FLUID MIXING OF COSMETIC FORMULATIONS 353 ,•? J•_•] II IIII1 •l•II• ] I JillIll !]! 1[ ....... • .................. •.. ]]•i • [ IllIll I [ J J 111[•• Illll I I I II1[ • • • [ •l [ [ I ••,:'•,TU•,•,• •'-•-- ,•l I I•11111 • • •EuRV•B•I,•I•I • • •,,•,H [[•LI • II11111 I I II1•',• I I I[llllJ ] I[I]1111 r I : •'•1 ' i II !Jill I-• •4•'.. • • ',',II',J', ', ', ', ',',', f:[rl [ I I[1111 • I [•T c ..... [ • , ,• r ] -• [[[][[[ [ [ [][][]•'C••ED•R•EN I IO IO t IO • IO 4 I0 9 [ io 6 D"N• D IMPELLER DIAMETER .).L N IMPELLER ROTATIONAL SPEED e LIQUID DENSITY J.L LIQUID VlSGOSITY P POWER g GRAVITY GONSTANT Figure 18.--Reynolds number-power number curve. For continuous processing, full scale equipment may either be a series of separate tanks with a mixer in each tank, or a compartmented column similar to that shown in Figs. 16 and 17. The particular column illustrated is 12 inches in diameter with six stages and was designed primarily for pilot plant studies. Pilo.t studies are often conducted on a batch basis to get an idea of the mechanism of the operation. If these studies lead to the conclusion that continuous multistage apparatus is required, then pilot planting in con- tinuous multistage apparatus is proper. IMPELLER POWER CHARACTERISTICS Power characteristics of impellets are important in specifying the com- plete design of the mixer. The impeller type is normally chosen by the size of the unit to be used and the type of operation. Once this is chosen, the fluid viscosity must be determined. At this point, the three variables remaining are the power, speed of the impeller and diameter of the impeller. Only two of these latter variables are independent. Thus, we have two independent choices to use in designing the mixers for the process job outlined in preceding sections. The mixer will be specified for the maxi- mum loading condition to be encountered. Taking the propeller for example, the power characteristics can be plotted on a curve which shows a Power Number versus a Reynolds Number (Fig. 18). At a given speed and diameter, the shape of this curve indicates the change in power with increase in viscosity as we move from right to

354 JOURNAL OF THE SOCIETY OF COSMETIC CHEMISTS left. Thus, as viscosity increases, the power remains relatively constant for any wide change in viscosity and then begins to increase. For a turbine type unit at a given speed and diameter, the power re- mains constant with viscosity increases up to about 50 c.p. It then drops slightly, rising again to its water value at about 50,000 c.p. Power rises rapidly with further increases in viscosity. There is a large latitude in the viscosity that may be accommodated by a turbine type unit without overloading the unit. The power drawn at the high viscosity stage must be sufficient to give the proper mixing action. Where the curve shown in Fig. 18 is fiat, the power varies with the cube of the speed and the fifth power of the impeller diameter, p o: N•D • (7) Thus, small changes in speed and diameter cause large changes in power consumption, and normally should not be done without consulting formulas to see what effect it will have on the power drawn by the unit. Additional data for various impeller types are given by Rushton, Cosrich and Everett (4). TABLE 2--NoM ENCLA'rUP, E Cp = specific heat N = rotational speed D = turbine diameter N• = Reynolds Number, D2No/i• d = tube diameter P = power Fo = gravitational force • = impeller flow Fi = inertia force •p = impeller flow at constant power F0 = surface tension R = process resttit Fv = viscous force T -- tank diameter H = impeller head o = density h = film heat transfer /• = viscosity k = thermal conductivit? SCALE-UP Similarity relationships are of great value in scale-up calculations. Since the fluid regime controls the mixing result, we can use the laws of hydraulic similarity. Two types of similarity are important, geometric similarity and dynamic similarity. Geometric similarity is the first requirement in all scale-up procedures. Dynamic similarity deals with fluid forces. There are four of these fluid forces. The first of these forces is inertia force, Ft. This is the force we apply to the system. It is related to the power, speed and diameter of the impeller. The other three forces are the ones which resist the accomplishment of our process, namely viscosity, F,, surface tension, Fo, and gravity Fa. One of these forces are usually negligible if baffles are used in the tank.