Standard Methods for Examination of Water & Wastewater_5 ppt

Rotational mixers aremixers that use a rotating element to effect the agitation; pneumatic mixers are mixersthat use gas or air bubbles to induce the agitation; and hydraulic mixers are

Mixing and Flocculation

Mixing is a unit operation that distributes the components of two or more materialsamong the materials producing in the end a single blend of the components Thismixing is accomplished by agitating the materials For example, ethyl alcohol andwater 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 puttingthem 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 waterand wastewater: rotational, pneumatic, and hydraulic mixers Rotational mixers aremixers that use a rotating element to effect the agitation; pneumatic mixers are mixersthat use gas or air bubbles to induce the agitation; and hydraulic mixers are mixers thatutilize for the mixing process the agitation that results in the flowing of the water

particles through a very slow agitation of the water suspending the particles Theagitation 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 flocculatorsused 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 determinethe 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 rotationagitates the water and mixes the chemicals with it The paddles used in this setupare, 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 thedriven fluid is along the axis of rotation These impellers are similar to the impellersused in propeller pumps treated in a previous chapter Small propellers turn at around1,150 to 1,750 rpm; larger ones turn at around 400 to 800 rpm If no slippage occursbetween 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 isone 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 twistinginduces the axial motion

6

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

Figure 6.2(a)1 is a guarded propeller, so called because there is a circular plate ringencircling the impeller The ring guides the fluid into the impeller by constraining theflow to enter on one side and out of the other Thus, the ring positions the flow for anaxial 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 tothe other Pitching in this case is locating the paddles at distances apart Three orfour paddles may be pitched on a single shaft; two and four-pitched paddles beingmore common The paddles are not twisted as are the propellers Paddles are socalled if their lengths are equal to 50 to 80% of the inside diameter of the vessel inwhich 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 ofthe 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)2and 6.2(c)3 are straight and curve-bladed turbines They both have six blades Theturbine 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 Foragitators mounted concentric with the horizontal cross section of the vessel in whichthe mixing is occurring, the current generated by the tangential push travels in aswirling motion around a circumference; the current generated by the radial pushtravels toward the wall of the vessel, whereupon it turns upward and downward Theswirling motion does not contribute to any mixing at all and should be avoided Thecurrents 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 Thefrequency of this return of the fluid in agitators is called the circulation rate Thisrate must be of such magnitude as to sweep all portions of the vessel in a reasonableamount 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.

Vortex

Swirl

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 willsimply move in a circumference unless it is broken As the tangential velocity isincreased, the mass of the swirling fluid tends to pile up on the wall of the vesseldue 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 vesselwall 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, andputting 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 vesselcenter 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 buteccentric to it, so the circumferential path needed to form the swirl would no longerexist, 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 clearthat swirls and vortices would also be avoided in this kind of configuration Figure6.4c 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

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 isdetermined The product of force and velocity is power Given a parcel of wateradministered a push (force) by the blade, the parcel will move and hence attain avelocity, 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 willcease The power that the parcel had acquired will therefore simply be dissipated

as it overcomes the friction imposed by surrounding parcels of water Power pation is power lost due to frictional resistance and is equal to the power given to

dissi-it by the agdissi-itator

Let us derive this power dissipation by dimensional analysis Recall that indimensional analysis pi groups are to be found that are dimensionless The powergiven to the fluid should be dependent on the various geometric measurements ofthe 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 geometricmeasurements are concerned, they have now been rendered dimensionless Thesedimensionless ratios are called shape factors

Refer to Figure 6.5 As shown, there are seven geometric measurements: W, thewidth 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 thebottom of the vessel; and D a, the diameter of the impeller The corresponding shapefactors are then S1=W/D a, S2 =L/D a, S3 =J/D a, S4=H/D a, S5=D t/D a, and S6=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.

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,

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/(FΤ2

/L4)(1/Τ )2

To solve for Π3, write [D a /g] = (L)/(L/T2

) = T 2 To eliminate T2, write

[(D a /g) ] = T2

{1/T2

} = 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 thiscondition, P o = P/N3

ρ will simply be a function of Re = ρN /µ and a function

of Fr = D a N2/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

of Re and the relationship simply becomes

- Fr called the Froude number

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 cmfrom the bottom of the vessel The tank is filled to a depth of 2.0 m and is mixingalum 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 tooperate the mixer?

Solution:

Therefore,

TABLE 6.1 Values of Power Coefficients

Propeller (square pitch, three blades) 1.0 0.001 Propeller (pitch of 2, three blades) 1.1 0.004

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

2 - 1

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

6.2 CRITERIA FOR EFFECTIVE MIXING

As the impeller pushes a parcel of fluid, this fluid is propelled forward Because ofthe inherent force of attraction between molecules, this parcel drags neighboringparcels along This is the reason why fluids away from the impeller flows even ifthey were not actually hit by the impeller This force of attraction gives rise to theproperty of fluids called viscosity

Visualize the filament of fluid on the left of Figure 6.6 composed of severalparcels 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 isactually 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 thefilament at the left Because of the nonuniform motion, the velocity at the bottom ofthe 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 Designatethis as ωz If Ψz is the torque of the rotating fluid, then in the x direction, the power P x is

Parcel taken from filament

The torque Ψz 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 Thisforce 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 Underlaminar 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 ofeffective mixing, however, the conditions may be assumed laminar and base thecriterion 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 τx = 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 averagevelocity 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 Table6.2 shows some criteria values that have been found to work in practice using the

TABLE 6.2 Criteria Values for Effective Mixing

<10 10–20 20–30 30–40 40–130

4000–1500 1500–950 950–850 850–750 750–700

parameters t o and ⋅ t o is the detention time of the vessel Thus, as shown in this

table, the detention time to be allowed for in mixing is a function G.

The power dissipation treated above is power given to the fluid To get the primemover power P b (the brake power), P must be divided by the brake efficiency η; and

to get the input power from electricity P i, P b must be divided by the motor efficiency M.

6.3 PNEUMATIC MIXERS

Diffused aerators may also be used to provide mixing The difference in densitybetween the air bubbles and water causes the bubbles to rise and to quickly attainterminal rising velocities As they rise, these bubbles push the surrounding waterjust as the impeller in rotational mixers push the surrounding water creating a pushingforce This force along with the rising velocity creates the power of mixing It isevident 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 thenumber of bubbles formed

Figure 6.7 shows designs of diffusers used to produce bubbles In a, air is forcedthrough 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

perfora-tions The sizes of the bubbles would depend on how small or large the holes are A

photo-graph of coarse bubbles is shown in b while a photophoto-graph 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 P REDICTION OF N UMBER OF B UBBLES AND R ISE V ELOCITY

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 Q a times its detention time t o 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 is the average rise velocity of the bubbles

FIGURE 6.7 Bubble diffuser designs.

Open pipe (c)

Perforated pipe

(d)

Fine bubble (e)

Saran wrapped tube (f)

Diffused aeration schematic

(g)

Ql

Qa

Qe

and is the net average upward velocity of the water, then the average total risevelocity of the bubbles is + and t o = h/( + )

Let the average volume of a single bubble at the surface of the vessel be

and let the influent absolute pressure of the air in Q i be P i In order to accurately

compute the number of bubbles, Q i should be corrected so its value would correspond

to at the surface when the pressure becomes the atmospheric pressure P a Sincepressure 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 i /P a )Q i Thus, the number

of bubbles n formed from a rate of inflow of air Q i is

We may want to perform the dimensional analysis ourselves, but the procedure

is similar to the one done before In other words, is first to be expressed as afunction of the variables affecting its value: = f(g, µ, ρl, ρg, σ, ) ρg is themass density of the gas phase (air) Each of the variables in this function is thenbroken down into its fundamental dimensions to find the number of referencedimensions Once the number of reference dimensions have been found, the number

of pi dimensionless variables can then be determined These dimensionless variablesare then found by successive eliminations of the dimensions of the physical variablesuntil the number of pi dimensionless ratios are obtained

The final equations are as follows:

ρl

9µ - Re<2

v b 0.33g0.76 ρl

µ

 

 0.52( )r 1.28 2 Re 4.02G1

2,214 –

<

<

=

(6.20)

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

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 v b going upward F B is the buoyant force acting on the bubble as a result

of the volume of water it displaces; F g is the weight of the bubble As the bubblemoves upward, it is resisted by a drag force exerted by the surrounding mass of

water; this force is the drag force F D As the bubbles emerge from the diffusers, theyquickly 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 F B −

F g − F D = 0 and F D = F B − F g

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

force The F D on the right is the reactive force to the F D on the left This force hasthe same action on the surrounding water as the impeller has on the water in thecase of the rotational mixer It pushes the opposing surrounding water with a force

F D traveling at an average speed of The product of this force and the velocitygives the power of dissipation Calling the average volume the bubble attains as

FIGURE 6.8 Bubble free-body diagrams to illustrate power dissipation.

Re 3.10G1

0.25 –

it rises in the water column, γl the specific weight of the water, and γg the specific

weight of the air, the power dissipated for n bubbles [Equation (6.16)] or simply the

power dissipation in the vessel is

(6.21)

The value of the volume of a single bubble V b varies as the bubble rises in the

water column By the inverse relationship of pressure and volume, V b at any depthmay be expressed in terms of , the average volumes of the bubbles at the surface

The pressure upon V b at depth h is P a + hγ l Thus, V b = [P a /(P a + hγ l)] The value

of may then be derived by integrating over the depth of the vessel as follows:

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

a rapid-mix tank used to rapidly mix an alum coagulant in a water treatment plantwas found to be 6.28 m3 with a power dissipation of 3.24 hp Assume air is beingprovided at a rate of 1.12 m3 per m3 of water treated and that the detention time ofthe tank is 2.2 min Calculate the pressure at which air is forced into the diffuser.Assume barometric pressure as 101,300 N/m2,the depth from the surface to thediffuser as 2 m, and the temperature of water as 25°C

6.4 HYDRAULIC MIXERS

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 theenergy 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 thefigure Hydraulic-jump mixers are designed as rectangular in cross section

FIGURE 6.9 Hydraulic-jump mixer.

y1

y2y

2

1 Sluice gate

Control volume

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 h f can be calculated The product of

rate of flow Q and specific weight γ is weight (force) per unit time If this product

is multiplied by h f the result is power Thus,

(6.24)The determination of the mixing power of hydraulic mixers is therefore reduced tothe 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 adhoc measure, the values for rotational mixers (Table 6.2) may be used

6.4.2 M IXING P OWER FOR H YDRAULIC J UMPS

Refer to the hydraulic jump schematic of Figure 6.9 The general energy equationmay be applied between points 1 and 2 producing

(6.25)

V is the velocity at the indicated points; y is the depth; g is the acceleration due to

gravity; and h f 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, V 1 = g/y1 and V 2 = g/y2 Substituting this in Equation (6.25), simplifying, and

∑ is the summation of forces acting at the faces of the cross sections at points 1

and 2; t is the time; is the velocity vector; ρ is the mass density of water; is the

V1 2

–

2gy1 2

y2 2

volume of the domain of integration; 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, ∑ = P1A1 − P2A2 , where P

is the pressure at the respective points; A is the area normal to the pressure; and 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 γ Thus, ∑ becomes During operation, the mixer is at steady state; hence,

Substituting all these into Equation (6.27), noting that only the x

direction is to be considered, and simplifying,

(6.28)

Fr1 is the Froude number at point 1 = v1/

Equation (6.26) may now be substituted into the general equation for mixingpower Equation (6.24) The mixing power for hydraulic jumps is then

(6.29)

W is the width of the channel As mentioned before, P hydJump must have a value that

corresponds to the value of G that is correct for effective mixing.

6.4.3 V OLUME AND D ETENTION T IMES OF H YDRAULIC -J UMP M IXERS

Referring to the bottom of Figure 6.9, let L be the length of the hydraulic jump.Then the volume jump of the hydraulic jump is simply the volume of the trapezoidalprism of volume Thus,

(6.30)

The detention time t o is then

(6.31)

The length L of the hydraulic jump is measured from the front face of the jump

to a point on the surface of the flow immediately after the roller as shown in Figure 6.9