As the impeller pushes a parcel of fluid, this fluid is propelled forward. Because of the inherent force of attraction between molecules, this parcel drags neighboring parcels along. This is the reason why fluids away from the impeller flows even if they were not actually hit by the impeller. This force of attraction gives rise to the property of fluids called viscosity.
Visualize the filament of fluid on the left of Figure 6.6 composed of several parcels strung together end to end. The motion induced on this filament as a result of the action of the impeller may or may not be uniform. In the more general case, the motion is not uniform. As a result, some parcels will move faster than others. Because of this difference in velocities, the filament rotates. This rotation produces a torque, which, coupled with the rate of rotation produces power. This power is actually the power dissipated that was addressed before. Out of this power dissipation, the criteria are derived for effective mixing.
Refer to the right-hand side of Figure 6.6. This is a parcel removed from the filament at the left. Because of the nonuniform motion, the velocity at the bottom of the parcel is different from that at the top. Thus, a gradient of velocity will exist. Designate this as Gz. From fluid mechanics, Gz = limAy^0 ^ = duldy, where u is the fluid velocity in the x direction. As noted, this gradient is at a point, since Ay has been shrunk to zero. If the dimension of Gz is taken, it will be found to have per unit time as the dimension. Thus, Gz is really a rate of rotation or angular velocity. Designate this as 0)z. If ¥z is the torque of the rotating fluid, then in the x direction, the power Px is
The torque ¥z is equal to a force times a moment arm. The force at the bottom face Fbot or the force at the upper face Fup in the parcel represents this force. This force is a force of shear. These two forces are not necessarily equal. If they were, then a couple would be formed; however, to produce an equivalent couple, each of these forces may be replaced by their average: (Fbot + Fup)/2 = Fx. Thus, the couple in the x direction is Fx Ay. This is the torque ¥z.
The flow regime in a vessel under mixing may be laminar or turbulent. Under laminar conditions, Fx may be expressed in terms of the stress obtained from Newton's law of viscosity and the area of shear, Ashx = A xAz. Under turbulent conditions, the stress relationships are more complex. Simply for the development of a criterion of effective mixing, however, the conditions may be assumed laminar and base the criterion on these conditions. If this criterion is used in a consistent manner, since it is only employed as a benchmark parameter, the result of its use should be accurate.
From Newton's law of viscosity, the shear stress Tx = ¡(du/dy), where 1 is the absolute viscosity. Substituting, Equation (6.12) becomes
Px = WzGz = Fx Ay Gz = TxAx Az AyGz = ¡(du/dy )Ax Az Ay Gz = lAVGz2
where V = Ax Az Ay, the volume of the fluid parcel element.
Although Equation (6.13) has been derived for the fluid element power, it may be used as a model for the power dissipation for the whole vessel of volume V. In this case, the value of Gx to be used must be the average over the vessel contents. Also, considering all three component directions x, y, and z, the power is P; the velocity gradient would be the resultant gradient of the three component gradients Gz, Gx, and Gy. Consider this gradient as G, remembering that this G is the average velocity gradient over the whole vessel contents. P may then simply be expressed as P = ¡AVG, whereupon solving for G
Various values of this G are the ones used as criteria for effective mixing. Table 6.2 shows some criteria values that have been found to work in practice using the
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