ISH Journal of Hydraulic Engineering
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CFD simulation model for mixing tank using multiple reference frame (MRF) impeller rotation Harshal Patil, Ajey Kumar Patel, Harish J. Pant & A. Venu Vinod To cite this article: Harshal Patil, Ajey Kumar Patel, Harish J. Pant & A. Venu Vinod (2018): CFD simulation model for mixing tank using multiple reference frame (MRF) impeller rotation, ISH Journal of Hydraulic Engineering, DOI: 10.1080/09715010.2018.1535921 To link to this article: https://doi.org/10.1080/09715010.2018.1535921
Published online: 25 Oct 2018.
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ISH JOURNAL OF HYDRAULIC ENGINEERING https://doi.org/10.1080/09715010.2018.1535921
CFD simulation model for mixing tank using multiple reference frame (MRF) impeller rotation Harshal Patila, Ajey Kumar Patelb, Harish J. Pantc and A. Venu Vinoda a Department of Chemical Engineering, National Institute of Technology Warangal, Warangal, India; bDepartment of Civil Engineering, National Institute of Technology Warangal, Warangal, India; cIsotope and Radiation Application Division, Bhabha Atomic Research Centre, Mumbai, India
ABSTRACT
In this work, CFD simulations have been conducted to investigate flow behaviour of water in fully baffled stirred tank with Rushton turbine as impeller. The aim of the paper is to develop CFD model of such tank and optimize the dimensions of inner rotating fluid zone for MRF model. The best inner rotating fluid zone is found when the simulation results for mean radial velocity, mean tangential velocity and mean axial velocity are reasonably matched with literature data of Wu and Patterson (1989). The global flow parameters such as flow number and power number predicted by CFD matched quite well with the data of Wu and Patterson (1989) and Rushton et al. (1950). The developed CFD model is able to predict key phenomenon of circulating loops inside the stirred tank. The trend of turbulence energy dissipation rate obtained from CFD simulations is in good agreement with the literature data of Wu and Patterson (1989).
ARTICLE HISTORY
Received 29 April 2018 Accepted 10 October 2018 KEYWORDS
CFD; Stirred tank; Rushton turbine; multiple reference frame (MRF); turbulence model
Abbreviations: CFD: Computational Fluid Dynamics; LDA: Laser Doppler Anemometry; PIV: Particle Image Velocimetry; LSI: Laser Sheet Illumination; MRF: Multiple Reference Frame; RANS: Reynolds averaged Navier–Stokes; LES: Large Eddy Simulation; DNS: Direct Numerical Simulation; SIMPLE: Semi Implicit Method for Pressure Linked Equations; RMSE: Root Mean Squared Error; PRESTO: Pressure Staggering Option
1. Introduction Mixing tanks are extensively used in the process industries such as chemical, pharmaceutical, food, oil and biochemical as well as in municipal and industrial wastewater treatment plants. Based on their applications, these are referred as mixing tanks, stirred tanks, agitated tanks and aeration tanks. Depending upon the purpose of application carried out in the mixer such as blending of miscible liquids, solid suspension, dispersion of gas into liquid, heat and mass transfer enhancement etc., the choice of tank geometry and type of impeller varies widely (Ciofalo et al. 1996). In order to get good product quality with optimum economical way, it is important to know the degree of mixing, performance and behavior inside the tanks. Hence, it is important to investigate agitation hydrodynamics inside the mixing tank (Yapici et al. 2008). The large amount of energy is required for a mixing process which causes major expenses. The cost of poor mixing from a multinational chemical company was estimated at $100 million per year in 1993 (Paul et al. 2004). Day by day the cost of energy is increasing, so it is essential to do the optimum design analysis of mixing tanks. On account of extensive use of stirred vessels in process industries, major price saving is possible with minor reduction in operational costs. Even though continuous efforts have been done to enhance the performance of different impellers with consideration of their pumping capacity and level of mixing attained, a variety of impellers being evolved and lead into virtual applications (Dewan et al. 2006). The flow structure inside the stirred tank has been investigated with sophisticated flow measurement techniques such as CONTACT Ajey Kumar Patel © 2018 Indian Society for Hydraulics
[email protected]
laser Doppler anemometry (LDA), particle image velocimetry (PIV) and laser sheet illumination (LSI). Even though these experimental studies have measured flow field accurately, these techniques were neither economical nor practical because the best choice for geometry of tank and impeller type varies depending on the purpose of operation carried out in stirred tank (Yapici et al. 2008). Rotating impellers generate high turbulence and complex three-dimension flow structure as the flow induced by them interacts with baffles mounted on wall of tank. It is very difficult to understand flow structure due to complexity in flow generated. With the consideration of the issues mentioned above, in recent years computational fluid dynamics (CFD) tool has been extensively used to understand such complex behavior inside stirred tanks. In the literature, various approaches have been adopted to investigate the flow pattern generated in stirred tank. In the past, flow simulations have been done with ‘black box’ approach, in that impeller region was excluded from computational domain in stirred tank (Brucato et al. 1998). Even though this technique gives the successful prediction of flow field, this technique requires the experimental data including turbulence quantities while this data is available only for few vessel geometries (Brucato et al. 1998; Ranade 1995; Ranade and Dommeti 1996). Different techniques have been proposed for the modeling of stirred tank with rotating impellers to overcome the flaws pointed in the ‘black box’ method. The sliding mesh technique was the first approach emanated by Luo et al. (1993) and same path was adopted by Bakker et al. (1997) for the flow modeling of pitched blade turbine. In the sliding mesh approach, the
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stirred vessel is segmented in two sections, the first one impeller containing section and second section contains the volume of liquid, vessel wall, vessel base and the baffles. The grid generated in the impeller occupied section moves with impeller; however, the grid in vessel section remains in stationary condition. Sliding mesh technique is more beneficial as it does not require any boundary conditions acquired from experiments. On the account of computational expenses, sliding mesh approach is more costly for the startup problems and this approach is not economical for design purpose (Dewan et al. 2006). The second optimistic approach, multiple reference frame (MRF), was put forward by Luo et al. (1994) for modeling of impeller rotation. In this approach, stirred vessel is partitioned into two frames, i.e. a moving frame and stationary frame. The moving reference frame encapsulates the impeller and the flow confined by it, and the stationary frame involves the vessel, the baffles and the flow outside moving frame. MRF approach was used by Naude et al. (1998) for modeling of baffled stirred tank with propeller rotor; the results obtained for time averaged computational mean velocities were found to be in good agreement with LDA measurements. The third approach is the snapshot flow model which was developed by Ranade and Dommeti (1996). This model was suitable to predict the flow around the impeller and did not require the experimental data. Further, they applied this model in the CFD work of stirred tank with pitched blade turbine and validated the results with experimental LDA measurement of Ranade and Joshi (1990). The snap shot model as well as MRF model are steadystate models but the later model is widely adopted by various researches in comparison with former model (Deglon and Meyer 2006; Coroneo et al. 2011; Brucato et al. 1998). Snapshot model does not consider impeller baffle interactions which considerably affect the accuracy of flow field predictions in baffled stirred tanks. But MRF method properly accounts the impeller baffle interactions by dividing into rotating zone containing impeller and stationary zone containing baffles. Further snap shot model is not widely validated throughout the entire flow domain in any stirred tank as compared to MRF model (Joshi et al. 2011). Thus, MRF technique is computationally less expensive and does not require any experimental data (Dewan et al. 2006; Khopkar et al. 2004). Accuracy in prediction of fluid turbulence is major challenge in the CFD modeling for stirred tank. Three different ways have been documented in the literature: (1) Reynolds averaged Navier–Stokes (RANS) equations, (2) direct numerical simulation (DNS) and (3) large eddy simulations (LES). In RANS approach, the instantaneous flow variables are decomposed into the mean and fluctuating components. This is the most widely used approach for engineering problems (Khopkar et al. 2004; Li et al. 2004) as it is computationally more economical compared to DNS and LES. In order to solve the RANS equations, it requires appropriate turbulence model. Every turbulence models are having their own advantages and disadvantages based on particular investigation. Different turbulence models have been applied to study the flow in stirred tanks (Murthy and Joshi 2008; Singh et al. 2011; Gimbun et al. 2012). The instantaneous flow variables are solved and no modeling is required in the case of DNS. In RANS model, all scales from smallest to largest are
resolved. In LES, smallest scales are modeled and large scales are resolved. Following the literature, MRF impeller rotation model and RANS turbulence model are suitable for modelling of stirred tank. Due to high speed of impeller rotation, the flow variations are very sharp near the impeller. The effective region for sharp variation in flow is 1.5 times of blade height above and below the impeller disc and D/2 away from the impeller tip which is reported by Lee and Yianneskis (1994). Following this concept, same dimensions have been adopted for inner rotating fluid zone in MRF technique in the CFD modeling of stirred tank by Deglon and Meyer (2006). As concerned with the use of MRF technique for impeller rotation model, no proper method is available in the literature for how the dimensions of inner rotating zones are considered. With this consideration of the issue for the selection of inner rotating fluid zone, this work has been done to find the optimal dimension of inner rotating fluid zone by varying diameter and height of inner rotating fluid zone. Also, only few works have been done to develop the CFD model to characterize the hydrodynamics inside the stirred or mixing tank which will be ultimately useful for optimal design of tanks. The optimal inner rotating fluid zone is considered where simulation results for velocity predictions such as tangential velocity, radial velocity and axial velocity were found to be in reasonable agreement with literature data available by Wu and Patterson (1989). Also, the efficiency of optimal MRF zone is investigated by comparing the power number predictions at various Reynolds numbers with the classical experimental results of Rushton et al. (1950). Moreover, the hydrodynamic characteristics such as eye of recirculation region, radial pumping capacity and turbulence dissipation rate are validated against literature data.
2. Tank configuration and impeller geometry The stirred tank configuration used for simulation in this study is same as Wu and Patterson (1989) as shown in Figure 1. The dimensions used are tank diameter T = 0.27 m and height of water in tank H = T. The diameter of turbine used is D = 0.093 m and placed at distance of one third from the bottom of tank. The water is used as working fluid for the system with density (ρ) of 998.2 kg/m3 and dynamic viscosity (μ) of 0.001003 kg/m-s at 25°C. The speed of impeller (N) was kept at 200 rpm for finding the optimal dimensions of inner rotating zone.
3. Computational methodology Finite volume method is used to solve the governing flow equations. Ansys Fluent 14.5 software package is used for modeling of stirred tank. The flow domain is divided into small volumes and mesh is created using meshing tool available in Ansys. An unstructured tetrahedral mesh is produced for fluid region while for rotor and baffles structured hexahedral mesh has been used as shown in Figure 2. The boundary conditions adopted for CFD model of stirred tank is shown in Figure 3. Momentum and turbulence quantities are discretized by second-order scheme. Pressure staggering option scheme is adopted for pressure. The velocity and pressure are coupled
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Figure 1. Dimensions of Rushton turbine stirred tank.
(a)
(c)
(b)
Figure 2. Meshing for different parts of CFD model of the stirred tank (a) fluid domain, (b) rotor and baffles and (c) inner rotating fluid zone.
Tank top (symmetry)
Tank wall (stationary wall)
Baffled wall (stationary wall)
Inner rotating zone
Figure 3. Boundary conditions adopted for CFD model.
through the semi-implicit method for pressure linked equations algorithm. To attain better convergence and stability in the solution, the under relaxation factor was kept at lower value. The convergence criterion is kept at 10−4 for continuity, velocity and turbulence quantities, and the convergence graph is shown in Figure 4.
4. Model equations and turbulence models 4.1. Model equations Flow behavior inside stirred tank is solved by discretized governing equations. The governing equations are the Navier–Stokes equations, which solve the mass and
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Figure 4. Convergence graph.
momentum conservation equations and provide solution for flow variables such as velocity and pressure. The continuity or mass conservation equation is given by @u @v @w þ þ ¼0 @x @y @z
(1)
The momentum conservation equations for incompressible flow are given by 2 Du @p @ u @2u @2u ρ ¼ ρgx þμ þ þ (2) Dt @x @x2 @y2 @z2 2 Dv @p @ v @2v @2v ¼ ρg þ μ þ þ ρ y Dt @y @x2 @y2 @z2 2 Dw @p @ w @2w @2w ¼ ρgz þμ þ þ ρ Dt @z @x2 @y2 @z2
(7) μt @ 2 þ ρc1 s2 σ 2 @xj 22 pffiffiffiffiffiffiffi ρc2 kþ v2 (8)
@ @ @ ðρ 2 Þ þ ρ 2 uj ¼ @t @xj @xj
(3) where, (4)
where u; v and w are the fluid velocities in the x; y and z directions, respectively, and p is the local pressure. 4.2. Turbulence models RANS equation is used for turbulence model; in this flow; variables in the instantaneous form are decomposed into mean and fluctuating components. Substituting the decomposed form of flow variables in the continuity and momentum equation and taking a time average yields the ensemble averaged continuity and momentum equations. The modified equations can be written as @ρ @ þ ðρui Þ ¼ 0 @t @xi
stresses. The realizable k-ε model solves the two additional equations for kinetic energy of turbulence and dissipation energy rate; the details of the equations are as follows: μt @k @ @ @ ðρkÞ þ ρkuj ¼ μþ þ Gk ρ 2 @t @xj @xj σ k @xj
(5)
@ @ @p @ @ui @uj 2 @ul ðρui Þ þ ρui uj ¼ þ μ þ δij @xj @xi 3 @xl @t @xj @xi @xj @ 0 i ρui uj þ @xj (6)
The above equations are the RANS equations. The additional 0 term ui uij represents the effects of turbulence and this is called as Reynolds stresses. It is the job of turbulence model to compute the value of Reynolds stresses. The realizable k-ε model is used as turbulence model to compute the Reynolds
c1 ¼ max 0:43
μþ
pffiffiffiffiffiffiffiffiffiffi η k ; η ¼ s ; s ¼ 2sij sij ηþ5 2
(9)
In Equation (7) Gk representsthe generation of k due to mean velocity gradients, μt is the turbulent viscosity. The model constant values are c12 ¼ 1:44; c2 ¼ 1:9; σ k ¼ 1 and σ 2 ¼ 1:2 (ANSYS Fluent 14.5 Theory Guide 2013). As explained earlier, the dimensions of inner rotating zones need to be specified in MRF technique for CFD modeling of stirred tank. Inner rotating fluid zones with varying dimensions considered for simulations are shown in Table 1 and the corresponding geometric model for CFD simulation is shown in Figure 5.
5. Results and discussion In the literature, the validity of CFD model is tested by comparing the hydrodynamic parameters with experimental data inside the stirred tanks. In this study, qualitative and quantitative comparison of CFD results with experimental Table 1. Fluid zone numbers and their dimensions. Inner rotating fluid zone number 1 2 3 4 5 6 7
Height (m) 0.0392 0.0410 0.0429 0.0503 0.0540 0.0578 0.0615
Diameter (m) 0.0930 0.1023 0.1116 0.1488 0.1674 0.1860 0.2046
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Zone 2
Zone 1
Zone 4
Zone 5
5
Zone 3
Zone 6
Zone 7
Figure 5. Diagrammatic representation of inner rotating fluid zones.
literature has been done in terms of velocity fields and global flow parameters inside the tank.
5.1. Prediction of mean velocities In this section, CFD results are predicted in terms of mean radial, tangential and axial velocities normalized with the impeller tip speed. The measurement has been done at radial distance of 5 cm and at an angle of 45°. The results are plotted in terms of dimensionless parameters as follows: Z ¼
2z Ur Uθ Uz ; Ur ¼ ; Uθ ¼ ; Uz ¼ w Utip Utip Utip
(10)
where Utip ¼ πND In the literature, experimental studies of Wu and Patterson (1989) have been widely used for the comparison of velocity fields. Hence, for current work, same literature data is referred for the comparison of results. The comparison between CFD predictions and literature data of mean velocity fields at different inner rotating zone is depicted in Figures 6, 8 and 10. At plane z ¼ 0, the value of radial and tangential velocities reached to maximum, as there is sharp variation in flow fields due to continuous rotation of impeller. CFD slightly under-predicted the normalized radial velocities but matches the profiles qualitatively with reported data. This under-prediction of radial velocity could be observed because k–ε turbulence model may not
accurately predict the flow behavior in the impeller region (Basavarajappa et al. 2015). The peak value of mean radial velocity predicted by CFD at zone 6 is in good agreement with literature data compared to other zones (Figure 6). Mean tangential velocities are well predicted by CFD as shown in Figure 8. The axial location of peak tangential velocity is shifted below as compared with experiments. The trend of tangential velocity observed at zone 6 is in good agreement with literature data compared to other zones (Figure 8). In order to find the best inner rotating fluid zone among all seven different inner rotating fluid zones under consideration, the CFD results of mean radial, mean tangential and mean axial velocities are quantitatively measured in terms of root mean squared error (RMSE) and correlation coefficient. RMSE and correlation coefficient are the basic statistical indicators which are commonly used by researchers for quantitative evaluation of model predictions. RMSE measures the difference between the values predicted by the CFD model and the values observed from the literature data. Further, the correlation coefficient is estimated which describes the strength of linear relationship between CFD model predicted values and literature data. The smallest value of RMSE and the largest value of correlation coefficient represent the good match between the CFD predictions and experimental data (Barnston 1992). The dimensions of inner rotating zones are increased from zone 1 to zone 7 as shown in Table 1. Zone 1 is very close to impeller tip and zone 7 is near to baffles. From Figure 7 the
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2.5 2 1.5
Wu and Patterson (1989)
CFD predictions at zone 1
CFD predictions at zone 2
CFD predictions at zone 3
CFD predictions at zone 4
CFD predictions at zone 5
CFD predictions at zone 6
CFD predictions at zone 7
1 0.5
Z* 0 -0.5 -1 -1.5 -2 -2.5 -0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Ur* Figure 6. Comparison of mean radial velocity profiles between literature data and CFD predictions for different inner rotating fluid zones.
Root mean squared error
Correlation coefficient
(a) 1.2 0.933 0.951 0.938 0.956 0.961 0.911
1 0.8 0.6 0.4
0.227
0.2
0 zone 1 zone 2 zone 3 zone 4 zone 5 zone 6 zone 7
(b) 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0
0.356
0.151 0.142
0.17 0.112 0.112 0.106
zone 1 zone 2 zone 3 zone 4 zone 5 zone 6 zone 7
Inner rotating zones
Inner rotating zones
Figure 7. Correlation coefficient (a) and root mean squared error (b) between CFD predictions and literature data of radial velocity at different inner rotating fluid zones.
2.5
Wu and Patterson (1989)
2 1.5
CFD predictions at zone 1
CFD predictions at zone 2
CFD predictions at zone 3
CFD predictions at zone 4
CFD predictions at zone 5
CFD predictions at zone 6
CFD predictions at zone 7
1 0.5
Z*
0 -0.5 -1 -1.5 -2 -2.5 -0.1
0
0.1
0.2
0.3
0.4
U
0.5
0.6
0.7
0.8
*
Figure 8. Comparison of mean tangential velocity profiles between literature data and CFD predictions for different inner rotating fluid zones.
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RMSE in radial velocity goes on decreasing from zone 1 to zone 6 and it increases at zone 7. Similarly the correlation coefficient for radial velocity increases from zone 1 to zone 6 and decreases at zone 7. The best value of RMSE and correlation coefficient in radial velocity for different zones in comparison with literature data is found at zone 6. Similarly, the best value of RMSE and correlation coefficient for tangential velocity is found at zone 6 as shown in Figure 9. But in the case of axial velocity, slight deviations are found at zone 6 as shown in Figure 10 and the corresponding RMSE and the correlation coefficient are shown in Figure 11 . As the Rushton turbine is the radial flow impeller, the flow variations predominantly happen due to radial and tangential velocities. The CFD results at zone 6 are found to qualitatively and quantitatively best match with literature data and hence the dimensions considered at zone 6 are optimal dimensions for CFD modeling of
stirred tank. Further, the zone 6 is considered to investigate the radial pumping capacity, power number and turbulent dissipation rate.
5.2. Location of eye of recirculating loops For qualitative comparison of CFD predictions over experimental results, the key phenomenon recirculating loops in radial flow stirred tank is considered. Velocity vector found in current work has been plotted at r-z plane as shown in Figure 12(a). We can see that the radial flow pattern is developed by Rushton impeller. The radial jet stream from impeller goes toward the walls and it splits into two regions, one of them circulates toward the upper region and other toward the lower region and both finally comes to impeller region. These two streams create two circulation loops, one loop is above impeller and other is below
Root mean squared error
Correlation coefficient
(a) 1.2 0.951 0.949 0.962 0.966 0.977 0.942
1 0.8 0.6 0.4 0.17
0.2
7
0 zone 1 zone 2 zone 3 zone 4 zone 5 zone 6 zone 7
(b) 0.4
0.356
0.3 0.2 0.088
0.1
0.1204 0.114
0.092 0.085
0.11
0 zone 1 zone 2 zone 3 zone 4 zone 5 zone 6 zone 7
Inner rotating zones
Inner rotating zones
Figure 9. Correlation coefficient (a) and root mean squared error (b) between CFD predictions and literature data of tangential velocity at different inner rotating fluid zones.
2.5 2 1.5
Wu and Patterson (1989)
CFD predictions at zone 1
CFD predictions at zone 2
CFD predictions at zone 3
CFD predictions at zone 4
CFD predictions at zone 5
CFD predictions at zone 6
CFD predictions at zone 7
1 0.5 0
Z*
-0.5 -1 -1.5 -2 -2.5 -0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
Uz* Figure 10. Comparison of mean axial velocity profiles between literature data and CFD predictions for different inner rotating fluid zones.
(b)
1
0.936 0.917 0.912
0.868
0.8 0.6
0.47
0.4 0.2
0.09
0.0905
0 zone 1 zone 2 zone 3 zone 4 zone 5 zone 6 zone 7
Inner rotating zones
Root mean squared error
Correlation coefficient
(a) 1.2
0.1
0.091
0.08 0.06
0.061 0.057 0.047
0.057 0.041
0.04
0.04
0.02 0 zone 1 zone 2 zone 3 zone 4 zone 5 zone 6 zone 7
Inner rotating zones
Figure 11. Correlation coefficient (a) and root mean squared error (b) between CFD predictions and literature data of axial velocity at different inner rotating fluid zones.
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the impeller. The structure of eye of recirculating loops found in current work (Figure 12(a)) matches with the experimental literature data of Ranade and Joshi 1990 as shown in Figure 12(b).
which can significantly affect the pumping number predictions (Deglon and Meyer 2006). The results of Rushton et al. (1950) show slight deviations from the CFD results as well as results of Wu and Patterson (1989) due to the variations in impeller geometry, operating conditions and measurement techniques (Wu and Patterson 1989).
5.3. Impeller pumping capacities Flow number is one of the global flow characteristic which is generally defined in experimental and computational works to validate their results. The flow number defines rate of liquid circulation in mixing tanks. The flow number is calculated by normalizing radial pumping capacity with ND3 . Here, in Equation 11, radial pumping capacities are obtained by integrating the mean radial velocities over the axial positions z1 = −2 to z2 = 2. The speed of impeller was kept at 200 rpm. The trend of flow number versus radial position is shown in Figure 13. The radial pumping capacity increases with increase in radial distance from the impeller tip due to the fluid entrainment.
5.4. Prediction of power number For account of process economics, the power required for stirring is the key factor. In this study, the effect of Reynolds number on power consumption is studied. The power consumption is defined in terms of dimensionless power number (Equation 12) where Pis the power given by impeller to liquid for agitation. The torque (τ) required to calculate the power consumption is extracted from CFD results.
z2
Qr ¼ 2πr ò Ur dz
NRe
(11)
z1
The percentage deviation in pumping number values at most of the radial locations is less than 15%. This shows a good match of CFD results with that of experimental results of Wu and Patterson (1989). But the pumping number near the impeller shows large variations from the experimental results and this may be due to the limitations of RANS approach as well as k-ε turbulence model in predicting turbulence quantities (Deglon and Meyer 2006). Further, the thickness of blade and impeller disc is not revealed in literature of Wu and Patterson (1989)
P where P ¼ 2πNτ; ρN 3 D5 ND2 ρ ¼ μ
power number ðNp Þ ¼
Figure 14 shows the log–log plot of power number versus Reynolds number. The Reynolds number is varied from 0.5 to 60,768. In the laminar range, where the Reynolds number is less than 20, the variation in power number is linear with Reynolds number. In turbulent regime, where the Reynolds number is more than 10,000, there is no variation in power number compared to Reynolds number. The CFD predictions of power number in the laminar regime match closely with experimental results of Rushton et al. (1950) with percentage deviation less
(a)
(b)
Eye of upper circulation loop
Eye of upper circulation loop Impeller jet stream Eye of lower circulation loop
Impeller jet stream
Figure 12. (a) Velocity vector plot at N = 200 rpm (current work). (b) Velocity vector in r-z plane (Ranade and Joshi 1990).
2.5 CFD predictions
Wu and Patterson (1989)
Rushton et al. (1950)
2
1.5
Qr /ND3
1
0.5
0 1
1.2
1.4
1.6
r/R Figure 13. Profiles of radial pumping capacity in the impeller stream.
(12)
1.8
2
2.2
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than 12% for all the Reynolds numbers. But the average percentage deviation in the power number predictions between CFD model and experimental results increases to 25% in the turbulent regime. This deviation is partly due to limitations of k-ε turbulence model as well as variations in the thickness of blades used in experiments as well as in CFD modeling. It is already found that the thickness of blades has considerable influence on power number predictions (Deglon and Meyer 2006).
dissipation is found in the impeller stream region. This difference is observed due to the limitations of the RANS which may not capture the small eddies in this region and hence their effect on dissipation rate.
5.5. Turbulence energy dissipation rate
6. Conclusions
Turbulent intensity is high in the region containing impeller or impeller stream. There are two energies, turbulent kinetic energy and turbulence dissipation energy, which are produced from power of impeller rotation and other energies are negligible (Wu and Patterson 1989). Figure 15 shows the comparison of experimental literature data and CFD predicted normalized local distribution of turbulence energy dissipation rate. Maximum turbulence energy dissipation rate is observed at middle part of impeller stream, and in this region, turbulence intensities are sharp. The percentage deviation in the CFD predicted turbulent energy dissipation rate over the literature data is found to be less than 23%. The maximum percentage change in turbulent energy
The CFD model for stirred tank with Rushton has been developed and steady-state simulations have been conducted. Based on the comparison of root mean squared error and correlation coefficient in the predictions of normalized mean velocities, zone 6 having diameter of 0.186 m and height of 0.0578 m was found to be optimal for CFD modeling of stirred tank. With the optimized zone, predictions of flow parameters from CFD model show good match with experimental results. The percentage deviation in the power number predictions is less than 12% in the laminar regime while it increases to 25% in the turbulent regime. But the pumping number predictions at various radial locations show a percentage deviation less than 15% in
T ¼
2 2 N where 2¼ turbulent energy dissipation rate Utip (13)
1000 Rushton et al. (1950)
CFD predictions
100
Np 10
1 0.1
1
10
100
1000
10000
100000
NRe Figure 14. Comparison of power number of CFD predictions with literature data.
2.5 CFD predictions
Wu and Patterson (1989)
2 1.5 1 0.5
Z*
0 -0.5 -1 -1.5 -2 -2.5 -5
0
5
10
T* Figure 15. Distribution of normalized turbulence energy dissipation rate.
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comparison with experimental results. Moreover, axial profile of normalized turbulent energy dissipation rate was found to match the experimental results with the relative difference of 23%. Thus, optimized MRF zone predicts power number and pumping number under various Reynolds numbers and radial locations in reasonably accurate manner.
Notation H T D c l w b N u; v; w r; θ; z p C12 ; C2 ; σ k ; σ 2 k z r R Utip Ur Uθ Uz Np P NRe
Height of water in tank, m Tank diameter, m Impeller diameter, m Impeller clearance, m Blade length, m Blade width, m baffle width, m Impeller rotational speed, rpm Fluid velocities in the x; y and z directions respectively, ms−1 Radial, tangential, axial The local pressure, Nm−2 Model constants used in the equations Turbulent kinetic energy, m2s−2 Axial coordinates, m Radial coordinates, m Impeller radius, m Impeller tip speed, m s−1 Mean radial velocity, m s−1 Mean tangential velocity, m s−1 Mean axial velocity, m s−1 Power number = ρNP3 D5 Power, J s−1 2 Reynolds number = NDμ ρ
Greek symbols µ ϵ Ρ
Viscosity of water, kg m−1s−1 Turbulence energy dissipation rate, m2s−3 Density of water, kg m−3
Acknowledgments Financial support from Board of Research in Nuclear Science (BRNS), Government of India, is greatly acknowledged [sanction no. 35/14/32/ 2014-BRNS, 23 May 2014).
Disclosure statement No potential conflict of interest was reported by the authors.
Funding This work was supported by the Board of Research in Nuclear Sciences, Government of India [sanction no. 35/14/32/2014-BRNS, 23 May 2014].
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