Drops Reynolds number

Figure 6-60 gives the drag coefficient as a function of bubble or drop Reynolds number for air bubbles in water and water drops in air, compared with the standard drag curve for rigid spheres. Information on bubble motion in non-Newtonian liquids may be found in Astarita and Apuzzo (AIChE J., 11, 815-820 [1965]) Calderbank, Johnson, and Loudon (Chem. Eng. Sci., 25, 235-256 [1970]) and Acharya, Mashelkar, and Ulbrecht (Chem. Enz. Sci., 32, 863-872 [1977]). 

Whether in turbulent flow viscous or inertial forces are predominant depends on the drop Reynolds number Redr (Table 5.1), where the... 

Nusselt number number of samples number of striations Prandtl number pressure drop Reynolds number correlation coefficient distance... 

This effect is, however, only pronounced at a drop Reynolds number range between 200 and 600. 

Emulsions of the types SE (O/W) and DE (W/O/W) with different initial secondary mean drop sizes were passed through differently sized nozzles (liquid caps, diameters 0.5, 0.7, 1.0, 2.5 mm) at GLR = 0 (no gas flow). From this, X503 Nozzie/J so,3, iniaai was plotted versus the related nozzle Reynolds number / ei,Nozzie or secondary drop Reynolds number Reix>rop [(23.5) and (23.6)] with Rei,Drop = / ij ozzie (x5o,3/dn) and denoting the nozzle diameter as demonstrated in Fig. 23.15 (top). 

Pressure Drop. The prediction of pressure drop in fixed beds of adsorbent particles is important. When the pressure loss is too high, cosdy compression may be increased, adsorbent may be fluidized and subject to attrition, or the excessive force may cmsh the particles. As discussed previously, RPSA rehes on pressure drop for separation. Because of the cychc nature of adsorption processes, pressure drop must be calculated for each of the steps of the cycle. The most commonly used pressure drop equations for fixed beds of adsorbent are those of Ergun (143), Leva (144), and Brownell and co-workers (145). Each of these correlations uses a particle Reynolds number (Re = G///) and friction factor (f) to calculate the pressure drop (AP) per... 

This equation is appHcable for gases at velocities under 50 m/s. Above this velocity, gas compressibiUty must be considered. The pitot flow coefficient, C, for some designs in gas service, is close to 1.0 for Hquids the flow coefficient is dependent on the velocity profile and Reynolds number at the probe tip. The coefficient drops appreciably below 1.0 at Reynolds numbers (based on the tube diameter) below 500. 

La.mina.r Flow Elements. Each of the previously discussed differential-pressure meters exhibits a square root relationship between differential pressure and flow there is one type that does not. Laminar flow meters use a series of capillary tubes, roUed metal, or sintered elements to divide the flow conduit into innumerable small passages. These passages are made small enough that the Reynolds number in each is kept below 2000 for all operating conditions. Under these conditions, the pressure drop is a measure of the viscous drag and is linear with flow rate as shown by the PoiseuiHe equation for capilary flow ... 

Flow Past Bodies. A fluid moving past a surface of a soHd exerts a drag force on the soHd. This force is usually manifested as a drop in pressure in the fluid. Locally, at the surface, the pressure loss stems from the stresses exerted by the fluid on the surface and the equal and opposite stresses exerted by the surface on the fluid. Both shear stresses and normal stresses can contribute their relative importance depends on the shape of the body and the relationship of fluid inertia to the viscous stresses, commonly expressed as a dimensionless number called the Reynolds number (R ), EHp/]1. The character of the flow affects the drag as well as the heat and mass transfer to the surface. Flows around bodies and their associated pressure changes are important. 

Flow Past Deformable Bodies. The flow of fluids past deformable surfaces is often important, eg, contact of Hquids with gas bubbles or with drops of another Hquid. Proper description of the flow must allow for both the deformation of these bodies from their shapes in the absence of flow and for the internal circulations that may be set up within the drops or bubbles in response to the external flow. DeformabiUty is related to the interfacial tension and density difference between the phases internal circulation is related to the drop viscosity. A proper description of the flow involves not only the Reynolds number, dFp/p., but also other dimensionless groups, eg, the viscosity ratio, 1 /p En tvos number (En ), Api5 /o and the Morton number (Mo),giJ.iAp/plG (6). 

The pressure drop accompanying pipe flow of such fluids can be described in terms of a generalized Reynolds number, which for pseudoplastic or dilatant fluids takes the form ... 

The transition from laminar to turbulent flow occurs at Reynolds numbers varying from ca 2000 for n > 1 to ca 5000 for n = 0.2. In the laminar region the Fanning friction factor (Fig. 2) is identical to that for Newtonian fluids. In the turbulent region the friction factor drops significantly with decreasing values of producing a family of curves. 

Eurther research on convective transport under low Reynolds number, quasicontinuum conditions is needed before the optimal design of such a micro heat exchanger is possible. The cooling heat exchanger is usually thermally linked to a relatively massive substrate. The effects of this linkage need to be explored and accurate methods of predicting the heat-transfer and pressure-drop performance need to be developed. 

Droplet trajectories for limiting cases can be calculated by combining the equations of motion with the droplet evaporation rate equation to assess the likelihood that drops exit or hit the wall before evaporating. It is best to consider upper bound droplet sizes in addition to the mean size in these calculations. If desired, an instantaneous value for the evaporation rate constant may also be used based on an instantaneous Reynolds number calculated not from the terminal velocity but at a resultant velocity. In this case, equation 37 is substituted for equation 32 ... 

In order to select the pipe size, the pressure loss is calculated and velocity limitations are estabHshed. The most important equations for calculation of pressure drop for single-phase (Hquid or vapor) Newtonian fluids (viscosity independent of the rate of shear) are those for the deterrnination of the Reynolds number, and the head loss, (16—18). 

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. 

Noncircular Channels Calciilation of fric tional pressure drop in noncircular channels depends on whether the flow is laminar or tumu-lent, and on whether the channel is full or open. For turbulent flow in ducts running full, the hydraulic diameter shoiild be substituted for D in the friction factor and Reynolds number definitions, Eqs. (6-32) and (6-33). The hydraiilic diameter is defined as four times the channel cross-sectional area divided by the wetted perimeter. For example, the hydraiilic diameter for a circiilar pipe is = D, for an annulus of inner diameter d and outer diameter D, = D — d, for a rectangiilar duct of sides 7, h, Dij = ah/[2(a + h)].T ie hydraulic radius Rii is defined as one-fourth of the hydraiilic diameter. 

The hydrauhc diameter method does not work well for laminar flow because the shape affects the flow resistance in a way that cannot be expressed as a function only of the ratio of cross-sectional area to wetted perimeter. For some shapes, the Navier-Stokes equations have been integrated to yield relations between flow rate and pressure drop. These relations may be expressed in terms of equivalent diameters Dg defined to make the relations reduce to the second form of the Hagen-Poiseulle equation, Eq. (6-36) that is, Dg (l2SQ[LL/ KAPy. Equivalent diameters are not the same as hydraulie diameters. Equivalent diameters yield the correct relation between flow rate and pressure drop when substituted into Eq. (6-36), but not Eq. (6-35) because V Q/(tiDe/4). Equivalent diameter Dg is not to be used in the friction factor and Reynolds number ... 

For condensing vapor in vertical downflow, in which the hquid flows as a thin annular film, the frictional contribution to the pressure drop may be estimated based on the gas flow alone, using the friction factor plotted in Fig. 6-31, where Re is the Reynolds number for the gas flowing alone (Bergelin, et al., Proc. Heat Transfer Fluid Mech. Inst., ASME, June 22-24, 1949, pp. 19-28). 

Porous Media Packed beds of granular solids are one type of the general class referred to as porous media, which include geological formations such as petroleum reservoirs and aquifers, manufactured materials such as sintered metals and porous catalysts, burning coal or char particles, and textile fabrics, to name a few. Pressure drop for incompressible flow across a porous medium has the same quahtative behavior as that given by Leva s correlation in the preceding. At low Reynolds numbers, viscous forces dominate and pressure drop is proportional to fluid viscosity and superficial velocity, and at high Reynolds numbers, pressure drop is proportional to fluid density and to the square of superficial velocity. 

Eo = Eotvos number = gA dVc Re = Reynolds number = du /[L Ap = density difference between the phases p = density of continuous liquid phase d = drop diameter [L = continuous liquid viscosity surface tension u = relative velocity... 

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