Liquid-Phase Models

There is no significant difference between the two extremes of condensate mixing (i.e., 

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A liquid-phase model for the excess Gibbs energy provides... 

P(I) cols 1-50 initial parameters for 1=1,LL LL is determined by the liquid-phase model used. 

NUMBER OF PARAMETEPSt DETERMINED Y CHOICE OF LIQUID-PHASE MODEL... 

INDICATOR FOR THE TYPE OF LIQUID-PHASE MODEL TO EE USED... 

Liquid-Phase Models. Theoretical models of the liquid state are not as well established as those for gases consequently, the development of general equations for the description of liquid-phase equilibrium behavior is not far advanced. Cubic equations of state give a qualitative description of liquid-phase equilibrium behavior, but do not generally yield good quantitative results (3). For engineering calculations, equations and estimation techniques developed specifically for liquids must normally be used. 

So far, the atomistic modeling on oxidation of CO and methanol has been aimed to elucidate mechanisms for (1) the bifunctional effect, in which the unique catalytic properties of each of the elements in the alloy combine in a synergetic fashion to yield a more active surface and (2) the ligand or electronic effect, in which the interaction between dissimilar atoms yield alters electronic states and hence results in a more active catalytic surface. In parallel to the study on the OER, study of oxidation of CO and methanol has seen a progress from vapor phase models to liquid phase models. However, polymer cluster has not been involved in the ab initio models. 

For low pressures (a few atmospheres and lower) we can apply the ideal gas model for gases and ideal mixture models for liquids. This formulation is very common in reactor technology. In some cases at higher pressures, the pressure effect on the gas phase is important. A suitable model for these systems is to use an EOS for the gas phase, and an ideal mixture model for liquids. However, in most situations at low pressures the liquid phase is more non-ideal than the gas phases. Then we will rather apply the ideal gas law for the gas phase, and excess properties for liquid mixtures. For polar mixtures at low to moderate pressures we may apply a suitable EOS for gas phases, and excess properties for liquid mixtures. All common models for excess properties are independent of pressure, and cannot be used at higher pressures. The pressure effect on the ideal (model part of the) mixture can be taken into account by the well known Poynting factor. At very high pressures we may apply proper EOS formulations for both gas and liquid mixtures, as the EOS formulations in principle are valid for all pressures. For non-volatile electrol3d es, we have to apply a suitable EOS for gas phases and excess properties for liquid mixtures. For such liquid systems a separate term is often added in the basic model to account for the effects of ions. For very dilute solutions the Debye-Htickel law may hold. For many electrolyte systems we can apply the ideal gas law for the gas phase, as the accuracy reflected by the liquid phase models is low. 

This presentation is limited to two models, the Murphree plate efficiency and the modified Murphree plate efficiency, and their applications to columns in the service of separating both binary and multicomponent mixtures. For convenience of application, these efficiencies are restated in terms of the vaporization plate efficiency. Definitions of the vaporization point and plate efficiency follow immediately and the Murphree plate efficiencies are defined in a subsequent section as they arise in the development of the perfectly mixed liquid phase model. 

Modified Murphree Plate and Point Efficiencies for the Perfectly Mixed Liquid Phase Model... 

In Table 13-5 the product distributions obtained experimentally are compared with those found by use of perfect plates and by use of the perfectly mixed liquid phase model [Eq. (13-51)]. The experimental results show that the plate behavior of the actual column was better than perfect plates. The observed behavior can be explained on the basis of the existence of a concentration gradient in the direction of flow of the liquid across each plate which was neglected in the perfect plate and perfectly mixed liquid phase models. 

For Vb constant but >) = 1, the general equations yield, for the gas phase, equation (8-183) again, and for variable bubble volume equations (8-176) to (8-178) again. Substituting into the liquid-phase model, (8-184), gives... 

The liquid phase model proposed below considers the mass transfer inside the droplet and the changes in liquid phase properties due to the temperature and composition changes. In derivation of the following equations, it has been assumed that liquid circulation is absent, the droplet surface is at local thermodynamic equilibrium state, momentum, energy and mass transfer are spherically symmetric within the droplet, and the two liquids (fuel and water) are immiscible. With these assumptions, the conservation equations for the total mass, mass of water, and the energy equation are written as follows [14] ... 

We further refer to a model as a gas phase model if the electrostatics corresponds to the moments of the monomer, or as a liquid phase model if the electrostatics results in a permanent dipole moment of a molecule that is significantly greater than the monomer moment. Interaction centers other than the oxygen and hydrogen sites are introduced in several models. The most common site, called the M site in this chapter, is located on the bisector of intramolecular H-O-H angle at a distance 5 from the oxygen site toward the hydrogens. 

Since the mass transfer process involves two or more phases (see Sect. 3.6), the interacted liquid-phase model is convenient for the process simulation by CMT. In applying this model, all parameters involved, such as /, w,p, fe, s, p, a, t, T, T, fex Sx C, C, D, A, fee, c in the model equation are denoted to liquid phase. 

The liquid-phase-gas-phase-interaction model (interacted liquid-phase model) accompanied with c — Sc model as described in preceding sections were used to predict the concentration distribution and compared with the experimental data as shown in Fig. 3.5. 

The number of equations needed for two-fluid model is more than that of the following interacted liquid-phase model and requires more computer capacity with the risk of harder convergence. In practice, for instance, the distillation simulation by some authors [28, 29] neglected the turbulent equations of vapor phase to simplify the simulation. 

Liquid phase under interaction of gas phase (interacted liquid phase) modeling form... 

In this modeling form, abbreviated as interacted liquid phase model, the liquid phase is considered as the system to be concerned aiming to obtain the transport information of the liquid phase. The dispersed phase is considered as the sm-roundings. The action of the dispersed phase (bubbles) on the liquid phase is treated as the external forces acting on the system (liquid phase). Thus, the evaluation of source term Su in Navier—Stokes equation of liquid phase should cover all the acting forces by the dispersed gas phase to the liquid phase. Such model can reduce the number of model equations and computer load. Computation shows that whether the interaction source term Su is properly considered, the final simulated result is substantially equal to that using two-fluid model (Fig. 3.7). 

For the detailed expression of the gas-liquid interacting forces in the interacted liquid-phase model, the soiuce term Su, involving gravitational force and interacting forces as given by Wang, is shown below ... 

Fig. 3.8 Comparison between simulations using two-fluid model and interacted liquid-phase model for a sieve tray Black diamond Experimental data [30], Dashed line Two-phase model simulation by Gesit [28], Solid line Interacted liquid-phase model simulation by Wang) [31], a upstream profile for 2l = 6.94 x 10 m /s and Fs = 1.015, b downstream profile for...

In our practice, the application of interacted liquid-phase model is successful in simulating liquid-gas (vapor) two-phase processes, such as distillation, absorption, and adsorption, as given in subsequent chapters. 

Two-phase model Interacted liquid phase Model... 

Sun et al. [4—6] and Li et al. [7] simulated this column using interacted liquid-phase modeling form with the assumption that the liquid density on a simulated single tray is constant, but for the multitray simulation, the density should be changed tray by tray. 

The interacted liquid-phase modeling form is employed for present simulation. The simplified assumptions of constant liquid fraction jSl and density Pl on a tray are applied. 

The model equation for packed column comprises CFD equation set and mass transfer equation set. UnUke the tray column, the porosity of packed column is non-uniformly distributed, and the liquid fraction should be retained in the model equations. The interacted liquid-phase model equations are as follows Overall mass conservation... 

Liquid phase models Assuming only liquid phase flow... 

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