Choice of an Optimal Agitated Vessel for the Drawdown of Floating

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The choice of an optimal agitated vessel for the drawdown of floating solids Ryszard Wójtowicz Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/ie500604q • Publication Date (Web): 15 Aug 2014 Downloaded from http://pubs.acs.org on August 17, 2014

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Industrial & Engineering Chemistry Research

The choice of an optimal agitated vessel for the drawdown of floating solids Ryszard Wójtowicz* Institute of Thermal and Process Engineering, Division of Industrial Equipment, Cracow University of Technology, Al. Jana Pawła II 37, 31-864 Cracow, Poland. KEYWORDS : Solid-liquid mixing; Multiphase reactors; Stirred vessel; Vibromixer; Reciprocating disc; Drawdown of floating solids.

ABSTRACT: The paper presents experimental results of the drawdown of floating solids in agitated vessels. The effect of an agitator type, its location in the tank and dispersed phase properties on minimum values of agitator speed and power requirements for floating solids suspension was investigated. The results were processed as dimensionless correlations using the Froude number and modified Reynolds number. A comparison was made between classical stirred tanks (with single or dual impeller systems) and a vibromixer agitated by a single, reciprocating disc. The tests showed that given an appropriate selection of disc diameter and off-bottom clearance, the power consumption for drawdown of floating solids in a vibromixer can be halved, compared to a classical stirred tank.

1. INTRODUCTION Solid-liquid multiphase systems with particles of solids with density lower than that of liquid particles (ρsρl) or emulsions . Many papers on dispersion of floating solids are only reports from experiments, often limited to one specific problem, e.g. apparatus type or one of its constituent elements, impellers or baffles, etc. Inevitably, the scope of such papers is quite limited. The literature lacks comprehensive review and comparison papers that would compare several mixing vessels types which could have drawn conclusions useful in dispersion of light particles. There are few papers describing universal correlations of the effect of many process parameters on the optimisation and selection of industrial mixing vessels design.

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Vibromixers with up-and-down moving agitators provide an interesting alternative to classical stirred tanks with 24,26-29 single or dual impeller systems . Preliminary results of research into dispersion of light particles were 30 reported in . The investigations were conducted on a relatively narrow range of parameters and focused mainly on determining average impeller speed guaranteeing drawdown and dispersion of solids. However, the findings were so promising that the range of the study has been widened to include power requirements of the process. Subsequently, the performance of a vibromixer was compared with that of classical stirred tanks with single or dual impeller systems.

3. EXPERIMENTAL Mixing equipment. A suspension of light solid particles was produced in three different agitated vessels; two mechanical stirred tanks with one (Fig. 1a) or two rotating impellers (Fig. 1b) on a single shaft, and in a vibromixer with a solid, reciprocating disc of a varying diameter (Fig. 1c). All the three agitated vessels had cylindrical tanks with a flat bottom of the same internal diameter of T=0.286 m. The tanks were always filled to the constant height of H=T. The vessels with rotating impellers were additionally equipped with four flat baffles of full length and standard width B=T/10.

Figure 1. Investigated stirred tanks and agitators: a) stirred tank with a single impeller, b) stirred tank with a dual impeller system, c) vibromixer, d) Rushton turbine RT, e) down-pumping pitched blade turbine (α=45º) PBT(D), f) up-pumping pitched blade turbine (α=135º) PBT(U), g) reciprocating disc.

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Impellers were selected to generate different flows. One impeller produced a typical radial flow - Rushton turbine RT (Fig. 1d) while the other two impellers generated axial flow - down-pumping pitched blade turbine (α=45º) PBT(D) (Fig. 1e) and up-pumping pitched blade turbine (α=135°) PBT(U) (Fig. 1f). The turbines had standard geometry: diameter D=T/3, blade height a=D/5 and blade width: b=D/4 (Rushton turbine) or b≈D/2.5 (pitched blade turbines). The Rushton turbine also had standard disc dimensions: diameter dt=2/3D, thickness gt≈D/30.

dp = 3

6 ms (1)

πρ s

where: ms is the mean mass of solids. The mass fraction of solids varied in the range of cm=0.02÷0.15. All the analysed agitated vessels produced a suspension containing the same types of light particles, to ensure reliability of results.

In the single impeller (Fig. 1a) the impeller off-bottom clearance varied in the range h=(1÷2)D. The dual impeller system (Fig. 1b) always consisted of identical impellers of the same type: RT-RT, PBT(D)PBT(D) or PBT(U)-PBT(U). The lower impeller was always positioned at the same height h=D , whereas the upper impeller was positioned at the distance from the lower one of ∆h=(0.5÷1)D. The vibromixer (Fig. 1c) was fitted with solid (without perforation) disc with varying diameters: D1=0.260; D2=0.238; D3=0.220 and D4=0.204 m, selected so that T/D1..4=1.1÷1.4. The thickness of the discs was gd=0.003 m. The disc off-bottom clearance was in the range h=(0.25÷0.75)H and vibration amplitude in the range A=0.02÷0.05 m was changed at spacing intervals 0.005 m.

Solids used. Distilled water (ρl=998 kg/m , ηl=0.001 Pa·s 3

(at 20˚C)) was used in the experiments as a continuous phase and three different types of solids with density smaller than water density were used as the dispersed phase: granulated polyethylene MALEN FGNX (Fig. 2a) and recycled granulated plastics, varying in shape and properties, PLAST 1 (Fig. 2b) and PLAST 2 (Fig. 2c).

Figure 3. Shape and dimensions of particles: a) MALEN, b) PLAST 1, c) PLAST 2.

Table 1. Characteristics of solid particles Solids

MALEN PLAST 1 PLAST 2

hs [mm]

4 4.13 5.27

ds [mm] l1 [mm] 4.35

l2 [mm] 3.52 2.33 3.41 2.42

dp [mm]

ms [g]

ρs 3 [kg/m ]

4.11 3.46 4.67

0.0412 0.0197 0.0484

916.7 883.3 906.4

Criteria for drawdown of light particles (nmin, fmin, Pmin, Pv min). Criteria used and recommended by most 4-17 researchers were used to determine the suspension moment. They are identical with the Zwietering criterion 31 used for classical suspensions with settling particles.

Figure 2. Solids used in the experiments: a) MALEN, b) PLAST 1, c) PLAST 2.

The shape and properties of used solids are shown and listed respectively in Fig.3 and Tab.1. The equivalent diameter of solid particles dp was calculated from a formula 3 recommended in relation:

The moment of full dispersion of particles and suspension formation was defined as a state of such minimum impeller speed nmin or disc vibration frequency fmin at which solid particles do not remain at the liquid surface longer than 1-2 s, are pulled into the liquid and get dispersed throughout the whole volume of the tank forming a uniform suspension. The layer of particles floating on the liquid surface completely disappears. The state was determined visually increasing impeller speed or disc frequency (after each 5 1/min for impellers and at 0.1 Hz step for reciprocating agitators) until the state of suspension has been reached. To minimise errors in determining precise values of nmin, fmin, every measurement was repeated three times and results were averaged. The same pro16, 17 cedure was followed in . Final correlations presented in this paper estimate minimum impeller speeds with the



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accuracy of ±5 1/min and minimum disc frequency with ±0,1 Hz. Selected examples of further dispersion stages of light particles in a liquid, spanning from the initial state (n=0 or f=0) to a pre-defined boundary state of full dispersion(n=nmin or f=fmin), are presented in Fig. 4 and in en-

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closed video clips (see Supporting Information). As can be seen, despite totally different flow regimes generated by impellers, the process of dispersing floating solids and their drawdown into the liquid proceeded in a very similar manner while the moment of suspension formation was easy to identify.

Figure 4. Successive stages of drawdown of floating solids from n=0 (f=0) to n=nmin (f=fmin): single Rushton turbine RT, h=d (top row); dual impeller system RT-RT, ∆h=d (middle row); reciprocating agitator D1=0.260 m, h=0.5H, A=0.04 m (bottom row); (PLAST 1).

Minimum mixing power Pmin (Pv min), defined as power input necessary to bring about a suspension, was the second examined parameter.

uct of the total force acting on the disc F(t) and the instantaneous disc velocity wmin(t):

For impellers, the minimum power consumption was calculated (Eq.2) on the basis of measurements of an impeller speed and torque on the shaft. The impeller speed was measured with photoelectric sensor Lutron DT-2236 and torque on the shaft using a non-contact method and an inductive torque meter Vibro-Meter TG0,2 B.

Pv min = f min

Pmin = 2πnmin (M n − M 0 )

(2)

where: Mn – is a torque on the shaft during liquid mixing, M0 – is a torque on the shaft during idle run. The calculations of minimal power input for light particles suspension formation in the vibromixer are more complex. It was calculated as a time integral of the prod-

1 / f min

∫ F ( t )w

min

( t )dt

(3)

0

The force function was determined using a piezoelectric sensor Kistler 9301B, mounted in the vibrating rod. For description of instantaneous disc velocity the following equation was used:

A   wmin ( t ) ≡ 2πAf min  sin( 2πf min t ) + sin( 4πf min t ) 2L  

(4)

where: L is the total connecting-rod length (L=0.25 m), fmin is the minimal disc frequency measured by photoelectric sensor Lutron DT-2236. Similarly to classical stirred tanks with impellers, power requirements were measured in an empty tank, without


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any liquid. The effective mixing power was the difference between that determined during liquid mixing and that measured during idle run. Other details of measuring 24, power input for the vibromixer are reported elsewhere 32, 33.

Result interpretation methodology and dimensionless numbers. The quantitative analysis of dispersion of floating solids was conducted using physical modelling results, processed with statistical methods. The values of constants and exponents in correlations were determined with Statistica 9.1 and the Levenberg–Marquardt algorithm (nonlinear estimation). The loss function was estimated using the least-squares method. Estimation precision was assessed by calculating the coefficient of determination (R), the proportion of explained variance (E.V.) and the mean relative error ( ∆ ). In each calculation distribution of residues, correlation between experimental and estimated values and significance of determined coefficients were double checked. The following dimensionless numbers were used in physical modelling: •

for stirred tanks with one or two turbine impellers:

- to express minimum impeller speed (nmin), the Froude number was used to indicate the influence of gravity on forces significant in the process of dispersing light solids:

Frmin =

2 nmin D g

(5)

- to express minimum mixing power in correlations (Pmin), a modified Reynolds number was used:

RePmin / V

 P =  min  V

•

4 2  d p ρl    3  ηl 

1/ 3

(6)

for the vibromixer with a reciprocating disc:

- to express minimum disc frequency (fmin), the Froude number defined for vibro-mixing as:

Frv min =

wmin gD

(7)

where: w min mean disc velocity for frequency fmin: (8)

wmin = 4 Afmin

- and in power requirement correlations (Pv min) a modified Reynolds number was used:

RePv min / V

 Pv min  d p 4 ρl 2   =   3   V  ηl 

1/ 3

(9)

The Froude number was used by many of authors 4,8,12,13,20,34 for the drawdown of floating solids data processing. The modified Reynolds number was created on the basis 35 of Kolmogoroff theory . It assumes that dispersion of solid particles is caused by turbulence in a small volume,

with an assumption that particle size is much larger than 35,36 an internal turbulence scale . The modified Reynolds number with a power per unit volume term gives proposed correlations a physical meaning, regardless of a mixing equipment (classic stirred tank, vibromixer). This 37,38 number was used earlier, e.g. in .

4. RESULTS AND DISCUSSION Stirred tank with a single impeller. The investigated impellers generated different flow regimes. Measurements showed that during dispersion of light particles in stirred tanks with a single impeller both impeller type and impeller off-bottom clearance play a decisive role. The size of suspended particles, their physical properties and mass fraction have a minor influence. The lowest impeller speed (Figs. 5 a,b) required to form a suspension was determined for the Rushton turbine producing radial flow out of all the investigated impellers, at their off-bottom clearance of h≤1.5d. By contrast, the highest impeller speeds at the whole range of off-bottom clearances were recorded for the down-pumping pitched blade turbine PBT(D), generating axial flow. For the uppumping impeller with pitched blades results were different. At the off-bottom clearance of h≤1.5d, the minimum impeller speed was slightly higher than that required for the Rushton turbine. However, when the impeller was positioned higher, closer to the surface of phase separation, upward liquid jets intensified circulation in the upper part of the tank, which led to faster pull of particles from the surface into the liquid. In this case, suspension was formed more quickly and minimum impeller speeds were the lowest compared to all the other agitators (Fig. 5c). It should be emphasised, though, that the effectiveness of mixing during dispersion of light particles does not only depend on the minimum required impeller speed (nmin), but most of all on power requirement of the process. Not every impeller with low minimum speed has low power requirements. This is due to varied effectiveness of energy transfer depending on impeller type. Consequently, the author’s primary criterion of impeller performance was minimum mixing power and not minimum impeller speed required for the state of complete suspension. Only such comparison can be used to determine which impeller, at which position in the tank is most useful in drawdown of floating solids. For stirred vessels with a single impeller, the lowest power requirements during dispersion of light particles was recorded for the up-pumping pitched blade turbine PBT(U) (Figs. 5d-f). The minimum power Pmin for this impeller was the lowest regardless of its off-bottom clearance, type of suspended particles and their mass fraction. The turbine’s high performance, compared to other agitators, can be particularly observed at large off-bottom clearance (h≈2d) (Fig. 5f). The down-pumping pitched blade turbine PBT(D) had the highest power requirements, which were approxi-



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mately 1.5 to 3.5 times larger than the up-pumping pitched blade turbine (Figs. 5d-f). Minimum power requirements for the third investigated impeller, the Rushton turbine, were found to be between a)

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those of pitched blade turbines, but even here performance increase was observed when off-bottom clearance was larger. d)

b)

e)

c)

f)

Figure 5. Changes of a minimum impeller speed nmin (left column) and minimum power per unit volume Pmin/V (right column) in a stirred tank with a single impeller for different impeller off-bottom clearances h: a) h=D, b) h=1.5D, c) h=2D, e) h=D, f) h=1.5D, g) h=2D (MALEN).


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The above experimental results are generally consistent 3, 7, 12, 17. with findings reported by other authors The quantitative analysis of the effect of process parameters on conditions required for suspending light particles was conducted using the following correlations: - for minimum impeller frequency:

Frmin

h = C1 ⋅   D

a1

 dp    D

b1

(10)

- for minimum mixing power Pmin:

RePmin / V

1/ 3

h = C2 ⋅   D

nmin~dp-0.191 -

-0.102

(12)

which for one impeller is presented in Fig. 6.

c1

 ∆ρ  d    ρ  cm  l 

1

 Pmin  d p 4 ρ l 2  =    3  V  ηl 

A quantitative and qualitative analysis of particle size provides interesting results. The smaller particles, the more difficult it is to suspend them in a liquid. The relation between minimum impeller speed required to form a suspension and particle size can be given by:

a2

 dp  D

  

b2

c2

 ∆ρ  d2    ρ  cm  l  (11)

The values of constants and exponents from Equations (10) and (11) for all the impellers are presented in Tables 2 and 3. The equations are valid respectively for Frmin = 0.300.71 (RT); 0.60-1.73 (PBT(D)); 0.26-1.12 (PBT(U)) (eq.10) and for RePmin/V = 467-790 (RT); 453-920 (PBT(D)); 304923 (PBT(U)) (eq.11) in the same range of process parameters: h/D=1-2; dp/D=0.0364-0.0491; ∆ρ/ρl=0.082-0.115 and cm=0.02-0.15. The determined values confirmed a tendency which was described earlier that impeller type and its off-bottom clearance play a major role in the unit’s performance.

The tendency described by Eq. (12) is not consistent with findings reported earlier by other authors who maintain that drawdown of larger particles requires higher impeller 7 34 speeds and nmin~dp0.032 or nmin~dp0.31 . However, the authors had conducted their studies on substantially smaller particles, with average dimensions in the range dp=0.2-0.7 mm and dp=0.3-0.9 mm which may have caused discrepancies. For larger particles, with sizes comparable with those investigated in the present study, the literature offers similar relations with negative exponent values. They can 39, 40 be even lower than those in Eq. (12), e.g. -0.52 . 7, 34

(higher mixing power necThe tendency reported in essary for drawdown of larger particles) was, however, observed in our vibromixer and it is described later in the paper.

This is most evident for the up-pumping pitched blade turbine PBT(U). The size, density and mass fraction of solid particles play a minor role (Tab. 2, Tab. 3). Table 2. Values of constants and exponents in Eq. (10) Impeller

C1 [-]

a1 [-]

b1 [-]

c1 [-]

d1 [-]

Determination coefficient R [-]

Variance explained V.E. [%]

Mean value of relative error

Rushton turbine RT

1.027

-0.279

-0.102

0.208

0.042

0.94

88.5

3.9

Down-pumping pitched blade turbine PBT(D) Up-pumping pitched blade turbine PBT(U)

0.972

-0.448

-0.191

0.170

0.043

0.92

85.7

5.2

0.909

-0.585

-0.133

0.112

0.051

0.91

83.2

8.9

∆ [%]

Table 3. Values of constants and exponents in Eq. (11) Impeller

C2 [-]

a2 [-]

b2 [-]

c2 [-]

d2 [-]




Rushton turbine RT

7.671·10

4

-0.264

1.291

0.241

0.042

0.97

94.1

3.3

4

-0.351

1.221

0.260

0.0213

0.96

92.7

3.9

-0.637

1.316

0.016

0.039

0.93

85.2

8.8

Down-pumping 6.799·10 pitched blade turbine PBT(D) 4 Up-pumping 4.914·10 pitched blade turbine PBT(U)


∆ [%]


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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7e). This may be due to the fact that two impellers do not create one circulation loop but two separate loops with a 3 visible reverse flow which instead of optimising the process slows it down. A comparison of the effectiveness of the dual impeller system with one impeller in optimal position for suspending light particles (h=2D) leads to interesting conclusions. The results of comparative analysis are shown in Fig. 7. The introduction of an additional impeller, even for the largest distance between two impellers (∆h=2D), does not improve performance. Minimum impeller speed values for RT-RT and PBT(U)-PBT(U) systems with two impellers were always higher than those for the systems with a single impeller (Fig. 7a,c). However, the down-pumping PBT(D)-PBT(D) system with two impellers had values only slightly lower than those for the system with one impeller (Fig. 7b). Figure 6. Influence of dp/D ratio, mass fraction of solids cm and impeller off-bottom clearance h on the value of

Frmin

(Stirred tank with a single impeller, up-pumping pitched blade turbine PBT(U)).

Stirred tank with a dual impeller system. At the successive stage of the tests, a second impeller was added to the stirred vessel. Both impellers were positioned in different distances to each other on a commonly shared powered shaft. The aim of the experiments was to determine whether or not the second impeller improved performance. The operation of a system consisting of two impellers was assessed, as for the stirred vessel with one impeller, by determining minimum impeller speed (nmin) required to form a suspension and power requirements (Pmin). Fig. 7 shows changes of minimum speed and minimum mixing power required for dispersion of solid particles in the system. For comparison purposes, an additional curve was added to the graph illustrating the performance of a single impeller in optimal position (h=2D) (section 4.1). Experiments showed a marked influence of the second impeller on particle dispersion. It significantly changes flow regime in the apparatus and induces additional circulation loops. Consequently, the mechanism of light particle drawdown and dispersion is noticeably different. Liquid flow in the tank mostly depends on the distance between impellers (∆h) as experiments showed that changes in this parameter may affect the process in a varied way (Fig. 7). In RT-RT and PBT(U)-PBT(U) systems (Figs.7a,c) the smaller impeller off-surface clearance the lower impeller minimum speed and the faster suspension is formed. Similar tendencies were also formulated for minimum mixing power Pmin (Fig. 7d,f). Another tendency was observed, though, for two downpumping PBT(D)-PBT(D) axial impellers. While an increase of the distance between impellers markedly worsened performance, it was also associated with higher minimum impeller speed (Fig. 7b) and mixing power (Fig.

This tendency is more marked when analysing mixing power (Figs. 7d-e). Power requirements for all the systems with two impellers were higher than those for one impeller, and the discrepancies for the PBT(U)-PBT(U) system amounted to four-fold differences (Fig. 7f). Experiments confirmed that the introduction of an additional impeller did not improve performance. Contrary to that, power requirements increased without any improvement of particle drawdown and dispersion effectiveness. In dual impeller systems, power requirements were always higher than those in a system with one impeller regardless of impeller model and particle type. The results were averaged as dimensionless correlations, as was the case with results determined for the single impeller. The only difference is that instead of impeller off-bottom clearance (h), the distance between impellers was used (∆h):

Frmin

RePmin / V

 ∆h  = C3 ⋅    D  P =  min  V

a3

 dp    D

4 2  d p ρl    3  ηl 

b3

1/ 3

c3

 ∆ρ  d    ρ  cm  l 

3

 ∆h  = C4 ⋅    D

(13)

a4

 dp  D

  

b4

c4

 ∆ρ  d4    ρ  cm  l  (14)

The values of constants and exponents in Equations (13) and (14) for different systems are presented respectively in Tables 4 and 5. The equations are valid respectively for: Frmin = 0.31-0.6 (TTP-TTP); 0.42-1.33 (PBT(D)-PBT(D)); 0.35-0.66 (PBT(U)-PBT(U)) (eq.12) and RePmin/V = 540-810 (RT-RT); 566-903 (PBT(D)-PBT(D)); 567-805 (PBT(U)-PBT(U)) (eq.13) in the same range of process parameters: ∆h/D=0.5-1; dp/D=0.0364-0.0491; ∆ρ/ρl=0.082-0.115 and cm=0.02-0.15.


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a)

d)

b)

e)

c)

f)

Figure 7. Changes of a minimum impeller speed nmin (left column) and minimum power per unit volume Pmin/V (right column) in a stirred tank with a dual impeller system for different distance between impellers ∆h: a) RT-RT, b) PBT(D)-PBT(D), c) PBT(U)-PBT(U), e) RT-RT, f) PBT(D)-PBT(D), g) PBT(U)-PBT(U) (MALEN).



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Page 10 of 16

Table 4. Values of constants and exponents in Eq. (13) Dual impeller system

C3 [-]

a3 [-]

b3 [-]

c3 [-]

d3 [-]




Rushton turbine – Rushton turbine RT-RT Down-pumping pitched blade turbine pitched blade turbine PBT(D)-PBT(D) Up-pumping pitched blade turbine pitched blade turbine PBT(U)-PBT(U)

1.096

-0.052

-0.177

0.403

0.049

0.91

82.3

3.6

0.913

0.233

-0.352

0.362

0.096

0.95

91.1

3.6

0.958

-0.242

-0.027

0.126

0.042

0.89

86.7

7.3

∆ [%]

Table 5. Values of constants and exponents in Eq. (14) Dual impeller system

C4 [-] 4

Rushton turbine – 2.297·10 Rushton turbine RT-RT 4 Down-pumping 2.219·10 pitched blade turbine pithed blade turbine PBT(D)-PBT(D) 4 Up-pumping 1.935·10 pitched blade turbine pitched blade turbine PBT(U)-PBT(U)

a4 [-]

b4 [-]

c4 [-]

d4 [-]




-0.041

0.943

0.187

0.062

0.96

92.4

3.5

0.297

0.819

0.291

0.051

0.92

84.3

7.7

-0.232

1.061

0.064

0.046

0.95

90.4

5.2

Regression coefficient values confirmed the earlier described influence of distance between impellers - increase of ∆h leads to decrease of nmin and Pmin for RT-RT and PBT(U)-PBT(U) dual impeller system with the reverse true for PBT(D)-PBT(D) impellers.

Figure 8. Influence of density modulus ∆ρ/ρl, mass fraction of solids cm and distance between impellers ∆h on the value of

Frmin (Stirred tank with a dual impeller system RT-RT).

∆ [%]

Compared to the single impeller, the influence of phase density differences is slightly higher (Fig. 8), while the effect of mass fraction of the solid phase is comparable. The influence of particle size on suspension formation is more varied; it is marked and significant for PBT(D)PBT(D) impellers and relatively insignificant for uppumping PBT(U)-PBT(U) systems. However, in all systems, as was the case for the single impeller, drawdown of smaller particles requires higher impeller speeds and higher power inputs.

Vibromixer. Liquid flow regime in a vibromixer is different from that generated by one or two impellers. A reciprocating disc generates a specific and more complex flow. The up-and-down reciprocating motion produces strong liquid pulsations, large-scale axially symmetric ring vortices and intensive liquid flow through the gap between the 29, 41 disc and the tank wall. CFD numerical simulations and measurements using Particle Image Velocimetry 42 (PIV) demonstrated that axial flow is a major flow pattern in vibromixers, with radial flow shortly to follow. Circular flow, common in stirred tanks with impeller systems, is much weaker in units with a reciprocating disc. Particles are actively drawdown into the liquid practically across the whole free space of a vibromixer owing to disc-induced strong axial flow, contrary to circular flow with central vortex in classical stirred tanks.


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The specific radial-axial flow generated by a reciprocating disc seems to be therefore particularly useful for drawdown and dispersing light particles initially floating on the liquid surface. This is, without doubt, a great advantage of vibromixers used for dispersed systems.

Minor influence is exerted by other process parameters, including agitator off-bottom clearance (Fig. 11), particle diameter, changes of phase density difference, and mass fraction of the solid phase.

It is worth noting that regardless of disc amplitude, total drawdown and dispersion of light solids always occurs at constant mean agitator velocities w min . This tendency was shown in Fig. 9.

Figure 11. Influence of h/D ratio and disc diameter on the value of Frv min (Vibromixer, A=0.035m, cm=0.1, MALEN).

Figure 9. Mean disc velocity w min corresponding to fmin as a function of disc amplitude A (Vibromixer, D1=0.260 m (T/D1=1.1), h=0.5H).

The pattern was found to hold for all the investigated reciprocating discs, their positions and dispersed particles. A similar tendency was observed during emulsion formation in a vibromixer, which was reported elsewhere 24 . Measurements showed that, as for classical suspen43

sions , disc mean velocity w min required for dispersion of light particles mainly depends on its diameter (Fig. 10).

The results of light particles dispersion in the vibromixer were determined using dimensionless numbers defined in (7) and (9). My own correlations (15) and (16) take a slightly different form for stirred tanks and vibromixers. The latter form additionally includes the term T/D, which considers the effect of disc diameter: a5

Frv min

ReP

v min

 dp  D

  

/V

c6

T   h  = C5 ⋅     D D

b5

 dp    D

 Pv min  d p 4 ρ l 2   =   3   V  η l  d6

1/ 3

c5

d5

 ∆ρ  e    ρ  cm  l 

5

a6

T   h  = C6 ⋅     D D

(15)

b6

(16)

 ∆ρ    cm e ρ  l 

6

The values of constants and exponents in Equations (14) and (15) are presented respectively in Tables 6 and 7. The correlations are valid for: Frv min = 0,002-0,1 (eq.15) and RePvmin/V = 97-870 (eq.16), in the following range of process parameters: T/D=1-1.4; h/D=0.28-1.05; dp/D=0.13-0.23; ∆ρ/ρl=0.082-0.115 and cm=0.02-0.15.

Figure 10. Influence of disc-to-tank diameter ratio T/D and mass fraction of solids cm on the value of Frv min (Vibromixer, A=0.035m, h=0.5H, MALEN).

Regression coefficient values demonstrate that changes of disc off-bottom clearance in a vibromixer affect the process, similarly to classical stirred tanks where changes of impeller clearance have the same effect. Similar correlations are true for the effect of phase density changes and mass fraction of the solid phase, particularly compared to the dual impeller system. The effect of suspended particle size in the vibromixer has a different tendency to that observed in the stirred tanks. The drawdown of larger particles in the vibromixer with a reciprocating disc requires, unlike in the stirred tanks, higher power inputs and mean disc velocities:



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(17)

w min ~dp0.48

Page 12 of 16

That is why drawdown and dispersion of particles with larger size requires more power input.

It may be due to the fact that the mechanism of solids drawdown in vibromixers is different from that in classical stirred vessels. A reciprocating disc does not generate a central vortex and solids drawdown is induced by axial flow. Therefore, the major problems are surface tension and buoyant forces that have to be overcome. Buoyant forces are to a large extend dependent on particle size.

In classical stirred vessels, particles tend to form local agglomerates in the vicinity of walls, behind baffles and near the shaft that remain on the surface and are hard to pull inside the liquid. The tendency was confirmed in my own experiments. The smaller the particles, the harder they are to drawdown. This is why the reverse is true for classical stirred vessels.

Table 6. Values of constants and exponents in Eq. (15) C5 [-]

a5 [-]

b5 [-]

c5 [-]

d5 [-]

e5 [-]




1.298

3.853

-0.277

0.481

0.485

0.069

0.94

89.3

8.9

∆ [%]

Table 7. Values of constants and exponents in Eq. (16) C6 [-] 4

7.313·10

a6 [-]

b6 [-]

c6 [-]

d6 [-]

e6 [-]




0.354

-0.338

1.381

0.081

0.027

0.86

76.3

13.3

Performance comparison of the analysed agitated vessels. Effectiveness plays a decisive role in the selection of a concrete mixing system. This decision has a bearing on process costs and performance. A precise comparison of the effectiveness of the investigated mixing vessels would be quite difficult owing to design differences, different agitator geometry, operation regime and criteria numbers. Performance comparison of the analysed agitated vessels was conducted on the basis on power requirement analysis for the process in question. An example for a comparison of power required to disperse particles is shown in Fig. 12.

∆ [%]

Discs with varying diameters were used in the vibromixer. The single up-pumping pitched blade turbine PBT(U) positioned h=2D off the bottom was most effective in the stirred tanks. As can be seen, the effectiveness of the vibromixer is different compared to that of a classical stirred tank, depending on disc diameter and its off-bottom clearance. However, for two discs with the largest diameters (D1=0.260 [m] and D2=0.238[m]), even at the clearance of h=0.5H, power inputs required to disperse floating solids are markedly lower than those determined for the PBT(U) impeller. It is practically noticeable for the whole range of particle mass fraction. A more in-depth comparison of all the investigated discs and particle types was conducted by determining power ratio:

Pv min / V Pmin PBT ( U ) / V

=K

(18)

where: Pv min/V – minimum power per unit volume for vibromixer, Pmin PBT(U)/V – minimum power per unit volume for uppumping pitched blade turbine at h=2D. The mean values of constant K are presented in Table 8.

Figure 12. Comparison of discs to impeller efficiency (Vibromixer, A=0.035m, h=0.5H; Single up-pumping pitched blade turbine PBT(U), h=2D; MALEN).

For discs with the two largest diameters (D1 and D2), for the height in tank h=(0.5÷0.75)H, the vibromixer is much more effective than a classical stirred vessel with the uppumping pitched blade turbine PBT(U). Mean values of power requirements for dispersion of particles in the vibromixer, with disc diameter of D2=0.238 m, were lower


Page 13 of 16

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by approximately 11÷30% (h=0.5H) and 17÷21% (h=0.75H) than those for rotating impeller. For the largest disc diameter of (D1=0.260 m), the differences were even larger and amounted respectively to 23÷39%, at its height of h=0.5H and even to 43÷50% when its position to the tank bottom was h=0.75H.

The above analysis of power requirements for the investigated agitated vessels clearly shows that the vibromixer with the solid, single reciprocating disc is the more effective mixing piece of equipment in forming suspensions of light particles than the classical stirred vessel with a single or dual impeller system.

Table 8. Mean values of constant K in Eq. (18) (configurations for which vibromixer is more effective than classical stirred tank with an impeller are bold printed) Disc

Disc clearance h=0.25H

D1=0.260 [m]

D2=0.238 [m]

D3=0.220 [m]

D4=0.204 [m]

MALEN

PLAST 1

PLAST 2

2.347

1.702

1.287

h=0.5H

0.613

0.769

0.669

h=0.75H

0.566

0.550

0.499

h=0.25H

2.927

2.337

2.196

h=0.5H

0.701

0.886

0.874

h=0.75H

0.792

0.827

0.812

h=0.25H h=0.5H h=0.75H

3.383 1.214 1.081

3.725 1.790 1.541

3.097 1.267 1.204

h=0.25H

3.923

4.522

4.052

h=0.5H

1.746

2.712

1.938

h=0.75H

1.573

3.177

1.754

Conclusions. The present study has demonstrated that the agitated vessels with turbine impellers or reciprocating agitators are effective apparatuses in forming suspensions of light particles. Their effectiveness depends on a range of parameters, but most of all on agitator type and its/their position(s) inside the vessel.

Power input analysis of the process revealed that mixing effectiveness did not increase following the introduction of the second, additional impeller. Power requirements of the system with two impellers were always higher than those of the single impeller system regardless of impeller type and type of dispersed particles.

The conducted experiments demonstrated that:

3) The mechanism of dispersing light particles in the vibromixer was different than that in classical stirred vessels since the reciprocating disc mainly produced axial circulation which seems to be particularly effective for solids drawdown inside the liquid. Light particles are always dispersed at constant, mean disc velocities regardless of its amplitude. The experiments also revealed that mean agitator velocity and minimum power input required for a state of suspension mainly depend on disc diameter. Other process parameters, including disc offbottom clearance, particle diameter, density difference and mass fraction of the solid phase play a minor role.

1) In the stirred tank with a single turbine impeller, minimum impeller speed nmin and equivalent minimum power input Pmin, required for drawdown and dispersion of floating solids mainly depend on impeller type and its offbottom clearance. The physical parameters and properties of dispersed particles and their mass fraction play a minor role. Based on power input assessment for the process, the up-pumping pitched blade turbine PBT(U) was found to be the most effective out of all the investigated single impellers regardless of their off-bottom clearance. The up-pumping pitched blade turbine high performance, compared to other impellers, is particularly noticeable for its relatively large off-bottom clearance. The downpumping pitched blade turbine PBT(D) had the highest power requirements. 2) The conducted experiments demonstrated the marked effect of the second, additional impeller on dispersion of light particles in the vessel. Minimum impeller speed values required for the state of suspension were determined for two RT-RT impellers whereas the lowest minimum power input values were recorded for the PBT(U)PBT(U) impeller system. It was found during the experiments that both values declined when the distance between the impellers increased.

4) The experiments conducted in a wide range of process and design parameters revealed that the vibromixer with a solid disc with diameter T/D=1.1÷1.2 and with off-bottom clearance in the range h=(0.5÷0.75)H is the most effective apparatus to form suspensions of light particles. Power input estimation demonstrated that for this configuration of the vibromixer with a reciprocating disc its power requirements may be even two-times lower than those of a classical stirred tank with a up-pumping pitched blade turbine PBT(U) at optimal off-bottom clearance of h=2D.



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Page 14 of 16

ASSOCIATED CONTENT

t

= time, s

Supporting Information

T

= internal tank diameter, m

V

= overall suspension tank/vibromixer, m3

V.E.

proportion of variance explained, %

w

= disc velocity, m s-1

w min

= mean disc velocity corresponding with minimum disc frequency fmin , m s-1

To view video clips illustrating drawdown of solids in the investigated agitated vessels, go to http:// Example graphs of disc velocity function, force acting on the disc and power input function for the vibromixer are available in the Supporting Information.

AUTHOR INFORMATION

volume

Corresponding Author

Greek Letters

* E-mail: [email protected]

α

blade angle, °

∆

mean value of relative error, %

NOMENLATURE a

= blade height, m

∆ρ

density difference (∆ρ=ρl-ρs), kgm-3

a ..e1..6

= exponents in Equations (1)÷(6)

η

dynamic viscosity, Nsm-2

A

= disc amplitude, m

ρ

density, kgm-3

b

= blade width, m

Indices

B

= baffle width, m

C1..6

= constants in Equations (1)÷(6)

cm

= mass fraction of solids

dp

= equivalent diameter of solids, m

dt

= Rushton turbine disc diameter, m

D

= impeller diameter, m

D1..4

= disc diameter, m

Dimensionless Numbers

fmin

= minimal disc frequency for light solids suspension formation, Hz

Frmin =

F

= force acting on the disc, N

g

= gravitational acceleration, m s-2

gd

= disc thickness, m

gt

= Rushton turbine disc thickness, m

h

= impeller clearance, disc off-bottom clearance, m

∆h

= distance between impellers for dual impeller systems, m

in

stirred

denotes liquid phase

l min

minimum value

max

maximum value

s

denotes solid phase

v

denotes vibro-mixing

Frv min =

RePmin / V

ReP

v min

= minimum value Froude number

2 nmin D g

= minimum value of Froude number for vibromixer

wmin gD

 P =  min  V

/V

of

4 2  d p ρl    3  ηl 

= minimum value of modified Reynolds number

1/ 3

 Pv min  d p 4 ρl 2   =   3   V  ηl 

1/ 3

= minimum value of modified Reynolds number for vibromixer

H

= liquid height, m

L

= total connecting-rod length, m

ms

= mean mass of solids, kg

Impellers

nmin

= minimum impeller speed for light solids suspension formation, s-1

RT

= Rushton turbine

PBT(D)

= down-pumping pitched blade turbine

Mn

= torque on the shaft during liquid mixing, Nm

PBT(U)

= up-pumping pitched blade turbine

M0

= torque on the shaft during idle run, Nm

RT-RT

Pmin/V

= minimum power per unit volume, W m-3

= Rushton turbine - Rushton turbine dual impeller system

Pv

= minimum power per unit volume for vibromixer, W m-3

PBT(D)PBT(D)

= down-pumping pitched blade turbine down-pumping pitched blade turbine dual impeller system

PBT(U)PBT(U)

= up-pumping pitched blade turbine - uppumping pitched blade turbine dual impeller system

min/V

Pmin

= minimum power input for light solids suspension formation in a stirred tank, W

Pv min

= minimum power input for light solids suspension formation in a vibromixer, W

R

= determination coefficient


Page 15 of 16

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Choice of an Optimal Agitated Vessel for the Drawdown of Floating

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