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Slurry Mixing with Impellers: Part 1, Theory and Previous Research T. R. CtJMaV* Mixing is essential for effective aerobic or anaerobic treatment of farm slurries, but can add significantly to the fixed and variable costs of such treatments. Rotating impellers are frequently used for mixing but their efficiency depends strongly on the relationships between their diameter, shape and speed. Previous research studies have attempted to explore these relationships with agricultural slurries in laboratory and pilot-scale mixing experiments, but none has used volumes greater than i m 3. Hence substantial scale-up factors are required to apply the results to full-size systems which may be 1000 m3 or more, and therefore slurry mixer design based on present research findings is susceptible to large errors. To provide a better basis for mixer scale-up, the appropriate theories of impeller design are discussed in this paper, and various alternative expressions for scale-up of impellers are compared. The problems in determining the impeller Reynolds number are also discussed and the most appropriate method is identified for large-scale mixing experiments. Also, a novel expression is derived to assess the effectiveness of slurry mixers. 1. Introduction All farm animals produce wastes containing solids and therefore mixing is essential in most slurry handling and treatment operations. 1 The energy required for this can be considerable, in fact it has been estimated that mixing requires about twice the energy needed for land spreading, per unit volume of slurry. 2 Consequently, cheaper mixing can significantly reduce the costs of most livestock production. Slurry mixing can be achieved with a variety of impeller designs, but efficiency and power requirements depend on the impeller configuration. Generally, there are three directions in which an impeller can move surrounding liquid: upwards, downwards and radially outwards (Fig. 1). In addition, any rotating impeller will impart some circumferential movement to the liquid. However, this can be regulated by fixing radiallyaligned baffles some distance from the impeller. Other design variations may include draught-tubes a n d / o r multiple impellers (Fig. 2). Draught-tubes prevent the radial motion of liquid away from the impeller and therefore create a mainly "up-and-down" flow pattern. This may lead to more effective resuspension of solid particles because flow velocities are greater near the base of the mixing vessel, but in agricultural waste slurries, and especially in fibrous cattle slurries, draught tubes can cause a greater risk of fouling and blockages. Multiple impellers are commonly used in industry for mixing in tall slender tanks (e.g. Ht/> 2Dt), where a single impeller may be insufficient to agitate all parts of the tank contents. This is especially important for viscous liquids and slurries. Although dual-impeUer systems have been studied experimentally for mixing agricultural wastes, 3 it is uncommon for these slurries to be stored in slender tanks and so the usefulness of multiple impellers in this area is limited. Other dual-impeller systems comprising different impeller designs operating together can however be useful where both mixing and aeration are required. 4"~ * Welfare Science Division, AFRC Engineering, Wrest Park, Silsoe, Bedford MK45 4HS Received 5 October 1987; accepted in revised form 26 August 1989

157 0021-8634/90/030157 + 17 $03.00/0

© 1990 The British Society for Research in Agricultural Engineering

158

SLURRY MIXING WITH IMPELLERS, 1

Notation a s

Ar mr'

b B Bi c

const. Ci di

Dp Ot

Fr g Gr' h H Ht i

J, I, ko kl K Ks l le Li

nominal solid particle size, m Archimedes number Archimedes number defined by Reference (26) scale-up exponent on (di/Dt) constant, Eqn (19) number of blades on impeller solids concentration, kg/kg a general constant of any value impeller clearance above tank bottom, m impeller diameter, m internal diameter of pipe, m tank diameter, m impeller Froude number gravitational acceleration, m/s 2 modified Grashof number height co-ordinate in tank, measured from bottom, m liquid depth, m tank depth, m sample designation, dimensionless liquid phase superficial velocity, m/s width of baffles, m constant constant pseudoplastic consistency coefficient, Pa sn Metzner-Otto impeller constant, Pa s2 a general linear dimension, m eddy length dimension, m length of impeller blades, m

M~ Mixing Index Chen's Mixing Index n pseudoplastic consistency index ns number of samples N impeller rotational speed, rev/s Nc critical impeller speed, rev/s No arbitrary reference number No impeller power number Noc impeller power number appropriate to critical mixing P~ impeller power, W Pic impeller power for critical mixing, W Pm mixer power, W Ps specific power input or specific power dissipation, W/m 3 p~s specific power input per unit volume of solids,

Ml(chen )

W/m 3

Ps~ specific power input or specific power dissipation in tank 1, W/m 3

P~2 specific power input or specific power dissipation in tank 2, W/m 3

R~ number of baffles Rei impeller Reynolds number Re[ generalized impeller Reynolds number Rei~ critical impeller Reynolds number for mixing Re~, generalized Reynolds number for pipeflow Rept particle Reynolds number S dimensionless number representing geometrical changes

159

T. R. C U M B Y

Si pitch of impeller, m T~ impeller torque, N m Tic impeller torque for critical mixing, N m TS(av) average (fully mixed) concentration of solids in the slurry, kg/kg TSo) local (sample) concentration of solids in the slurry, kg/kg u a general velocity, m/s v, liquid volume, m 3 01 volume of tank 1, m 3 v2 volume of tank 2, m 3 Vp superficial liquid velocity in pipeflow, m/s way time-averaged value of the superficial slip velocity, m/s W~ width of impeller blades, m Zo thickness of settled slurry layer, m 7 shear rate, sYav impeller average shear rate, s- 1

/z ~a /~1 p p~ Ps /5c "['av q~

volumetric scale-up exponent for specific power requirements viscosity, Pa s apparent viscosity, Pa s liquid viscosity, Pa s density, kg/m 3 liquid density, kg/m 3 solids density, kg/m 3 overall mean slurry density, kg/m 3 shear stress, Pa impeller average shear stress, Pa volume fraction of solids in slurry, m3/m 3

m

q t Z X

e V

constant exponents, Eqn (1)

W

J

b r

c._

C~

d]

rm? t 11

tj

Radial impeller

t Axial downflow impeller

Axial upflow impeller

Fig. 1. The three types of flow from rotating impellers

160

SLURRY

MIXING

WITH

IMPELLERS,

1

C

1

<.j Draught- tube mixer

Multiple impeller

mixer

Fig. 2. Draught tubes and multiple impellers When mixing a given volume of slurry in a vertical cylindrical tank, the ratio of the tank diameter to the impeller diameter is also important. Smaller impellers obviously have a smaller pumping capacity per revolution than do larger ones, and so they create less mixing. However, as they require less power at a given speed, correct choice of impeller diameter is necessary to optimize the mixing process. Since most agricultural waste slurries have a relatively low viscosity when compared with heavy industrial sludges and pastes, normal chemical engineering practice indicates that satisfactory mixing should be achievable with "remote clearance impellers ''e (i.e. those where Dr~4 ~

2. The relationships between power and speed for mixing impellers When designing any impeller mixing system, an accurate estimation of its power/speed curve is essential for the selection of a power unit that will achieve the expected mixing performance. Moreover, selection of an incorrect power unit will result either in unnecessarily high purchase costs or in increased failures caused by over-stressed operation. Where the necessary data can be obtained by full-scale experiment then this is obviously the best design strategy to adopt, but where experiments are limited to smaller-scale systems, then analytical techniques are required to enable the results to be scaled-up.

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161

Li

t7 i I

F-1 Ji Dt

Fig. 3. Impeller and tank dimensions

Rushton et al. 7 defined three groups of variables which can affect fluid motion during mixing: (a) The linear dimensions that describe the geometrical boundary conditions and the shape of the tank and impeller. These include: tank diameter (Dr), impeller diameter (di), tank depth (Ht), impeller clearance above tank bottom (Ci), impeller pitch (Si), blade length (Li), blade width (Wi), baffle width (Ji), number of blades (Bi) and number of baffles (Ri) (Fig. 3). (b) The properties of the liquid, particularly density and viscosity. (c) The kinematic and dynamic characteristics of the flow, such as velocity, (defined as the peripheral velocity or tip speed of the impeller), power input and the force of gravity. Rushton et al. ~ applied the Buckingham Pi Theorem a of dimensional analysis to the above collection of variables to obtain the expression: N~ = const. ( R e ~ ) ' ~ ( F r ) q ( D t / d ~ ) t ( H t / d ~ ) * ( C d d ~ ) x ( S d d ~ ) ~ ( L d d ~ )

( Wd d~)"(J~/ d~)i( B~/ No)b( n~/ No) ~

"

(1)

162

SLURRY

MIXING

WITH

IMPELLERS,

1

where: 2~rNT~ N o = power number = -pN3d -

Re~ = impeller Reynolds number = and

Fr = Froude number -

(2) pd2.,N #

di N2 g

(3) (4)

The significance of Eqn (1) is that if a value can be predicted for the power number from the other terms, then this will immediately define the power/speed characteristic of the impeller. The last nine terms of Eqn (1) define the geometric boundary conditions of the system. Therefore, if the expression is used to compare results from systems where all of the geometric ratios remain constant, then the expression can be simplified to: Np = const. (Rei)m(Fr) q

(5)

If various impellers of the same geometry but differing sizes are used in a common tank, the tank/impeller diameter ratio becomes significant: Np = const. (Rei)m(Fr)q(Dt/di) t

(6)

The Reynolds and Froude numbers describe kinematic conditions. The Reynolds number, being a ratio of inertial to viscous forces indicates when either one or the other of these predominates, or when both have effect. The Froude number indicates the extent to which gravity and radial acceleration of the liquid (leading to changes in shape of the liquid surface) influence the system. Thus in baffled conditions the prevention of vortexing causes (Fr) q to tend to unity. 7"s Hence: Np = const. (Rei) m

(7)

or, if various impellers of the same geometry but differing sizes are used in the same tank,

N0

(Dt/di)~ - const. (Rei) m

(8)

In baffled tanks at very high Reynolds numbers (e.g. greater than 10 4) mixing is dominated by inertial effects, and viscosity no longer influences the power number, thus N 0 becomes almost constant. In unbaffied tanks at Reynolds numbers above about 300, the power n u m b e r is affected by vortexing and thus the Froude number becomes important. 7 T o predict power requirements under these conditions, data are required which enable the power number to be compared with the Froude number at constant values of Reynolds number. 1° Thus, in any impeller mixing experiment it is desirable to determine the impeller Reynolds number (Rei). This is simple for Newtonian liquids in which viscosity remains constant, but becomes more difficult when non-Newtonian behaviour is encountered.

3. The effects of the non-Newtonian properties of agricultural slurries on impeller performance

3.1. The non-Newtonian characteristics of slurries Calculation of the impeller Reynolds number for agricultural slurries is complex because these materials are generally, pseudoplastic. 11-17 This characteristic can be

T. R. C U M B Y

163

described conveniently by a power-law relationship between shear stress and shear rate: r = Ky"

(9)

where K (the consistency coefficient) and n (the consistency index) are constant for the liquid concerned. It is also possible to define an "apparent viscosity" /~a such that: r = ~,

(10)

#x~,= f ( r )

(11)

where Hence by eliminating • from Eqns (9) and (10): IXa = K y ( " - I )

(12)

Provided K and n are known, the apparent viscosity may be used to calculate the Reynolds number for a power-law pseudoplastic fluid at a particular shear rate. If the shear rate is known, as in a smooth cylinder rotary viscometer, then K and n can be determined, but in an impeller mixer, the shear rate cannot be obtained analytically because of the complex nature of the flow. Therefore, empirical methods are necessary and as will be shown below, these methods are based on analysis of the performance of impellers in laminar flow regimes. 3.2. The M e t z n e r - O t t o technique f o r determining impeller shear rate and apparent viscosity

A technique for evaluation of impeller shear rate was developed by Metzner and Otto, TM who proposed that the average shear rate at the impeller was directly proportional to the rotational speed of the impeller: Y~,v = K s N

(13)

They verified this assumption by experiment with various impellers in the laminar and transitional flow regimes (1 < Re~ <400), and by the argument that in the laminar flow regime, a multi-bladed impeller approximates to a smooth cylinder, for which Eqn (13) is true (in an infinite volume of liquid). Using this assumption, K~ and apparent viscosity can be determined via an iterative procedure which is described in detail elsewhere. TM Because the procedure of Metzner and Otto TM is quite involved, other authors have suggested simplifications.TM The starting point was Eqn (13) and in addition, a Reynolds number was defined as: pNd 2 Rei -

/Za

(14)

where #,, = r,,,,/r,,,,

(15)

These expressions were combined with the functional relationship for a power law fluid: T = Ky"

(16)

pN(2-")d~ Rei = K(K~)~,,_,)

(17)

to obtain the result:

However, this method still requires laminar flow data for the determination of K~.

164

SLURRY MIXING WITH IMPELLERS,

3.3.

1

The analogy between flow around an impeller and flow in a pipe

An alternative approach to calculating the impeller Reynolds number was developed by Calderbank and Moo-Young. 20 They based their argument on an earlier relationship for determining a generalized Reynolds number for the flow of non-Newtonian liquids in pipes: 21

Re,p=[Dpj,p(8j;Dp)(l-')][

4n ]" (3n + 1)

(18)

By analogy, they proposed that a corresponding expression could be written for agitators such that:

Re:=[d2N(~N)°-")][

4n

]"

(19)

(3n + 1) Values of the constant B were determined by experiment. 2° These results suggested values of 11-6 for turbines and 11-0 for four-bladed propellers, where D,/d~ was greater than 1.5: this was indeed the case in the present work, and therefore these values were assumed here. Although this assumption may have introduced some error into the calculation because in fact six blades were used on all of the impellers, the error is likely to be small since both values are close to the theoretical value for a smooth cylinder, namely, 4st. B will tend towards this figure as the number of impeller blades increases, and so the error will be less than 9%.

4. Previous research on slurry mixing

4.1. Experimental studies There has been very little fundamental research in the field of livestock slurry mixing with impellers. The first significant study~ investigated the relationships between power consumption and rheological properties for single, six-bladed disc turbines and single, three-bladed marine propellers in beef cattle slurry of 8.6% dry matter. This work was confined to a small-scale cylindrical, baffled tank of 0.3 m diameter and 0.03 m 3 volume. The study included both radial flow impellers (disc turbines) and axial flow impellers (propellers). Different sizes of each were used sequentially in the same tank, giving Dt/d i ratios from 4-76 to 2-94 with disc turbines and from 5-00 to 2.94 with propellers. Mixing effects were not measured directly but it was argued that as impeller speed is increased, the point at which the slurry becomes completely mixed corresponds to the attainment of a plateau in the observed value of the power number, and hence to a critical value of impeller Reynolds number (Reic). It was suggested that with a turbine, complete mixing occurs When Rei = 1200, and with a propeller when Rei = 2300. This prediction related to one impeller/tank configuration, namely, Dt/di = 3.85 and was extended to larger systems (up to 1000 m3) to predict mixing power requirements (Table 1). The investigation was not extended to other Dt/di values. The large reductions in specific power requirements predicted from this work were later disproved in another study: 3 beef cattle slurries were mixed in four geometrically similar, cylindridal baffled vessels with volumes ranging between 0.00525 and 0.359 m 3, but with constant Dt/d~ ratios of about 3.0 and Ht/Dt ratios of 1.0. The criterion for complete mixing of the slurry was when the local solids concentration was within +5% of the average solids concentration at all points throughout the tank, and this was expressed in

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165

Table 1 Minimum power requirements for slurry mixiug Specific power, W / m 3 Slurry volume, m 3

Turbine

Propeller

1 5 10 50 100 500 1000

20.6 7-76 5.09 1-92 1.26 0.47 0.31

22-6 8-50 5.58 1-26 1.38 0-52 0.34

terms of a mixing index: Local solids concentration x 100% M~(chc,) - Bulk average solids concentration

(20)

The choice of this particular criterion to define "complete mixing" appears to have been quite arbitrary, but in the absence of any experimental data to indicate the required degree of mixing for satisfactory treatment and/or handling of slurry, this choice is justified. Six-bladed disc turbines and three-bladed marine propellers were used again, but two of each were mounted on the shaft in each experiment. This study showed that the previous assumptions were incorrect and that the critical impeller Reynolds number increased with mixing volume. It was concluded that the minimum specific power for dual propeller mixing decreased with increasing volume according to a power-law such that:

Ps2J

L~J2J

The scale-up exponent (¢) was found to be an empirical function of the power-law consistency index of the slurry. = -(0.219 - 0.052n)

(22)

Hence, for consistency indices of 0.9 and 0.3, the scale-up exponents were - 0 . 1 7 and -0.20, respectively. For mixing using two turbines on one shaft, a similar power-law scale-up relationship was proposed, the exponent being given by: = -(0-212 - 0.029n)

(23)

Thus, the scale-up exponents were respectively: -0-186 and -0-203 for the same consistency indices. Eqns (22) and (23) show that the scale-up exponents were relatively insensitive to rheological properties and hence to solids content. Indeed, this result is to be expected since in fully developed turbulent flow viscous effects should be negligible. 4.2.

Scale-up

No other significant experimental investigations of slurry mixing have been noted, and in particular, no fundamental work has been reported involving large-scale systems approaching practical sizes. However, there is a substantial amount of literature available

166

S L U R R Y M I X I N G WITH I M P E L L E R S , 1

concerning the scale-up of mixers in chemcial and industrial processes and it is possible to draw some parallels with these. In particular, these studies indicate the effects of particle size and density, showing in general that larger and more dense particles increase the mixing power requirement. However, most of these analyses are concerned with dispersions of uniformly-sized particles and since agricultural waste slurries contain particles with a complex range of sizes, shapes and densities, 17 present knowledge is insufficient to enable a fully analytical treatment of slurry mixing. However, some contrasting analytical approaches are described below which represent the main points of the published work. Initially, it is assumed that full geometrical similarity is maintained on scale-up and that the necessary conditions to achieve mixing in a given impeller/tank/slurry combination can be characterized by a critical impeller speed (N¢), i.e. the speed at which all of the solid particles in the slurry just become suspended, for example: 23 Pl)a~3] °5 p-~4/---3 j

[g(Ps-

N¢ = const. L

(24)

Hence for a given slurry in a particular mixing vessel: N¢ = const. (di) -2j3

(25)

This result was also determined separately by others 44 and was verified experimentally in tanks of 0.48m and 4-26m diameter. Other alternative expressions have also been proposed, for example: as [g(ps-

N¢ = const. L

p,)O2t"] °5

-pTd~ ~:i~

J

(26)

hence for a particular slurry in a cylindrical vessel: N¢ = const. (Dr) ~'°5(di)-1"55

(27)

and since geometrical similarity is assumed: N¢ = const. (di) -°'5

(28)

Musil and Vlk as developed an alternative expression based on the assumption that the power number does not vary with impeller speed during turbulent mixing, and that it could be used in the expression: Pi = N p p e N 3 d~

(29)

where/3e is the mean slurry density, expressed as: Po = Pl + ~ ( P ~ - P,) (30) Hence they derived an expression for a critical impeller Reynolds number, and in a subsequent experimental study 27 mixing with a 45 ° pitched blade impeller the following correlation was made: F , ,n dipiD2zo], [ H ] Rei¢ =/~tr/Xept _-ZZ--_7---7./lOll-z-/ + k o ( D t / d i ) ~3 k

where

a s p e [Zol J

(31)

LUtJ

Ncd~p,

Rei¢ - - -

A r ' - a3gpt(Ps - PO

/~t2

(32) (33)

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167

and Rept = particle Reynolds number for sedimentation _

aswavpt

(34)

/zl The term for the thickness of the settled slurry layer, z0, is a constant and therefore its numerical value is incorporated into the dimensionless constants ko and k~. However, the z0 term is required inside the first bracket of the RHS of Eqn (3.1) only to keep the bracket dimensionally consistent with the rest of the equation. Thus the dimensionless IZol term is introduced to eliminate the numerical effect of z0 in the expression. This suggests an unsound application of dimensional analysis and therefore the work must be regarded as entirely empirical. However, from the data presented, it could be shown that: Nc = const. (l~,/pO(di)-s/3(O,) 213

(35)

and by geometrical similarity: N¢ = const. (di) -I

(36)

Thus the literature variously suggests that for geometrically-similar scale-up, the critical impeller speed (N=) is inversely proportional to impeller diameter raised to a power ranging from 0-5 to 1.0. It is possible to relate these findings to the scale-up of specific power requirements for mixing in the following way: at the critical impeller speed, P~c= 2~tNcT~c

(37)

2~rN~T~ ~ ~

(38)

Also,

N~

pNcdi

and therefore substituting for Tic from Eqn (38) into Eqn (37) gives

P~ = Nr~ oN3~d~

(39)

Hence, substituting from Eqn (25) for N~

P~ = N~ o (const. )[ ( di)- ~3]3d~ P~c= N~o(const.) d3

(40) (41)

and dividing by the liquid volume to obtain the specific power dissipation,

es = N ~ o ( c o n s t . ) ( d ~ / v , )

(42)

If constant geometric ratios are maintained during scale-up, then

(d3/v,) = (const.)

(43)

Ps = N~(const.)p

(44)

and so, suggesting that the specific power needed to attain critical mixing is not affected by the scale of the mixer. Applying similar analyses to the results of Rieger and Ditl, ~ and to those of Musil et al.~71eads to the alternative equations: P~ = N~(const.)p(di) 1/2

(45)

Ps = N~( const. )p( di)- ~(p.t/O03

(46)

and

SLURRY MIXING WITH IMPELLERS, l

168

Table 2 Comparison of published scale-up exponents for solids suspension with impellers Label in Fig. 4

Scale-up exponent ~

1

0.167

2

0.087

3 4

0.067 0.000

5 6

-0-033 -0.083

Sources(s)

Comments

Kneule and Weinspach3° Rieger and Ditl~ Ditl and Rieger sl

Equal Froude number

Pavlushenko et al. ~ Judata3 Moro and Tejima~a Burman et al. 24 Weisman and Efferding~ Kolar~ Gibbon and Attwood ~ Rieger and Ditlzs

7

-0-093

Gates et al. 37 Chapman et alfl a

8 9

-0.10 -0.137

10

-0.17

Einenkel and Mersmann3a Weidmann~ Weinspach41 Chena

11

-0-183

12

-0-203

Zweitering'm Nienow'm Chena

13

-0-222

Baldi et al. "~

14 15 16

-0.233 -0.257 -0.333

17

-0.607

Hobler~ Muller and Todtenhaupt'~ Bourne and Sharma47 Musil and VIk2s'z7 Chen and Hashimoto ~

Suspension of large particles (as~It >I 32) Constant P~

Suspension of small particles (as/l~ < 32) Manufacturers' rule Empirical result 0.3-1.83 m diameter tank

Agricultural slurries [Eqn (22), n = 0-9] Empirical result based on dimensional analysis Agricultural slurries [Eqn (22), n = 0.3] Particle suspension is mainly due to eddies of (t~ = as) Constant tip speed Constant Reynolds number, agricultural slurries

T h e expressions q u o t e d in E q n s (45) a n d (46) r e p r e s e n t the u p p e r a n d lower e x t e n t s within the l i t e r a t u r e for e s t i m a t i o n of m i x i n g p o w e r r e q u i r e m e n t s o n scale-up. M a n y o t h e r a u t h o r s have a d d r e s s e d this p r o b l e m a n d their results have b e e n well r e v i e w e d elsewhere. 2 a ~ T h e s e results are s u m m a r i z e d in T a b l e 2 t o g e t h e r with s o m e o t h e r s , a n d are c o m p a r e d with C h e n ' s a ' ~ results by r e a r r a n g e m e n t of E q n (21): Ps = ( c o n s t . ) ( d 3) ~

(47)

T h e c o m p a r i s o n is also m a d e graphically in F i g . 4 which shows that C h e n ' s a e x p e r i m e n t a l data agree with the scale-up line of (~ = - 0 . 1 8 3 ) p r o d u c e d by two o t h e r i n d e p e n d e n t authors. I n view of this close a g r e e m e n t , the findings of these a u t h o r s will also be used to indicate the effect of c h a n g i n g the ( D t / d i ) ratio. U s i n g again the c o n c e p t of critical i m p e l l e r s p e e d , the c o r r e l a t i o n of Z w e i t e r i n g ~ a n d

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169

101

j

l

u k.-

o n,. 10-1

,

i

,

,

,

,

,t

10 °

i

,

,

I

,

Ratio

,i

,

101

. . . . . . .

10 2 of volumes,

.16I 10 3

(v2/vl)

Fig. 4. Comparison of preoious scale-up relationships for specific power and experimental data ( 0 ) from Chen a for dual marine propellers in beef cattle slurry. Annotations are defined in Table 2 Nienow 44 can Nienow44 canbebeexpressed expressed as: as: S

Nc -

0-1 0.2[g(Ps-Pl))°'45cO-13 /zt as ~ Pl pO.XdO.85

(48)

where S is a dimensionless number accounting for the effects of: (a) impeller type, (b) impeller clearance above the tank bottom (Ci) and (c) (Dr~dO ratio. It has been shown by experiment that S = const. (Dt/di) b

(49)

where b = 0.82 for propellers ~ and 1-3 for disc turbines. 44 Hence, to show the effect of changing impeller diameter for a given slurry in a given size of tank, Eqn (48) may be expressed as: Nc = const. (Ot/di)b(di) -°'85

(50)

Thus, substituting Eqn (50) into Eqn (29) and summing the exponents of d~: Pi~ = const. (di) (2"45-3b)

(51)

= const. (di) -°°1 for propellers, and

(52)

= const. (di) -1"45 for disc turbines

(53)

Experimental data 44"44 show that these relationships are valid provided that (Dt/di)> 2 and (DJCi) < 6.

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5. Criteria for m i x i n g

For convenient analysis and interpretation of the results from mixing experiments, it is useful to characterize the degree of mixing of the entire tank contents in terms of a single variable. Moreover, although the lack of data to indicate the required degree of mixing for satisfactory handling and/or slurry treatment still persists, the definition of a single variable would simplify the future establishment of this knowledge. This is not possible using Chen's a Mixing Index, because this only indicates the state of mixing at each of several sample locations. Instead, the single variable must be a function of several local measurements of dry matter. To derive such a variable it is first necessary to consider the changes that occur in the tank during a transition from unmixed to fully mixed conditions. For typical pig slurry, where most of the dry matter is more dense than water, local dry matter concentrations in the upper levels are small in the unmixed state and increase gradually with increasing depth until the top of the settled solids layer is encountered, where the concentration is much greater. Consequently, a selection of samples taken from all depths includes values both higher and lower than the average concentration to be expected if the tank were fully mixed. A Mixing Index must therefore indicate this uneven distribution by responding to the "scatter" of the data. As the contents become more evenly distributed by mixing, the "scatter" becomes less, and as the completely mixed condition is approached, the Mixing Index must be sufficiently sensitive to indicate the small changes that will occur in this region. For these reasons, a Mixing Index was defined in this study as: 1

MI =

i=n5

TS~av)(ns- 1) i=~ ~ (TS~av)-

TS(i)) 2

(54)

and is expressed on logarithmic coordinates to enhance sensitivity in the region of near-complete mixing. Eqn (54) has been derived from a previous expression to quantify mixing4a and is essentially an adaption of the well-known expression for estimating population variance from a sample of results, but includes the (TStav))2 term to normalize the result. By definition, a system is completely mixed when the local solids concentration is the same at every point in the tank. Under these conditions, the Mixing Index has a value of zero. In practice, such conditions may require a power input beyond that absolutely necessary for the process in hand. Instead, a lesser degree of mixing may be acceptable, needing less power. For example, a variation of +10% in the local solids concentration may be acceptable, (i.e. if the fully-mixed dry matter content is 5% by weight, this tolerance would be met if all samples contained between 4-5 and 5-5%). There is a direct Table 3 Relationship between mixing tolerance and Mixing Index (where n, >> 1, so that (n,) (n, -- 1))

Mixing tolerance (+ %)

Mixing Index

0-5 1.0 2-0 5-0 10.0

0.025 × 10 - 3 0.10 × 10 - 3 0.40 x 10-3 2-50 x 10-3 10-00 × 10-3

T. R. CUMBY

171

mathematical relationship between this permitted tolerance and the Mixing Index (Table

3). It can be seen from Table 3 that the + 5 % mixing criterion used by Chen 3 corresponds to a Mixing Index of 2.5 × 10 -3. 6. Conclusions

(1) There has been little fundamental research in the field of livestock slurry mixing. The most significant published research ~ has examined only tank volumes of up to 0.359 m 3, concluding that marine propeller mixers needed less power than did the disc turbines, and that a 100-fold increase in mixing volume reduced the minimum specific power for complete mixing by about 55% (for both impeller types and at every dry matter content). (2) The scale-up exponent relating the specific power necessary to achieve critical mixing to the diameter of the mixing impeller found from Chen's a work compares very well with a value derived from other published work relating to different liquids. ~'4a In slurry, the exponent ranged from - 0 - 5 1 to - 0 . 6 1 , compared with the other value of - 0 . 5 5 . (3) In large-scale impeller mixing experiments where laminar flow data may not be obtained, the most appropriate method of calculating the impeller Reynolds number, avoiding the problems caused by non-Newtonian flow characteristics of slurry, is to use the expression proposed by Calderbank and Moo-Young: 2°

Re[

[d~N(BN)(t-n'][ L

k

4n ]"

JL(an+ 1)J

(19)

(4) A novel mixing index may be defined for slurry as: 1

i=,s

M , - TS~v~(ns- 1) ,=, ~ (TS~v~- T&i~)2

(54)

Acknowledgement

The author wishes to thank Dr N. K. H. Slater of the Unilever Research Laboratory, Vlaardingen, The Netherlands, for his helpful comments regarding the planning, execution and analytical stages of this work. References

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