Homogeneity assumption

The brackets symbolize fiinction of, not multiplication.) Smce there are only two parameters, and a, in this expression, the homogeneity assumption means that all four exponents a, p, y and S must be fiinctions of these two hence the inequalities in section A2.5.4.5(e) must be equalities. Equations for the various other thennodynamic quantities, in particular the singidar part of the heat capacity Cy and the isothemial compressibility Kp may be derived from this equation for p. The behaviour of these quantities as tire critical point is approached can be satisfied only if... 

The pseudo-homogeneous assumption means that both the solid and fluid phases are are considered a single phase. Therefore, we avoid considering mass and heat transfer from and to the catalytic pellets. This model assumes that the conqionent concentrations and the temperature in the pellets are the same as those in the fluid phase. This assumption is approximated when the catalyst pellet is small and mass and heat transfer between the pellets and the fluid phase are rapid. The reaction rate for this model, called the global reaction rate, includes heat and mass transfer. If heat and mass transfer are made insignificant, then the reaction rate is called the intrinsic reaction rate. 

The concept of viscosity has been introduced and developed with homogeneous fluids, Newtonian or not. In case of an emulsion some phenomena can take place at the scale of a drop size, and the homogeneity assumption is no longer valid. For instance, it is known that in most cases of fluid transport in the presence of surfactant, there is some slipping velocity resulting from the electrical double-layer effect called the streaming potential (86). 

The pseudo-homogeneous assumption reduces the complexity of the model implementation and resolution, but of course this model category gives less accurate results with respect to heterogeneous one. 

The analysis here is mainly inspired by the papers of Leyvraz (1986) and van Dongen and Ernst (1988), referenced in Chapter 5. The adaptation of the arguments of van Dongen and Ernst is readily made to the case in which the homogeneity assumption is replaced by condition (6.2.5) required for self-similarity and has been accomplished by Wright and Ramkrishna (1992). 

A tentative answer to this question may be sought from characteristic parameters, namely the reaction time tR an intraphase mass transfer time tj) = 1 / Z) or 16/<0 where 1 = 1/a is the reciprocal of the specific interfacial area of the dispersion, the diffusivity and 6 the equivalent diffu-sional film thickness the heat transfer time tj = 1 /a or 16/a where a is the heat diffusivity the adiabatic temperature rise J = (- AH)Cq/pCp. From these parameters the following criteria may be tentatively proposed, Local concentration and temperature gradients are negligible and the pseudo-homogeneous assumption is valid if... 

ETC is an important parameter describing the thermal behavior of packed beds with a stagnant or dynamic fluid and has been extensively investigated experimentally and theoretically in the past. Various mathematical models, including continumn models and microscopic models, have been proposed to help solve this problem, but they are often limited by the homogeneity assumption in a continuum model (Wakao and Kaguei, 1982 Zehner and Schliinder, 1970) or the simple assumptions in a microscopic model... 

Next, in order to draw results that do not depend either on M-the number of structure variables nor on iV-the number of chemicals/molecules involved in a custom QSAR study, one assumes dealing with the same dispersion of the observed activity as well as for each descriptor (the so-called homogeneous assumption, 0 = 0 = U likely to be valid when... 

In pseudo-homogeneous models, the process gas and catalyst are assumed to be at the same temperature and to be almost in contact. The pseudo-homogeneous assumption simplifies mass-transfer modelling, since external and internal diffusion are not considered explicitly. An effectiveness factor is applied to reaction rates to model the lower concentration of reactants at the catalyst sites. Since the process gas and catalyst are assumed to be at the same temperature, an overall heat-transfer coefficient can be used to describe heat transfer from the inner tube wall to the catalyst and process gas. 

In Table 12.3 some fixed-bed reactor models for the MSR reaction are reported. As shown, several one-dimensional fixed-bed reactor models make pseudo-homogeneous assumptions (Murty and Murthy, 1988 Singh and Saraf, 1979 Yu et al., 2006) and other heterogeneous assumptions (Plehiers... 

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