Reynolds number function

Figure 11.5.a-2 Reynolds number function versus porosity... 

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. 

The power number depends on impeller type and mixing Reynolds number. Figure 5 shows this relationship for six commonly used impellers. Similar plots for other impellers can be found in the Hterature. The functionality between and Re can be described as cc Re in laminar regime and depends on p. N in turbulent regime is constant and independent of ]1. 

The drag coefficient has different functionalities with particle Reynolds number Ri in three different regimes (Fig. 14), which results in the following expressions (1). 

Using this simplified model, CP simulations can be performed easily as a function of solution and such operating variables as pressure, temperature, and flow rate, usiag software packages such as Mathcad. Solution of the CP equation (eq. 8) along with the solution—diffusion transport equations (eqs. 5 and 6) allow the prediction of CP, rejection, and permeate flux as a function of the Reynolds number, Ke. To faciUtate these calculations, the foUowiag data and correlations can be used (/) for mass-transfer correlation, the Sherwood number, Sb, is defined as Sh = 0.04 S c , where Sc is the Schmidt... 

Fig. 4. Mathcad simulations (Cp = 5000 mg/L) as a function of Reynolds number for a NaCl solution (a) concentration polarization (CP), and (b) (-...

Dimensional analysis (qv) shows that is generally a function of the particle Reynolds number ... 

Suppose that an experiment were set up to determine the values of drag for various combinations of O, p, and ]1. If each variable is to be tested at ten values, then it would require lO" = 10, 000 tests for all combinations of these values. On the other hand, as a result of dimensional analysis the drag can be calculated by means of the drag coefficient, which, being a function of the Reynolds number Ke, can be uniquely determined by the values of Ke. Thus, for data of equal accuracy, it now requires only 10 tests at ten different values of Ke instead of 10,000, a remarkable saving in experiments. 

The dimensionless relations are usually indicated in either of two forms, each yielding identical resiilts. The preferred form is that suggested by Colburn ran.s. Am. In.st. Chem. Eng., 29, 174—210 (1933)]. It relates, primarily, three dimensionless groups the Stanton number h/cQ, the Prandtl number c Jk, and the Reynolds number DG/[L. For more accurate correlation of data (at Reynolds number <10,000), two additional dimensionless groups are used ratio of length to diameter L/D and ratio of viscosity at wall (or surface) temperature to viscosity at bulk temperature. Colburn showed that the product of the Stanton number and the two-thirds power of the Prandtl number (and, in addition, power functions of L/D and for Reynolds number <10,000) is approximately equal to half of the Fanning friction fac tor//2. This produc t is called the Colburn j factor. Since the Colburn type of equation relates heat transfer and fluid friction, it has greater utility than other expressions for the heat-transfer coefficient. 

For smooth pipe, the friction factor is a function only of the Reynolds number. In rough pipe, the relative roughness /D also affects the friction factor. Figure 6-9 plots/as a function of Re and /D. Values of for various materials are given in Table 6-1. The Fanning friction factor should not be confused with the Darcy friction fac tor used by Moody Trans. ASME, 66, 671 [1944]), which is four times greater. Using the momentum equation, the stress at the wall of the pipe may be expressed in terms of the friction factor ... 

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 ... 

Probably Fl is a function of particle Reynolds number and concentration, but Fig. 6-33 gives Durand s empirical correlation for Fl as a function of particle diameter and the input, feed volume fraction solids, Cs = QsKQs + Ql)- The form of Eq. (6-145) may be derived from turbulence theory, as shown by Davies (Chem. Eng. Sci., 42, 1667-1670 [1987]). 

However, the transition Reynolds number depends on free-stream turbulence and may range from 3 X 10 to 3 X lO ". The laminar boundary layer thickness 8 is a function of distance from the leading edge ... 

The drag coefficient for rigid spherical particles is a function of particle Reynolds number, Re = d pii/ where [L = fluid viscosity, as shown in Fig. 6-57. At low Reynolds number, Stokes Law gives 24... 

Nonsplierical Rigid Particles The drag on a nonspherical particle depends upon its shape and orientation with respect to the direction of motion. The orientation in free fall as a function of Reynolds number is given in Table 6-8. 

The drag coefficients for disks (flat side perpendicular to the direction of motion) and for cylinders (infinite length with axis perpendicular to the direclion of motion) are given in Fig. 6-57 as a Function of Reynolds number. The effect of length-to-diameter ratio for cylinders in the Newton s law region is reported by Knudsen and Katz Fluid Mechanics and Heat Transfer, McGraw-Hill, New York, 1958). 

UfQ = terminal velocity of a single sphere (infinite dilution) c = volume fraction sohd in the suspension n = function of Reynolds number Re = dpUto /[L as given Fig. 6-58... 

Heat Transfer In general, the fluid mechanics of the film on the mixer side of the heat transfer surface is a function of what happens at that surface rather than the fluid mechanics going on around the impeller zone. The impeller largely provides flow across and adjacent to the heat-transfer surface and that is the major consideration of the heat-transfer result obtained. Many of the correlations are in terms of traditional dimensionless groups in heat transfer, while the impeller performance is often expressed as the impeller Reynolds number. 

FIG. 20-38 Newton number as a Function of Reynolds number for a horizontal stirred bead mill, with fluid alone and with various filling fractious of 1-mm glass beads [Weit and Schwedes, Chemical Engineering and Technology, 10(6), 398 04 (1987)]. (N = power input, W d = stirrer disk diameter, m n = stirring speed, 1/s i = liquid viscosity, Pa-s Qj = feed rate, mVs.)... 

Determination of friction factors for some fluid flow applications can involves a trial-and-error procedure because the friction factor is not a simple function of the Reynolds number. Process engineers, therefore, refer to a Moody chart that has been developed using the following relationships ... 

Coefficient A and exponent a must be evaluated experimentally. Experiments have shown that A and a are themselves functions of the Reynolds number. Equation 47 shows that the resistance force increases with increasing velocity. If the force field (e.g., gravity) has the same potential at all points, a dynamic equilibrium between forces P and R develops shortly after the particle motion begins. As described earlier, at some distance from its start the particle falls at a constant velocity. If the acting force depends on the particle location in space, in a... 

Coefficient A and exponent a can be evaluated readily from data on Re and T. The dimensionless groups are presented on a single plot in Figure 15. The plot of the function = f (Re) is constructed from three separate sections. These sections of the curve correspond to the three regimes of flow. The laminar regime is expressed by a section of straight line having a slope P = 135 with respect to the x-axis. This section corresponds to the critical Reynolds number, Re < 0.2. This means that the exponent a in equation 53 is equal to 1. At this a value, the continuous-phase density term, p, in equation 46 vanishes. 

Figure 7-15 shows plots of Pumping number Nq and Power number Np as functions of Reynolds number Np for a pitched-blade turbine and high-efficiency impeller. Hicks et al. [8] further introduced the scale of agitation, S, as a measure for determining agitation intensity in pitched-blade impellers. The scale of agitation is based on a characteristic velocity, v, defined by... 

Figure 7-15. Power number and Pumping number as functions of Reynolds number for a pitched-blade turbine and high-efficiency impeller. (Source Bakker, A., and Gates L. , Properly Choose Mechanical Agitators for Viscous Liquids," Chem. Eng. Prog., pp. 25-34, 1995.)...

To allow for the effect of roughness one can use the results of empirical tests in ducts that have been artificially roughened with particles glued on the surface. This approach allows roughness levels to be determined as a function of the particle diameter k. The following friction factor equation has been derived for large Reynolds numbers ... 

This is an ultimate case, when the friction factor is no longer a function of the Reynolds number and is a function of roughness the pressure loss is now Ap tv", where w is the fluid velocity in the duct. The surface roughness of typical manufactured ductworks varies between the values of a theoretically fully smooth duct and an artificially roughened one. Accordingly the pressure loss varies between Ap w -w and f =/ (Re, roughness). 

Fully developed nonisothermal flow may also be similar at different Reynolds numbers, Prandtl numbers, and Schmidt numbers. The Archimedes number will, on the other hand, always be an important parameter. Figure 12.30 shows a number of model experiments performed in three geometrically identical models with the heights 0.53 m, 1.60 m, and 4.75 m." Sixteen experiments carried out in the rotxms at different Archimedes numbers and Reynolds numbers show that the general flow pattern (jet trajectory of a cold jet from a circular opening in the wall) is a function of the Archimedes number but independent of the Reynolds number. The characteristic length and velocity in Fig. 12.30 are defined as = 4WH/ 2W + IH) and u = where W is... 

Experience gives values of f for various geometries, but St is found to be a weak function of Reynolds number, so in practice there is relatively little variation in cooling efficiency (0.6 < tjc[Pg.72]

Figure 2.2 Drag coejficients for the sphere as a function of particle Reynolds number...

The drag coefficient also depends on shape and 0(, but in addition, because drag is partially due to friction, and frictional effects in a flow arc governed by a powerful dimensionless quantity called Reynolds number, then Cu is also a function of the Reynolds number. Re ... 

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