Comparison of Power Number for Paddle Type Impellers by Three Methods

Journal of Chemical Engineering of Japan, Advance Publication doi: 10.1252/jcej.11we115; released online on July 13, 2011 Comparison of power number for paddle type impellers by three methods

Ming-hui XIE1,2, Guo-zhong ZHOU1*, Jian-ye XIA2*, Chen ZOU1, Pei-qing YU1 and Si-liang ZHANG2 1. Wenzhou Great wall Mixer Design Institute, Wenzhou, 325019, China 2. State Key Laboratory of Bioreactor Engineering, College of Biotechnology, East China University of Science and Technology, Shanghai 200237, China

Keyword: Power number; empirical correlations; Computational Fluid Dynamics (CFD); Paddle impeller;

Power numbers in a baffled stirred tank with paddle type impellers were measured over a range of Reynolds numbers from laminar to turbulent flow. Impellers include two-blade flat paddle impeller, 45° two-blade pitched paddle impeller, pitched blade turbine (PBT, 45° four-blade) and double-stage pitched blade turbine, with impeller-to-tank diameter ratio of 0.5-0.6. The measured power numbers were compared with other two methods, the prediction by using correlations and Computational Fluid Dynamics (CFD). Power number correlations showed that the estimated values from the empirical correlation of Nagata, (1975) close agreed with the measured ones in the laminar region, the deviation was large in the transitional and turbulent regions. The power numbers from the correlation of Kamei et al., (1996)and Hiraoka et al., (1997) got good agreement from laminar to turbulent regions. The power number in stirred tank was also simulated by CFD method. The numerical results showed satisfactory agreement with the experimental data over a wide range of Reynolds numbers.

Introduction As shown in Fig.1, the stirred tank used is a flat bottom cylindrical tank of inner diameter D=0.58 m, the liquid Mixing and stirred tanks are widely used in many height H=D for one-stage and H=1.2D for double-stage. industries, such as chemical, biochemical, food and Four baffles of width BW =D/10, were installed with an off- watertreatment, etc. Due to lack of accurate calculation of wall clearance 0.008 m. The ratio of impeller to tank power consumption, many industrial mixing operations are diameter d/D is 0.5-0.6, impeller off-bottom clearance C is inefficient and energy wasteful. The power number is the 0.35D for one-stage, 0.3D for double-stage. The detailed basic parameter to design an agitator. And, it also influence parameters for impellers are shown in Table 1. The the equipment invest and running cost. Recently, less experimental procedure has been described in detail research paper concerns on impeller power number. elsewhere (Chen et al., 2010) . The impeller Reynolds number Paddle and turbine impellers are commonly used in is computed as Re=Nd2ρ/µ, where N is rotational speed, ρ is industrial mixing operations. There are many literatures liquid density and µ is liquid viscosity. The power related to how to obtain the power consumption. To estimate consumption P is observed from measurements of the shaft power consumption, the correlation of Nagata, (1975) has torque M and rotational speed N, as: traditionally been used, which is developed for two-blade P=2πNM (1) paddle impeller. Kamei et al., (1996) and Hiraoka et al., and impeller power number is computed as: 3 5 (1996) developed a new correlation of power consumption NP =P/ρN d (2) for paddle, anchor and helical ribbon impellers. Computional Fluid Dynamics (CFD) technique are being used as a B substitute for experiments to get power consumption and W mixing characteristics (Shekhar and Jayanti, 2002; Suzukawa et al.,2006; Driss et al., 2006). One advangtage with the CFD- b based prediction methods is quick and easy to perform h various scale’s modeling. Shekhar and Jayanti (2006) have H studied the power number of eight-blade paddle impeller in a C d unbaffled vessel by using CFD methods, they found that the simulated data were lower than the Nagata correlations. The purpose of present study is to compare the power D number of different impellers by experiments, empirical Fig.1 Schematic diagram of experimental set-up correlations and CFD method. In this paper, two-blade flat paddle impeller (2-FP), two-blade pitched (45°) paddle Table 1 Detailed parameters of impellers impeller (2-PP), four-blade pitched (45°) blade turbine (4- Number Name d/D b/d C/D h/D PBT) and double-stage 4-PBT, were used. 1 2-FP 0.6 0.109 0.35 2 2-PP 0.6 0.109 0.35 1. Research methods 3 4-PBT 0.6 0.092 0.35 1.1 Experiments double-stage 4 0.5 0.083 0.30 0.6 4-PBT

Correspondence concerning this article should be addressed to 1.2 Correlation method G. Z. Zhou (E-mail address: [email protected]) or J.Y. Xia (E-mail address: [email protected])

Journal of Chemical Engineering of Japan, Advance Publication doi: 10.1252/jcej.11we115; released online on July 13, 2011 Predicting impeller power number through empirical impeller and about 670,000 tetrahedral cells for double-stage correlations is commonly used due to its simplicity and time impeller. In order to achieve high resolution, the grids were saving. Nagata correlation, Kamei and Hiraoka correlation refined near the impeller, rotating interface and tank wall. (see Table2,3) are being used for paddle impellers of various geometry. In this paper, we supposed that when Re≤200 is Solution method in laminar flow region and the power number was calculated All the calculation were carried out with commercially by the equations under unbaffled condition. When Re>200 available software FLUENT6.3, which is a general purpose the power number was calculated by the equations under computer program using a finite volume method. The fully baffled or baffled condition. Multiple Reference Frame (MRF) was employed for the flow of rotating impeller to stationary baffles. The pressure- Table 2 Nagata correlation of power number velocity coupling was handled by SIMPLE algorithm, for paddle impeller second-order upwind accurate scheme was used for the Unbaffled condition discretization method. The laminar model was chosen for 0.66 0.66 p NP0=A/Re+B[(1000+1.2Re )/(1000+3.2Re )] laminar flow (Re≤200), the standard k-epsilon turbulent (H/D)(0.35+b/D)sin1.2θ model was chosen for transitional and turbulent flows. A = 14+(b/D)[670(d/D-0.6)2+185] 2 B =10[1.3-4(b/D-0.5) -1.14(d/D)] 2. Results and Discussion p =1.1+4(b/D)-2.5(d/D-0.5)2-7(b/D)4 Re =Nd2ρ/µ 2.1 Power number of 2-FP Fully baffled condition (Nagata, 1975) The NP of 2-FP from CFD, Nagata correlation, Kamei NPmax is calculated through NP0 calculation formula under and Hiraoka correlation and experiments are compared in unbaffled condition, where Re is substituted by Reθ. Fig. 2. It can be observed that every method predicts the right 4(1-sinθ) 2 Reθ=10 25(d/D-0.4) /(b/D)+{(b/D)/[0.11(b/D)-0.0048]} trend of the power number curve, NP decreases as Reynolds Baffled condition (Wang et al., 2002) number increases, and then try to become to constant in the 1.2 2 (NPmax-NP)/( NPmax-NP0) =[1-2.9(BW/D) nb] turbulent region. But there are also deviations at different regions for the three methods. In the laminar region, power Table 3 Kamei and Hiraoka correlation of power number numbers from the three methods agree well with experiment for paddle impeller data. Nagata, Kamei and Hiraoka correlation values are Unbaffled condition approximately 10% larger than measured ones. There is good 4 2 3 2 agreement between Kamei and Hiraoka correlation values NP0 = {[1.2π β ]/[8d /(D H)]}f -1 1/m m and experiment data in the transitional region (Reynolds f = CL/ReG + Ct{[(Ctr/ReG) + ReG] + (f∞/Ct) } number from 300 to 2000). While Nagata correlation give ReG = {[πηln(D/d)]/(4d/βD)}Re 2 1/3 higher values and CFD gives lower predictions. In the CL = 0.215ηnp(d/H)[1-(d/D) ]+1.83(bsinθ/H)(np/2sinθ) 1.19 -7.8 -7.8 -1/7.8 turbulent region, simulated values and correlation values are Ct = [(1.96X ) + (0.25) ] m = [(0.71X0.373)-7.8 + (0.333)-7.8]-1/7.8 about 10-15% lower than experimental data. -3.24 -1.18 -0.74 For 2-FP, Kamei and Hiraoka correlation could give Ctr = 23.8(d/D) (bsinθ/D) X 0.308 good results over a range of Reynolds numbers from laminar f∞ = 0.0151(d/D) Ct 0.7 1.6 to turbulent flow. Nagata correlation and CFD will provide X = γnp bsin θ/H β = 2ln(D/d)/[(D/d)-(d/D)] accurate predictions in the laminar and turbulent regions. γ = [ηln(D/d)/(βD/d5)]1/3 0.611 0.52 2 20 20 η = 0.711{0.157 + [np ln(D/d)] }/{ np [1-(d/D) ]} Fully baffled condition experiments CFD simulation 0.7 1.3 0.7 10 10 Flat paddle: NPmax = 10(np b/d) (np b/d) ≦ 0.54 Nagata 0.7 0.7 = 8.3(np b/d) 0.54＜(np b/d) ≦ 1.6 Kamei and Hiraoka 0.7 0.6 0.7 = 10(np b/d) 1.6＜(np b/d) 0.9 0.7 1.6 Pitched paddle: NPmax = 8.3(2θ/π) (np bsin θ/d) Np Np Baffled condition -3 -1/3 NP = [(1+x ) ]NPmax 0.8 0.72 0.2 x = 4.5(Bw/D)nb /{(2θ/π) NPmax }+ NP0/NPmax 1 1 The power number correlations for two-stage referred to

Design Handbook for Practical Subject of Chemical 0.5 0.5 Engineering (PartII) (Wang et al., 2002). In this investigation, 100 101 102 103 104 105 the power number for double-stage 4-PBT was twice the Re power number of single impeller. Fig. 2 Comparison of power number of 2-FP obtained by three methods 1.3 Numerical method Computational grids 2.2 Power number of 2-PP The grids of tank domain and impeller domain were respectively generated with the help of the commercial The NP of 2-PP from CFD，Nagata correlation, Kamei software GAMBIT 2.3. Total tank was modeled, the grid and Hiraoka correlation and experiments are compared in density was about 550,000 tetrahedral cells for one-stage Fig. 3. In the laminar region, Nagata correlation and Kamei and Hiraoka correlation give larger Np than the experimental

Journal of Chemical Engineering of Japan, Advance Publication doi: 10.1252/jcej.11we115; released online on July 13, 2011 data, about 23% and 15% respectively. While the CFD results 2.4 Power number of double-stage 4-PBT agree well with the experiment data. In the transitional region The NP of 4-PBT from CFD，Nagata correlation, Kamei and low Reynolds number turbulent region, the three methods and Hiraoka correlation and experiments are compared in all overpredict the power number, especially the Nagata Fig.5. Like the results before, the three methods can give very correlation. But in high Reynolds number turbulent region, good prediction of Np for double-stage 4-PBT in the laminar the three methods give close data with other methods and region. Kamei and Hiraoka correlations values are about 15% predict lower Np than the experimental data, about 5~10%. larger than the experimental data. For Nagata correlations

values and simulation data, the deviation is about 5% 20 20 compared with experiment data. For Reynolds number lower experiments 10 CFD simulation 10 than 200, Nagata correlations could be used for double-stage Nagata 4-PBT, but At high Reynolds number, Np from Nagata Kamei and Hiraoka correlations is nearly 50~80% higher than experiment data. So, this method could not be used in the present experimental conditions. In the turbulent region, Kamei and Hraoka Np Np correlations and CFD predictions correspond well with the experiment data. 1 1

50 50 experiments CFD simulation 0.3 0.3 100 101 102 103 104 105 Nagata Kamei and Hiraoka Re 10 10 Fig. 3 Comparison of power number of 2-PP

Np obtained by three methods Np

2.3 Power number of 4-PBT Nagata correlation is usually used for two-blade impellers whereas the present study used a four-blade pitched 1 1 blade turbine. According to Nagata’s suggestion, the four- 0.6 0.6 blade PBT was changed to a two-blade pitched paddle which 0 1 2 3 4 5 10 10 10 10 10 10 has the equivalent blade area. Re As shown in Fig.4, Kamei and Hiraoka correlation has Fig. 5 Comparison of power number of double-stage the largest deviation with other methods in the laminar 4-PBT obtained by three methods region. The deviation is about 10~20% compared with experiment data. The other three agree well with each other. Conclutions However, in the transitional region, Nagata correlation gives higher values compared with other methods. The results are Power numbers of paddle and turbine impellers in a about 10~30% higher than experimental data. In the turbulent baffled stirred tank were measured over a wide range of region, simulated values and correlation values are about 10- Reynolds numbers from laminar to turbulent flow regions. 15% lower than experimental data. Therefore, in the present Different correlations results and CFD results were compared experimental conditions, the Nagata correlations with with experimental data. equivalent blade area method are useful in laminar region, Nagata correlation for 2-FP, 2-PP and 4-PBT impellers but there are large deviations in transitional and turbulent agreed well with the experimental data in laminar and regions. turbulent regions, but there is a large deviation in the transitional region. For and double stage 4-PBT in a baffled 30 30 agitated tank, it can be used just for the laminar region. experiments The power numbers from Kamei and Hiraoka correlation CFD simulation were consistent with the experimental data from laminar to 10 Nagata 10 Kamei and Hiraoka turbulent regions for all the impellers researched. Therefore, Kamei and Hiraoka correlation can easily be used for predicting accurately power number for the paddle or turbine

Np Np impellers under baffled condition. The power numbers from CFD method agreed well with measured data over a wide range of Reynolds numbers. So, 1 1 CFD can be used not only as a tool for impeller flow field research, but also a very useful tool for power consumption research. 0.4 0.4 100 101 102 103 104 105 Nomenclature Re Fig. 4 Comparison of power number of 4-PBT BW baffle width, m b height of impeller blade, m obtained by three methods C clearance between bottom to impeller, m D tank diameter, m

Journal of Chemical Engineering of Japan, Advance Publication doi: 10.1252/jcej.11we115; released online on July 13, 2011 d impeller diameter, m f friction factor H liquid height, m NP power number NP0 power number in unbaffled condition NPmax power number in fully baffled condition N impeller rotational speed nb number of baffle plates np number of impeller blades P power consumption, W Re Reynolds number ReG modified Reynolds number M shaft toque, N·m θ angle of impeller blade µ liquid viscosity, Pa·s ρ liquid density, kg/m3

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