# Correlation of Power Consumption for Several Kinds of Mixing Impellers

Authors: Haruki Furukawa; Yoshihito Kato; Yoshiro Inoue; Tomoho Kato; Yutaka Tada; Shunsuke Hashimoto
Journal: International Journal of Chemical Engineering
 (2012)

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## Abstract

The authors reviewed the correlations of power consumption in unbaffled and baffled agitated vessels with several kinds of impellers, which were developed in a wide range of Reynolds numbers from laminar to turbulent flow regions. The power correlations were based on Kamei and Hiraoka's 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.


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## Body

## 1. Introduction

Mixing vessels is widely used in chemical, biochemical, food, and other industries. Recently scientific approaches were developed by Inoue and Hashimoto [1, 2]. On the other hand, the power consumption is the most important factor to estimate mixing performance and to design and operate mixing vessels.To estimate the power consumption, the correlation of Nagata et al. [3] has traditionally been used. However, this correlation was developed for two-blade paddle impellers, which do not always have the same numerical values of power consumption 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 more accurate than Nagata’s.However, the new correlation also cannot reproduce the power consumption for other types of impellers. The propeller- and Pfaudler-type impellers are used for low-viscosity liquid and solid-liquid suspensions, and the propeller type has been widely used in vessels ranging from portable type to large tanks. Kato et al. [7] developed a new correlation of power consumption for propeller- and Pfaudler-type impellers based on the correlations of Kamei and Hiraoka.The power consumption for an anchor impeller was measured by Kato et al. [8, 9] in a wide range of Reynolds numbers from laminar to turbulent flow regions. In the laminar region, the power number of the anchor was reproduced by the correlations of Nagata and Kamei et al. by considering the anchor as a wide paddle impeller. In the turbulent region, it was reproduced by the correlation of Kamei et al. without the correction of the parameters.In this paper, the authors reviewed the power correlations developed by authors in unbaffled and baffled mixing vessels with several kinds of impellers.


## 2.  Experimental

Figure1 shows a photograph of mixing impellers used in this work. Figure 2 shows the geometry of impellers with symbols. The mixing vessel used is shown in Figure 3. The vessels for the measurement of power consumption are flat-bottom cylindrical ones of inner diameter D=185 and 200 mm. Three kinds of baffled conditions were mainly used: unbaffled, four baffles of BW=D/10 (i.e., the standard baffled condition), and fully baffled. The baffles were plate type. The paddle, pitched paddle, Rushton turbine, propeller, and pfaudler impellers were symmetrically set up at one-half the level of the liquid depth (C/H=0.5). The pfaudler, anchor, and helical-ribbon impellers were set up slightly above the bottom (bottom clearance of 1 mm).Figure 1
Photograph of several kinds of mixing impellers.Geometry of mixing impellers.
(a)
Paddle(b)
Pitched paddle(c)
Rushton(d)
Propeller(e)
Pfaudler(f)
Anchor(g)
Helical ribbonFigure 3
Geometry of mixing vessel.For the measurement of the power consumption, the liquids used were desalted water and varying starch-syrup solutions (μ=0.003–13Pa·s). The liquid was filled to the height equal to the vessel diameter (H=D).The power consumptionP(=2πnT) was measured with the shaft torque T and rotational speed n by using two types of torque meter (ST-1000 and ST-3000, Satake Chemical Equipment Mfg., Ltd.). The range of rotational speed was from 60 to 540 rpm to avoid a large vortex at the center of the free surface of the liquid.


## 3. Results and Discussion

### 3.1. Paddle Impeller

The correlation equations of Table1 were developed by Kamei et al. [4, 5] for paddle and Rushton turbine impellers. The comparison between experimental values and calculated values were shown in Figure 4. The correlation equations in Table 1 reproduced the experimental results better than Nagata’s correlation equations for the wide range of impeller. The correlation equations for the pitched paddle impeller were shown in Table 2. The correlation equations also reproduced experimental values, which were not shown in a figure here.Table 1
Correlation equations of power number for paddle (a), Rushton turbine (c) and anchor (f).
Unbaffled conditionNP0={[1.2π4β2]/[8d3/(D2H)]}ff=CL/ReG+Ct{[(Ctr/ReG)+ReG]-1+(f∽/Ct)1/m}mRed=nd2ρ/μReG={[πηln(D/d)]/(4d/βD)}RedCL=0.215ηnp(d/H)[1-(d/D)2]+1.83(bsinθ/H)(np/2sinθ)1/3Ct=[(1.96X1.19)-7.8+(0.25)-7.8]-1/7.8m=[(0.71X0.373)-7.8+(0.333)-7.8]-1/7.8Ctr=23.8(d/D)-3.24(bsinθ/D)-1.18X-0.74f∽=0.0151(d/D)Ct0.308X=γnp0.7bsin1.6θ/Hβ=2ln(D/d)/[(D/d)-(d/D)]γ=[ηln(D/d)/(βD/d)5]1/3η=0.711{0.157+[npln(D/d)]0.611}/{np0.52[1-(d/D)2]}Baffled conditionNP=[(1+x-3)-1/3]NPmaxx=4.5(Bw/D)nB0.8/NPmax0.2+NP0/NPmaxFully baffled conditionNPmax={10(np0.7b/d)1.3(np0.7b/d)≤0.548.3(np0.7b/d)0.54<(np0.7b/d)≤1.610(np0.7b/d)0.61.6<(np0.7b/d)Table 2
Correlation equations of power number for pitched paddle (b).
Unbaffled  conditionNP0={[1.2π4β2]/[8d3/(D2H)]}ff=CL/ReG+Ct{[(Ctr/ReG)+ReG]-1+(f∽/Ct)1/m}mRed=nd2ρ/μReG={[πηln(D/d)]/(4d/βD)}RedCL=0.215ηnp(d/H)[1-(d/D)2]+1.83(bsinθ/H)(np/2sinθ)1/3Ct=[(1.96X1.19)-7.8+(0.25)-7.8]-1/7.8m=[(0.71X0.373)-7.8+(0.333)-7.8]-1/7.8Ctr=23.8(d/D)-3.24(bsinθ/D)-1.18X-0.74f∽=0.0151(d/D)Ct0.308X=γnp0.7bsin1.6θ/Hβ=2ln(D/d)/[(D/d)-(d/D)]γ=[ηln(D/d)/(βD/d)5]1/3η=0.711{0.157+[npln(D/d)]0.611}/{np0.52[1-(d/D)2]}Baffled conditionNP=[(1+x-3)-1/3]NPmaxx=4.5(BW/D)nB0.8/{(2θ/π)0.72NPmax0.2}+NP0/NPmaxFully baffled conditionNPmax=8.3(2θ/π)0.9(np0.7bsin1.6θ/d)Figure 4
Power diagram of paddle impeller (- - - [3]; — [4, 5]).


### 3.2. Propeller and Pfaudler Impellers

The blades of the propeller and pfaudler impellers do not have sharp edges. The laminar termCL in Table 1 can be used without modification. Because the deviation from correlations to measured values in the turbulent region was large, the turbulent terms Ct and m in Table 1 were modified by Kato et al. [7] based on fitting with the modified Reynolds number ReG and the friction factor f, as follows:(1)Ct=[(3X1.5)-7.8+(0.25)-7.8]-1/7.8,m=[(0.8X0.373)-7.8+(0.333)-7.8]-1/7.8.Figures5 and 6 show the values correlated by the equations in Table 3 and the measured ones. The same equations can be used for the propeller- and the pfaudler-type impellers, regardless of the clearance between the vessel bottom and impeller.Table 3
Correlation equations of power number for propeller (d) and pfaudler (f).
Unbaffled  conditionNP0={[1.2π4β2]/[8d3/(D2H)]}ff=CL/ReG+Ct{[(Ctr/ReG)+ReG]-1+(f∽/Ct)1/m}mRed=nd2ρ/μReG={[πηln(D/d)]/(4d/βD)}RedCL=0.215ηnp(d/H)[1-(d/D)2]+1.83(bsinθ/H)(np/2sinθ)1/3Ct=[(3X1.5)-7.8+(0.25)-7.8]-1/7.8m=[(0.8X0.373)-7.8+(0.333)-7.8]-1/7.8Ctr=23.8(d/D)-3.24(bsinθ/D)-1.18X-0.74f∽=0.0151(d/D)Ct0.308X=γnp0.7bsin1.6θ/Hβ=2ln(D/d)/[(D/d)-(d/D)]γ=[ηln(D/d)/(βD/d)5]1/3η=0.711{0.157+[npln(D/d)]0.611}/{np0.52[1-(d/D)2]}Baffled conditionNP=[(1+x-3)-1/3]NPmaxx=4.5(BW/D)nB0.8/{(2θ/π)0.72NPmax0.2}+NP0/NPmaxFully  baffled  conditionNPmax=6.5(np0.7bsin1.6θ/d)1.7Figure 5
Power diagram of propeller impeller (— calc;⚪  NP0; ∆ NP).Figure 6
Power diagram of pfaudler-type impeller (— calc;⚪  NP0; ∆ NP).


### 3.3.  Anchor Impeller

The anchor impeller was used for the high-viscosity liquid normally. In this work, power consumption for an anchor impeller was measured in a wide range of Reynolds numbers from laminar to turbulent flow regions. In the laminar region, the power number of the anchor was reproduced by the correlations of Nagata and Kamei et al. by considering the anchor as a wide paddle impeller. In the turbulent region, it was reproduced by the correlation of Kamei et al. [4] without the correction of the parameters as shown in Figure 7 [8]. If a large vortex was generated in a turbulent mixing vessel, the experimental values of power number were larger than the calculated ones.Figure 7
Power diagram of anchor impeller (— calc;⚪  NP0).


### 3.4.  Helical-Ribbon Impeller

As an example applied to a helical-ribbon impeller the correlation by Kamei et al. [4] was shown in Figure 8. Since it is the special case, the values of the equations for fitting the experimental values were shown in Table 4. It is the first time to show the power correlation of the helical-ribbon impeller over the wide range of Reynolds numbers.Table 4
Correlation equations of power number for helical ribbon (g).
NP0={[1.2π4β2]/[8d3/(D2H)]}f=16.0ff=CL/ReG+Ct{[(Ctr/ReG)+ReG]-1+(f∽/Ct)1/m}mRed=nd2ρ/μReG={[πηln(D/d)]/(4d/βD)}Red=0.0388RedCL=1.00,Ct=0.100,m=0.333,Ctr=2500f∽=0.00683β=2ln(D/d)/[(D/d)-(d/D)]=0.999η=0.538Figure 8
Power diagram of helical ribbon-impeller (— calc;⚪  NP0).


### 3.5.  Pitched Paddle and Propeller Impellers with Partial Baffle

The effects of the baffle length on the power consumption of a mixing vessel with several impellers were studied as shown in Figure3. The power number was generally correlated with the baffle length, the number of baffles, and the baffle width [9]. The power number of the pitched paddle and the propeller impellers was correlated with only the correction as follows,(2)NP=[(1+x-3)-1/3]NPmax⁡x=4.5(BW/D)nB0.8(hB/H){(2θ/π)0.72NPmax⁡0.2}+NP0NPmax⁡,
the results were shown in Figure 9.Figure 9
Correlation of power number for propeller with partial baffle (— calc;⚪  exp.).


### 3.6.  Comparison of Power Number of Propeller Impeller with Rushton’s Data

Figure10 shows the comparison of power number of a propeller impeller by Rushton et al. [10] with correlation in Table 3. The correlation equations in Table 3 also reproduced the data which other researchers measured well. As mentioned above, the correlation reproduced in this work has a possibility of correlating the power number of the mixing impellers of all geometry. It is the first time to compare between the data of Rushton’s and the power correlation over the wide range of Reynolds number.Figure 10
Comparison of power number for propeller with Rushton’s data (— calc;⚪, ∆ Rushton’s).


## 3.1. Paddle Impeller

The correlation equations of Table1 were developed by Kamei et al. [4, 5] for paddle and Rushton turbine impellers. The comparison between experimental values and calculated values were shown in Figure 4. The correlation equations in Table 1 reproduced the experimental results better than Nagata’s correlation equations for the wide range of impeller. The correlation equations for the pitched paddle impeller were shown in Table 2. The correlation equations also reproduced experimental values, which were not shown in a figure here.Table 1
Correlation equations of power number for paddle (a), Rushton turbine (c) and anchor (f).
Unbaffled conditionNP0={[1.2π4β2]/[8d3/(D2H)]}ff=CL/ReG+Ct{[(Ctr/ReG)+ReG]-1+(f∽/Ct)1/m}mRed=nd2ρ/μReG={[πηln(D/d)]/(4d/βD)}RedCL=0.215ηnp(d/H)[1-(d/D)2]+1.83(bsinθ/H)(np/2sinθ)1/3Ct=[(1.96X1.19)-7.8+(0.25)-7.8]-1/7.8m=[(0.71X0.373)-7.8+(0.333)-7.8]-1/7.8Ctr=23.8(d/D)-3.24(bsinθ/D)-1.18X-0.74f∽=0.0151(d/D)Ct0.308X=γnp0.7bsin1.6θ/Hβ=2ln(D/d)/[(D/d)-(d/D)]γ=[ηln(D/d)/(βD/d)5]1/3η=0.711{0.157+[npln(D/d)]0.611}/{np0.52[1-(d/D)2]}Baffled conditionNP=[(1+x-3)-1/3]NPmaxx=4.5(Bw/D)nB0.8/NPmax0.2+NP0/NPmaxFully baffled conditionNPmax={10(np0.7b/d)1.3(np0.7b/d)≤0.548.3(np0.7b/d)0.54<(np0.7b/d)≤1.610(np0.7b/d)0.61.6<(np0.7b/d)Table 2
Correlation equations of power number for pitched paddle (b).
Unbaffled  conditionNP0={[1.2π4β2]/[8d3/(D2H)]}ff=CL/ReG+Ct{[(Ctr/ReG)+ReG]-1+(f∽/Ct)1/m}mRed=nd2ρ/μReG={[πηln(D/d)]/(4d/βD)}RedCL=0.215ηnp(d/H)[1-(d/D)2]+1.83(bsinθ/H)(np/2sinθ)1/3Ct=[(1.96X1.19)-7.8+(0.25)-7.8]-1/7.8m=[(0.71X0.373)-7.8+(0.333)-7.8]-1/7.8Ctr=23.8(d/D)-3.24(bsinθ/D)-1.18X-0.74f∽=0.0151(d/D)Ct0.308X=γnp0.7bsin1.6θ/Hβ=2ln(D/d)/[(D/d)-(d/D)]γ=[ηln(D/d)/(βD/d)5]1/3η=0.711{0.157+[npln(D/d)]0.611}/{np0.52[1-(d/D)2]}Baffled conditionNP=[(1+x-3)-1/3]NPmaxx=4.5(BW/D)nB0.8/{(2θ/π)0.72NPmax0.2}+NP0/NPmaxFully baffled conditionNPmax=8.3(2θ/π)0.9(np0.7bsin1.6θ/d)Figure 4
Power diagram of paddle impeller (- - - [3]; — [4, 5]).


## 3.2. Propeller and Pfaudler Impellers

The blades of the propeller and pfaudler impellers do not have sharp edges. The laminar termCL in Table 1 can be used without modification. Because the deviation from correlations to measured values in the turbulent region was large, the turbulent terms Ct and m in Table 1 were modified by Kato et al. [7] based on fitting with the modified Reynolds number ReG and the friction factor f, as follows:(1)Ct=[(3X1.5)-7.8+(0.25)-7.8]-1/7.8,m=[(0.8X0.373)-7.8+(0.333)-7.8]-1/7.8.Figures5 and 6 show the values correlated by the equations in Table 3 and the measured ones. The same equations can be used for the propeller- and the pfaudler-type impellers, regardless of the clearance between the vessel bottom and impeller.Table 3
Correlation equations of power number for propeller (d) and pfaudler (f).
Unbaffled  conditionNP0={[1.2π4β2]/[8d3/(D2H)]}ff=CL/ReG+Ct{[(Ctr/ReG)+ReG]-1+(f∽/Ct)1/m}mRed=nd2ρ/μReG={[πηln(D/d)]/(4d/βD)}RedCL=0.215ηnp(d/H)[1-(d/D)2]+1.83(bsinθ/H)(np/2sinθ)1/3Ct=[(3X1.5)-7.8+(0.25)-7.8]-1/7.8m=[(0.8X0.373)-7.8+(0.333)-7.8]-1/7.8Ctr=23.8(d/D)-3.24(bsinθ/D)-1.18X-0.74f∽=0.0151(d/D)Ct0.308X=γnp0.7bsin1.6θ/Hβ=2ln(D/d)/[(D/d)-(d/D)]γ=[ηln(D/d)/(βD/d)5]1/3η=0.711{0.157+[npln(D/d)]0.611}/{np0.52[1-(d/D)2]}Baffled conditionNP=[(1+x-3)-1/3]NPmaxx=4.5(BW/D)nB0.8/{(2θ/π)0.72NPmax0.2}+NP0/NPmaxFully  baffled  conditionNPmax=6.5(np0.7bsin1.6θ/d)1.7Figure 5
Power diagram of propeller impeller (— calc;⚪  NP0; ∆ NP).Figure 6
Power diagram of pfaudler-type impeller (— calc;⚪  NP0; ∆ NP).


## 3.3.  Anchor Impeller

The anchor impeller was used for the high-viscosity liquid normally. In this work, power consumption for an anchor impeller was measured in a wide range of Reynolds numbers from laminar to turbulent flow regions. In the laminar region, the power number of the anchor was reproduced by the correlations of Nagata and Kamei et al. by considering the anchor as a wide paddle impeller. In the turbulent region, it was reproduced by the correlation of Kamei et al. [4] without the correction of the parameters as shown in Figure 7 [8]. If a large vortex was generated in a turbulent mixing vessel, the experimental values of power number were larger than the calculated ones.Figure 7
Power diagram of anchor impeller (— calc;⚪  NP0).


## 3.4.  Helical-Ribbon Impeller

As an example applied to a helical-ribbon impeller the correlation by Kamei et al. [4] was shown in Figure 8. Since it is the special case, the values of the equations for fitting the experimental values were shown in Table 4. It is the first time to show the power correlation of the helical-ribbon impeller over the wide range of Reynolds numbers.Table 4
Correlation equations of power number for helical ribbon (g).
NP0={[1.2π4β2]/[8d3/(D2H)]}f=16.0ff=CL/ReG+Ct{[(Ctr/ReG)+ReG]-1+(f∽/Ct)1/m}mRed=nd2ρ/μReG={[πηln(D/d)]/(4d/βD)}Red=0.0388RedCL=1.00,Ct=0.100,m=0.333,Ctr=2500f∽=0.00683β=2ln(D/d)/[(D/d)-(d/D)]=0.999η=0.538Figure 8
Power diagram of helical ribbon-impeller (— calc;⚪  NP0).


## 3.5.  Pitched Paddle and Propeller Impellers with Partial Baffle

The effects of the baffle length on the power consumption of a mixing vessel with several impellers were studied as shown in Figure3. The power number was generally correlated with the baffle length, the number of baffles, and the baffle width [9]. The power number of the pitched paddle and the propeller impellers was correlated with only the correction as follows,(2)NP=[(1+x-3)-1/3]NPmax⁡x=4.5(BW/D)nB0.8(hB/H){(2θ/π)0.72NPmax⁡0.2}+NP0NPmax⁡,
the results were shown in Figure 9.Figure 9
Correlation of power number for propeller with partial baffle (— calc;⚪  exp.).


## 3.6.  Comparison of Power Number of Propeller Impeller with Rushton’s Data

Figure10 shows the comparison of power number of a propeller impeller by Rushton et al. [10] with correlation in Table 3. The correlation equations in Table 3 also reproduced the data which other researchers measured well. As mentioned above, the correlation reproduced in this work has a possibility of correlating the power number of the mixing impellers of all geometry. It is the first time to compare between the data of Rushton’s and the power correlation over the wide range of Reynolds number.Figure 10
Comparison of power number for propeller with Rushton’s data (— calc;⚪, ∆ Rushton’s).


## 4. Conclusions

The authors reviewed the power correlations developed by authors. The new correlation equations of power number, based on the correlation of Kamei and Hiraoka, were developed for several kinds of mixing impellers, and it was shown that the calculated values of the power number agreed very closely with the measured ones. In future work, this correlation will be expanded to other impellers.



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