Reynolds number hquid

For hquid systems v is approximately independent of velocity, so that a plot of JT versus v provides a convenient method of determining both the axial dispersion and mass transfer resistance. For vapor-phase systems at low Reynolds numbers is approximately constant since dispersion is determined mainly by molecular diffusion. It is therefore more convenient to plot H./v versus 1/, which yields as the slope and the mass transfer resistance as the intercept. Examples of such plots are shown in Figure 16. 

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. 

Both wetted-sensor and clamp-on Doppler meters ate available for Hquid service. A straight mn of piping upstream of the meter and a Reynolds number of greater than 10,000 ate generally recommended to ensure a weU-developed flow profile. Doppler meters ate primarily used where stringent accuracy and repeatabiHty ate not requited. Slurry service is an important appHcation area. 

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). 

A numerical study of the effect of area ratio on the flow distribution in parallel flow manifolds used in a Hquid cooling module for electronic packaging demonstrate the useflilness of such a computational fluid dynamic code. The manifolds have rectangular headers and channels divided with thin baffles, as shown in Figure 12. Because the flow is laminar in small heat exchangers designed for electronic packaging or biochemical process, the inlet Reynolds numbers of 5, 50, and 250 were used for three different area ratio cases, ie, AR = 4, 8, and 16. 

The pumping number is a function of impeller type, the impeller/tank diameter ratio (D/T), and mixing Reynolds number Re = pND /p.. Figure 3 shows the relationship (2) for a 45° pitched blade turbine (PBT). The total flow in a mixing tank is the sum of the impeller flow and flow entrained by the hquid jet. The entrainment depends on the mixer geometry and impeller diameter. For large-size impellers, enhancement of total flow by entrainment is lower (Fig. 4) compared with small impellers. 

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). 

Viscous Drag. The velocity, v, with which a particle can move through a Hquid in response to an external force is limited by the viscosity, Tj, of the Hquid. At low velocity or creeping flow (77 < 1), the viscous drag force is /drag — SirTf- Dv. The Reynolds number (R ) is deterrnined from... 

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). 

Pressure drop due to hydrostatic head can be calculated from hquid holdup B.]. For nonfoaming dilute aqueous solutions, R] can be estimated from f i = 1/[1 + 2.5(V/E)(pi/pJ ]. Liquid holdup, which represents the ratio of liqmd-only velocity to actual hquid velocity, also appears to be the principal determinant of the convective coefficient in the boiling zone (Dengler, Sc.D. thesis, MIT, 1952). In other words, the convective coefficient is that calciilated from Eq. (5-50) by using the liquid-only velocity divided by in the Reynolds number. Nucleate boiling augments conveclive heat transfer, primarily when AT s are high and the convective coefficient is low [Chen, Ind Eng. Chem. Process Des. Dev., 5, 322 (1966)]. 

Forveiy thin hquids, Eqs. (14-206) and (14-207) are expected to be vahd up to a gas-flow Reynolds number of 200 (Valentin, op. cit., p. 8). For liquid viscosities up to 100 cP, Datta, Napier, and Newitt [Trans. In.st. Chem. Eng., 28, 14 (1950)] and Siems and Kauffman [Chem. Eng. Sci, 5, 127 (1956)] have shown that liquid viscosity has veiy little effec t on the bubble volume, but Davidson and Schuler [Trans. Instn. Chem. Eng., 38, 144 (I960)] and Krishnamiirthi et al. [Ind. Eng. Chem. Fundam., 7, 549 (1968)] have shown that bubble size increases considerably over that predic ted by Eq. (14-206) for hquid viscosities above 1000 cP. In fac t, Davidson et al. (op. cit.) found that their data agreed veiy well with a theoretical equation obtained by equating the buoyant force to drag based on Stokes law and the velocity of the bubble equator at break-off ... 

Power Consumption of Impellers Power consumption is related to fluid density, fluid viscosity, rotational speed, and impeller diameter by plots of power number (g P/pN Df) versus Reynolds number (DfNp/ l). Typical correlation lines for frequently used impellers operating in newtonian hquids contained in baffled cylindri-calvessels are presented in Fig. 18-17. These cui ves may be used also for operation of the respective impellers in unbaffled tanks when the Reynolds number is 300 or less. When Nr L greater than 300, however, the power consumption is lower in an unbaffled vessel than indicated in Fig. 18-17. For example, for a six-blade disk turbine with Df/D = 3 and D IWj = 5, = 1.2 when Nr = 10. This is only about... 

There are well-estabhshed empirical correlations for stirrer power rcquircrnenls [6, 7]. Figure 7.8 [6] is a log-log plot of the power number for imgassed hquids versus the stirrer Reynolds number (Re). These dimensionless numbers are defined as follows ... 

L T ), and v is the kinematic viscosity ofthe hquid (L" T ). Ihe dimensionless groups include N/v) = Reynolds number (Re) d N /g) = Froude number (Fr) and (Q/N d = aeration number (Na), which is proportional to the ratio of the superficial gas velocity with respect to the tank cross section to the impeller tip speed. 

When estimating the stirrer power requirements for non-Newtonian liquids, correlations of the power number versus the Reynolds number (Re see Figure 7.8) for Newtonian hquids are very useful. In fact, Figure 7.8 for Newtonian liquids can be used at least for the laminar range, if appropriate values of the apparent viscosity are used in calculating the Reynolds number. Experimental data for various non-Newtonian fluids with the six-blade turbine for the range of (Re) below 10 were correlated by the following empirical Equation 12.1 [1] ... 

Although they used droplets with diameters of 2 mm and more, the work of Park et alP is interesting on account of the fact that they used four different substrates and four different hquids. They observed the impact of droplets of distilled water, n-Octane, n-Tetradecane or n-Hexadecane onto glass slides, sihcon wafers, HMDS (Hexamethyl dishazane) coated sihcon wafers or Teflon, for Reynolds numbers from 180 to 5513 and Weber numbers from 0.2 to 176. A model was constructed to predict the maximum spreading ratio, which is the ratio of the maximum spreading diameter to the initial droplet s diameter, for low impact velocities. 

Since the rise or fall of liquid droplets is interfered with by lateral flow of the hquid, the diameter of the drum should be made large enough to minimize this adverse effect. A rule based on the Reynolds number of the phase through which the movement of the liquid drops occurs is proposed by Flooper and Jacobs (1979). The Reynolds number is Df,up/fi, where Dh is the hydrauhc diameter and u is the linear velocity of the continuous phase. The rules are ... 

Open Channel Flow For flow in open channels, the data are largely based on experiments with water in turbulent flow, in channels of suflFicient roughness that there is no Reynolds number effect. The hydraulic radius approach may be used to estimate a friction factor with which to compute friction losses. Under conditions of uniform flow where hquid depth and cross-sectional area do not vary significantly with position in the flow direction, there is a balance between gravitational forces and wall stress, or equivalently between frictional losses and potential energy change. The mechanical energy balance reduces to = g(zi — z. In terms of the friction factor and hydraulic diameter or hydraulic radius,... 

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