Models for the Pressure Drop

As mentioned, some of the models for cyclone pressure drop are based on a consideration of the dissipative loss in the cyclone while many are purely empirical. 

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The model for the pressure drop analysis is depicted in Fig. 9 in which multiple microchannels with an abrupt entrance and exit are shown. The equations to be presented in this section apply to this model and are not restricted to a particular geometry of the microchannel cross section. 

The consideration of the pressure drop over the monoliths containing a variety of CPSI (cells per in ) for the modeling of honeycomb reactor may be required, since Ap of the reactor strongly depends on CPSI of monolith. Eqn. (7) for the pressure drop of the honeycomb was employed to develop the reactor model describing the performance of the honeycomb fabricated in the present work [8]. and Ke indicate contraction and expansion loss coefficient at the honeycomb inlet and outlet, respectively and o is the ratio of free flow area to frontal area. 

To account for the differences in machine configuration and to better explain the pressure-drop observations (e.g., lower pressure drop with onset of liquid flow), a model based on conservation of mass and momentum, in particular gas angular momentum, was developed (40). This model divided the pressure drop into four increments that included the gas inlet to the machine housing, the rotor, the eye of the rotor, and the gas exit nozzle from the machine. 

Otherwise the model must be modified. For example, eqn. (119) can be supplemented by the equations accounting for the pressure drop of the system. 

Several models are used for the pressure drop and fluid flow correlation, but the most commonly used for pipes and fittings is simply ... 

Recommendations The reader should consult Refs. 11 and 45 for the pressure drop calculations. The Hutton and Leung model should be verified with additional experimental data obtained with small particles and hydrocarbon systems. 

The model for total pressure drop will be discussed in a later section however, for a comparison of two different cell configurations we need to write the expression for the pressure drop due to gas flow through open channels, namely. 

In addition to the above, the model for the catalyst bed module should include equations for the pressure drop across the bed. Although these equations are reasonably simple and their solution as a part of the model equation is a straightforward exercise, they are of crucial importance for without appreciation of the pressure drop consideration fast increases in reactor productivity can be theoretically estimated using the model with very fine particles. Obviously, this is not practically possible because of the excessive pressure drop associated with fine particles. In fact, it is the excessive pressure drop associated with small catalyst pellets that necessitates the use of relatively large catalyst particles in fixed beds. The use of these relatively large particles in turn, is the reason behind the existence of diffusional resistances and thus all the complexities associated with reliable modelling of industrial fixed bed catalytic reactors. 

The authors modeled the microchannel heat sink, subject to an impinging jet, as a porous medium. Based on their experimental results, they suggested correlatiOTis for the pressure drop across a microchannel heat sink subject to an impinging jet as well as its thermal resistance as follows ... 

Using Q = 2 Qy + Q ) and by deriving and substituting as above with Model III, one obtains the differential equation for the pressure drop ... 

It is seen that in order to obtain the equations for the one-dimensional model, the two radial dispersion terms must be removed. This results in the proper mass balance equation, whereas the flux boundary condition at the tube wall must be included in the heat balance equation to obtain the conventional one-dimensional model. The equation for the pressure drop is in both cases an ordinary first-order differential equation which will be discussed in Section 3.3.4. 

Fig. 20.10 Examples for the pressure drop of wall-flow particulate filter as function of the soot load. Lab pressure drop on 2" samples with Printex U soot. Filter with Fe-ZSM-5 model coating...

A general correlation for the pressure drop through fixed beds of spherical particles, based on a discrete particle model corrected for particle interaction, was proposed by Barnea and Mednick (1978). They extended the standard Cp versus Re curve for single spheres to multiparticle systems by incorporating proper functions of the volumetric particle concentration. The modified drag coefficient and Reynolds number they suggested are... 

Analogous to the bubble point pressure, the reseal pressure can be defined from a simplification of the general 3D YT.E for the pressure drop across a curved L/V interface embedded within the 3D space of the mesh. Consider the L/V interface formed within the LAD mesh screen as shown in Eigures 3.16 and 3.17. Retaining assumptions 1-4 from the bubble point model in Section 3.2.2, the following additional assumptions are required to solve the reseal pressure ... 

A study of pressure drop in right angles and elliptical bends by Morikawa et al. (1978) suggests a form for the pressure drop similar to Schuchart s, This model does not predict other investigators data with much accuracy, although Mason and Smith s (1973) data on fine particles are more closely predicted than by other correlations. 

A similar equation can be set up for the pressure drop. The combined model now contains K+1 coupled parabolic partial differential equations and one ordinary first order differential equation. They are solved by discretization in the radial direction by use of the orthogonal collocation method, and integration of the resulting set of coupled first order differential equations by use of a semi-implicit Runge-Kutta method. With this model and the used solution method, one can now concentrate on the effective transport properties given in PeH, Pcm and the wall heat transfer coefficient, with the latter being the most important parameter for design. 

Spencer-Smith, J. L. The physical Basis of Clothing Comfort, Part 3 Water Vapor Transfer Through Dry Clothing Assembhes, Clothing Res. J. 1977,5, 82-100. Spilman, L. Goren, S. Model for Predicting Pressure Drop and Filtration Efficiency in Fibrous Media, Environ. Sci. Techno. 1968, 2, TV -2%1. 

Models for cyclone pressure drop sometimes spring from models for the flow pattern and are based on an estimation of the actual dissipative losses in the cyclone others are purely empirical. 

Let us predict the separation efficiency of the cyclone reported in Ap>-pendix 4. A. As for the pressure drop, we have different models at our disposal and this is advantageous in that we are not dependent upon just one model for our predictions. If, for the case of X50, various model predictions were to differ by, say, 50%, we would want to allow for this variation and interpret the results accordingly—not becoming overly attached to any one model s predictions. 

As we found for separation performance described above, i e-similarity is not critical for the pressure drop, either. In Chap. 4 we found that many of the empirical models for cyclone pressure drop only contain the ratio of inlet to outlet areas, implying that Eu will be the same between geometrically similar cyclones, irrespective of f e-similarity. Obviously, as was the case for separation efficiency, this is only valid when Re is high enough that the friction factor is essentially independent of Re. This should come as no real surprise since the same situation holds true for most flow devices (such as pipes, elbows, orifices, contractions and expansions, etc.) that operate in fully developed turbulent flow. In such cases, pressure loss can be characterized by the formula ... 

Solving for the pressure drop over the industrial prototype, we obtain a value of 1400 Pa. We note that, even though the gas density was lower in the commercial unit, the overall pressure drop for the commercial unit increased, relative to the model, due the increase in inlet velocity. 

For the pressure drop they compared not only the models of Smolik and Muschelknautz with their experimental results, but also the empirical models of Briggs (1946) ... 

Fig. 9.B.2. The experimental results for the pressure drop in the pressurized fluidized bed combustion cyclone of Gil, Romeo and Cortes plotted together with the model predictions of Smolik, Briggs and Baskakov, respectively...

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