Model for pressure drop

Rao, V. G., and Drinkenburg, A. A. H., A model for pressure drop in two-phase gas-liquid down-flow through packed columns. AIChEJ. 31, 1010-1018 (1985). 

R.A. Holub, M.P. Dudukovic and P.A. Ramachandran, A phenomenological model for pressure drop, liquid hold-up and flow regimes transition in gas-liquid trickle flow, Chem. Engng. Science, 47 (1992) 2343-2348. 

Although the models for pressure drop have a basis in theory, all are fit empirically to data generated from specific equipment. Application of these models may not be relevant to machines of different configurations and packing types (9). 

Figure 7 Flow model for pressure drop calculations, (a) Entry and exit losses (b) pressure drop through clean channel (c) pressure drop through cell wall (d) pressure drop through sooted channel. (Courtesy of Society of the Automotive Engineers.)...

A schematic of the flow pattern used to represent annular flow is shown in Figure 10. A preliminary model for pressure drops in the annular flow regime for the circular tubes under consideration here was reported in Garimella et al. [27], followed by the more detailed model [28] described below. For the development of this model the following assumptions were made steady flow, equal pressure gradients in the liquid and gas core at any cross section, uniform thickness of the liquid film and no entrainment of the liquid in the gas core. The measured pressure drops were used to compute the Darcy form of the interfacial friction factor to represent the interfacial shear stress as follows ... 

Development of additional models for pressure drop in noncircular channels, and for heat transfer coefficients and transition criteria based on nondimensional parameters is underway. This integrated approach using flow visualization, pressure drop and heat transfer measurements, and analytical modeling, is yielding a comprehensive understanding of condensation in microchannels. 

Pinna, D. Tronconi, E. Tagliabue, L. High interaction regime Lockhart-Martinelli model for pressure drop in trickle-bed reactors. Am. Inst. Chem. Eng. J. 2001, 47, 19. 

Summary of various models for pressure drops tlirough a porous packed bed... 

Pressure-Driven Single-Phase Liquid Flows, Fig. 9 Model for pressure drop analysis of multiple-microchannel systems... 

Attou A, Ferschneider G. A simple model for pressure drop and liquid hold-up in packed-bed bubble reactors. Chem. Eng. Sci. 1999 54 139. 

There are several approaches to developing predictive models for pressure drop and energy losses in dilute-phase pneumatic conveying. The German literature, for instance, uses an expression proposed by Barth (1954) that includes a lift term in the horizontal analysis, while the U.S. approach is prone to lump this effect into the frictional term. 

Obviously the predictions of the models differ considerably. While the models of Stairmand and Shepherd and Lapple agree reasonably well, the Casal/Mar-tinez and Barth models differ by almost a factor of two. In the following chapter we shall put the models for pressure drop to a test we shall compare their predictions of the effect of cyclone length on pressure drop with experiment. 

For pressure drop and holdup in inclined pipe with upward or downward flow, see Beggs and Brill ]. Pet. Technol, 25, 607-617 [1973]) the mechanistic model methods referenced above may also be apphed to inchned pipes. Up to 10° from horizontal, upward pipe inclination has httle effecl on holdup (Gregory, Can. J. Chem. Eng., 53, 384-388 [1975]). 

Structured packings maintain mass-transfer performance with minimum penalty for pressure drop [108]. Two models are presented for calculating pressure drop (1) Bravo-Rocha-Fair [111] and (2) Stichlmair-Bravo-Fair [112]. Each method is qmte involved with rather complex equations to calculate the factor to ultimately calculate a pressure drop. The authors [108] recommend for design using... 

It will be noted that for a Newtonian fluid (n = 1) equation 5.11 gives — 1 for all values of b. In other words, the pressure drop will be unaffected by air injection provided that the liquid flow remains laminar. In practice, because of losses not taken into account in the simplified model, the pressure drop for a Newtonian fluid always increases with air injection. 

Fig. 5.27 Model-predicted pressure drops normalized with experimentally measured pressure drops for a circular test section (Triplett et al. 1999b). Model predictions represent the homogeneous wall friction model. Reprinted from Triplett et al. (1999b) with permission...

Change the model and program to account for a linear pressure profile, allowing for pressure drop through the reactor. 

The model is a significant improvement over the Lockhart and Martinelli correlations for pressure drop and holdup (discussed in Sec. 3.5.3). A severe limitation of the model, however, is the dependency on the empirical expression for Cf j [Eq. 3-126]. This expression is based on air-water data only, and has not been shown to apply to other systems. 

For greater concentrations of fine particles the suspension is more likely to be non-Newtonian, in which case the viscous properties can probably be adequately described by the power law or Bingham plastic models. The pressure drop-flow relationship for pipe flow under these conditions can be determined by the methods presented in Chapters 6 and 7. 

Contributions to pressure drop have also been studied by lattice Boltzmann simulations. Zeiser et al. (2002) postulated that dissipation of energy was due to shear forces and deformational strain. The latter mechanism is usually missed by capillary-based models of pressure drop, such as the Ergun equation, but may be significant in packed beds at low Re. For a bed of spheres with N — 3, they found that the dissipation caused by deformation was about 50% of that... 

A similar equation to that of Eq. (43) was proposed by Bankoff (B6) on the basis of a bubble-flow model for vertical flow. His derivations are discussed in the following section (Section V, B). Finally, it should be mentioned that the momentum exchange model of Levy (L4), and the slip-ratio model of Lottes and Flinn (L7) are more readily applied for the determination of void fractions than for pressure drops. In general, these methods seem to give rather poorer accuracy than those already discussed. 

The reactor is modeled by three partial differential equations component balances on A and B [Eqs. (6.1) and (6.2)] and an energy balance [Eq. (6.3) for an adiabatic reactor or Eq. (6.4) for a cooled reactor]. The overall heat transfer coefficient U in the cooled reactor in Eq. (6.4) is calculated by Eq. (6.5) and is a function of Reynolds number Re, Eq. (6.6). Equation (6.7) is used for pressure drop in the reactor using the friction factor /given in Eq. (6.8). The dynamics of the momentum balance in the reactor are neglected because they are much faster than the composition and temperature dynamics. A constant... 

The particle model This model attributes pressure drop to friction losses due to drag of a particle. The preeence of liquid reduces the void fraction of the bed and also increases the particle dimensions. Ergun (94) applied this model for single-phase flow (e.g., fixed and fluidized beds). Stichlmair et al. (95) successfully extended this model to correlate pressure drop and flood for both random and structured packings. Their correlation is complex and requires some additional validation, but is the most fundamental correlation available. 

As discussed in the previous section, the work of Coleman and Garimella [22] identified several other regimes and patterns however, for pressure drop model development, it will be shown that this broad categorization suffices. In the absence of other valid transition criteria for phase-change flow in small hydraulic diameter circular channels, these criteria were also assumed to apply for circular channels of equivalent diameters under consideration here. 

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