8/12/2019 Design Blade

1/7

Hindawi Publishing CorporationInternational Journal of Chemical EngineeringVolume 2012, Article ID 106496, 6 pagesdoi:10.1155/2012/106496

Research ArticleCorrelation of Power Consumption forSeveral Kinds of Mixing Impellers

Haruki Furukawa, 1 Yoshihito Kato, 1 Yoshiro Inoue, 2 Tomoho Kato, 1 Yutaka Tada, 1 and Shunsuke Hashimoto 2

1 Department of Life and Materials Engineering, Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya-shi, Aichi 466-8555, Japan

2Division of Chemical Engineering, Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama-cho,Toyonaka-shi, Osaka 560-8531, Japan

Correspondence should be addressed to Yoshihito Kato, kato.yoshihito@nitech.ac.jp

Received 28 November 2011; Revised 26 January 2012; Accepted 3 February 2012

Academic Editor: See-Jo Kim

Copyright 2012 Haruki Furukawa et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The authors reviewed the correlations of power consumption in unba ffl ed and baffl ed agitated vessels with several kinds of impellers, which were developed in a wide range of Reynolds numbers from laminar to turbulent ow regions. The power correla-tions were based on Kamei and Hiraokas expressions for paddle and pitched paddle impellers. The calculated correlation values

agreed well with experimental ones, and the correlations will be developed the other types of impellers.

1. Introduction

Mixing vessels is widely used in chemical, biochemical, food,and other industries. Recently scientic approaches were de-veloped by Inoue and Hashimoto [ 1, 2]. On the other hand,the power consumption is the most important factor to esti-mate mixing performance and to design and operate mixingvessels.

To estimate the power consumption, the correlation of Nagata et al. [3] has traditionally been used. However, thiscorrelation was developed for two-blade paddle impellers,which do not always have thesame numerical valuesof powerconsumption as those of multiblade impellers. Kamei et al.[4, 5] and Hiraoka et al. [ 6] developed the new correlation of the power consumption of paddle impellers, which is moreaccurate than Nagatas.

However, the new correlation also cannot reproduce thepower consumption for other types of impellers. The pro-peller- and Pfaudler-type impellers are used for low-viscosity liquid and solid-liquid suspensions, and the propeller typehas been widely used in vessels ranging from portable typeto large tanks. Kato et al. [7] developed a new correlation of

power consumption for propeller- and Pfaudler-type im-pellers based on the correlations of Kamei and Hiraoka.

Thepower consumption for an anchor impeller wasmea-sured by Kato et al. [8, 9] in a wide range of Reynolds num-bers from laminar to turbulent ow regions. In the laminarregion, the power number of the anchor was reproduced by the correlations of Nagata andKameiet al. by considering theanchor as a wide paddle impeller. In the turbulent region, it

was reproduced by the correlation of Kamei et al. without thecorrection of the parameters.In this paper, the authors reviewed the power correlations

developed by authors in unba ffl ed and baffl ed mixing vesselswith several kinds of impellers.

2. Experimental

Figure 1 shows a photograph of mixing impellers used in thiswork. Figure 2 shows the geometry of impellers with sym-bols. The mixing vessel used is shown in Figure 3. The vesselsfor the measurement of power consumption are at-bottomcylindrical ones of inner diameter D = 185 and 200 mm.

8/12/2019 Design Blade

2/7

8/12/2019 Design Blade

3/7

International Journal of Chemical Engineering 3

104

104

102

102

102

100

100

100 10 2

N P

0.447 0.447 6

0.524 0.131 4

0.585 0.146 2

d/D b/D

0.146 80.585

n p

Red

Figure 4: Power diagram of paddle impeller (- - - [ 3]; [4, 5]).

Table 1: Correlation equations of power number for paddle (a),Rushton turbine (c) and anchor (f).

Unba ffl ed conditionN P 0 = { [1.2 4 2] / [8d 3 / (D2H )]} f f = C L / ReG + C t {[(C tr / ReG) + ReG]

1 + ( f /C t )1 /m }m

Red = nd 2 /ReG = { [ ln(D/d )] / (4d/D)}Red

C L = 0.215n p(d/H )[1 (d/D )2] + 1.83(b sin /H )(n p / 2sin )1 / 3

C t = [(1.96 X 1.19) 7.8 + (0 .25) 7.8]

1 / 7.8

m = [(0.71 X 0.373) 7.8 + (0 .333) 7.8] 1 / 7.8

C tr = 23.8(d/D ) 3.24(b sin /D ) 1.18 X 0.74

f = 0.0151(d/D )C 0.308t

X = n0.7 p bsin1. 6/H

= 2 ln(D/d ) / [(D/d ) (d/D )] = [ ln(D/d ) / ( D/d )5]1 / 3

= 0.711{0.157 + [n p ln(D/d )]0.611} / {n0.52 p [1 (d/D )2]}

Ba ffl ed conditionN P = [(1 + x 3)

1 / 3]N P max x = 4.5(Bw /D)n0.8B /N 0.2P max + N P 0 /N P max

Fully ba ffl ed condition

N P max =10(n0.7 p b/d )1.3 (n0.7 p b/d ) 0.548.3(n0.7 p b/d ) 0.54 < (n0.7 p b/d ) 1.6

10(n0.7 p b/d )0.6 1.6 < (n0.7 p b/d )

104

104

102

102

102

100

100

100

10 2

Red ()

N P

( )

C/H = 0.25

C/H = 0.5

d/D = 0.429

b/d = 0.253

n p = 3

N P

( ) = / 4

Figure 5: Power diagram of propeller impeller ( calc; N P 0; N P ).

d/D = 0 .559

b/d = 0.126

n p = 3

= / 2

C/H = 0.25

C/H = 0.5

104

104

102

102

102

100

100

100

10 2

N P

( )

N P

( )

Red ()

Figure 6: Power diagram of pfaudler-type impeller ( calc; N P 0;N P ).

3.3. Anchor Impeller. The anchor impeller was used for thehigh-viscosity liquid normally. In this work, power con-sumption for an anchor impeller was measured in a widerange of Reynolds numbers from laminar to turbulent ow regions. In the laminar region, the power number of theanchor was reproduced by the correlations of Nagata andKamei et al. by considering the anchor as a wide paddle im-peller. In the turbulent region, it was reproduced by thecorrelation of Kamei et al. [ 4] without the correction of theparameters as shown in Figure 7 [8]. If a large vortex wasgenerated in a turbulent mixing vessel, the experimentalvalues of power number were larger than the calculated ones.

8/12/2019 Design Blade

4/7

4 International Journal of Chemical Engineering

104

104

102

102100

100

N P

( )

10 2

d/D = 0 .892

b/d = 1

n p = 2

Red ()

Figure 7: Power diagram of anchor impeller ( calc; N P 0).

Table 2: Correlation equationsof power number forpitched paddle(b).

Unba ffl ed conditionN P 0 = { [1.2 4 2] / [8d 3 / (D2H )]} f f = C L / ReG + C t {[(C tr / ReG) + ReG]

1 + ( f /C t )1 /m }m

Red = nd 2 /ReG = { [ ln(D/d )] / (4d/D)}Red

C L = 0.215n p(d/H )[1 (d/D )2] + 1.83(b sin /H )(n p / 2sin )1 / 3C t = [(1.96 X 1.19)

7.8 + (0 .25) 7.8] 1 / 7.8

m = [(0.71 X 0.373) 7.8 + (0 .333) 7.8] 1 / 7.8

C tr = 23.8(d/D ) 3.24(b sin /D ) 1.18 X 0.74

f = 0.0151(d/D )C 0.308t X = n0.7 p bsin

1.6/H = 2 ln(D/d ) / [(D/d ) (d/D )] = [ ln(D/d ) / ( D/d )5]

1 / 3

= 0.711{0.157 + [n p ln(D/d )]0.611} / {n0.52 p [1 (d/D )2]}

Ba ffl ed conditionN P = [(1 + x 3)

1 / 3]N P max

x = 4.5(BW /D)n0.8B / {(2/ )0.72N 0.2P max } + N P 0 /N P max Fully ba ffl ed conditionN P max = 8.3(2/ )0.9(n0.7 p bsin

1.6/d )

3.4. Helical-Ribbon Impeller. As an example applied to ahelical-ribbon impeller the correlation by Kamei et al. [ 4]was shown in Figure 8. Since it is the special case, the valuesof the equations for tting the experimental values wereshown in Table 4. It is the rst time to show the powercorrelation of the helical-ribbon impeller over the wide rangeof Reynolds numbers.

Table 3: Correlation equations of power number for propeller (d)and pfaudler (f).

Unba ffl ed conditionN P 0 = { [1.2 4 2] / [8d 3 / (D2H )]} f f = C L / ReG + C t {[(C tr / ReG) + ReG]

1 + ( f /C t )1 /m }m

Red = nd 2

/ReG = { [ ln(D/d )] / (4d/D )}Red

C L = 0.215n p(d/H )[1 (d/D )2] + 1.83(b sin /H )(n p / 2sin )1 / 3

C t = [(3 X 1.5) 7.8 + (0.25) 7.8] 1 / 7.8

m = [(0.8 X 0.373) 7.8 + (0.333) 7.8] 1 / 7.8

C tr = 23.8(d/D ) 3.24(b sin /D ) 1.18 X 0.74

f = 0.0151(d/D )C 0.308t X = n0.7 p bsin

1.6/H = 2 ln(D/d ) / [(D/d ) (d/D )] = [ ln(D/d ) / ( D/d )5]

1 / 3

= 0.711{0.157 + [n p ln(D/d )]0.611} / {n0.52 p [1 (d/D )2]}

Ba ffl ed conditionN P = [(1 + x 3) 1 / 3]N P max

x = 4.5(BW /D)n0.8B / {(2/ )0.72N 0.2P max } + N P 0 /N P max

Fully ba ffl ed conditionN P max = 6.5(n0.7 p bsin

1.6/d )1.7

Table 4: Correlation equations of power number for helical ribbon(g).

N P 0 = { [1.2 4 2] / [8d 3 / (D2H )]} f = 16.0 f f = C L / ReG + C t {[(C tr / ReG) + ReG]

1 + ( f /C t )1 /m }m

Red = nd 2 /ReG = { [ ln(D/d )] / (4d/D )}Red = 0.0388Red

C L = 1.00, C t = 0.100, m = 0.333, C tr = 2500 f = 0.00683

= 2ln(D/d ) / [(D/d ) (d/D )] = 0.999 = 0.538

3.5. Pitched Paddle and Propeller Impellers with Partial Ba ffl e.The eff ects of the baffl e length on the power consumption of a mixing vessel with several impellers were studied as shownin Figure 3. The power number was generally correlatedwith the baffl e length, the number of ba ffl es, and the baffl ewidth [ 9]. The power number of the pitched paddle and thepropeller impellers was correlated with only the correction asfollows,

N P = 1 + x 3 1 / 3 N P max

x = 4.5(BW /D)n0.8B (hB /H )

(2/ )0.72N 0.2P max + N P 0

N P max ,

(2)

the results were shown in Figure 9.

3.6. Comparison of Power Number of Propeller Impeller withRushtons Data. Figure 10 shows the comparison of powernumber of a propeller impeller by Rushton et al. [ 10] withcorrelation in Table 3. The correlation equations in Table 3

8/12/2019 Design Blade

5/7

International Journal of Chemical Engineering 5

104

102

102100

100

N P

( )

10 2

d/D = 0.919

b/d = 0.853

n p = 1

Red ()

sin = { 1 + ( d/s)2 } 0.5

= 0 .148

Figure 8: Power diagram of helical ribbon-impeller ( calc; N P 0).

101

100

10010 1

10 1

nB

4

3

2

1

BW /D

0.05

0.1

0.13

0.145

hB /H 00.10.20.30.40.50.60.70.80.91

N P

/ N P m a x

( )

4.5(Bw /D)n0.8B (hB /H ) / (2 / )0.72 N 0.2P max + N P 0 /N P max ()

101

Figure 9: Correlation of power number for propeller with partialbaffl e ( calc; exp.).

also reproduced the data which other researchers measuredwell. As mentioned above, the correlation reproduced inthis work has a possibility of correlating the power numberof the mixing impellers of all geometry. It is the rst timeto compare between the data of Rushtons and the powercorrelation over the wide range of Reynolds number.

4. Conclusions

The authors reviewed the power correlations developed by authors. The new correlation equations of power number,

102

102

101

100

100

10 1104 106

d/D = 0.25, = / 6N P 0N P

N P

( )

Re ()

Figure 10: Comparison of power number for propeller withRushtons data ( calc; , Rushtons).

based on the correlation of Kamei and Hiraoka, were de-veloped for several kinds of mixing impellers, and it wasshown that the calculated values of the power number agreedvery closely with the measured ones. In future work, this cor-relation will be expanded to other impellers.

Nomenclature

b: Height of impeller blade (m)C : Clearance between bottom and impeller (m)D: Vessel diameter (m)d : Impeller diameter (m) f : Friction factor ()

H : Liquid depth (m)hB: Baffl e length (m)N P : Power number ( = P/n3d 5) ()N P 0: Power number in unba ffl ed condition ()N P max : Power number in fully ba ffl ed condition ()n: Impeller rotational speed ()nB: Number of baffl e plates ()n p: Number of impeller blades ()P : Power consumption (W)Red : Impeller Reynolds number ( = nd 2 /) ()ReG: Modied Reynolds number ()T : Shaft torque (N m) : Angle of impeller blade ()

: Liquid viscosity (Pa s) : Liquid density (kg/m 3).

References

[1] Y. Inoue andS. Hashimoto, Analysis of mechanismof laminaruid mixing in 3-D mixing tank based on streakline lobes,Kagaku Kogaku Ronbunshu, vol. 36, no. 4, pp. 355365, 2010.

[2] Y. Inoue, Invariant structure and variable properties in lam-inar uid mixing model based on strealines, Kagaku KogakuRonbunshu , vol. 37, pp. 211222, 2011.

[3] S. Nagata, T. Yokoyama, and H. Maeda, Studies on the powerrequirement of paddle agitators in cylindrical vessels, KagakuKogaku, vol. 20, pp. 582592, 1956.

8/12/2019 Design Blade

6/7

8/12/2019 Design Blade

7/7

Submit your manuscripts athttp://www.hindawi.com