## Pneumatic Mixers

Diffused aerators may also be used to provide mixing. The difference in density between the air bubbles and water causes the bubbles to rise and to quickly attain terminal rising velocities. As they rise, these bubbles push the surrounding water just as the impeller in rotational mixers push the surrounding water creating a pushing force. This force along with the rising velocity creates the power of mixing. It is evident that pneumatic mixing power is a function of the number of bubbles formed. Thus, to predict this power, it is first necessary to develop an equation to predict the number of bubbles formed.

Figure 6.7 shows designs of diffusers used to produce bubbles. In a, air is forced through a ceramic tube. Because of the fine opening in the ceramic mass, this design produces fine bubbles. In d, holes are simply pierced into the pipe creating perforations. The sizes of the bubbles would depend on how small or large the holes are. A photograph of coarse bubbles is shown in b while a photograph of fine bubbles is shown is e. The figure in c is simply an open pipe where air is allowed to escape; this produces large bubbles. The figure in f is a saran wrapped tube; this produces fine bubbles.

6.3.1 Prediction of Number of Bubbles and Rise Velocity

The number of bubbles formed is equal to the volume of air in the vessel divided by the average volume of a single bubble. The volume of air in the vessel is equal to the rate of inflow of air Qa times its detention time to. The detention time, in turn, is equal to the depth of submergence h (see Figure 6.7g) divided by the average total rise velocity of the bubbles. If vb is the average rise velocity of the bubbles

and Vl is the net average upward velocity of the water, then the average total rise velocity of the bubbles is Vb + Vl and to = h/(Vb + Vl).

Let the average volume of a single bubble at the surface of the vessel be Vbo and let the influent absolute pressure of the air in Qt be P. In order to accurately compute the number of bubbles, Qt should be corrected so its value would correspond to Vbo at the surface when the pressure becomes the atmospheric pressure Pa. Since pressure and volume are in inverse ratio to each other, the rate of inflow of air corrected to its value at the surface of the vessel is then (P;/Pa)Q;. Thus, the number of bubbles n formed from a rate of inflow of air Qi is

Vbo Vbo

The average upward velocity of the liquid Vl is small in comparison with the rise velocity of the bubbles Vb and may be neglected. The equation then reduces to

VboVb Pa Vbo Vb

The rise velocities of bubbles were derived by Peebles and Garber (1953). Using the techniques of dimensional analysis as was used in the derivation of the power dissipation for rotational mixers, they discovered that the functionality of the rise velocities of bubbles can be described in terms of three dimensionless quantities: G1 = gf4/pla3, G2 = g(r)4(Vb)4p3i/a3, and Re = 2plVbr/f. Re is a Reynolds number; g is the acceleration due to gravity; f is the absolute viscosity of fluid; pt is the mass density of fluid; a is the surface tension of fluid; and r is the average radius of the bubbles. To give G1 and G2 some names, call G1 the Peebles number and G2 the Garber number.

We may want to perform the dimensional analysis ourselves, but the procedure is similar to the one done before. In other words, Vb is first to be expressed as a function of the variables affecting its value: Vb = f(g, f, pl, pg, a, r). pg is the mass density of the gas phase (air). Each of the variables in this function is then broken down into its fundamental dimensions to find the number of reference dimensions. Once the number of reference dimensions have been found, the number of pi dimensionless variables can then be determined. These dimensionless variables are then found by successive eliminations of the dimensions of the physical variables until the number of pi dimensionless ratios are obtained.

The final equations are as follows:

b 9f 9f

Vb = Q.33g076 f (r)128 2 < Re < 4.02 Gf214 (6.18)

n x0.50

vb = 1.35I P- 4.02GI2'214 < Re < 3.10G-025 (6.19)

vb = 1.53 (P) " 3.10GI0'25 < Re < G2 (6.20)

The mass density of air has been eliminated in Equation (6.17), because it is negligible.

### 6.3.2 Power Dissipation in Pneumatic Mixers

The left-hand side of Figure 6.8 shows the forces acting on the bubble having a velocity of vb going upward. FB is the buoyant force acting on the bubble as a result of the volume of water it displaces; Fg is the weight of the bubble. As the bubble moves upward, it is resisted by a drag force exerted by the surrounding mass of water; this force is the drag force FD. As the bubbles emerge from the diffusers, they quickly attain their terminal rise velocities. Thus, the bubble is not accelerated and application of Newton's second law of motion to the bubble simply results in FB -Fg - Fd = 0 and FD = FB - Fg.

The right side of Figure 6.8 shows the action of the bubble upon the surrounding water as a result of Newton's third law of motion: For every action, there is an equal and opposite reaction ^ For every force there is an equal and opposite reactive force. The FD on the right is the reactive force to the FD on the left. This force has the same action on the surrounding water as the impeller has on the water in the case of the rotational mixer. It pushes the opposing surrounding water with a force Fd traveling at an average speed of vb. The product of this force and the velocity gives the power of dissipation. Calling Vb the average volume the bubble attains as

it rises in the water column, y the specific weight of the water, and y the specific weight of the air, the power dissipated for n bubbles [Equation (6.16)] or simply the power dissipation in the vessel is

PaVboVb

The value of the volume of a single bubble Vb varies as the bubble rises in the water column. By the inverse relationship of pressure and volume, Vb at any depth may be expressed in terms of Vbo, the average volumes of the bubbles at the surface. The pressure upon Vb at depth h is Pa + hy. Thus, Vb = [Pal(Pa + hy)] Vbo. The value of Vb may then be derived by integrating Vb over the depth of the vessel as follows:

And,

' - - - PQk - P-V n (- r. v m, =><p-^y- pq n&ñ <6,3)

Because the specific weight of air yg is very much smaller than that of water yl, it has been neglected.

The power dissipation must be such that it causes the correct velocity gradient G. The literature have shown the criteria values for effective mixing in the case of rotational mixers. Values of G need to be determined for pneumatic mixers. As an ad hoc measure, however, the values for rotational mixers (Table 6.2) may be used.

Example 6.2 By considering the criterion for effective mixing, the volume of a rapid-mix tank used to rapidly mix an alum coagulant in a water treatment plant was found to be 6.28 m with a power dissipation of 3.24 hp. Assume air is being

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