Model homogeneous

The simplest way to compute line friction in two-phase flow is to 0.01 3.40 1.10 

The case of Example 6.12 will be solved with a van der Waals equation of steam. From the CRC Handbook of Chemistry and Physics (CRC Press, Boca Raton, FL, 1979), 

The integration is performed with Simpson s rule with 20 intervals. Values of V2 are assumed until one is found that makes 0=0. Then the pressure is found from the v dW equation  

Two trials and, a linear interpolation are shown. The value P2 = 18.44 bar compares with the ideal gas 17.13. 

The simplest way to compute line friction in two-phase flow is to adopt some kinds of mean properties of the mixtures and to employ the single phase friction eqnation. The main problem is the assignment of a two-phase viscosity. Of the number of definitions that have been proposed, that of McAdams et al. [Trans. ASME 

Dukler proposed a correlation based on the assumption that the gas and liquid flow at the same velocity (no slip) and that the properties of the fluid can be suitably averaged. 

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A fundamental difference exists between the assumptions of the homogeneous and porous membrane models. For the homogeneous models, it is assumed that the membrane is nonporous, that is, transport takes place between the interstitial spaces of the polymer chains or polymer nodules, usually by diffusion. For the porous models, it is assumed that transport takes place through pores that mn the length of the membrane barrier layer. As a result, transport can occur by both diffusion and convection through the pores. Whereas both conceptual models have had some success in predicting RO separations, the question of whether an RO membrane is truly homogeneous, ie, has no pores, or is porous, is still a point of debate. No available technique can definitively answer this question. Two models, one nonporous and diffusion-based, the other pore-based, are discussed herein. 

Kawahara et al. (2002) presented void fraction data obtained in a 100 pm micro-channel connected to a reducing inlet section and T-junction section. The superficial velocities are Uqs = 0.1-60m/s for gas, and fAs = 0.02-4 m/s for liquid. The void fraction data obtained with a T-junction inlet showed a linear relationship between the void fraction and volumetric quality, in agreement with the homogeneous model predictions. On the contrary, the void fraction data from the reducing section inlet experiments showed a non-linear void fraction-to-volumetric quality relationship ... 

This complex and structurally related molecules served as a functional homogeneous model system for commercially used heterogeneous catalysts based on chromium (e.g. Cp2Cr on silica - Union Carbide catalyst). The kinetics of the polymerization have been studied to elucidate mechanistic features of the catalysis and in order to characterize the potential energy surface of the catalytic reaction. 

Excel-auto homogenizer Model EX-10 equipped with a 250-mL centrifuge tube (Nihon Seiki Seisakusyo Co., Japan)... 

Temperature and concentration differences between gas and catalyst can be neglected to give a pseudo-homogeneous model,... 

Estimate the two-phase pressure drop though the tubes, due to friction. Use the homogenous model or another simple method, such as the Lochart-Martenelli equation see Volume 1, Chapter 5. 

At tube exit, pressure drop per unit lengths, using the homogeneous model homogeneous velocity = G/pm = 237/66.7 = 3.55 m/s Viscosity, taken as that of liquid, = 0.12 mN sm 2... 

The homogeneous model treats the mixture as a whole, and consequently the physical properties are represented by the average value of the mixture. This treatment assumes that the gas and liquid phases possess the same velocity (or the slip velocity is neglected). This model was used extensively in the past, because of its simplic-... 

In reality, the slip velocity may not be neglected (except perhaps in a microgravity environment). A drift flux model has therefore been introduced (Zuber and Findlay, 1965) which is an improvement of the homogeneous model. In the drift flux model for one-dimensional two-phase flow, equations of continuity, momentum, and energy are written for the mixture (in three equations). In addition, another continuity equation for one phase is also written, usually for the gas phase. To allow a slip velocity to take place between the two phases, a drift velocity, uGJ, or a diffusion velocity, uGM (gas velocity relative to the velocity of center of mass), is defined as... 

The model is considered a generalized homogeneous model (Boure, 1976), and it seems to be well suited to system codes describing complex systems, such as nuclear reactors, which cannot take into account a detailed separate-phase flow model because of their complexity. 

A steady homogeneous model is often used for bubbly flow. As mentioned previously, the two phases are assumed to have the same velocity and a homogeneous mixture to possess average properties. The basic equations for a steady one-dimensional flow are as follows. 

Using a homogeneous model proposed by Owens (1961) for low void fractions (a < 0.30) and high mass flux, as is usually encountered in a water-cooled reactor, the momentum change (or acceleration) pressure gradient term is obtained from... 

In a horizontal flow with a homogeneous model, UG — UL, thus... 

For cases of nonuniform velocity distribution, Kays and London (1958) suggested using momentum correction and energy correction factors in the above equations. However, these factors are very difficult to evaluate, so the homogeneous model is used here. 

One can view a quiescent molecular cloud as a one-dimensional PDR with % = 1. Here, instead of spherical shells representing outer and inner layers, one has one-dimensional slabs. The advantage of such shell models65 over homogeneous models of the inner portions of clouds is that the roles of the outer layers can be accounted for such roles are especially important for atoms (e.g. C) and radicals (e.g. OH, CH). Small dense clouds, known as translucent clouds, have particularly salient outer portions which should be included in models. 

The units of rv are moles converted/(volume-time), and rv is identical with the rates employed in homogeneous reactor design. Consequently, the design equations developed earlier for homogeneous reactors can be employed in these terms to obtain estimates of fixed bed reactor performance. Two-dimensional, pseudo homogeneous models can also be developed to allow for radial dispersion of mass and energy. 

This equation may be used as an appropriate form of the law of energy conservation in various pseudo homogeneous models of fixed bed reactors. Radial transport by effective thermal conduction is an essential element of two-dimensional reactor models but, for one-dimensional models, the last term must be replaced by one involving heat losses to the walls. 

Pseudo homogeneous models of fixed bed reactors are widely employed in reactor design calculations. Such models assume that the fluid within the volume element associated with a single catalyst pellet or group of pellets can be characterized by a given bulk temperature, pressure, and composition and that these quantities vary continuously with position in the reactor. In most industrial scale equipment, the reactor volume is so large compared to the volume of an individual pellet and the fraction of the void volume associated therewith that the assumption of continuity is reasonable. 

Pseudo homogeneous models require global rate expressions of the type discussed in Section... 

The term pseudo homogeneous model has been popularized by Froment (94, 95). 

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