Calculation Double Helical Ribbon PDF [PDF]

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Chinese Journal of Chemical Engineering, 16(5) 686—692 (2008)



Calculation of Metzner Constant for Double Helical Ribbon Impeller by Computational Fluid Dynamic Method* ZHANG Minge (张敏革)1, ZHANG Lühong (张吕鸿)1,**, JIANG Bin (姜斌)1,2, YIN Yuguo (尹玉国)1 and LI Xingang (李鑫钢)1,2 1 2



School of Chemical Engineering and Technology, Tianjin University, Tianjin 300072, China National Engineering Research Centre of Distillation Technology, Tianjin 300072, China



Abstract Using the multiple reference frames (MRF) impeller method, the three-dimensional non-Newtonian flow field generated by a double helical ribbon (DHR) impeller has been simulated. The velocity field calculated by the numerical simulation was similar to the previous studies and the power constant agreed well with the experimental data. Three computational fluid dynamic (CFD) methods, labeled I, II and III, were used to compute the Metzner constant ks. The results showed that the calculated value from the slop method (method I) was consistent with the experimental data. Method II, which took the maximal circumference-average shear rate around the impeller as the effective shear rate to compute ks, also showed good agreement with the experiment. However, both methods suffer from the complexity of calculation procedures. A new method (method III) was devised in this paper to use the area-weighted average viscosity around the impeller as the effective viscosity for calculating ks. Method III showed both good accuracy and ease of use. Keywords computational fluid dynamic, double helical ribbon impeller, non-Newtonian fluid, Metzner constant



1



INTRODUCTION



The mixing of liquids by mechanical agitation is one of the most commonly used operations in chemical and food industries as well as in polymerization applications [1, 2]. Of all close clearance impellers, the helical ribbon impeller, due to its high mixing efficiency [3] and its unique feature of producing axial flow in a stirred tank [4], is widely chosen in mixing highly viscous Newtonian fluids and non-Newtonian fluids. In industrial applications, power consumption is one of the most important factors concerning designers [5-8]. For Newtonian fluids, theoretical and experimental studies [9, 10] have shown that the power number Np is inversely proportional to the Reynolds number Re in the laminar flow: N p Re = K p



(1)



where ks, originally noted to be a proportional coefficient, is referred to as the Metzner constant. For shear thinning non-Newtonian fluids, the apparent viscosity ηa can be expressed by a power law model: n −1 ηa = K psu γeff



The power curve of non-Newtonian fluids obtained using the Metzner-Otto method is consistent with that of Newtonian fluids, which explains the fact that the Metzner-Otto method is a widely used method to design impellers for non-Newtonian fluids applications [12, 13]. As a result, the ks value has become a key factor to predict power consumption. ks is a function of impeller geometry. Its value is normally obtained experimentally using the Metzner method and Rieger-Novak method [10, 14, 15]. The Metzner method defines the Ren and Kpn by



Ren =



where Re is defined as Re =



ρ Nd 2 η



d 2 N 2− n ρ K psu



(2)



K pn = N p Ren = K p ksn −1



Given a certain type of impeller, Re can be calculated based on the fluid properties and the rotation speed of the impeller, then Np can be determined from the power curve. However, for non-Newtonian fluids, fluid viscosity varies along with shear rate. Metzner and Otto [11] defined Rea based on apparent (or effective) viscosity ηa:



where Kpn is a function of n [8], so that



Rea =



ρ Nd 2 ηa



(3)



and assumed that the effective shear rate γeff is proportional to the rotation speed of the impeller: γeff = ks N (4)



(5)



(6) (7)



1



⎛ K pn ⎞ n −1 ks = ⎜ (8) ⎜ K p ⎟⎟ ⎝ ⎠ The ks value can be directly calculated from Eq. (8), so this method is also referred to as direct calculation of ks. The Rieger-Novak method is denoted sometimes as the slop method. The ks value can be obtained from the slope of the straight line resulting from the plot of lnKpn versus ( 1 − n ), based on the linearized Eq. (8):



Received 2007-11-11, accepted 2008-07-10. * Supported by the Natural Science Foundation of Tianjin (07JCZDJC02600). ** To whom correspondence should be addressed. E-mail: [email protected]



ln K pn = ln K p − (1 − n ) ln ks



(9)



687



Chin. J. Chem. Eng., Vol. 16, No. 5, October 2008



The two aforementioned methods are both based on the Metzner concept and can obtain ks through measuring the power input of mixing in a stirred tank. From Eqs. (4) and (5), one can notice that the ks value can be acquired directly and easily using the calculated effective shear rate or the effective viscosity in a mixing process. However, no explicit methods for the calculation of effective shear rate or effective viscosity have been given since Metzner and Otto brought about the concept of effective shear rate in 1957 [11]. Fortunately, with the development of computer technology and computational fluid dynamic (CFD) method, it is feasible now to simulate the detailed mixing flow fields and determine the various performance parameters [16-19]. Shekhar and Jayanti [20] took the circumference-averaging shear rate at mid-height of the DHR impeller calculated by CFD as the effective shear rate to determine ks, however, the calculating steps of the method were complex. The calculating procedure of the slop method (experimental method) was also tedious. So a cost-effective and time-saving method to predict ks through the calculation of effective viscosity is proposed and compared with the above two methods in this work. 2 2.1



3



NUMERICAL METHOD



3.1



Calculation model



It has been difficult to simulate the flow field in a stirred tank because of the complex interactions between moving blades and stationary tank wall and baffle plates. Two methods, the multiple reference frames method (MRF) and the sliding mesh method, have been developed to solve the problem without requiring any empirical data [25]. The MRF method can be applied when the relative position between the impeller and baffle does not significantly affect the flow field due to weak interactions, whereas the sliding mesh method can be used in the case of strong interactions. The sliding mesh model requires more time for calculations [26]. No baffle was used in the studied tank in this paper. Therefore, the MRF model was chosen to simulate the flow field generated by DHR impeller in the stirred tank.



MATHEMATIC MODELS 3.2



Flow equations



In this research, the flow field generated by a double helical ribbon (DHR) impeller was studied. Assuming that the fluids were continuous and incompressible non-Newtonian fluids and the mixing process was carried out under a constant temperature. Numerical simulations were run in a steady state calculation mode. The mass and momentum conservation equations are: ∇⋅v = 0 (10)



(v ⋅ ∇ )v = f −



1



ρ



∇P +



ηa 2 ∇ v ρ



(11)



The governing equations were solved using the commercial flow solver Fluent 6.2 (ANSYS Inc.). The finite volume method and the second-order modified scheme were applied to discrete the control equations to algebra equations. 2.2



The consistency factor Kpsu was given as 10 Pa·sn, and the flow behavior index n of the non-Newtonian fluids was set between 0.52 − 1 .



Mixer configuration and mesh generation



As shown in Fig. 1, the mixer consisted of a mixing tank of 101 mm in both diameter (D) and height (H). The tank was equipped with a DHR impeller with both a diameter and height of 96 mm. Other important dimensional parameters of the impeller are shown in Table 1. The impeller was located 2.5 mm above the bottom of the vessel. In order to use the MRF model, the calculation area is segmented into two zones, impeller area (Zone 1) and tank wall area (Zone 2). Steady-state calculations were performed with a rotating reference frame in Zone 1 and a stationary reference frame in Zone 2. The unstructured tetrahedral mesh was generated in impeller region and structured hexahedral mesh in tank wall region. Adaptive



Rheological model



For shear-thinning non-Newtonian fluids with no or negligible elasticity, some researchers [21-23] used the power law equation as the rheological model. Convenient for engineering calculations with sufficient accuracy, the power law equation is simple and easy to use. The power law model is given by



ηa = K psu γ n −1 = K psu ⎡⎣ 0.5tr(γ )2 ⎤⎦



n −1



(12)



where 0.5tr(γ )2 is the second shear rate tensor invariant. In this study, several fictitious power law fluids were investigated according to Kelly and Gigas [24].



Figure 1 Table 1



The structural sketch of stirred tank



The dimensional parameters of DHR impeller



d/m



s:d



L:d



w:d



c:d



0.096



1.0



1.0



0.1



0.025



688



Chin. J. Chem. Eng., Vol. 16, No. 5, October 2008



(a) X = 0 (b) Z = 50 mm (c) Near the impeller tip Figure 2 Illustration of a computational mesh at the X＝0 plane of the stirred tank, at the Z＝50 mm of the stirred tank and near the impeller tip



mesh refinement had been applied to the edge of the impeller where the higher shear strain rate was produced. Grid independence was verified by demonstrating that additional requirement on mesh size near the impeller surface did not change the calculated power number by more than 2% (Table 2). From Table 2, it can be seen that the maximal mesh size of 1 mm normal to the impeller surface (Grid 3) is proper. The generated mesh is shown in Fig. 2, where X, Y, Z are the three directions of the Cartesian coordinate system which make the bottom center of the stirring shaft as the coordinate origin. Interpolation was used at the interface of two zones as the solutions progressed. Table 2 Mesh independence and selection － (Kpsu＝10 Pa·sn, n＝0.8, N＝2.5 r·s 1) Grid



Total meshes



1



342756



Relative Max. mesh size NP defined normal to impeller in Eq. (14) deviation for by CFD power number surface/mm 1.5



44.45



4 4.1



RESULTS AND DISCUSSION Characteristics of flow field



Flow patterns enhanced by DHR have been studies by many investigators, such as Bourne and Butler [27], Carreau et al. [6]. However, Delaplace et al. [4] did a survey on existing literature and found that, in spite of the different DHR impeller geometries, the primary circulation patterns are approximately the same, namely the liquid between the blades and the wall flows downwards, inwards along the bottom, upwards in the core near the shaft and radically outwards near the liquid surface of the tank. It showed contrary flow pattern when the impeller rotated reversely. From Fig. 3 (a), we can see the basic flow patterns on the Y = 0 plane of the stirred tank simulated in this work is consistent with the conclusion in



-



2



789779



1.2



45.44



2.23%



3



1515566



1.0



44.90



1.19%



4



1984809



0.8



44.36



1.21%



3.3 Boundary conditions



No slip boundary conditions were imposed at the solid walls of the tank and the impeller, while the free surface at the top of the vessel was treated as a flat, shear free boundary. In this study, considering that the fluid flow in the stirred tank is in the laminar region, the free surface assumption is realistic. The rotational speeds of the impeller under the stirring condition with various fluid properties were reported in Table 3.



(a) Calculated by CFD in this work (Kpsu＝10 Pa·sn, n＝0.8, N＝2.5 r·s－1)



Table 3 Description of the fluids and rotational speed of the impeller No.



Kpsu/Pa·sn



n



1



10



0.52



1



0.92



2



10



0.6



1.5



1.63



3



10



0.7



2



2.27



4



10



0.8



2.5



2.77



5



10



0.9



3



3.09



6



10



1



3.5



3.23



N/r·s



－1



Ren



(b) Calculated by Yao et al. [28] (Newtonian fluid, η＝ 20 Pa·s, ρ＝1400 kg·m－3, N＝0.25 r·s－1) Figure 3 Velocity vector map in Y＝0 plane of the stirred tank



Chin. J. Chem. Eng., Vol. 16, No. 5, October 2008



Ref. [4], and there is a secondary circulation near the impeller region which is similar to the calculation results of Yao et al. [28] with different geometrical dimensions [Fig. 3(b)]. 4.2



Power consumption



To verify the simulation model quantitatively, we compared experimental results by Rieger and Novak [10] with simulation predictions on the same experimental equipment. For the non-Newtonian fluids, the power consumption P and the power number Np of the mixing process were calculated from Eqs. (13) and (14): (13) P = 2πNτ Np =



P ρ N 3d 5



689



post-simulation-processing methods are discussed on such a basis to determine ks. 4.3.1 Method I-slop method Wang et al. [30] found that, when ks is independent of the flow behavior index n, the slope method, which can avoid exponential operation, can be applied to predict ks more accurately than direct calculation method. In this paper, the power consumption was acquired through CFD data-processing and ks was predicted through the slope method. The curve of lnKpn versus (1 − n ) for DHR impeller was plotted in Fig. 5. It was determined by the line slope that the value of ks was 39.6.



(14)



where τ can be obtained through data-processing by Fluent 6.2 after calculation convergence:



τ = ∑ i ( Δp )i Ai ri



(15)



where the summation in Eq. (15) is carried over all the volumes having the impeller as one of the boundary. Then the mixing power constant Kpn of the non-Newtonian fluids can be calculated according to Eqs. (6) and (7). The mixing power constant calculated from the CFD was compared with the experimental data of non-Newtonian fluids in the DHR impeller system, as shown in Fig. 4. It can be seen that two sets of data satisfactorily agreed with one another with an average error of 6.32%, a value acceptable for engineering design.



Figure 5 The relationship between lnKpn and ( 1 − n ) of DHR impeller (N＝1-3.5 r·s－1)



4.3.2 Method II When conducting numerical simulation to predict the Metzner number ks of DHR impeller in pseudo plastic fluid, Shekhar and Jayanti [20] drew the curve of circumferentially averaged local shear rate versus the radial profile at the mid-height of the impeller to find the maximal value under each rotational speed. Based on this, the curve of the maximal shear rate value versus rotational speed was plotted. As a result, the value of ks was obtained through measuring the slope of the line. In this work, the same method was adopted to predict ks for analysis. The curve of the maximal circumferentially-averaged shear rate and rotational speed was shown in Fig. 6. The line slope, known as the ks value, was 33.6. The curve of circumferentially averaged shear rate versus radius at each rotational speed was omitted.



Figure 4 Experimental vs. CFD-computed power constant Kpn for DHR impeller (Kpsu＝10 Pa·sn, N＝1-3.5 r·s－1) ▲ CFD-computed; ● experimental [10]



4.3



CFD solution of ks



Previous research [29] has proven that the Metzner number ks of weak non-Newtonian fluids with the flow behavior index greater than 0.4 for helical ribbon impeller is related to the structure and dimensions of the impeller, but that it is independent of the consistency factor Kpsu and the flow behavior index n. In this study, ks was a constant given that the fluids were weak non-Newtonian fluids and the dimensions of the helical ribbon impeller used in the experiment were not changed. Therefore, three



Figure 6 The relationship between maximal circumference-averaging shear rate in the middle-height of impeller and rotational speed (Kpsu＝10 Pa·sn, n＝0.52-1)



It can be seen that two drawing steps were necessary to obtain ks. In addition, the maximal values of



690



Chin. J. Chem. Eng., Vol. 16, No. 5, October 2008



shear rate under each rotational speed need to be read out from the first curve to plot the second one in this method. 4.3.3 Method III Due to the convenience of the Fluent software to calculate area-weighted average viscosity on the impeller, the method to predict ks through calculating area-weighted average viscosity as the effective viscosity was investigated in this paper. The feasibility of this method was analyzed and described in the next paragraph. After the numerical simulation converges, the area-weighted average viscosity can be obtained directly through calculating the facet values of viscosity on the DHR impeller in Fluent software： 1 ηav = ∑ j μ j | A j | (16) A Here, the area-weighted average viscosity was taken as the effective viscosity. Then the effective shear rate can be calculated using Eq. (5). The curve of the effective shear rate versus the rotational speed was plotted in Fig. 7. The results proved that the effective shear rate had a linear relationship with the speed of rotation (R2＝0.996), a relationship that was assumed by Metzner and Otto [11]. So this method names the slope value, 34.8, as the value of ks. This method is clear and concise according to the Metzner and Otto assumption.



and 8.5%, respectively. Yet calculation procedures of these two methods were quite complicated. The method III, however, utilized the area-weighted average viscosity as the effective viscosity in the mixing process and thus was easy to use. Also, the calculation error was only 5.4％. In some literature [15, 31], the effective shear rate and the effective viscosity are also referred to as the average shear rate and the average viscosity, respectively. Therefore, the effective viscosity and the effective shear rate, to some extent, are close to the average viscosity and the average shear rate in the entire stirred tank during the mixing process. However, according to Metzner and Otto [11], the effective shear rate must also be proportional to the rotational speed. Fig. 8 illustrates the distribution of viscosity in the X＝0 section in the stirred tank for the non-Newtonian fluid (Kpsu＝10 Pa·sn, n＝0.52) at the rotation speed of － 1 r·s 1. Results showed that the viscosity in the stirred tank ranged from 0.39 to 8.96 Pa·s with only a little change. Therefore, the area-weighted average viscosity around the impeller, to some extent, can represent the average viscosity in the whole stirred tank. Also the effective shear rate calculated from the area-weighted average viscosity in Eq. (5) was proportional to the rotational speed and thus the area-weighted average viscosity around the DHR impeller can be taken as the effective viscosity. Therefore, ks predicted by method III was consistent with the experimental data. Similarly, the maximal circumferential-average shear rate on the DHR impeller is the average shear rate in the motion region of the DHR impeller (method II ), so it is not only approximated to the average shear rate in the whole stirred tank, but also proportional to the rotational speed. Thus ks predicted by method II agree with the experimental data, too. The difference is that the post treatment steps of method II is more complicated.



Figure 7 The relationship between effective shear rate and rotational speed (Kpsu＝10 Pa·sn, n＝0.52-1)



Simple and clear, this method only needed one calculation step and one plotting procedure to predict the value of ks. Table 4 compares the values of ks predicted by the three aforementioned methods with the experimental data. It can be seen that the ks values calculated using method I and method II agreed with the experimental data with calculation errors of 7.8％ Table 4



Comparison of ks values by CFD solution and experiment



Method



ks



R2



Rieger and Novak [10]



36.73 ± 1.45



-



method I



39.6



0.990



method II



33.6



0.986



method III



34.8



0.996



Figure 8 Apparent viscosity distribution on X＝0 section of the agitated tank for non-Newtonian fluid (Kpsu＝10 Pa·sn, n＝0.52, N＝1 r·s－1)



It seems that Method III might be used to predicted ks value for any close-clearance impeller in laminar region based on the analysis in this study, such as anchors, spirals, etc. However, for the other impellers or in the turbulent region, more effort is



Chin. J. Chem. Eng., Vol. 16, No. 5, October 2008



needed in order to draw a definitive conclusion on this subject. 5



CONCLUSIONS



ηa ηav ηj ρ τ



691



apparent(or effective) viscosity, Pa·s area-weighted average viscosity, Pa·s facet viscosity, Pa·s － fluid density, kg·m 3 torque, N·m



Subscripts



Using the CFD method, three-dimensional simulation of the flow field generated by a DHR impeller in non-Newtonian fluids was investigated. The velocity field calculated from the numerical simulation was similar to the previous studies. The power constant Kpn of non-Newtonian fluids with a flow behavior index n which is higher than 0.52 agreed well with the experimental data. Three methods were used to calculate the Metzner constant for post-simulation data processing. The ks values calculated from the slope method (method I) and method II agreed fairly well with the experimental data. However, the post-data processing were more complicated than that of method III. The area-weighted average viscosity around the DHR impeller, close to the average viscosity in the entire stirred tank in some degree and the shear rate calculated from power law equation is proportional to the rotational speed of the impeller, can be used as the effective viscosity in the stirred tank to predict ks (method III). This method generated satisfactory simulation results and simplified the calculation. The calculation error of method III is 5.4%, also lower than that of methods I and II. Therefore, the method III can be safely applied for DHR impeller to engineering calculations. NOMENCLATURE A Ai c D d f H i Kp Kpn Kpsu ks L N Np n P ΔP R2 Re Rea Ren ri s v w



γ γeff η



surface area of the impeller, m2 projected surface area of the surface element i, m2 impeller clearance, m stirred tank diameter, m impeller diameter, m － body force, m·s 2 stirred tank height, m surface element power constant for Newtonian fluids power constant for non-Newtonian fluids consistency factor, Pa·sn Metzner constant impeller height, m － impeller rotational speed, r·s 1 power number flow behavior index hydrodynamic press, Pa pressure difference, Pa correlation coefficient Reynolds number ( Re = ρ Nd 2 / η ) apparent Reynolds number ( Rea = ρ Nd 2 / ηa ) modified Reynolds number ( Ren = d 2 N 2−n ρ / K psu ) radial coordinate, m pitch, m － velocity, m·s 1 blade width, m － shear rate, s 1 － effective shear rate, s 1 viscosity, Pa·s



i, j



surface element



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13



14 15



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