Mixing and Flocculation Mixing is a unit operation that distributes the components of two or more materials among the materials producing in the end a single blend of the components. This mixing is accomplished by agitating the materials. For example, ethyl alcohol and water can be mixed by agitating these materials using some form of an impeller. Sand, gravel, and cement used in the pouring of concrete can be mixed by putting them in a concrete batch mixer, the rotation of the mixer providing the agitation. Generally, three types of mixers are used in the physical–chemical treatment of water and wastewater: rotational, pneumatic, and hydraulic mixers. Rotational mixers are mixers that use a rotating element to effect the agitation; pneumatic mixers are mixers that use gas or air bubbles to induce the agitation; and hydraulic mixers are mixers that utilize for the mixing process the agitation that results in the flowing of the water. Flocculation , on the other hand, is a unit operation aimed at enlarging small particles through a very slow agitation of the water suspending the particles. The agitation provided is mild, just enough for the particles to stick together and agglom- erate and not rebound as they hit each other in the course of the agitation. Floccu- lation is effected through the use of large paddles such as the one in flocculators used in the coagulation treatment of water. 6.1 ROTATIONAL MIXERS Figure 6.1 is an example of a rotational mixer. This type of setup is used to determine the optimum doses of chemicals. Varying amounts of chemicals are put into each of the six containers. The paddles inside each of the containers are then rotated at a predetermined speed by means of the motor sitting on top of the unit. This rotation agitates the water and mixes the chemicals with it. The paddles used in this setup are, in general, called impellers. A variety of impellers are used in practice. 6.1.1 T YPES OF I MPELLERS Figure 6.2 shows the various types of impellers used in practice: propellers (a), paddles (b), and turbines (c). Propellers are impellers in which the direction of the driven fluid is along the axis of rotation. These impellers are similar to the impellers used in propeller pumps treated in a previous chapter. Small propellers turn at around 1,150 to 1,750 rpm; larger ones turn at around 400 to 800 rpm. If no slippage occurs between water and propeller, the fluid would move a fixed distance axially. The ratio of this distance to the diameter of the impeller is called the pitch . A square pitch is one in which the axial distance traversed is equal to the diameter of the propeller. The pitching is obtained by twisting the impeller blade; the correct degree of twisting induces the axial motion. 6 TX249_frame_C06.fm Page 293 Friday, June 14, 2002 4:30 PM © 2003 by A. P. Sincero and G. A. Sincero 294 FIGURE 6.1 An example of a rotational mixer. (Courtesy of Phipps & Bird, Richmond, VA. © 2002 Phipps & Bird.) FIGURE 6.2 Types of impellers. (a) Propellers: (1) guarded; (2) weedless; and (3) standard three-blade. (b) Paddles: (1) pitched and (2) flat paddle. (c) Turbines: (1) shrouded blade with diffuser ring; (2) straight blade; (3) curved blade; and (4) vaned-disk. (a) (b) (c) 1 2 3 1 2 3 4 1 2 Shroud TX249_frame_C06.fm Page 294 Friday, June 14, 2002 4:30 PM © 2003 by A. P. Sincero and G. A. Sincero 295 Figure 6.2(a)1 is a guarded propeller, so called because there is a circular plate ring encircling the impeller. The ring guides the fluid into the impeller by constraining the flow to enter on one side and out of the other. Thus, the ring positions the flow for an axial travel. Figure 6.2(a)2 is a weedless propeller, called weedless, possibly because it originally claims no “weed” will tangle the impeller because of its two-blade design. Figure 6.2(a)3 is the standard three-blade design; this normally is square pitched. Figure 6.2(b)1 is a paddle impeller with the two paddles pitched with respect to the other. Pitching in this case is locating the paddles at distances apart. Three or four paddles may be pitched on a single shaft; two and four-pitched paddles being more common. The paddles are not twisted as are the propellers. Paddles are so called if their lengths are equal to 50 to 80% of the inside diameter of the vessel in which the mixing is taking place. They generally rotate at slow to moderate speeds of from 20 to 150 rpm. Figure 6.2(b)2 shows a single-paddle agitator. Impellers are similar to paddles but are shorter and are called turbines . They turn at high speeds and their lengths are about only 30 to 50% of the inside diameter of the vessel in which the mixing is taking place. Figure 6.2(c)1 shows a shrouded turbine. A shroud is a plate added to the bottom or top planes of the blades. Figures 6.2(c)2 and 6.2(c)3 are straight and curve-bladed turbines. They both have six blades. The turbine in Figure 6.2(c)4 is a disk with six blades attached to its periphery. Paddle and turbine agitators push the fluid both radially and tangentially. For agitators mounted concentric with the horizontal cross section of the vessel in which the mixing is occurring, the current generated by the tangential push travels in a swirling motion around a circumference; the current generated by the radial push travels toward the wall of the vessel, whereupon it turns upward and downward. The swirling motion does not contribute to any mixing at all and should be avoided. The currents that bounce upon the wall and deflected up and down will eventually return to the impeller and be thrown away again in the radial and tangential direction. The frequency of this return of the fluid in agitators is called the circulation rate . This rate must be of such magnitude as to sweep all portions of the vessel in a reasonable amount of time. Figure 6.3 shows a vaned-disk turbine. As shown in the elevation view on the left, the blades throw the fluid radially toward the wall thereby deflecting it up and down. FIGURE 6.3 Flow patterns in rotational mixers. Vor tex Swirl TX249_frame_C06.fm Page 295 Friday, June 14, 2002 4:30 PM © 2003 by A. P. Sincero and G. A. Sincero 296 The arrows also indicate the flow eventually returning back to the agitator blades— the circulation rate. On the right, the swirling motion is shown. The motion will simply move in a circumference unless it is broken. As the tangential velocity is increased, the mass of the swirling fluid tends to pile up on the wall of the vessel due to the increased centrifugal force. This is the reason for the formation of vortices. As shown on the left, the vortex causes the level of water to rise along the vessel wall and to dip at the center of rotation. 6.1.2 P REVENTION OF S WIRLING F LOW Generally, three methods are used to prevent the formation of swirls and vortices: putting the agitator eccentric to the vessel, using a side entrance to the vessel, and putting baffles along the vessel wall. Figure 6.4 shows these three methods of pre- vention. The left side of Figure 6.4a shows the agitator to the right of the vessel center and in an inclined position; the right side shows the agitator to the left and in a vertical position. Both locations are no longer concentric with the vessel but eccentric to it, so the circumferential path needed to form the swirl would no longer exist, thus avoiding the formation of both the swirl and the vortex. Figure 6.4b is an example of a side-entering configuration. It should be clear that swirls and vortices would also be avoided in this kind of configuration. Figure 6.4c shows the agitator mounted at the center of the vessel with four baffles installed on the vessel wall. The swirl may initially form close to the center. As this swirl FIGURE 6.4 Methods of swirling flow prevention. (a) (b) (c) Baffle Baffle TX249_frame_C06.fm Page 296 Friday, June 14, 2002 4:30 PM © 2003 by A. P. Sincero and G. A. Sincero 297 propagates toward the wall, its outer rim will be broken by the baffles, however, preventing its eventual formation. 6.1.3 P OWER D ISSIPATION IN R OTATIONAL M IXERS A very important parameter in the design of mixers is the power needed to drive it. This power can be known if the power given to the fluid by the mixing process is determined. The product of force and velocity is power. Given a parcel of water administered a push (force) by the blade, the parcel will move and hence attain a velocity, thus producing power. The force exists as long as the push exists; however, the water will not always be in contact with the blade; hence, the pushing force will cease. The power that the parcel had acquired will therefore simply be dissipated as it overcomes the friction imposed by surrounding parcels of water. Power dissi- pation is power lost due to frictional resistance and is equal to the power given to it by the agitator. Let us derive this power dissipation by dimensional analysis. Recall that in dimensional analysis pi groups are to be found that are dimensionless. The power given to the fluid should be dependent on the various geometric measurements of the vessel. These measurements can be conveniently normalized against the diameter of the impeller D a to make them into dimensionless ratios. Thus, as far as the geometric measurements are concerned, they have now been rendered dimensionless. These dimensionless ratios are called shape factors . Refer to Figure 6.5. As shown, there are seven geometric measurements: W , the width of the paddle; L , the length of the paddle; J, the width of the baffle; H , the depth in the vessel; D t , the diameter of the vessel; E , the distance of the impeller to the bottom of the vessel; and D a , the diameter of the impeller. The corresponding shape factors are then S 1 = W / D a , S 2 = L / D a , S 3 = J / D a , S 4 = H / D a , S 5 = D t / D a , and S 6 = E / D a . In general, if there are n geometric measurements, there are n − 1 shape factors. The power given to the fluid should also be dependent on viscosity µ , density ρ , and rotational speed N. The higher the viscosity, the harder it is to push the fluid, increasing the power required. A similar argument holds for the density: the denser FIGURE 6.5 Normalization of geometric measurements into dimensionless ratios. H E D a D t W L JJ TX249_frame_C06.fm Page 297 Friday, June 14, 2002 4:30 PM © 2003 by A. P. Sincero and G. A. Sincero the fluid, the harder it is to push it, thus requiring more power. In addition, the power requirement must also increase as the speed of rotation is increased. Note that N is expressed in radians per second. As shown in Figure 6.3, a vortex is being formed, raising the level of water higher on the wall and lower at the center. This rising of the level at one end and lowering at the other is has to do with the weight of the water. Because the weight of any substance is a function of gravity, the gravity g must enter into the functionality of the power given to the fluid. The shape factors have already been nondimension- alized, so we will ignore them for the time being and consider only the diameter D a of the impeller as their representative in the functional expression for the power. Letting the power be P , P = (6.1) Now, to continue with our dimensional analysis, let us break down the variables of the previous equation into their respective dimensions using the force-length-time (FLT) system as follows: By inspection, the number of reference dimensions is 3; thus, the number of pi variables is 6 − 3 = 3 (the number of variables minus the number of reference dimensions). Reference dimension is the smallest number of groupings obtained from grouping the basic dimensions of the variables in a given physical problem. Call the pi variables ⌸ 1 , ⌸ 2 , and ⌸ 3 , respectively. Letting ⌸ 1 contain P , write [ P / N ] = ( FL / T ) / ( 1/ Τ ) = FL to eliminate T. [ ] is read as “the dimensions of.” To eliminate L , write [ P / ND a ] = FL/L = F. To eliminate F, write [P/ND a (1/ ρ )] = F{1/(F Τ 2 /L 4 ) (1/ Τ ) 2 L 4 } = 1. Therefore, (6.2) To solve for Π 2 , write [ µ ] = L 2 (FT/L 2 ) = FT. To eliminate FT, write [ µ (1/ ρ N )] = FT{1/(FT 2 /L 4 )(1/T)L 4 } = 1. Thus, (6.3) Variable Dimensions, FLT P FL/T N 1/T only if N is in radians per unit time; a radian by definition is dimensionless D a L gL/T 2 ρ M/L 3 = FT 2 /L 4 , M is the dimension of mass in the MLT system µ M/LT = FT/L 2 ψ ND a g µ ,,,,() N 2 D a 4 Π 1 P N 3 ρ D a 5 P o called the power number== D a 2 D a 2 D a 4 Π 2 µ ρ ND a 2 Π 2 ⇒ ρ ND a 2 µ Re called the Reynolds number=== TX249_frame_C06.fm Page 298 Friday, June 14, 2002 4:30 PM © 2003 by A. P. Sincero and G. A. Sincero To solve for Π 3 , write [D a /g] = (L)/(L/T 2 ) = T 2 . To eliminate T 2 , write [(D a /g)] = T 2 {1/T 2 } = 1. Thus, (6.4) Including the shape factors and assuming there are n geometric measurements in the vessel, the functional relationship Equation (6.1) becomes (6.5) (6.6) For any given vessel, the values of the shape factors will be constant. Under this condition, P o = P /N 3 ρ will simply be a function of Re = ρ N / µ and a function of Fr = D a N 2 /g. The effect of the Froude number Fr is manifested in the rising and lowering of the water when the vortex is formed. Thus, if vortex formation is prevented, Fr will not affect the power number P o and P o will only be a function of Re. As mentioned before, the power given to the fluid is actually equal to the power dissipated as friction. In any friction loss relationships with Re, such as the Moody diagram, the friction factor has an inverse linear relationship with Re in the laminar range (Re ≤ 10). The power number is actually a friction factor in mixing. Thus, this inverse relationship for P o and Re, is (6.7) K L is the proportionality constant of the inverse relationship. Substituting the expres- sion for P o and Re, (6.8) At high Reynolds numbers, friction losses become practically constant. If the Moody diagram for flow in pipes is inspected, this statement will be found to be true. Agitators are not an exception. If vortices and swirls are prevented, at high Reynolds numbers greater than or equal to 10,000, power dissipation is independent of Re and the relationship simply becomes (6.9) K T is the constant. Flows at high Reynolds numbers are characterized by turbulent conditions. Substituting the expression for P o , (6.10) N 2 Π 3 D a N 2 g Fr called the Froude number== P N 3 ρ D a 5 φ ρ ND a 2 µ D a N 2 g S 1 S 2 … S n−1 ,,,,, = P N 3 ρ D a 5 ⇒ P o φ ρ ND a 2 µ D a N 2 g S 1 S 2 … S n−1 ,,,,, == φ Re, Fr, S 1 , S 2 …,S n−1 ,()= D a 5 D a 2 P o K L Re = P K L N 2 D a 3 µ = P o K T = P K T N 3 D a 5 ρ = TX249_frame_C06.fm Page 299 Friday, June 14, 2002 4:30 PM © 2003 by A. P. Sincero and G. A. Sincero K L and K T are collectively called power coefficients of which some values are found in Table 6.1. For Reynolds number in the transition range (10 < Re < 10,000), the power may be taken as the average of Eqs. (6.8) and (6.10). Thus, (6.11) Example 6.1 A turbine with six blades is installed centrally in a baffled vessel. The vessel is 2.0 m in diameter. The turbine, 61 cm in diameter, is positioned 60 cm from the bottom of the vessel. The tank is filled to a depth of 2.0 m and is mixing alum with raw water in a water treatment plant. The water is at a temperature of 25°C and the turbine is running at 100 rpm. What horsepower will be required to operate the mixer? Solution: Therefore, TABLE 6.1 Values of Power Coefficients Type of Impeller K L K T Propeller (square pitch, three blades) 1.0 0.001 Propeller (pitch of 2, three blades) 1.1 0.004 Turbine (six flat blades) 1.8 0.025 Turbine (six curved blades) 1.8 0.019 Shrouded turbine (six curved blades) 2.4 0.004 Shrouded turbine (two curved blades) 2.4 0.004 Flat paddles (two blades, D t /W = 6) 0.9 0.006 Flat paddles (two blades, D t /W = 8) 0.8 0.005 Flat paddles (four blades, D t /W = 6) 1.2 0.011 Flat paddles (six blades, D t /W = 6) 1.8 0.015 Note: For vessels with four baffles at wall and J = 0.1 D t From W. L. McCabe and J. C. Smith (1967). Unit Oper- ations of Chemical Engineering. McGraw-Hill, New York, 262. P K L N 2 D a 3 µ K T N 3 D a 5 ρ + 2 1 2 N 2 D a 3 K L µ K T ND a 2 ρ +()== Re ρ ND a 2 µ ρ 997 kg/m 3 N 100 2 π () 60 10.47 rad/s== == D a 0.61 m µ 8.5 10 −4 () kg/m ⋅ s== Re 997 10.47()0.61() 2 8.5 10 4– () 4.57 10 6 () turbulent== TX249_frame_C06.fm Page 300 Friday, June 14, 2002 4:30 PM © 2003 by A. P. Sincero and G. A. Sincero Therefore, 6.2 CRITERIA FOR EFFECTIVE MIXING 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 G z . From fluid mechanics, G z = = where u is the fluid velocity in the x direction. As noted, this gradient is at a point, since ∆y has been shrunk to zero. If the dimension of G z is taken, it will be found to have per unit time as the dimension. Thus, G z is really a rate of rotation or angular velocity. Designate this as ω z . If Ψ z is the torque of the rotating fluid, then in the x direction, the power P x is (6.12) FIGURE 6.6 A parcel of fluid acted upon by shear forces in the x direction. P K T N 3 D a 5 ρ From the table, K T 0.025== P 0.025 10.47() 3 0.61() 5 997()2416.15 N ⋅ m/s 2416.15 746 3.24 hp Ans==== lim ∆y→0 ∆u ∆y ∂ u/ ∂ y, P x Ψ z ω z Ψ z G z Ψ z ∂ u/ ∂ y=== TX249_frame_C06.fm Page 301 Friday, June 14, 2002 4:30 PM © 2003 by A. P. Sincero and G. A. Sincero The torque Ψ z is equal to a force times a moment arm. The force at the bottom face F bot or the force at the upper face F up 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: (F bot + F up )/2 = . Thus, the couple in the x direction is ∆y. This is the torque Ψ z . The flow regime in a vessel under mixing may be laminar or turbulent. Under laminar conditions, may be expressed in terms of the stress obtained from Newton’s law of viscosity and the area of shear, A shx = ∆ x∆z. 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 τ x = µ ( ∂ u/ ∂ y), where µ is the absolute viscosity. Substituting, Equation (6.12) becomes (6.13) where = ∆x∆z∆y, 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 . In this case, the value of G x 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 G z , G x , and G y . Consider this gradient as , remembering that this is the average velocity gradient over the whole vessel contents. P may then simply be expressed as P = µ ∆ , whereupon solving for (6.14) Various values of this 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 TABLE 6.2 Criteria Values for Effective Mixing t o , seconds , <10 10–20 20–30 30–40 40–130 4000–1500 1500–950 950–850 850–750 750–700 F x F x F x P x Ψ z G z F x ∆yG z τ x ∆x ∆z ∆yG z µ∂ u/ ∂ y()∆x ∆z ∆yG z µ ∆ V G z 2 == = = = V V G G V G G G P µ V = G G G s 1– TX249_frame_C06.fm Page 302 Friday, June 14, 2002 4:30 PM © 2003 by A. P. Sincero and G. A. Sincero [...]... TX249_frame_C 06. fm Page 324 Friday, June 14, 2002 4:30 PM 6. 29 6. 30 6. 31 6. 32 6. 33 6. 34 6. 35 6. 36 6.37 6. 38 6. 39 A suppressed rectangular is used to mix chemical in a wastewater treatment unit The power dissipation developed is 7423.5 N·m/s The length of the weir L and the height P were measured and found to be 2 m and 3 1 m, respectively The rate of flow through the weir is 0.3 m /s and HD = 2.5 m... density of water Surface tension of water One of the π variables in dimensional analysis Fluid torque in the z direction Angular velocity of paddle wheel Angular velocity in the z direction PROBLEMS 6. 1 6. 2 6. 3 6. 4 6. 5 6. 6 6. 7 6. 8 6. 9 6. 10 6. 11 A turbine with six curved blades is installed centrally in a baffled vessel The vessel is 2.0 m in diameter The turbine 61 cm in diameter is positioned 60 cm from... depth of 5 m and letting L represent the length of tank, 5 ( 5 )L = 771.53 ; L = 30. 86 m Ans 1 length of compartment no.1, the shortest = - ( 30. 86 ) 1 + 2 + 2 .6 1 = ( 30. 86 ) = 5.51 m 5 .6 Ans Width and depth of compartment no.1 = width and depth of compartment no 2 = width and depth of compartment no 3 = 5 m, as assumed Ans Length of compartment no 2, the middle = 2/5 .6( 30. 86) = 11.02... dissipation of the mixer is 3.24 hp 2 The pressure Pi at which air is forced into the diffuser is 2 56, 920.02 N/m The temperature of the water is 25°C and the barometric pressure is 2 101,300 N/m At what depth is the diffuser being placed at the bottom of the tank? © 2003 by A P Sincero and G A Sincero TX249_frame_C 06. fm Page 323 Friday, June 14, 2002 4:30 PM 6. 21 6. 22 6. 23 6. 24 6. 25 6. 26 6.27 6. 28 The... Sincero and G A Sincero TX249_frame_C 06. fm Page 322 Friday, June 14, 2002 4:30 PM 6. 12 6. 13 6. 14 6. 15 6. 16 6.17 6. 18 6. 19 6. 20 For the vessel in Problem 6. 3, at what detention time should it be operated for it to function effectively? At what rate of inflow should it be operated? For the vessel in Problem 6. 4, at what detention time should it be operated for it to function effectively? At what rate of inflow... Water Science Technol Proc 1997 Workshop on IAWQ-IWSA Joint Group on Particle Separation, July 1–2, Sapporo, Japan, 37, 10, 55 67 Elsevier Science Ltd., Exeter, England Wistrom, A and J Farrell (1998) Simulation and system identification of dynamic models for flocculation control Water Science Technol Proc 7th Int Workshop on Instrumentation, Control and Automation of Water and Wastewater Treatment and. .. Figure 6. 9 is used to mix alum in a water treatment plant The mixing power developed is 165 7.2 N·m/s and the width of the channel is 10 cm The rate of inflow 3 into the jump is 0.048 m /s The upstream depth of the jump is 5 cm and the temperature of mixing is 25°C Calculate the downstream depth of the jump A hydraulic-jump mixer similar to the one shown in Figure 6. 9 is used to mix alum in a water treatment. .. bottom of the tank is 2 2 m and the barometric pressure is 101,300 N/m What is the temperature of the water? A hydraulic-jump mixer similar to the one shown in Figure 6. 9 is used to mix alum in a water treatment plant The height of the bottom of the sluice gate to the bottom of the channel is 5 cm The mixing power developed is 165 7.2 N·m/s and the width of the channel is 10 cm Calculate the rate of inflow... the jump The temperature of the water is 25°C and the downstream depth of the jump is 0.94 m A hydraulic-jump mixer similar to the one shown in Figure 6. 9 is used to mix alum in a water treatment plant The height of the bottom of the sluice gate to the bottom of the channel is 5 cm The mixing power developed is 165 7.2 N·m/s and the width of the channel is 10 cm The 3 rate of inflow into the jump is... – 0.25 < Re < G 2 (6. 19) (6. 20) The mass density of air has been eliminated in Equation (6. 17), because it is negligible 6. 3.2 POWER DISSIPATION PNEUMATIC MIXERS IN 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 . the rotation of the mixer providing the agitation. Generally, three types of mixers are used in the physical chemical treatment of water and wastewater: rotational, pneumatic, and hydraulic. positioned 60 cm from the bottom of the vessel. The tank is filled to a depth of 2.0 m and is mixing alum with raw water in a water treatment plant. The water is at a temperature of 25°C and the. Sincero and G. A. Sincero (6. 19) (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