Reynolds number power

Recently, Celata et al. (1994c) modified Eq. (5-12) on the parameter C, together with a slight modification of the Reynolds number power, to give a more accurate prediction in the range of pressures below 5.0 MPa (725 psia). The modified equation is... 

Table I shows the three areas of consideration in mixer design. The first area is process design, which will be covered in detail in succeeding pages. Process design entails determining the power and diameter of the impeller to achieve a satisfactory result. The speed is then calculated by referring to the Reynolds number-power number curve, shown in Fig. 12. Such a curve allows trial-and-error calculations of the speed once the fluid properties, P, D, and the impeller design are known.

FIGURE 12 Reynolds number-power number curve for several impeller types D, impeller diameter N, impeller rotational speed p, liquid density 11, liquid viscosity P, power and g, gravity constant. 

Figure 25 shows a typical Reynolds number-Power number curve for different impellers. The important thing about this curve is that it holds true whether the desired process job is being done or not. Power equations have three independent variables along with fluid properties power, speed and diameter. There are only two independent choices for process considerations. 

If we compare equation (6-17) and (6-21), we see that the differences occur in the multiplier term (1.62 versus 0.023), Reynolds number power ( versus 0.8), and D/L power ( versus 0). The Prandtl number power (5) is the same for both. A phenomenological approach to these effects indicate that the change in flow from laminar to turbulent is reflected both in the larger power on the Reynolds number and the elimination of the D/L effect (transformation from layers to vortices and eddies). Also, the change in flow does not alter the Prandtl number power. 

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 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. 

Figure 6-40 shows power number vs. impeller Reynolds number for a typical configuration. The similarity to the friction factor vs. Reynolds number behavior for pipe flow is significant. In laminar flow, the power number is inversely proportional to Reynolds number, reflecting the dominance of viscous forces over inertial forces. In turbulent flow, where inertial forces dominate, the power number is nearly constant. 

Curve g is for disk flat-blade turbines operated in unbaffled vessels filled witb liquid, covered, so tbat no vortex forms. If baffles are present, tbe power characteristics at high Reynolds numbers are essentially tbe same as curve h for baffled open vessels, witb only a slight increase in power. 

Radial-flow impellers include the flat-blade disc turbine, Fig. 18-4, which is labeled an RlOO. This generates a radial flow pattern at all Reynolds numbers. Figure 18-17 is the diagram of Reynolds num-ber/power number curve, which allows one to calculate the power knowing the speed and diameter of the impeller. The impeller shown in Fig. 18-4 typically gives high shear rates and relatively low pumping capacity. 

Not only is the type of flow related to the impeller Reynolds number, but also such process performance characteristics as mixing time, impeller pumping rate, impeller power consumption, and heat- and mass-transfer coefficients can be correlated with this dimensionless group. 

Power consuiTmtion has also been measured and correlated with impeller Reynolds number. The velocity head for a mixing impeller can be calculated, then, from flow and power data, by Eq. (18-3) or Eq. (18-5). 

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... 

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.)... 

Impeller Reynolds number and equations for mixing power for particle suspensions are in Sec. 5. Dispersion of gasses into liquids is in Sec. 14. Usually, an increase in mechanical agitation is more effective than is an increase in aeration rate for improving mass transfer. 

Figure 3-8b. Turbine map. (Balje, O.E., A Study of Reynolds Number Effects in Turbomachinery, Journal of Engineering for Power, ASME Trans., Vol. 86, Series A, p. 227.)...

And introducing the ratio of accelerations, = ag/g, where indicates the relative strength of acceleration, ag, with respect to the gravitational acceleration g. This is known as the separation number. The LHS of equation 60 contains a Reynolds number group raised to the second power and the drag coefficient. Hence, the equation may be written entirely in terms of dimensionless numbers ... 

This chapter reviews the various types of impellers, die flow patterns generated by diese agitators, correlation of die dimensionless parameters (i.e., Reynolds number, Froude number, and Power number), scale-up of mixers, heat transfer coefficients of jacketed agitated vessels, and die time required for heating or cooling diese vessels. 

Experimental data on power eonsumption are generally plotted as a funetion of the Power number Np versus Reynolds number Np, that is by rearranging Equation 7-14. 

Figure 7-14. Power number versus Reynolds number oorrelation for oommon impellers. (Source Ruchton et at., Chem. Eng. Prog., 46, No. 8, 495, 1950. Reprinted with permission of AlChE. Copyright 1950. All rights reserved.)...

Rush ton et al. [10] investigated the effect of varying the tank geometrical ratios and the correlation of the Power number with Reynolds number. At high Reynolds number, it was inferred that. 

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.)...

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