ARTICLE pubs.acs.org/IECR
Local Momentum Transfer Process in a Wall Region of an Agitated Vessel Equipped with an Eccentric Impeller Magdalena Cudak and Joanna Karcz* Department of Chemical Engineering ,West Pomeranian University of Technology, Szczecin, Al. Piastow 42, 71-065 Szczecin, Poland ABSTRACT: The results of the experimental studies of local momentum transfer process in the region of the cylindrical wall of an agitated vessel equipped with an eccentric axial flow impeller are presented. The data for eccentrically located impellers both, propeller and HE 3 impeller, differing in the operating liquid pumping mode were compared. In total, 5120 experimental points for both impellers were obtained on the basis of the electrochemical experiments for each of the tested impeller. Axial and angular distributions of the shear rate and friction coefficient in the agitated vessel without baffles were found within the turbulent regime of the Newtonian liquid flow. Local shear stress, dynamic velocity and energy dissipated were also identified in this region of the wall. The results of the study show that the profiles of the transport coefficients depend strongly on the eccentricity e/R, axial and angular coordinates, liquid circulation imposed by axial flow impeller, and liquid turbulence in the agitated vessel. The distributions of the quantities which describe momentum transfer process in the region of the cylindrical wall of the agitated vessel as the function of the e/R, z/H, j/2π, and Re number were approximated mathematically for both impellers by means of eqs 6-9. Averaged values of the local quantities, obtained as a result of the numerical integration, were described by means eqs 12-17. The eqs 6-9 and 12-17 have no equivalent in the open literature concerning this subject.
1. INTRODUCTION In some industrial applications, eccentric unbaffled configuration of the agitated vessel may be used alternatively instead of the baffled vessel with the centric impeller.1 Eccentric configurations of the impellers in unbaffled vessels improve the agitation compared to those with the centric impellers. The asymmetry of the system with the off-centered impeller causes distortions in the distributions of the quantities, which describe momentum, mass and heat transfer processes.2 These disturbances affect the active or dead zones of the transfer processes in the agitated vessel. For this reason, quantitative information about local transfer processes in such geometrical systems has of importance. Recently, systems with an eccentrically located rotating element are intensively investigated theoretically, experimentally and numerically. Aref, Ottino, Hansen et al., Alvarez et al. theoretically and experimentally studied effect of the eccentric position of such element on the laminar flow of the liquid in the apparatus.3-7 Rivera et al.8 carried out numerical simulations of the liquid flow generated by eccentric impellers in an agitated vessel. Power consumption for eccentrically located impellers in the agitated vessels was investigated by Dyla- g and Brauer, Medek and Fort, King and Musket and Karcz et al.9-12 Agitation of the viscous Newtonian liquid within the laminar flow regime in an agitated vessel equipped with a dual shaft system was studied experimentally and numerically by Cabaret et al.13 Two Rushton turbines were mounted on each of the eccentrically located shaft in the vessel of inner diameter T = 0.21 m and liquid height H = 2T. Using a fast acid-base discoloration reaction the mixing times were determined in such system. The results of the studies show that the employing of the dual shaft mixing system reduce mixing time and can eliminate the flow r 2011 American Chemical Society
compartmentalization in such agitated vessel, comparing with the system with the centrally located single shaft with two Rushton turbines. Moreover, numerical simulation results show the effect of the rotating mode of the impellers (counter-rotating or corotating) on the agitation. Montante et al. experimentally and numerically studied effect of the shaft eccentricity on the hydrodynamics of unbaffled agitated vessels.14 The measurements were carried out in a cylindrical vessel of inner diameter T = 0.236 m equipped with a standard six bladed Rushton turbine. The particle image velocimetry (PIV) method and Reynolds Averaged Navier-Stokes (RANS) equation-based CFD simulations were used in the experiments and numerical computations, respectively. The RANS equations, coupled with the standard k-ε turbulence model or the Reynolds stress model (RSM), were solved. Comparing experimental and simulated mean flow fields, Montante et al. proved that calculations based on Reynolds-averaged Navier-Stokes equations are suitable for obtaining accurate results.14 Hall et al. investigated flow field velocities and mixing times using both two-dimensional PIV and planar Laser induced fluorescence (PLIF) techniques, respectively, in the small-scale unbaffled eccentric vessels of inner diameter T = 0.035, 0.045, or 0.06 m.15 The vessels were equipped with up-pumping six-bladed pitched blade turbines. The authors stated that the spatial distribution of turbulent kinetic energy within the vessels is independent of the vessel scale, but local values of this quantity are dependent on the vessel diameter. Hall et al. studied also Received: September 27, 2010 Accepted: February 22, 2011 Revised: February 17, 2011 Published: March 08, 2011 4140
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Figure 1. Positions of the measuring points on the cylindrical wall of the agitated vessel; (a) view of the arrangement; (b) angular positions; (c) axial positions for each quarter part of the vessel.
hydrodynamics in air-water system produced in a small unbaffled reactor of inner diameter T = 0.04 m, equipped with an eccentrically located, up-pumping six-bladed pitched blade turbine.16 Two different positions of a gas sparger in the agitated vessel were tested. In the first case, the sparger was located directly below the impeller. In the second one, it was situated opposite the impeller on the other side of the vessel. Hall et al. stated that the position of the sparger with respect to the impeller location significantly affected the mean flow field within the agitated vessel.16 The hydrodynamics of an unbaffled agitated vessel of inner diameter T = 0.29 m equipped with an eccentrically located Rushton turbine was investigated by Galletti and Brunazzi and Galletti et al..17,18 Using in the studies both Laser Doppler Anemometry (LDA) and flow visualization techniques, Galletti and Brunazzi17 observed two main vertical structures and identified flow instabilities in such geometrical system. Galletti et al. experimentally determined the effect of the impeller eccentricity and the impeller blade thickness on the features of the vortices, which dominate the flow in the agitated vessel equipped with the eccentrically located Rushton turbine.18
Cabaret et al. investigated the ability of eccentrically located shafts of the impellers in an unbaffled agitated vessel of inner diameter T = 0.21 m to generate effective gas-liquid mass transfer.19 The dual shaft system consisting of two off-centered shafts equipped with standard six blades Rushton turbines was tested. As a liquid phase low- and high viscosity Newtonian (water, aqueous solutions of glucose) and non-Newtonian liquids (CMC solutions) were used. Air was supplied with a ring sparger of diameter ds = 0.75D (where D is impeller diameter). It follows from this study that shaft off-centring is a good alternative to improve gas-liquid mass transfer in low- and high-viscosity Newtonian liquids. However, the off-centring of the shaft in the unbaffled agitated vessel have to be avoided when shear-thinning liquids are aerated in the agitated vessel. Cabaret et al. studied also the effect of shaft eccentricity on the mixing efficiency in an unbaffled agitated vessel of inner diameter T = 0.21 m.19 Radial flow six-blade Rushton turbine was used for agitation of the viscous Newtonian liquid (aqueous solution of corn syrup) within the laminar regime of the flow in the agitated vessel. The results of this study show that dimensionless mixing time is 4141
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Industrial & Engineering Chemistry Research not a constant value within the range of the performed experiments. Moreover, shaft eccentricity and Re number have significant effects on the destruction of the toroidal segregated regions surrounding the Rushton turbine. Karcz using computer-aided electrochemical method studied local momentum transfer process in a region of the wall of the agitated vessel.20 A fully baffled agitated vessel of inner diameter T = 0.3 m was filled with the electrolyte up to height H = T. The following turbine impellers, centrally located in the vessel (e/R = 0) on the height h = 0.33H from the flat bottom of the vessel, were tested: disk turbines (D = 0.33T, Z = 6; D = 0.5T, Z = 10), turbine with six straight blades (D = 0.33T, Z = 6, β = 90), and the uppumping flow pitched blade turbines (D = 0.33T, Z = 6, β = 45; D = 0.5T, Z = 8, β = 45). The measurements of the diffusion current on the vessel wall were carried out within the turbulent flow of the liquid. The distributions of the dimensionless shear rate (γ/n), friction coefficient f, shear stress τ and dynamic velocity v/ on the cylindrical wall of the vessel were presented graphically for the each type of the impeller tested. Averaged values of those quantities were computed as a result of the numerical integration. The dependences of the averaged values of the (γ/n)m, fm, τm, and vm/ as a function of the Re number were correlated by mean of the equations. Distributions of the friction coefficient f on the cylindrical wall of the unbaffled agitated vessel equipped with a centrically located propeller were studied by Karcz et al..12 Electrochemical measurements were carried out within the turbulent liquid flow in the vessel of inner diameter T = 0.3 m. The results were described mathematically in the form of the equation which relates local friction coefficient f, obtained for the centric position of the propeller, with the dimensionless axial and angular coordinates (z/H, j/2π) and Re number. Momentum transfer process in the region of the wall of an agitated vessel was experimentally studied by Cudak and Karcz.21-23 Graphical distributions of the shear rate on the vessel wall as a function of the impeller type (eccentrically located propeller, HE 3, A 315, Rushton or Smith turbines of diameter D = 0.33T) were determined. The dependences of the averaged dimensionless shear rate and friction coefficient on the Reynolds number and impeller eccentricity were approximated mathematically.21 Local momentum transfer was also analyzed experimentally for the system equipped with eccentric HE 3 impeller of diameter D = 0.5T.22 The local and averaged momentum transfer quantities were correlated mathematically as a function of the impeller eccentricity and modified Reynolds number ReP,M (where ReP,M = ((P/M)F3T4/η3) and M = VF). An effect of the impeller HE 3 diameter, impeller eccentricity and modified Reynolds number ReP,M on the averaged shear rate, friction coefficient, shear stress, and dynamic velocity was identified.23 The aim of the experimental studies was to analyze local momentum transfer process in the region of the wall of an agitated vessel equipped with an eccentric axial flow impeller. The results for eccentrically located impellers both, propeller and HE 3 impeller, differing in the operating liquid pumping mode were compared. Distributions of the shear rate and friction coefficient in the agitated vessel without baffles were found and quantitatively described. It is worth to notice that the results presented in this paper and concerning local momentum transfer process are obtained within the wide range of the measurements (change of the impeller speed, impeller eccentricity as well as axial and angular coordinates of the vessel wall) and have no equivalent in the open literature concerning this subject.
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2. EXPERIMENTAL SECTION Experimental studies of the local transport coefficients were carried out in an agitated vessel of inner diameter T = 2R = 0.3 m. Liquid height was equal to H = T. Axial flow impeller of diameter D = 0.33T was located vertically at the distance h = 0.33T from the flat bottom of the vessel. Centric (e/R = 0) and four eccentric (e/R 6¼ const; 0.13 < e/R < 0.53) positions of the propeller (S/D = 1) and HE 3 impeller were tested. Local momentum transfer in the region of the cylindrical wall of the agitated vessel was studied using computer aided electrochemical method. This experimental method was described in detail by Karcz and Abragimowicz, as well as Karcz et al.12-24 In this method, nickel circular cathodes with a diameter 4 mm were used as measuring sensors. Cathodic reduction of the potassium ferricyanide K3Fe(CN)6 was employed as a measuring system. A total, 128 electrochemical sensors identified by axial and angular coordinates z/H and j/2π, were built on the vessel wall (Figure 1). For a given quarter part of the cylindrical wall, measuring sensors were positioned in four columns I, II, III, and IV (Figure 1a). Arrangement of the electrochemical sensors for each quarter part of the vessel wall was shown in Figure 1c. In each measuring point (Figure 1), the measuring digital voltage signal was sampled 200 times. Diffusion current Id representative for a given measuring point was calculated automatically based of the averaged value from the sampling. Size of the sampling was chosen experimentally. The tests showed that representative locally averaged values for the measuring point, obtained from the 200 and 500 samples, do not differ statistically. Local shear rate γ and friction coefficient f near-wall region of the stirred tank were calculated from the formulas proposed by Wichterle et al.25 and Zak26 !3 Id γ¼ ð1Þ 2=3 5=3 2:156ze FDA CAo Rc and f ¼
τT 2 Fn2 D4
ð2Þ
where Id is measured diffusion current and shear stress τ on a wall is correlated with shear rate as τ ¼ ηγ
ð3Þ
Local values of the dynamic velocity v/ and energy dissipated ε were also computed from the following equations rffiffiffi τ ð4Þ v/ ¼ F ε¼
τγ F
ð5Þ
The measurements were carried out for different agitator speeds n within the turbulent regime of the Newtonian liquid flow in the agitated vessel. On the basis of the 5120 experimental data of the diffusion current, local values of the shear rate γ, friction coefficient f, shear stress τ, dynamic velocity v/, and energy dissipated ε were calculated. Two-dimensional profiles of these values were graphically created from the computer-processed experimental data. 4142
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Figure 2. Two-dimensional distributions of (a, b) the friction coefficient f and (c) dimensionless shear rate γ/n on the agitated vessel wall for centric or eccentric location of the propeller, (a) e/R = 0; (b, c) e/R = 0.4; Re = 4.2 104.
Figure 3. Two-dimensional distributions of (a, b) the friction coefficient f and (c) dimensionless shear rate γ/n on the agitated vessel wall for centric or eccentric location of the HE 3 impeller; (a) e/R = 0; (b, c) e/R = 0.4; Re = 4.2 104.
3. RESULTS AND DISCUSSION The distributions of the transport coefficients on the cylindrical wall, described by both dimensionless coordinates: axial z/H and angular j/2π, show that the zones of the more intensive momentum transfer occur in the un - baffled agitated vessel with the centrally located axial flow impellers. Generally, there are placed in the regions above the up-pumping flow propeller (Figure 2a), or below the down-pumping flow HE 3 impeller (Figure 3a). Moreover, an asymmetry of the distributions of the transport coefficients is observed on the level of the axial coordinate, corresponding to the height of the impeller in the vessel. These effects are related with the significant participation of the tangential liquid flow in the vessel without baffles. The impeller eccentricity affects the process of the momentum transfer very strongly. In detail, this effect is shown for impellers in Figures 2b, c (propeller) and 3b, c (HE 3 impeller), where two-dimensional distributions of the friction coefficient f (Figure 2b, 3b) and dimensionless shear rate γ/n (Figures 2c and 3c) are illustrated. With the increase of the eccentricity e/R, the zones of the high intensity of the momentum transfer replace at the vicinity of the cylindrical wall of the agitated vessel. This effect, depending on the type of the impeller used, reflects in the deformation and considerable asymmetry of the distributions of the transport coefficients in the region of the agitated vessel wall. 2520 local values of the diffusion current Id have been obtained on the basis of the electrochemical experiments for each of the tested impellers, i. e propeller and HE 3, respectively (in total, 5120 experimental points for both impellers). These data have been used to quantitative description of the dimensionless shear rate γ/n and friction coefficient f distributions as a function of the Re number, axial and angular coordinates (z/H, j/2π), as well as impeller eccentricity e/R. The following equations have been proposed:
For agitated vessel with the propeller ! 2 γ e e z j 0:52 ¼ C0 Re 1:82 - 0:52 þ 1 F1 F2 n R R H 2π ð6Þ f ¼ C0 Re
-0:46
! 2 e e z j þ 1 F1 F2 1:134 - 0:259 R R H 2π
ð7Þ For agitated vessel with the HE 3 impeller ! 2 γ e e z j - 0:866 ¼ C0 Re0:67 2:346 þ 1 F1 F2 n R R H 2π ð8Þ f ¼ C0 Re
-0:34
! 2 e e z j 2:231 - 0:849 þ 1 F1 F2 R R H 2π
ð9Þ In eqs 1-4, functions F1 and F2 are defined as follows ! 2 e1 !-1 z z z z F1 þ b1 ¼ a1 þ c1 1 þ d1 H H H H F2
j 2π
¼
a2
j 2π
2
þ b2
j þ c2 2π
!
1 þ d2
j 2π
ð10Þ e2 !-1 ð11Þ
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Table 1. Coefficients C0 and Functions F1 and F2 in eq 6; Data for the Propeller F1((z)/(H)) range of j
Co
1
(0;π/2>
2
(π/2; π>
157.42
1.454
3
(π;3π/2>
370.96
4
(3π/2;π>
0.0052
a1
b1
c1
F2((j)/(2π)) d1
e1
a2 -0.75
1.22
-2.165
1
1
-1.54
1.123
-2.079
1
1
-1.54
0.92
-1.894
1
1
-1.54
0.825
-1.818
1
1
-1.54
b2
4.276
d2
e2
(Δ (%)
0.717
1
1
1.28
10
-3.968
1
1
-2.48
10
-2.776
1
1
-4.35
12
647.6
1
1.1
15.51
13
d2
e2
(Δ (%)
1.964 -199.4
c2
Table 2. Coefficients Co and Functions F1 and F2 in eq 7; Data for the Propeller F1((z)/(H)) range of j
Co
a1
b1
c1
F2((j)/(2π)) d1
e1
a2
b2
c2
5
(0;π/2>
1.212
-2.156
1
1
-1.51
1.112
0.331
1
1
1.31
10
6
(π/2; π>
977.1
1.095
-2.051
1
1
-1.51
4.232
-3.936
1
1
-2.39
11
7 8
(π;3π/2> (3π/2;π>
2751.5 1.048
0.875 0.854
-1.851 -1.84
1 1
1 1
-1.51 -1.51
1.966 -0.848
-2.779 -0.05
1 1
1 -1
-4.46 0.09
13 13
e2
(Δ (%)
10.295
Table 3. Coefficients Co and Functions F1 and F2 in eq 8; Data for the HE 3 Impellera F1((z)/(H)) range of j
a
z/H
Co
a1
b1
F2((j)/(2π))
c1,d1
e1
9
(0;π/2>
(0; 1)
0.321
1.513
-2.393
1
-1.14
10
(π/2; π>
(0; 1)
0.0031
1.167
-2.101
1
-1.14
a2 4.876 -892.6
b2
c2, d2
-2.666
1
690.4
1
1,54 -0.15
13 11
11
(π;3π/2>
(0; 1)
61.91
1.484
-2.381
1
-1.14
1.93
-2.737
1
-4.89
13
12a
(3π/2;π>
(0.33; 1)
47.10
1.087
-2.002
1
-1.14
0.998
-1.991
1
-9.85
12
12 b
(3π/2;π>
(0; 0.33)
1.22
1.28
1
-0.66
85.37
1
-6.21
14
e2
(Δ (%)
0.059
-81.12
c1 = d1; c2 = d2.
Table 4. Coefficients Co and Functions F1 and F2 in eq 9; Data for the HE 3 Impeller F1((z)/(H)) range of j
z/H
Co
a1
b1
F2((j)/(2π))
c1,d1
e1
a2
13
(0;π/2>
(0; 1)
3.133
1.505
-2.386
1
-1.14
6.615
14
(π/2; π>
(0; 1)
4.725
1.24
-2.154
1
-1.14
-7.975
b2 -3.05 4.272
c2, d2 1
1.62
13
1
-0.91
11
15
(π;3π/2>
(0; 1)
626.2
1.484
-2.382
1
-1.14
1.979
-2.773
1
-4.87
12
16a
(3π/2;π>
(0.33; 1)
465.9
1.085
-2.001
1
-1.14
0.99
-1.983
1
-9.89
13
16b
(3π/2;π>
(0; 0.33)
-0.041
-1.347
1
-0.78
141.2
1
-6.34
16
0.519
-134.6
c1 = d1; c2 = d2
Coefficients C0 in eqs 6-9 and parameters ai, bi, ci, di, and ei in eqs 10 and 11 for each type of the impeller are collected in Tables 1-4, where mean relative errors Δ are also given. For example, experimental and calculated from eqs 6 and 7 values for the system with the propeller are compared in Figures 4 and 5. All the dependencies presented in Figures 4 and 5 show good approximation of the experimental data by means of eqs 6 and 7). Equations 6-11 describe the results of the experiments within the following range of the variables: agitator speed n [1/s] ∈ Æ4.2; 7.5æ, Reynolds number Re ∈ Æ3.5 104; 6.6 104æ, impeller eccentricity e/R ∈ Æ0; 0.53æ, dimensionless axial coordinate z/H ∈ (0; 1) and dimensionless angular coordinate j/2π ∈ Æ0; 1æ. It is worth to emphasize that the transport coefficients in the wall region for the centric
and eccentric location of the impellers in the unbaffled agitated vessel are identified using eqs 6-9. The effect of the turbulence of the liquid flow on the local dimensionless shear rate in the vicinity of the cylindrical wall for the agitated vessel equipped with the eccentric HE 3 impeller is stronger than that obtained for the agitated vessel with the eccentric propeller. It can be stated comparing values of the exponents at Re number in both eqs 6 and 8. Exponent value equal to 0.67 in eq 8 is higher than 0.52 in eq 1. Similar influence is observed in the case of the local friction coefficients for both agitated vessels. Negative exponent at Re number in eq 9 is higher (-0.34) than that obtained in eq 7, which is equal to -0.46. Two-dimensional distributions of the shear stress τ, dynamic velocity v/, and specific energy dissipated ε for the extremely 4144
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Figure 4. Comparison of the experimental and calculated values of the dimensionless shear rate γ/n in eq 6; data for the propeller agitator.
Figure 5. Comparison of the experimental and calculated values of the friction coefficient f in eq 7; data for the propeller agitator.
eccentric position (e/R = 0.53) of the propeller and HE 3 impeller in the agitated vessel are shown in Figures 6 and 7. The dependencies
τ = F(z/H, j/2π), v/ = F(z/H, j/2π) and ε = F(z/H, j/2π) are presented for the constant value of the Re number equal to 42 000. 4145
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Figure 6. Two-dimensional distributions of (a) the shear rate τ, (b) dynamic velocity v/, and (c) energy dissipated ε on the agitated vessel wall for eccentric position of the propeller, e/R = 0.53; Re = 4.2 104.
Figure 7. Two-dimensional distributions of (a) the shear rate τ, (b) dynamic velocity v/, and (c) energy dissipated ε on the agitated vessel wall for eccentric position of the HE 3 impeller, e/R = 0.53; Re = 4.2 104.
For the agitated vessel equipped with the propeller (Figure 6), higher values of the variables τ, v/, and ε have been obtained in the region of the wall corresponding to the axial coordinate z/H > 0.5. The region of the higher τ, v/, and ε values for the agitated vessel equipped with HE 3 impeller (Figure 7) has been identified on the part of the wall under the impeller, i.e., for z/H < 0.33. The τ, v/, and ε values increase when the dimensionless angular coordinate j/2π is increasing for both impellers. The shape of the axial profiles of the specific energy dissipated ε on the vessel wall depends on the type of the impeller used and its eccentricity e/R. Axial distributions of the ε = F(z/H)j=const obtained for the agitated vessel with propeller (Figure 8a, b) differ in the shape from the analogous distributions representing data for the agitated vessel equipped with the HE 3 impeller (Figure 8c, d). The differences are caused by opposite liquid circulation generated by both impellers, namely, the propeller acts in the up-flow circulation mode, whereas the HE 3 impeller operates in the downflow circulation mode. In the case of the agitated vessel equipped with the propeller, higher values of the specific energy dissipated are obtained for the zone located above the impeller height. Comparing the local values of the specific energy dissipated ε for the vessel equipped with HE 3 impeller, it can be observed that higher values of the ε correspond to the level situated on the impeller height or under the impeller. It is justified, taking into account direction of the liquid circulation to the bottom of the agitated vessel. The values of the energy dissipated ε at the position e/R = 0.4 higher than ones at e/R = 0.53 (Figure 8d) may be caused by the creation of the vortex from the impeller blade to the liquid surface. The examples of the distributions of the local values of the shear stress τ and dynamic velocity v/ (for different values of the axial dimensionless coordinates z/H) are illustrated in Figures 9-11. These distributions are strongly asymmetrical and depend not only
on the part of the vessel wall considered (angular coordinate), but also on the level of the axial coordinate z/H. In this case, the effects of the impeller type and liquid circulation generated in the vessel on the shape of the angular profiles have been also revealed. For the axial coordinate z/H equal to 0.33 or 0.25 (it corresponds to the impeller height and the level below the impeller), higher values of the parameters τ and v/ are characteristic for the vessel with the HE 3 impeller than ones obtained for the vessel with the propeller (Figures 9a, 10a, 10b, and 11a). These results reflect the effect of the direction of the liquid circulation imposed by the HE 3 impeller. When the axial coordinate increases (to the free surface of the liquid agitated) the situation is change. For the z/H > 0.33, higher values of the parameters τ and v/ correspond to the agitated vessel with the up-pumping flow propeller (Figures 9c, 10c, and 11c). In the case of the centrally located on the shaft both propeller and HE 3 impellers in the agitated vessel (Figures 9b and 11b), the distributions of the transport coefficients on the level corresponding to the impeller height (z/H = 0.33) are similar. Local values of the shear stress τ, dynamic velocity v/, and energy dissipated ε were numerically integrated for whole surface area of the cylindrical wall of the agitated vessel using trapeze method. In this way, averaged values of the τm, v/m, and εm were obtained as a functions of Re number and impeller eccentricity e/R. These effects were mathematically described in the form of the following equations: for the agitated vessel with the propeller τm ¼ 3:51 ! 2 e e -9 1:62 10 3 Re 0:891 - 0:0222 þ1 R R ð12Þ 4146
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Figure 8. Axial profiles of the energy dissipated in the wall region ε = f(z/H)j=const; (a, b) propeller; (c, d) HE 3 impeller; D = 0.33T; Re = 4.2 104; (, e/R = 0; Δ, e/R = 0.4; 9, e/R = 0.53.
Figure 9. Angular distributions of the shear stress τ = f(j) for the agitated vessel with centrally located agitator; e/R = 0; Re = 4.2 104; O, propeller; Δ, HE 3 impeller; (a) z/H = 0.25; (b) z/H = 0.33; (c) z/H = 0.67.
v/m ¼ 8:99
εm ¼ 7:142
!
-7
10 Re
0:87
2 e e -0:174 þ 0:46 þ1 R R
10
ð13Þ
-17
Re
3:03
! 2 e e -13:936 þ 10:439 þ1 R R ð17Þ
εm ¼ 1:903 10-17 Re3:21
! 2 e e 1:454 - 0:0826 þ1 R R ð14Þ
Equations 12 and 17 approximate the averaged results with mean relative error (6% within the following ranges of the impeller eccentricity e/R ∈ Æ0; 0.53æ and Re number Re ∈ Æ3.5 104; 6.6 104æ. Exponents at Re number in eqs 12 and 15 are lower than value 1.74 which was obtained by Karcz (1996) in the following equation
for the agitated vessel with HE 3 impeller τm ¼ 4:83 -9
10 Re
1:58
v/m ¼ 1:83 10-6 Re0:80
! 2 e e 0:392 þ 0:352 þ1 ð15Þ R R ! 2 e e 0:128 þ 0:203 þ1 ð16Þ R R
τm ¼ 6:166 10-9 Re1:74
ð18Þ
for the baffled agitated vessel with standard Rushton turbine at centered position. However, exponents at Re number in eqs 13 and 16 have similar values as that in equation (Karcz (1996)) v/m ¼ 1:937 10-6 Re0:88
ð19Þ
which was proposed for the baffled agitated vessel with centrally located standard Rushton turbine. 4147
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Industrial & Engineering Chemistry Research
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Figure 10. Angular distributions of the shear stress τ = f(j) for the agitated vessel with eccentrically located agitator; e/R = 0.53; Re = 4.2 104; O, propeller; Δ, HE 3 impeller; (a) z/H = 0.25; (b) z/H = 0.33; (c) z/H = 0.67.
Figure 11. Angular distributions of the dynamic velocity v/ = f(j) for the agitated vessel with eccentrically located agitator; e/R = 0.53; Re = 4.2 104; O, propeller; Δ, HE 3 impeller; (a) z/H = 0.25; (b) z/H = 0.33; (c) z/H = 0.67.
4. CONCLUSIONS The results of the experimental study of the local momentum transfer process within the region of the wall of the agitated vessel with the eccentrically located axial flow propeller or HE 3 impeller were obtained within the turbulent regime of the Newtonian liquid flow and wide range of the impeller eccentricity. These findings can be summarized as follows: Experimentally it was proved that the distributions of the dimensionless shear rate γ/n and friction coefficient f are strongly differentiated on the vessel wall significantly depending on the eccentricity e/R of the impeller shaft, position of the measuring point on the vessel wall (axial and angular coordinates z/H and j/2π) and Re number. Asymmetry of the profiles, which describe momentum transfer increases with the increase of the shaft eccentricity. Type of the axial flow impeller used and its liquid pumping mode affect the location of the zones of the higher or lower intensity of the transport processes near wall region of the vessel. The vortex from the impeller blade to the liquid surface was observed at the eccentricity e/R equal to 0.4. In this case, energy dissipated ε higher than that at the position e/R = 0.53 was identified. The distributions of the quantities which describe momentum transfer process in the region of the cylindrical wall of the agitated vessel as the function of the e/R, z/H, j/2π and Re number were approximated mathematically for both impellers by means of eqs 6-9. Averaged values of the local quantities, obtained as the result of the numerical integration, were described by means eqs 12-17. The eqs 6-9 and 12-17 have no equivalent in the open literature concerning this subject. Moreover, the results concerning local values of the quantities, which describe momentum transfer process, can be useful to verify the data obtained using numerical methods.
’ AUTHOR INFORMATION Corresponding Author
*Tel.: þ48 91 449 43 35. Fax: þ48 91 449 46 42. E-mail: Joanna.
[email protected].
’ ACKNOWLEDGMENT Financial support for the studies was provided partially by the National Grant Foundation (Grants KBN 7 T09C 029 21 and 4 T09C 065 25) ’ NOMENCLATURE CAo = concentration of the component A in an electrolyte, kmol/m3 D = agitator diameter, m DA = diffusion coefficient, m2/s ds = sparger diameter, m e = eccentricity, m F = Faraday’s constant f = friction coefficient H = liquid height in the agitated vessel, m h = off-bottom clearance of the agitator, m Id = diffusion current, A M = mass of liquid, kg n = impeller speed, 1/s P = power consumption, W R = radius of the agitated vessel (R = T/2), m Rc = radius of the circular electrode, m S = pitch of the propeller, m T = inner diameter of the agitated vessel, m V = volume of the liquid, m3 v/ = dynamic velocity, m/s 4148
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Industrial & Engineering Chemistry Research Z = number of impeller blades z = axial coordinate, m ze = number of the electrons taking part in the reaction Greek Letters
β = angle of the blade inclination to horizontal, deg γ = shear rate, 1/s ε = energy dissipated, W/kg η = dynamic viscosity of the liquid, Pas F = liquid density, kg/m3 τ = shear stress N/m2 j = angular coordinate, deg Subscripts
m = averaged value Re ¼
nD2 F ; η
ReP, M ¼
ðP=MÞF3 T 4 η3
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