Hydraulic mixers are mixers that use the energy of a flowing fluid to create the power dissipation required for mixing. This fluid must have already been given the energy before reaching the point in which the mixing is occurring. What needs to be done at the point of mixing is simply to dissipate this energy in such a way that the correct value of G for effective mixing is attained. The hydraulic mixers to be discussed in this chapter are the hydraulic-jump mixer and the weir mixer.
Figure 6.9 shows a hydraulic jump and its schematic. By some suitable design, the chemicals to be mixed may be introduced at the point indicated by "1" in the figure. Hydraulic-jump mixers are designed as rectangular in cross section.
6.4.1 Power Dissipation in Hydraulic Mixers
The power of mixing is simply power dissipation. In any hydraulic process, power or energy is dissipated through friction. Thus, the power of mixing in any hydraulic mixer can be determined if the fluid friction hf can be calculated. The product of rate of flow Q and specific weight y is weight (force) per unit time. If this product is multiplied by hf the result is power. Thus,
The determination of the mixing power of hydraulic mixers is therefore reduced to the determination of the friction loss.
For mixing to be effective, the power derived from this loss must be such that the G falls within the realm of effective mixing. As in pneumatic mixers, G values for hydraulic-jump mixers discussed in this section need to be established. As an ad hoc measure, the values for rotational mixers (Table 6.2) may be used.
6.4.2 Mixing Power for Hydraulic Jumps
Refer to the hydraulic jump schematic of Figure 6.9. The general energy equation may be applied between points 1 and 2 producing
V is the velocity at the indicated points; y is the depth; g is the acceleration due to gravity; and hf is the friction loss. The velocities may be expressed in terms of the flow q per unit width of the channel and the depth using the equation of continuity. Thus, V1 = g/y1 and V2 = g/y2. Substituting this in Equation (6.25), simplifying, and solving for hf, h (y2- y1)[ q2( y2 + yi) -2 gy1 y2] ((. ^
hf = ------2-----------1------------------2----2------2-1--------------------1-----2--- (6.26)
For all practical purposes, the depth y1 may be made equal to the distance from the bottom of the sluice gate to the bottom of the channel as shown in Figure 6.9. Thus, in design this parameter is known, of course, in addition to q. Using the equation of momentum, the value of y2 may be found, thus, solving hf .
As derived in any good book on fluid mechanics and as applied to the control volume indicated in Figure 6.9, the momentum equation is y F = d[ vpdv +[ vp(v ■ n)dA (6.27)
yF is the summation of forces acting at the faces of the cross sections at points 1 and 2; t is the time; v is the velocity vector; p is the mass density of water; V is the volume of the domain of integration; n is the unit normal vector to surface area A bounding the domain of integration. CV refers to the control volume.
Considering only the x direction in our analysis, X F = P1A11 - P2A2i, where P is the pressure at the respective points; A is the area normal to the pressure; and i is the unit vector in the x direction. The P's and the A's may be expressed in terms of the respective depths y and specific weight y. Thus, X F becomes 1- y1yy1i -1- y2yy2i.
During operation, the mixer is at steady state; hence, J-.¡CMVpdV = 0. §Vp(V ■ n)dA = -qpV11 + qpV2i. Substituting all these into Equation (6.27), noting that only the x direction is to be considered, and simplifying, y2 = y U1 + 8 Fri - 1 )
Fr1 is the Froude number at point 1 = V^Jgy^
Equation (6.26) may now be substituted into the general equation for mixing power Equation (6.24). The mixing power for hydraulic jumps is then
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