Dimensionless pressure drop

Friction Factor and Reynolds Number For a Newtonian fluid in a smooth pipe, dimensional analysis relates the frictional pressure drop per unit length AP/L to the pipe diameter D, density p, and average velocity V through two dimensionless groups, the Fanning friction factor/and the Reynolds number Re. 

Vfjp is the friction velocity and =/pVV2 is the wall stress. The friction velocity is of the order of the root mean square velocity fluctuation perpendicular to the wall in the turbulent core. The dimensionless distance from the wall is y+ = yu p/. . The universal velocity profile is vahd in the wall region for any cross-sectional channel shape. For incompressible flow in constant diameter circular pipes, = AP/4L where AP is the pressure drop in length L. In circular pipes, Eq. (6-44) gives a surprisingly good fit to experimental results over the entire cross section of the pipe, even though it is based on assumptions which are vahd only near the pipe wall. 

AP is the pressure drop, cm of water p and Pg are the density of the scrubbing liquid and gas respectively, g/cm L/g is the velocity of the gas at the throat inlet, cm/s QtIQg is the volumetric ratio of liquid to gas at the throat inlet, dimensionless It is the length of the throat, cm Coi is the drag coefficient, dimensionless, for the mean liquid diameter, evaluated at the throat inlet and d[ is the Sauter mean diameter, cm, for the atomized liquid. The atomized-liquid mean diameter must be evaluated by the Nuldyama and Tanasawa [Trans. Soc Mech Eng (Japan), 4, 5, 6 (1937-1940)] equation ... 

K Empirical proportionality constant for cyclone pressure drop or friction loss Dimensionless Dimensionless ... 

Generalized single phase pressure drop coirelations for pipes and packed beds have used complicated factors composed of groups (sometimes dimensionless) of physical property data and system geometry. ... 

This expression can be represented graphically in dimensionless form to simplify its use. It is generally expressed as the so-called filtration number , defined as follows E, = /iR, / 2APT3 jr x . The filtration number, E, is dimensionless and varies from zero at Rf = 0 to a large value when there is an increase in the viscosity of the sludge and Rf or a decrease in pressure drop, auxiliary time, specific cake resistance and the ratio of cake volume to filtrate volume. It may be assumed in practice that F(, = 0 to 10. If washing and drying times are constant and independent of filtration time, they may be added directly to the auxiliary time. In... 

The result is a modified Euler number. You can prove to yourself that the pressure drop over the particle can be obtained by accounting for the projected area of the particle through particle size, S, in the denominator. Thus, by application of dimensional analysis to the force balance expression, a relationship between the dimensionless complexes of the Euler and Reynolds numbers, we obtain ... 

There are two flow regimes corresponding to sonic (or choked) flow for liigher pressure drops and subsonic flow for lower pressure drops. The transition between the two flow regimes occurs at tlie dimensionless critical pressure ratio, Ter,I, which is related to tlie gas lieiit capacity ratio y via... 

Empirical proportionally constant for cy clone pressure drop or friction loss, dimensionless Constant for stationary vane separators, based on design... 

Vh = vapor velocity through valve holes, ft/sec P = tray aeration factor, dimensionless AP = tray pressure drop, in. liquid pvm = valve metal density, tj = tray deck thickness, in. 

Capacity factor based on tower area, ft/sec Capacity fector at flood, ft/sec Liquid gradient vapor load correction factor or Discharge coefficient (see accompanying table) or Gas phase loading factor, ft/sec. Equation 8-281 Eddy loss coefficient, dimensionless. Table 8-22 Wet cap pressure drop correction factor. Figure 8-115... 

Actual tray pressure drop, in. liquid Prandtl number dimensionless Fractional opening in the circumference or a valve or. Pi... 

P0 = Np = Power number, dimensionless, Equation 5-19 Ppew = Plate coil width, one plate, ft Ap = Pressure drop, psi AP0 = Pressure drop for open pipe, psi AP, = Static mixer pressure drop in turbulent flow, psi Q = Flow rate or pumping capacity from impeller, cu l t/sec, or Ls/1... 

A separated flow model for stratified flow was presented by Taitel and Dukler (1976a). They indicated analytically that the liquid holdup, R, and the dimensionless pressure drop, 4>G, can be calculated as unique f unctions of the Lockhart-Martinelli parameter, X (Lockhart and Martinelli, 1949). Considering equilibrium stratified flow (Fig. 3.37), the momentum balance equations for each phase are... 

For gas-liquid flows in Regime I, the Lockhart and Martinelli analysis described in Section I,B can be used to calculate the pressure drop, phase holdups, hydraulic diameters, and phase Reynolds numbers. Once these quantities are known, the liquid phase may be treated as a single-phase fluid flowing in an open channel, and the liquid-phase wall heat-transfer coefficient and Peclet number may be calculated in the same manner as in Section lI,B,l,a. The gas-phase Reynolds number is always larger than the liquid-phase Reynolds number, and it is probable that the gas phase is well mixed at any axial position therefore, Pei is assumed to be infinite. The dimensionless group M is easily evaluated from the operating conditions and physical properties. 

Figure 8. Experimental data of the dimensionless pressure drop per bubble as a function of capillary number for 1 and 2 mm diameter glass capillaries. The solid line denoted by E - 0 gives the theory of Bretherton.

The parameter X is the dimensionless solids contribution to the pressure drop ... 

Ishii and Murakami (1991) evaluated the CFB scaling relationships of Horio et al. (1989) using two cold CFB models. Solids flux, pressure drop, and optical probe measurements were used to measure a large number of hydrodynamic parameters to serve as the basis for the comparison. Fair to good similarity was obtained between the beds. Dependent hydrodynamic parameters such as the pressure drop and pressure fluctuation characteristics, cluster length and voidage, and the core diameter were compared between the two beds. The gas-to-solid density ratio was not varied between the beds. As seen in Table 7, the dimensionless solids flux decreased as the superficial velocity was increased because the solids flux was held constant. 

L/pj-A)(S/psA), liquid-solids velocity ratio, dimensionless Number of heat-transfer stages, dimensionless = hdp/kg, Nusselt number, dimensionless Pressure drop, gm-wt/cm2 = Cpu kg, Prandtl number, dimensionless = dpiipj U, Reynolds number, dimensionless S Mass velocity of solids, gm/cirf sec... 

O Dimensionless pressure drop in accelerative particle motion... 

The most satisfactory way of representing the relation between drag force and velocity involves the use of two dimensionless groups, similar to those used for correlating information on the pressure drop for flow of fluids in pipes. 

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