transport velocity. Like Equation (A8), both equations can be solved simultaneously by numerical integration or more sophisticated numerical method. 

Thermodynamic models are widely used for the calculation of equilibrium and thermophysical properties of fluid mixtures. Two types of such models will be examined cubic equations of state and activity coefficient models. In this chapter cubic equations of state models are used. Volumetric equations of state (EoS) are employed for the calculation of fluid phase equilibrium and thermophysical properties required in the design of processes involving non-ideal fluid mixtures in the oil and gas and chemical industries. It is well known that the introduction of empirical parameters in equation of state mixing rules enhances the ability of a given EoS as a tool for process design although the number of interaction parameters should be as small as possible. In general, the phase equilibrium calculations with an EoS are very sensitive to the values of the binary interaction parameters. 

It is well known that cubic equations of state have inherent limitations in describing accurately the fluid phase behavior. Thus our objective is often restricted to the determination of a set of interaction parameters that will yield an "acceptable fit" of the binary VLE data. The following implicit least squares objective function is suitable for this purpose... 

There are numerous sources of phase equilibrium data available that serve as a database to those developing or improving equations of state. References to these databases are widely available. In addition, new data are added mainly through the Journal of Chemical and Engineering Data and the journal Fluid Phase Equilibria. Next, we give data for two systems so that the reader may practice the estimation methods discussed in this chapter. 

Englezos, P., N. Kalogerakis and P.R. Bishnoi, "Estimation of Binary Interaction Parameters for Equations of State Subject to Liquid Phase Stability Requirements", Fluid Phase Equilibria, 53, 81-88, (1989). 

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