Simplified flow model

When two or more phases are present, it is rarely possible to design a reactor on a strictly first-principles basis. Rather than starting with the mass, energy, and momentum transport equations, as was done for the laminar flow systems in Chapter 8, we tend to use simplified flow models with empirical correlations for mass transfer coefficients and interfacial areas. The approach is conceptually similar to that used for friction factors and heat transfer coefficients in turbulent flow systems. It usually provides an adequate basis for design and scaleup, although extra care must be taken that the correlations are appropriate. 

After a discussion of the mobility control requirement using the simplified flow model, this section moves to a model with realistic water and oil relative permeability curves. Now the interstitial (connate) water saturation and residual oil saturation are 0.2. The endpoint relative permeabilities of oil and water are 0.85 and 0.3, respectively. The Corey exponents of relative permeabilities for oil and water are 2. Others are the same as those in the simplified model discussed earlier particularly, the initial water saturation is 0.5. Again, capillary and gravity are not included. 

The solution to the full set of conservation equations for mass, momentum and energy requires major computational resources and may become very costly. Instead, some assumptions are introduced to obtain simplified flow models. The most popular models are the generalized Hele-Shaw (GHS) model and the Barone-Caulk model. 

PRISIM embodies the IREP model of Arkansas 1. It includes extensive grapitivs of. simplified flow diagrams and relevant operating history from LERs (Licensee Event Reports required by Regulatory Guide 1.16) The plant model consists of 500 cutsets truncated by probabilities determined from normal operation. 

Horie and his coworkers [90K01] have developed a simplified mathematical model that is useful for study of the heterogeneous nature of powder mixtures. The model considers a heterogeneous mixture of voids, inert species, and reactant species in pressure equilibrium, but not in thermal equilibrium. The concept of the Horie VIR model is shown in Fig. 6.3. As shown in the figure, the temperatures in the inert and reactive species are permitted to be different and heat flow can occur from the reactive (usually hot) species to the inert species. When chemical reaction occurs the inert species acts to ther-... 

Miyauchi and Vermeulen (M7, M8) have presented a mathematical analysis of the effect upon equipment performance of axial mixing in two-phase continuous flow operations, such as absorption and extraction. Their solutions are based, in one case, upon a simplified diffusion model that assumes a mean axial dispersion coefficient and a mean flow velocity for... 

Gal-Or and Resnick (Gl) have developed a simplified theoretical model for the calculation of mass-transfer rates for a sparingly soluble gas in an agtitated gas-liquid contactor. The model is based on the average gas residencetime, and its use requires, among other things, knowledge of bubble diameter. In a related study (G2) a photographic technique for the determination of bubble flow patterns and of the relative velocity between bubbles and liquid is described. 

A first model of the calender nip flow has been presented by ArdichviUi. Further on Gaskefl presented a more precise and well-known model. Both models are very simplified, which yields that the flow is Newtonian and isothermal, and they predict that the nip force is inversely proportional with the clearance. Since mbber materials show a shear thinning behavior Ardichvilli s model seems not to be very realistic. The purpose of this section is to present a calender nip flow model based on the power law. The model is stiU being considered isothermal. Such a model was first presented by McKelvey. ... 

A simplified mathematical model was developed for the novel OCM reactor. One version of the model, presented here, describes batch operation. A second version addressing continuous flow operation will appear elsewhere [16]. 

In what follows, we present a simplified box-model that capture the essential of the chromatographic porous flow model with the only assumption that there is equilibration... 

Illustration Kinetics of dispersion the two-zone model. The models for agglomerate rupture when integrated with a flow model are useful for the modeling of dispersion in practical mixers, as was discussed for the case of drop dispersion. Manas-Zloczower, Nir, and Tadmor (1982), in an early study, presented a model for the dispersion of carbon black in rubber in a Banbury mixer (Fig. 34). The model is based on several simplifying assumptions Fragmentation is assumed to occur by rupture alone, and each rupture produces two equal-sized fragments. Rupture is assumed to occur... 

To simplify the treatment for an LFR in this chapter, we consider only isothermal, steady-state operation for cylindrical geometry, and for a simple system (A - products) at constant density. After considering uses of an LFR, we develop the material-balance (or continuity) equation for any kinetics, and then apply it to particular cases of power-law kinetics. Finally, we examine the results in relation to the segregated-flow model (SFM) developed in Chapter 13. 

The effect of temperature of the reactor surroundings on the reaction stability can be discussed via the simplified heat-flow model [171-173] shown in Figure 3.7, which is really an extension of Figure 3.1. Consumption of material is not considered in this figure. 

Chung, R M. (1969). A simplified statistical model for turbulent, chemically reacting shear flows. AIAA Journal 7, 1982-1991. 

A simplified flow chart for homology model building... 

Eq.(34) are a set of differential equations, which lead a flow in a four dimensional phase space R. This flow can be simplified to three dimensional phase space R when the dynamics of the jacket can be considered negligible respect to the reactor s dynamics. Putting dxA/dr = 0 the dimensionless jacket s temperature X4 can be eliminated from Eq.(34), and the simplified mathematical model of the reactor can be written as... 

The simplified flow analysis of Weibefs model A indicates undeveloped flow with a flat profile in the trachea for Reynolds numbers up to approximately 2,000. However, this does not consider disturbances produced by the rough walls, the eccentricity of the cross section, and the larynx. 

Other Springer model derivatives include those of Ge and Yi, ° van Bussel et al., Wohr and co-workers, and Hertwig et al. Here, the models described above are slightly modified. The model of Hertwig et al. includes both diffusive and convective transport in the membrane. It also uses a simplified two-phase flow model and shows 3-D distributions... 

In the case of Hollosep, it may be possible to employ a simplified module model because of its low concentration polarization, uniform feed water flow and low pressure drop. 

More recent experiments [62] concerning the viscous sublayer have shown a three-dimensional structure for turbulence near the wall. In a plane normal to the mean flow, counterrotating eddy pairs are involved (Fig. 6c), whereas in the direction of the mean flow, the motion is quasi-periodic (as described earlier). Since the wavelength along the mean flow is much larger than along the perimeter of the tube, a simplified bidimensional model may account only... 

In summary, all features of the liquid rocket engine combustion processes are extensively affected by injector design, and any simplified combustion model, in which the essential three-dimensional nature of the flow processes is ignored, can only be of qualitative significance. Nevertheless, these simplified models are useful in giving us some insight into the nature of the physicochemical phenomena that determine engine performance. In this connection, steady-state combustion rates and overall combustion efficiencies in propellant utilization are far less important practical problems than are control or elimination of instabilities, excessive heat transfer, and hard starts. 

Using a chemical kinetic model is one way to describe the chemistry in reacting flow modeling. The chemical kinetic model offers a comprehensive description of the chemistry, but it requires a larger computational effort than simplified chemical models. 

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