- Email: [email protected]
Large Eddy Simulations of Mixing Time in a Stirred Tank
Chinese J. Chem. Eng., 14(1) 1-7
(2006)
Large Eddy Simulations of Mixing Time in a Stirred Tank MIN Jim(N#) and GAO Zhengming($j E a )* College of Chemical Engineering, Beijing University of Chemical Technology, Beijing 100029, China
Abstract Large eddy simulations (LES) of mixing process in a stirred tank of 0.476m diameter with a h a r r o w blade hydrofoil CBY impeller were reported. The turbulent flow field and mixing time were calculated using LES with Smagorinsky-Lilly subgrid scale model. The impeller rotation was modeled using the sliding mesh technique. Better agreement of power demand and mixing time was obtained between the experimental and the LES prediction than that by the traditional Reynolds-averaged Navier-Stokes (FUNS) approach. The curve of tracer response predicted by LES was in good agreement with the experimental. The results show that LES is a reliable tool to investigate the unsteady and quasi-periodicbehavior of the turbulent flow in stirred tanks. Keywords large eddy simulations, subgrid scale model, mixing time, hydrofoil impeller
1 INTRODUCTION
ton turbine (DT6). They suggested that the quality of results was highly dependent on exact fluid dynamic Mechanically stirred tanks are widely used in many industrial processes. The flow structures in a computation, especially concerning turbulence modturbulently stirred tank are highly three-dimensional eling. Jaworski et uZ.'~]reported that the computed 095 and complex, and cover a wide range of spatial and was about two to three times the measured values in temporal scales. The fluid is circulated through tank stirred tanks with dual DT6, and suggested that this under the action of a revolving impeller. The vortices difference arose because the mass exchange between the four distinct axial-radial circulation loops was produced by impeller rotating, which retain their coherency over a significant distance into the bulk liquid, heavily underpredicted by CFD. are associated with high shear rates and strong turbuLES, first adopted in stirred tank by Eggle'61,was proved to be a good method to investigate turbulent lent activity. Therefore, they are essential to the mixing performance in the flow field. The mixing time, &, flows, unsteady and quasi-periodic behavior. Subsequently, Revstedt et ~ 1 . ' ~ is the time required to mix the added secondary fluid pointed ' out LES could prowith the contents of the vessel to a certain degree of vide details of the flow field that cannot be obtained uniformity, typically Og5meant to reach above 95% of with so-called Reynolds averaged equations and corthe final concentration. In any case the mixing time of responding models, and then Revstedt and Fuchs[*] a stirred tank is often used as an indication of its efsimulated the tank stirred by either two standard fectiveness. Knowledge of mixing time is required for Rushton impeller or two Scaba 6SRGT impellers. the optimum design of stirred tanks. Extensive exDerksen et aZ.[91used LES with Smagorinsky subgrid perimental studies on mixing time have been reported model, a Smagorinsky constant Cs=O. 12, to simulate a during the last 30 years"]. baffled stirred tank driven by a Rushton turbine at In the last two decades, progress has been Re=29000 and Derksen"" also simulated on the sinachieved in the computational fluid dynamics (CFD) gle-phase flow driven by a pitched blade impeller ussimulations of mixing process owning to the great ing LES with the standard Smagorinsky or structure progress in the computer technique. Ranade et ~ 1 . ' ~ ' function subgrid model. The author all focused on used numerical simulation to give detail on the flow studying the three-dimension velocity and the turbuand bulk mixing generated by a downflow-pitched lent kinetic energy in a stirred tank and also proved blade turbine in a fully baffled cylindrical vessel. the LES was a good tool to investigate turbulent flow Lunden et uZ.'~] and Schmalzriedt and re us^'^' simuin industrial application of practical importance. lated pulse tracer experiments by solving the material In this work, the LES with Smagorinsky-Lilly balance in three-dimensional flow field with a Rushsubgrid model was introduced firstly to simulate mixReceived 2005-04-19, accepted 2005-10-26.
* To whom correspondence should be addressed. E-mail:[email protected]
2
Chinese J. Ch. E. (Vol. 14, No.1)
ing process through monitoring the concentration of tracer to get mixing time in a tank stirred by h a r r o w blade hydrofoil CBY impeller["]. Comparison was made between LES and RANS.
2 PHYSICALAND COMPUTATIONALCONFIGURATION The stirred tank used in this work was a perspex vessel of 0.476m diameter with flat bottom and four baffles. Ambient tap water was used. A 3-narrow blade hydrofoil CBY impeller was used. The impeller speed N was 150, 180, 260 and 300r.min-'(the corresponding Re=pND2/v=9X lo4 to 1.8X lo5) respectively and the fluid flow was fully turbulent. The details of experimental apparatus were shown in Fig. 1.
T (a)
I "I
@)
Figure 1 View of (a) the stirred tank and (b) 3-narrow blade hydrofoil CBY impellers (T=476=; H/T=l.O; C/T=1/3; DITzO.4;w~IT=0.1)
3 EXPERIMENTAL 1395 was measured from changes in conductivity after the introduction of a small quantity of tracer (saturated potassium chloride solution). lOml of the tracer was added to the free surface of the liquid between two baffles. The detector was mounted at the position near the bottom of the tank opposite to the adding point. The output from the conductivity meter February, 2006
was acquired through an analogue filter and an amplifier with an A/D converter, and then stored for subsequent analysis. The measurements were repeated at least 5 times for each experimental condition to get the average mixing time. Impeller speed and shaft torque were measured by using an optic-electrical tachometer and a torque transducer respectively. The detailed description was reported in the other paper[12', but the only difference was that the present tank had a flat bottom rather than dished base[13].
4 MATHEMATICALMETHOD 4.1 The turbulence model The main difficulty associated with the simulation of turbulent flow in a stirred tank is the wide range of scales: ranging from small scale in the bulk of the tank to the trailing vortex structure associated with the impeller blade movement, and to the large re-circulation confined by the physical geometry of the tank. So the quality and accuracy of the simulation in a stirred tank are heavily relied on turbulence model. A very accurate prediction is possible by means of direct numerical simulation (DNS). In DNS, fluid motion down to the dissipative scale is resolved and it is therefore applicable only to relatively low Reynolds number flows and not applicable to industrial related applications. The Reynolds-averaged Navier-Stokes (RANS) model represents transport equations for the mean flow quantities only, with all the scales of the turbulence being modeled. The approach of permitting a solution of the mean flow variables greatly reduces the computational effort. If the mean flow is steady, the governing equations have no contain time derivatives and a steady-state solution can be obtained economically. The computational advantage is seen even in transient situations, since the time step will be determined by the global unsteadiness in the mean flow rather than by the turbulence. The Reynolds-averaged approach is generally adopted for practical engineering calculations,and uses models such as Spalart-Allmaras, k-E, k-u), RSM and their variants. The standard k-E model, RNG k-E model['41 and anisotropic algebraic Reynolds stress model[151,are the simplest model and could predict the fully turbulent flow field reasonable agreed with the experimental data. So the standard k-E model was chosen in present work to compare with LES.
Large Eddy Simulations of Mixing Time in a Stirred Tank
LES is the one somewhere between DNS and the RANS approach. Basically large eddies are resolved directly in LES, while small eddies are modeled[61.In LES, the governing equations employed for LES are obtained by filtering the time-dependent Navier-Stokes equations in either Fourier (wave-number) space or configuration (physical) space. The filtering process effectively filters out the eddies whose scales are smaller than the filter width or grid spacing used in the computation. The resulting equations thus govern the dynamics of large eddies. A filtered variable is defined by
@ ( X I = I n @ ( X ’ ) G ( X 7 X’)dX’
(1)
where D i s the fluid domain, and G is the filter function that determines the scale of the resolved eddies. In FLUENT, the finite-volume discretization itself implicitly provides the filtering operation: -
1 @ ( X )= - jn)(X’)dX’ V
X’€
a
(2)
where V is the volume of a computational cell. The filter function, G(X , X’) ,implied here is then G(X,X’) =
l/V,X’€ 0,X’E
a
a
(3)
Filtering the incompressible Navier-Stokes equations results in
a axi
*+--(p+o at
(4)
and
P-
at
(5)
.where Z, is the subgrid-scale stress defined by -
The subgrid-scale stresses resulting from the filtering operation are unknown, and require modeling. ’The most basic of subgrid-scale models was the Smagorinsky-Lilly model”], In the Smagorinsky-Lilly model, the eddy viscosity is modeled by
(7) where Ls is the mixing length for subgrid scales is the rate-of-strain tensor for
3
the resolved scale. In FLUENT, LS is computed using
4 = min(Kd, c , v ~ ’ ~ )
(8)
where K is the Von Karman constant, d is the distance to the closest wall, V is the volume of the computational cell, CS is the Smagorinsky constant. It is set to 0.1 in this work, which was found to yield the best results for a wide range of flows“61.
4.2 Hybrid meshes For the mixing tank simulations performed, the computational mesh is made up of two parts: an inner rotating cylindrical volume enclosing the turbine, and an outer, stationary volume containing the rest of the tank. Mesh strategy adopted was the technique of combination with the structured and unstructured g~id‘*~-’~]. The impeller region was divided into unstructured tetrahedral cells and densified to get more accurate description of impeller. And the rest, the bulk of the tank, was treated with the multi-block method, in which the hexahedral cells were used to decrease cost of calculation. So in RANS the size of cells was chosen 4mm in inner part and 8 mm in the outer region. But considering the blade thickness[61and the Taylor micro it was chosen 2mm and 4mm in LES. The total grid nodes numbers are 1044356 in LES and 373775 in RANS, as shown in Fig.2.
4.3 Impeller description The main difficulty in simulating stirred tanks is the accurate representation of impeller action. The sliding mesh (SMf211 and multiple reference frame (MRF) techniques[221are two effective ways dealing with the effects of impeller, and are implemented in the commercial software packages such as FLUENT. The MRF method, in which the region of impeller and impeller steam is described in as the rotating reference frame and a stationary frame or mesh is selected for the flow outside the impeller region. In the SM technique, which is in fact a transient method, two grids are generated one rotating with the impeller and one stationary representing the bulk tank. Under the same conditions, the sliding mesh method requires much more computing time, in which the data are transferred over the interface between the two grids. Because the MRF provides a reasonable model of the time-averaged flow and the SM could compute the unsteady and transient flow field. The SM method was used by LES in this work. Chinese J. Ch. E. 14(1) l(2006)
Chinese J. Ch. E. (Vol. 14, No.1)
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(a) Half tank
(b) Horizontal surface
(c) Horizontal impeller
Figure 2 View of the mesh in the stirred tank
4.4 LES procedure The first step was to calculate continuity and velocity equations using the standard k-E model, and the results were reported in elsewhere. After the flow field is somewhat converged, the result was used as an initial condition for LES simulation. The simulation chose the second-order implicit formulation for temporal discretization and the central-differencing February, 2006
scheme for spatial. The solution of continuity and velocity equations continued until the flow became statistically steady through monitoring the torque of impeller. And the second step was that the equation for concentration of tracer was solved in the time domain to get the mixing time. The tracer injection is assumed not to affect the flow. Therefore, the separation of moment and tracer balance equations is presumed to reduce the computational effort. Based on the physical coordinates of tracer adding location, the adding tracer in simulation was done over several cells near it to ensure that the mass of tracer in simulation was the same as that of experiment. The concentration of tracer was initialized as 1 in the adding region, and in the rest region as 0. And the monitoring point was set at lOmm above bottom of tank and 50mm from the wall of tank, the same as the position of the detector in experiment amid two baffles. The LES was run on XEON dual processors (Pentium IV) machines (Dell) with 1Gb of memory, 2GHz clock frequency and a LINUX operating system. The simulation for one agitator speed flow field done on six processors in parallel took about 4 weeks to get at least 20 revolutions to ensure the velocity field statistic steady, and for concentration field it took about 3 weeks.
5 RESULTS AND DISCUSSION 5.1 The flow field and concentration distribution of tracer The velocity and concentration field gotten through RANS and LES were drawn in Figs.3 and 4. In a real stirred tank,there are large vortices and resulting macro-instabilities, which promote tracer mass exchange through this boundary. The standard k-E model can not fully account for this phenomenon. But LES could catch the details of vortices even down to small Taylor and Kolmogorov micro scale[231due to sufficiently small mesh size in simulation. So LES could give the more accurate result for the instantaneous flow than RANS. As can be seen from Figs.3 and 4, comparing with that gotten from RANS, the boundary of circulation loops in LES was broken and irregular, and a lot of large and small vortices occurred in the bulk region. The mass exchange between the vortices and even in the whole tank could be resolved accurately. The mixing process can be seen directly from the contour of concentration.
Large Eddy Simulations of Mixing Time in a Stirred Tank 8.42 X lo4 8.00 X l o 4 7.58X lo4 7.16~ 10.~ 6.74X lo4 6.32X lo4 5.90X lo4 5.48X lo4 5.05 X lo4 4.63 X lo4 4.21 X lo4 3.79X lo4 3.37x lo4 2.95 X lo4 2.53 X lo4 2.1 1x lo4 1.69X104 1.27 X lo4 8.51 x 10.~ 4.30 X l o 5 9.89 x 10"
2.97 2.82 2.67 2.52 2.37 2.23 2.08 1.93 1.78 1.63 1.48 1.34 1.19 1.04 8.90 X 10'' 7.42 X 10.' 5.93 x 10'' 4.45 x 10'' 2.97 X 10.' 1.48 X 10.' 0
8.42 X l o 4 8.00 X lo4 7.58X l o 4 7.16X lo4 6.74X1O4 6.32 X lo4 5.90X l o 4 5.48 X lo4 5.05 X l o 4 4.63 X lo4 4.21 X lo4 3.79 x 10.~ 3.37x lo4 2.95 X lo4 2.53 X l o 4 2.1 1x lo4 1.69 X lo4 1.27X l o 4 8.51 X l o 5 4.30 X l o 5 9.89 X l o 7
2.97 2.82 2.67 2.52 2.37 2.23 2.08 1.93 1.78 1.63 1.48 1.34 1.19 1.04 8.90X 10'' 7.42 X 10.' 5.93 x 10-I 4.45 x lo-' 2.97X 10.' 1.48X 10" 0
Figure 4 Contour of concentration distribution of tracer (vertical in middle of two baffles, N=30Or.mi1-', k4s)
01) LES Figure 3 Instantaneous contour of velocity magnitude (vertical in middle of two baffles, N=300rmin-')
5.2
5
t
1
A
The response curve of tracer
The normalized curves of tracer response of experimental, LES and RANS are shown in Fig. 5. Good agreement can be seen from the Fig.5 between the experimental and LES prediction. The shape of response curve of tracer is largely dependent on the large vortices in the stirred tank. LES could catch the different scales of vortices especially the large vortices, which resulting in the good prediction of response curve of tracer. The RANS approach only accounts for the small vortices and couldn't predict the reasonable response curve of tracer, as shown in the Fig.5.
Comparison of mixing time between numerical simulation and experimental The torque, M , exerted by the impeller was cal(culated from the CFD analysis method used by Ja-
0
5
15
10
t, s
Figure 5 The response curves of tracer in LES and RNAS (N=260 r.min-l, all normalized) RANS -experimental; ------ LES;
------
5.3
rnrorski et a ~ . ' ~n] .e power per unit volume 6;imulation is calculated by
PV
in
Pv = 2 n : M N / V
(9)
The comparisons of power demand between experimental and numerical simulations based on LES and RANS are shown in the Fig.6. Good agreement can be seen between the experimental and LES or Chinese J. Ch. E. 14(1) l(2006)
Chinese J. Ch. E. (Vol. 14, No.1)
6
RANS, though LES offers a slightly better prediction. The relative errors of power demand for LES and RANS are 6.18% and 9.30% respectively. The comparison of mixing time between experimental and numerical simulations based on LES and RANS is shown in Fig.7. Much better agreement can be seen between the experimental and LES prediction and than RANS prediction. The relative errors of mixing time for LES and RANS are 9.26% and 16.93% respectively. As mentioned above, the LES could predict more accurate fluid field and different scale of vortices, resulting in the better agreement between the experimental and LES prediction.
150
I
I
I
200
250
300
.
Figure 6 Comparison of the power per unit volume between simulation results and experiment data .experimental; -LES; ------ RANS I
6 CONCLUSIONS Large eddy simulations of the flow field and mixing time in a stirred tank of 0.476m diameter with a 3-narrow blade hydrofoil CBY impeller have been performed. The unsteady and quasi-periodic behavior of the turbulent flows in the stirred tank was well captured, which indicates that LES is an accurate approach for simulating this kind of complex flow situations. The curve of tracer response predicted by LES is in good agreement with the experimental. RANS approach fails to predict the reasonable response curve. The relative errors of power demand between the experiment and numerical simulation for LES and RANS are 6.18% and 9.30% respectively. The relative errors of mixing time between the experiment and numerical simulation for LES and RANS are 9.26% and 16.93% respectively.
C
D G
H K LS M N PV PO Re
18 16
14 v1
Y
(T / D ) 2 (10) And the coefficient k for experimental, LES and RANS are 5.1,4.8 and 4.4 respectively.
CS
k
12
T t
V W, Tij
10
-
100
P”, w m”
Figure 7 Comparison of the mixing time between simulation results and experiment data .experimental; -LES; ------ RANS
The following empirical correction equation for liquid mixing time Om in a stirred tank proposed by Ru~zkowski[~~’ and Grenville et aZ.[261was used in this work February, 2006
=k (
NOMENCLATURE
N, r m i d
g
NO,
p, Q
the Smagorinsky constant clearance between impeller and base, m impeller diameter, m filter function liquid height of the vessel, m Von Karman constant mixing length, m torque, N m impeller speed, rmin-’ power per unit volume, ~ . m - ~ power number (Po=PvVlpN3Ds) Reynolds number vessel diameter, m time, s volume of liquid, m3 width of baffle, m subgrid-scale stress, s-l eddy viscosity, Pas-‘ fluid domain mixing time, s
REFERENCES Mezaki, R., Mochizuki, M., Ogawa, K., Engineering Data on Mixing, Elsevier Science, Amsterdam, 85-115 (2000). Ranade, V.V., Bourne, J. R., Joshi, J.B., “Fluid mechanics and blending in agitated tanks”, Chem. Eng. Sci., 46, 1883-1893 (1991).
Large Eddy Simulations of Mixing Time in a Stirred Tank 3
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Lunden, M., Stenberg, O., Anderson, B., “Evaluation of a method for measuring mixing time using numerical simulation and experimental data”, Chem. Eng. Commun., 139, 115-136 (1995). Schmalzriedt, S., Reuss, M., “Application of computational fluid dynamics to simulations of mixing and biotechnical conversion processes in stirred tank bioreactors”, In: Proc. 9th Eur. Conf. Mixing, Paris, 171-178 (1997). Jaworski, Z., Bujalski, W., Otiomo, N., “CFD study of homogenization with dual Rushton turbines comparison with experimental results”, Trans. IChemE, 78A, 327333 (2000). Eggels, J.GM., “Direct and large-eddy simulation of turbulent fluid flow using the Lattice-Boltzmann scheme”, Int. J . Heat Fluid Flow, 17(3), 307-323 (1996). Revsted, J., Fuchs, L., Tragardh, C., “Large eddy simulation of the turbulent flow in a stirred reactor”, Chem. Eng. Sci., 53(24), 4041-4053 (1998). Revsted, J., Fuchs, L., “Influence of impeller type on the flow structure in a stirred reactor”, A K h E J., 46(12), 2373-2382 (2000). Derksen, J., Harry, E., Akker, V., “Large eddy simulations on the flow driven by a Rushton turbine”, AIChE J., 45(2), 209-221 (1999). Derksen, J., “Assessment of large eddy simulations for agitated flows”, Trans. IChemE, 79A, 824-830 (2001). Hou, S., Wang, Y., Zhang, Z., Shi, L., “Turbulence flow generated by axial impeller”, J. Chem. Ind. Eng. (China), 51(2), 259-263 (2000). (in Chinese) Gao, Z., Niu, G., Shi, L., Smith, J. M., “Mixing in stirred tanks with multiple hydrofoil impellers”, In: Proc. 11th Eur. Conf. Mixing, Bamberg, 14-17 (2003). Gao, Z., Shi, L., “Effect of temperature on gas hold-up in aerated stirred tanks”, Chinese J. Chem. Eng., 11(2), 204-207 (2003). Zhou, G., “Experimental and numerical study on fluid dynamics and mixing process in stirred tank”, Ph. D. Thesis, Beijing Univ. Chem. Tech., Beijing (2002). (in Chinese) Sun, H., Wang, W., Mao, Z., “Numerical simulation of the whole three-dimensional flow in a stirred tank with anisotropic algebraic stress model”, Chinese J . Chem.
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Eng., 10(1), 15-24 (2002). Yeoh, S., Papadakis, G., Lee, K., “Large eddy simulation of turbulent flow in rushton impellers stirred reactor with a sliding-deforming mesh methodology”, In: Proc. 11th Eur. Conf. Mixing, Bamberg, 39-46 (2003). Naude, I., Xuereb, C., Bewand, J., “Direct prediction of the flows induced by a propeller in an agitated vessel using an unstructured mesh”, Can. J. Chem. Eng., 76, 631-40 (1998). Lu, Z.Y., Liao, Y., Qian, D.Y., “Large eddy simulations of a stirred tank using the Lattice Boltzmann method on a nonuniform grid’’, J . Comput. Phys., 181, 675-704 (2002). Bakker, A., Lame, M., Elizabeth, M., “The use of large eddy simulation to study stirred vessel hydrodynamics”, In: Roc. 10th Eur. Cod. Mixing, Elsevier Science B.V., 247-254 (2000). Costes, J., Couderc, J., “Study by laser Doppler anemometry of the turbulent flow: Influence of the size of the units- 11. Spectral analysis and scales of turbulence”, Chem. Eng. Sci., 43,2765-2772 (1988). Luo, J., Gosman, A., Issa, R., Middleton, J., Fitzgerald, M., “Full flow field computation of mixing in b a e d stirred reactors”, Trans. IChemE., 71A, 342 - 344 (1993). Oshinowo, L., Jaworski, Z., Dyster, K. N., “Predicting the tangential velocity field in stirred tanks using the multiple reference frames model with validation by LDA measurements”, In: Proc. 10th Eur. Cod. Mixing, Delft, 281-288 (2000). Hartmann, H., Derksen, J., “Assessment of large eddy and RANS stirred tank simulations by means of LDA”, Chem. Eng. Sci., 59,2419-2432 (2004). Jaworski, Z., Wyszynski, M., Moore, I., Nienow, A., “Sliding mesh CFD-A predictive tool in stirred tank design”, Proc. IMechE., 211, Part E, 149-156 (1997). Ruszkowski, S., “A rational method for measuring blending performance and comparison of different impeller types”, In: Proc. 8th Eur. Conf. Mixing, 283-291 (1994). Grenville, R., Ruszkowski, S., Garred, E., “Blending of miscible liquids in the turbulent and transional regimes”, In: 15th NAMF Mixing Conf., Banff, Canada, 1-5 (1995).
Chinese J. Ch. E. 14(1) l(2006)
(2006)
Large Eddy Simulations of Mixing Time in a Stirred Tank MIN Jim(N#) and GAO Zhengming($j E a )* College of Chemical Engineering, Beijing University of Chemical Technology, Beijing 100029, China
Abstract Large eddy simulations (LES) of mixing process in a stirred tank of 0.476m diameter with a h a r r o w blade hydrofoil CBY impeller were reported. The turbulent flow field and mixing time were calculated using LES with Smagorinsky-Lilly subgrid scale model. The impeller rotation was modeled using the sliding mesh technique. Better agreement of power demand and mixing time was obtained between the experimental and the LES prediction than that by the traditional Reynolds-averaged Navier-Stokes (FUNS) approach. The curve of tracer response predicted by LES was in good agreement with the experimental. The results show that LES is a reliable tool to investigate the unsteady and quasi-periodicbehavior of the turbulent flow in stirred tanks. Keywords large eddy simulations, subgrid scale model, mixing time, hydrofoil impeller
1 INTRODUCTION
ton turbine (DT6). They suggested that the quality of results was highly dependent on exact fluid dynamic Mechanically stirred tanks are widely used in many industrial processes. The flow structures in a computation, especially concerning turbulence modturbulently stirred tank are highly three-dimensional eling. Jaworski et uZ.'~]reported that the computed 095 and complex, and cover a wide range of spatial and was about two to three times the measured values in temporal scales. The fluid is circulated through tank stirred tanks with dual DT6, and suggested that this under the action of a revolving impeller. The vortices difference arose because the mass exchange between the four distinct axial-radial circulation loops was produced by impeller rotating, which retain their coherency over a significant distance into the bulk liquid, heavily underpredicted by CFD. are associated with high shear rates and strong turbuLES, first adopted in stirred tank by Eggle'61,was proved to be a good method to investigate turbulent lent activity. Therefore, they are essential to the mixing performance in the flow field. The mixing time, &, flows, unsteady and quasi-periodic behavior. Subsequently, Revstedt et ~ 1 . ' ~ is the time required to mix the added secondary fluid pointed ' out LES could prowith the contents of the vessel to a certain degree of vide details of the flow field that cannot be obtained uniformity, typically Og5meant to reach above 95% of with so-called Reynolds averaged equations and corthe final concentration. In any case the mixing time of responding models, and then Revstedt and Fuchs[*] a stirred tank is often used as an indication of its efsimulated the tank stirred by either two standard fectiveness. Knowledge of mixing time is required for Rushton impeller or two Scaba 6SRGT impellers. the optimum design of stirred tanks. Extensive exDerksen et aZ.[91used LES with Smagorinsky subgrid perimental studies on mixing time have been reported model, a Smagorinsky constant Cs=O. 12, to simulate a during the last 30 years"]. baffled stirred tank driven by a Rushton turbine at In the last two decades, progress has been Re=29000 and Derksen"" also simulated on the sinachieved in the computational fluid dynamics (CFD) gle-phase flow driven by a pitched blade impeller ussimulations of mixing process owning to the great ing LES with the standard Smagorinsky or structure progress in the computer technique. Ranade et ~ 1 . ' ~ ' function subgrid model. The author all focused on used numerical simulation to give detail on the flow studying the three-dimension velocity and the turbuand bulk mixing generated by a downflow-pitched lent kinetic energy in a stirred tank and also proved blade turbine in a fully baffled cylindrical vessel. the LES was a good tool to investigate turbulent flow Lunden et uZ.'~] and Schmalzriedt and re us^'^' simuin industrial application of practical importance. lated pulse tracer experiments by solving the material In this work, the LES with Smagorinsky-Lilly balance in three-dimensional flow field with a Rushsubgrid model was introduced firstly to simulate mixReceived 2005-04-19, accepted 2005-10-26.
* To whom correspondence should be addressed. E-mail:[email protected]
2
Chinese J. Ch. E. (Vol. 14, No.1)
ing process through monitoring the concentration of tracer to get mixing time in a tank stirred by h a r r o w blade hydrofoil CBY impeller["]. Comparison was made between LES and RANS.
2 PHYSICALAND COMPUTATIONALCONFIGURATION The stirred tank used in this work was a perspex vessel of 0.476m diameter with flat bottom and four baffles. Ambient tap water was used. A 3-narrow blade hydrofoil CBY impeller was used. The impeller speed N was 150, 180, 260 and 300r.min-'(the corresponding Re=pND2/v=9X lo4 to 1.8X lo5) respectively and the fluid flow was fully turbulent. The details of experimental apparatus were shown in Fig. 1.
T (a)
I "I
@)
Figure 1 View of (a) the stirred tank and (b) 3-narrow blade hydrofoil CBY impellers (T=476=; H/T=l.O; C/T=1/3; DITzO.4;w~IT=0.1)
3 EXPERIMENTAL 1395 was measured from changes in conductivity after the introduction of a small quantity of tracer (saturated potassium chloride solution). lOml of the tracer was added to the free surface of the liquid between two baffles. The detector was mounted at the position near the bottom of the tank opposite to the adding point. The output from the conductivity meter February, 2006
was acquired through an analogue filter and an amplifier with an A/D converter, and then stored for subsequent analysis. The measurements were repeated at least 5 times for each experimental condition to get the average mixing time. Impeller speed and shaft torque were measured by using an optic-electrical tachometer and a torque transducer respectively. The detailed description was reported in the other paper[12', but the only difference was that the present tank had a flat bottom rather than dished base[13].
4 MATHEMATICALMETHOD 4.1 The turbulence model The main difficulty associated with the simulation of turbulent flow in a stirred tank is the wide range of scales: ranging from small scale in the bulk of the tank to the trailing vortex structure associated with the impeller blade movement, and to the large re-circulation confined by the physical geometry of the tank. So the quality and accuracy of the simulation in a stirred tank are heavily relied on turbulence model. A very accurate prediction is possible by means of direct numerical simulation (DNS). In DNS, fluid motion down to the dissipative scale is resolved and it is therefore applicable only to relatively low Reynolds number flows and not applicable to industrial related applications. The Reynolds-averaged Navier-Stokes (RANS) model represents transport equations for the mean flow quantities only, with all the scales of the turbulence being modeled. The approach of permitting a solution of the mean flow variables greatly reduces the computational effort. If the mean flow is steady, the governing equations have no contain time derivatives and a steady-state solution can be obtained economically. The computational advantage is seen even in transient situations, since the time step will be determined by the global unsteadiness in the mean flow rather than by the turbulence. The Reynolds-averaged approach is generally adopted for practical engineering calculations,and uses models such as Spalart-Allmaras, k-E, k-u), RSM and their variants. The standard k-E model, RNG k-E model['41 and anisotropic algebraic Reynolds stress model[151,are the simplest model and could predict the fully turbulent flow field reasonable agreed with the experimental data. So the standard k-E model was chosen in present work to compare with LES.
Large Eddy Simulations of Mixing Time in a Stirred Tank
LES is the one somewhere between DNS and the RANS approach. Basically large eddies are resolved directly in LES, while small eddies are modeled[61.In LES, the governing equations employed for LES are obtained by filtering the time-dependent Navier-Stokes equations in either Fourier (wave-number) space or configuration (physical) space. The filtering process effectively filters out the eddies whose scales are smaller than the filter width or grid spacing used in the computation. The resulting equations thus govern the dynamics of large eddies. A filtered variable is defined by
@ ( X I = I n @ ( X ’ ) G ( X 7 X’)dX’
(1)
where D i s the fluid domain, and G is the filter function that determines the scale of the resolved eddies. In FLUENT, the finite-volume discretization itself implicitly provides the filtering operation: -
1 @ ( X )= - jn)(X’)dX’ V
X’€
a
(2)
where V is the volume of a computational cell. The filter function, G(X , X’) ,implied here is then G(X,X’) =
l/V,X’€ 0,X’E
a
a
(3)
Filtering the incompressible Navier-Stokes equations results in
a axi
*+--(p+o at
(4)
and
P-
at
(5)
.where Z, is the subgrid-scale stress defined by -
The subgrid-scale stresses resulting from the filtering operation are unknown, and require modeling. ’The most basic of subgrid-scale models was the Smagorinsky-Lilly model”], In the Smagorinsky-Lilly model, the eddy viscosity is modeled by
(7) where Ls is the mixing length for subgrid scales is the rate-of-strain tensor for
3
the resolved scale. In FLUENT, LS is computed using
4 = min(Kd, c , v ~ ’ ~ )
(8)
where K is the Von Karman constant, d is the distance to the closest wall, V is the volume of the computational cell, CS is the Smagorinsky constant. It is set to 0.1 in this work, which was found to yield the best results for a wide range of flows“61.
4.2 Hybrid meshes For the mixing tank simulations performed, the computational mesh is made up of two parts: an inner rotating cylindrical volume enclosing the turbine, and an outer, stationary volume containing the rest of the tank. Mesh strategy adopted was the technique of combination with the structured and unstructured g~id‘*~-’~]. The impeller region was divided into unstructured tetrahedral cells and densified to get more accurate description of impeller. And the rest, the bulk of the tank, was treated with the multi-block method, in which the hexahedral cells were used to decrease cost of calculation. So in RANS the size of cells was chosen 4mm in inner part and 8 mm in the outer region. But considering the blade thickness[61and the Taylor micro it was chosen 2mm and 4mm in LES. The total grid nodes numbers are 1044356 in LES and 373775 in RANS, as shown in Fig.2.
4.3 Impeller description The main difficulty in simulating stirred tanks is the accurate representation of impeller action. The sliding mesh (SMf211 and multiple reference frame (MRF) techniques[221are two effective ways dealing with the effects of impeller, and are implemented in the commercial software packages such as FLUENT. The MRF method, in which the region of impeller and impeller steam is described in as the rotating reference frame and a stationary frame or mesh is selected for the flow outside the impeller region. In the SM technique, which is in fact a transient method, two grids are generated one rotating with the impeller and one stationary representing the bulk tank. Under the same conditions, the sliding mesh method requires much more computing time, in which the data are transferred over the interface between the two grids. Because the MRF provides a reasonable model of the time-averaged flow and the SM could compute the unsteady and transient flow field. The SM method was used by LES in this work. Chinese J. Ch. E. 14(1) l(2006)
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(a) Half tank
(b) Horizontal surface
(c) Horizontal impeller
Figure 2 View of the mesh in the stirred tank
4.4 LES procedure The first step was to calculate continuity and velocity equations using the standard k-E model, and the results were reported in elsewhere. After the flow field is somewhat converged, the result was used as an initial condition for LES simulation. The simulation chose the second-order implicit formulation for temporal discretization and the central-differencing February, 2006
scheme for spatial. The solution of continuity and velocity equations continued until the flow became statistically steady through monitoring the torque of impeller. And the second step was that the equation for concentration of tracer was solved in the time domain to get the mixing time. The tracer injection is assumed not to affect the flow. Therefore, the separation of moment and tracer balance equations is presumed to reduce the computational effort. Based on the physical coordinates of tracer adding location, the adding tracer in simulation was done over several cells near it to ensure that the mass of tracer in simulation was the same as that of experiment. The concentration of tracer was initialized as 1 in the adding region, and in the rest region as 0. And the monitoring point was set at lOmm above bottom of tank and 50mm from the wall of tank, the same as the position of the detector in experiment amid two baffles. The LES was run on XEON dual processors (Pentium IV) machines (Dell) with 1Gb of memory, 2GHz clock frequency and a LINUX operating system. The simulation for one agitator speed flow field done on six processors in parallel took about 4 weeks to get at least 20 revolutions to ensure the velocity field statistic steady, and for concentration field it took about 3 weeks.
5 RESULTS AND DISCUSSION 5.1 The flow field and concentration distribution of tracer The velocity and concentration field gotten through RANS and LES were drawn in Figs.3 and 4. In a real stirred tank,there are large vortices and resulting macro-instabilities, which promote tracer mass exchange through this boundary. The standard k-E model can not fully account for this phenomenon. But LES could catch the details of vortices even down to small Taylor and Kolmogorov micro scale[231due to sufficiently small mesh size in simulation. So LES could give the more accurate result for the instantaneous flow than RANS. As can be seen from Figs.3 and 4, comparing with that gotten from RANS, the boundary of circulation loops in LES was broken and irregular, and a lot of large and small vortices occurred in the bulk region. The mass exchange between the vortices and even in the whole tank could be resolved accurately. The mixing process can be seen directly from the contour of concentration.
Large Eddy Simulations of Mixing Time in a Stirred Tank 8.42 X lo4 8.00 X l o 4 7.58X lo4 7.16~ 10.~ 6.74X lo4 6.32X lo4 5.90X lo4 5.48X lo4 5.05 X lo4 4.63 X lo4 4.21 X lo4 3.79X lo4 3.37x lo4 2.95 X lo4 2.53 X lo4 2.1 1x lo4 1.69X104 1.27 X lo4 8.51 x 10.~ 4.30 X l o 5 9.89 x 10"
2.97 2.82 2.67 2.52 2.37 2.23 2.08 1.93 1.78 1.63 1.48 1.34 1.19 1.04 8.90 X 10'' 7.42 X 10.' 5.93 x 10'' 4.45 x 10'' 2.97 X 10.' 1.48 X 10.' 0
8.42 X l o 4 8.00 X lo4 7.58X l o 4 7.16X lo4 6.74X1O4 6.32 X lo4 5.90X l o 4 5.48 X lo4 5.05 X l o 4 4.63 X lo4 4.21 X lo4 3.79 x 10.~ 3.37x lo4 2.95 X lo4 2.53 X l o 4 2.1 1x lo4 1.69 X lo4 1.27X l o 4 8.51 X l o 5 4.30 X l o 5 9.89 X l o 7
2.97 2.82 2.67 2.52 2.37 2.23 2.08 1.93 1.78 1.63 1.48 1.34 1.19 1.04 8.90X 10'' 7.42 X 10.' 5.93 x 10-I 4.45 x lo-' 2.97X 10.' 1.48X 10" 0
Figure 4 Contour of concentration distribution of tracer (vertical in middle of two baffles, N=30Or.mi1-', k4s)
01) LES Figure 3 Instantaneous contour of velocity magnitude (vertical in middle of two baffles, N=300rmin-')
5.2
5
t
1
A
The response curve of tracer
The normalized curves of tracer response of experimental, LES and RANS are shown in Fig. 5. Good agreement can be seen from the Fig.5 between the experimental and LES prediction. The shape of response curve of tracer is largely dependent on the large vortices in the stirred tank. LES could catch the different scales of vortices especially the large vortices, which resulting in the good prediction of response curve of tracer. The RANS approach only accounts for the small vortices and couldn't predict the reasonable response curve of tracer, as shown in the Fig.5.
Comparison of mixing time between numerical simulation and experimental The torque, M , exerted by the impeller was cal(culated from the CFD analysis method used by Ja-
0
5
15
10
t, s
Figure 5 The response curves of tracer in LES and RNAS (N=260 r.min-l, all normalized) RANS -experimental; ------ LES;
------
5.3
rnrorski et a ~ . ' ~n] .e power per unit volume 6;imulation is calculated by
PV
in
Pv = 2 n : M N / V
(9)
The comparisons of power demand between experimental and numerical simulations based on LES and RANS are shown in the Fig.6. Good agreement can be seen between the experimental and LES or Chinese J. Ch. E. 14(1) l(2006)
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RANS, though LES offers a slightly better prediction. The relative errors of power demand for LES and RANS are 6.18% and 9.30% respectively. The comparison of mixing time between experimental and numerical simulations based on LES and RANS is shown in Fig.7. Much better agreement can be seen between the experimental and LES prediction and than RANS prediction. The relative errors of mixing time for LES and RANS are 9.26% and 16.93% respectively. As mentioned above, the LES could predict more accurate fluid field and different scale of vortices, resulting in the better agreement between the experimental and LES prediction.
150
I
I
I
200
250
300
.
Figure 6 Comparison of the power per unit volume between simulation results and experiment data .experimental; -LES; ------ RANS I
6 CONCLUSIONS Large eddy simulations of the flow field and mixing time in a stirred tank of 0.476m diameter with a 3-narrow blade hydrofoil CBY impeller have been performed. The unsteady and quasi-periodic behavior of the turbulent flows in the stirred tank was well captured, which indicates that LES is an accurate approach for simulating this kind of complex flow situations. The curve of tracer response predicted by LES is in good agreement with the experimental. RANS approach fails to predict the reasonable response curve. The relative errors of power demand between the experiment and numerical simulation for LES and RANS are 6.18% and 9.30% respectively. The relative errors of mixing time between the experiment and numerical simulation for LES and RANS are 9.26% and 16.93% respectively.
C
D G
H K LS M N PV PO Re
18 16
14 v1
Y
(T / D ) 2 (10) And the coefficient k for experimental, LES and RANS are 5.1,4.8 and 4.4 respectively.
CS
k
12
T t
V W, Tij
10
-
100
P”, w m”
Figure 7 Comparison of the mixing time between simulation results and experiment data .experimental; -LES; ------ RANS
The following empirical correction equation for liquid mixing time Om in a stirred tank proposed by Ru~zkowski[~~’ and Grenville et aZ.[261was used in this work February, 2006
=k (
NOMENCLATURE
N, r m i d
g
NO,
p, Q
the Smagorinsky constant clearance between impeller and base, m impeller diameter, m filter function liquid height of the vessel, m Von Karman constant mixing length, m torque, N m impeller speed, rmin-’ power per unit volume, ~ . m - ~ power number (Po=PvVlpN3Ds) Reynolds number vessel diameter, m time, s volume of liquid, m3 width of baffle, m subgrid-scale stress, s-l eddy viscosity, Pas-‘ fluid domain mixing time, s
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