On the Direct Measurement of Turbulence Energy Dissipation in

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Ind. Eng. Chem. Res. 2003, 42, 7006-7016

On the Direct Measurement of Turbulence Energy Dissipation in Stirred Vessels with PIV Sandro Baldi and Michael Yianneskis* Experimental and Computational Laboratory for the Analysis of Turbulence, Division of Engineering, King’s College London, Strand, London WC2R 2LS, U.K.

The mean flow and turbulence fields and the turbulence energy dissipation rate () in a vessel stirred by a hydrofoil impeller were studied using particle image velocimetry (PIV). The vessel diameter was 100 mm, and it was stirred by an impeller of diameter D ) T/3. The impeller Reynolds number was 40 000 to ensure fully turbulent flow in the vessel. Particular attention was paid to the effect of spatial resolution on the estimation of , which was determined directly from its definition, i.e., by measuring the instantaneous spatial gradients of the Reynolds stresses. The data provide new information on the quantification and distribution of in stirred vessels and indicate that estimates of dissipation based on a constant length scale across the vessel and the distribution of the turbulence levels may underestimate the maximum and overestimate the minimum levels of present in a stirred vessel. 1. Introduction Stirred vessels are employed in ∼25% of all process engineering operations. The determination of the distribution of the dissipation rate of the turbulence kinetic energy () is of paramount importance for the optimization of fluid mixing processes in such vessels. Knowledge of the maximum and minimum values of and of its distribution in the impeller stream and bulk flow in the vessel is required to ensure that processes such as the break-up of drops, particles, and bubbles in liquidliquid, solid-liquid, and gas-liquid mixing, respectively, are effectively achieved and the desirable product quality is obtained. Energy dissipation is consequently a key variable for mixing in such processes, as well as micromixing dependent ones, and relevant information is required for a number of practical design rules, as process design may fail if relevant data are missing or are estimated from integral parameters such as the power input. Although a large amount of information has already been obtained on the mean flow and turbulence levels in stirred tanks with LDA and other methods (see, for example, Yianneskis et al.,1 Stoots and Calabrese,2 Schaefer et al.,3 and Wernersson and Tragardh4), the measurement of the dissipation rate remains a challenging task, primarily in view of the small scales at which dissipation takes place. Consequently, most approaches employed to date for the determination of have made use of approximate relations, assuming locally isotropic turbulence and equations of the general form (see, for example, Kresta5)

) Au′3/L

(1)

where A is a constant, u′ is the turbulence level in the main flow direction, and L is a characteristic length scale, often assumed to be a tenth of the impeller * To whom correspondence should be addressed. Tel: +44 (0)207 848 2428. Fax: + 44 (0)207 848 2932. E-mail: [email protected]; [email protected].

diameter (D). Clearly, such approaches are only approximate, as they assume that the length scale is constant throughout the vessel and that most terms contributing to the energy dissipation equation can be neglected. Kresta5 compared also different schemes to estimate dissipation and found that close to the impeller (but not in the trailing vortices) turbulence may be approximated as locally isotropic, and therefore, a fixed length scale and a single rms velocity component can be used for its estimation. Although the above relation has been of much use for process design and provides useful estimates, and it can yield approximately correct volume-averaged energy dissipation values, such results cannot provide reliable local values of as both assumptions are of limited validity and better ways to determine accurately are necessary. Jaworski and Fort6 applied an energy balance to small volumes of the flowing liquid in order to estimate the local energy dissipation. Wernersson and Tragardh4 estimated the energy dissipated in stirred vessels using a method based on Kolmogorov’s hypothesis that if the local Reynolds number is large enough, the local energy dissipation can be estimated from the slope of the energy spectrum (see Pope7). Turbulence power spectrum methods are often based on equations such as (2) below, which is valid for isotropic turbulence

∫

) 15ν κ12E1(κ1)dκ1

(2)

where E1(κ1) is the one-dimensional power spectrum and κ is the wavenumber. Stoots and Calabrese2 calculated an approximation of the energy dissipation using the deformation rates given by mean velocity gradients. Other methods employ dimensional considerations of the form (Hinze8)

u′2 λ

) 15ν

(3)

where ν is kinematic viscosity, λ is the microscopic turbulence length scale, and u′ is the rms velocity in

10.1021/ie0208265 CCC: $25.00 © 2003 American Chemical Society Published on Web 11/18/2003

Ind. Eng. Chem. Res., Vol. 42, No. 26, 2003 7007

the main flow direction or

) A′

q3/2 L

(4)

where A′ is an empirical constant, L is the integral length scale, and q is the turbulence kinetic energy. Lee and Yianneskis9 evaluated the energy dissipation in the impeller stream of a Rushton turbine using the relations

)

0.85k3/2 (3Λx2)1/2

(5)

(u′2)3/2 Λx

(6)

)

Figure 1. CAD diagram of the hydrofoil impeller.

where k is the turbulence kinetic energy, u′ is the mainstream rms velocity, and Λx is the integral length scale. Only relatively recently efforts have been made to determine directly using PIV methods. The PIV technique allows for a totally different approach in calculating the energy dissipation. Instantaneous velocity data in adjacent points in space are available from PIV analysis, thus permitting the direct calculation of turbulence energy dissipation through the determination of the instantaneous velocity gradients. Sheng et al.10 developed a large eddy PIV method for dissipation rate estimation, based on an assumption of dynamic equilibrium between the spatial scales that can be resolved by PIV and the sub-grid scales. Sharp and Adrian11 studied the effects of isotropic assumptions and spatial resolution in PIV on the estimation of in a vessel stirred by a Rushton turbine. Piirto et al.12 and Saarenrinne et al.13 analyzed the correlation between energy dissipation, spatial resolution, and length scales using PIV. It should be noted that none of the aforementioned works has reported the distribution of over all scales across a stirred vessel in fully turbulent flow based on direct measurement of the instantaneous velocity gradients. PIV, as a whole-field technique, can provide also useful information of flow structures such as trailing vortices which have been analyzed with single-point techniques such as LDA by many authors in the past. However, PIV can provide information on the development of the trailing vortices across the entire vessel and through phase-resolved measurements their structure and importance for mixing processes can be better quantified. The objectives of the present work were to assess the advantages and shortcomings of direct measurement of in stirred vessels and thus to enable its accurate quantification and distribution across the flow field, as well as to obtain a more detailed understanding of the mean flow and turbulence structures present in such vessels. 2. Flow Configuration, Instrumentation and Techniques A standard configuration cylindrical vessel of diameter T ) 100 mm was used in this study. The vessel was equipped with four equispaced wall-mounted baffles of width B ) 0.1 T; the liquid column height was H ) T. The vessel was constructed from acrylic plastic

(Plexiglass) and was located inside a square acrylic container filled with distilled water in order to minimize the refraction of the laser sheet over the outer cylindrical surface of the vessel. The vessel was filled with distilled water seeded with neutrally buoyant 10 µm sized hollow silver coated glass spheres. The vessel was located on a traverse that allowed movement in the vertical and the two horizontal directions. The impeller used was a three-bladed hydrofoil axial impeller with a diameter of D ) T/3 described in detail by Fentiman et al.14 and shown in Figure 1. A transparent acrylic lid was positioned in the vessel, at a height equal to the liquid column height H, to prevent entrainment of air bubbles into the flow from the free surface of the liquid. Measurements were taken at an impeller rotational speed of 2165 rpm corresponding to a impeller Reynolds number (Re) of 40 000 and to a velocity at the tip of the blades Vtip ) 3.78 m/s. The origin of the coordinate system used is the center of the base of the vessel with U h z, U h r and uz′, ur′ referring to the mean and root-meansquare (rms) velocities in the axial (z) and radial (r) coordinate directions, respectively. An OFS Ltd. PIV system was employed for the experiments. An argon-ion laser provided the light sheet. A timing box was used to trigger a Sensicam digital camera and a Bragg cell to pulse the laser light. The pulsed laser illuminated a 1 mm thick vertical sheet passing through the center of the vessel and midway between two baffles (in the θ ) 0° plane). The axial and radial velocity components were measured. For each set of measurements, 500 pairs of images were obtained, evenly distributed over all phases, so that ensemble-averages over the entire 360° of revolution were calculated. The image pairs were grabbed at a frequency of 1-2 Hz, depending on the resolution employed. The resolution of the camera was changed according to the size of the area over which measurements were taken: it was increased from 13 pixels/mm (lowest spatial resolution) to 72 pixels/mm (highest spatial resolution). With the lowest resolution case the area covered by the camera picture was one-half of the entire vessel, while with the highest resolution case it comprised an area 10 mm wide and 3.5 mm high below the impeller region. Considering an image resolution varying from 13 to 72 pixels/mm depending on the image size, a practically negligible time uncertainty, a software interpolation accuracy of 0.1 pixels, and a calibration mapping accuracy of 1 pixel, the error in the velocity measurements was found to vary from 0.03 to 0.06Vtip. Each interrogation cell comprised 64 × 64 pixels with an overlapping of 32 pixels for the cross

7008 Ind. Eng. Chem. Res., Vol. 42, No. 26, 2003

Figure 2. 360° ensemble-averaged velocity vector map in the θ ) 0° plane.

correlation processing step, 32 × 32 pixels with an overlapping of 16 pixels for the subsequent adaptive cross correlation processing step, and when necessary, 16 × 16 pixels with an overlapping of 8 pixels for the last adaptive cross-correlation step. 3. Results 3.1. Mean Flow and Turbulence Fields. The first part of this study comprised ensemble-averaged measurements of the mean flow and turbulence fields to identify their distribution across the vessel and enable comparisons with earlier works employing LDA and to confirm and assess the PIV data obtained. The ensemble averaged mean flow and turbulence fields across one-half of the vessel were analyzed over an image area that covered the plane illuminated by the laser sheet, that is: r/T ) 0.0-0.5 and z/T ) 0.00.7. The resolution of the camera was 13 pixels/mm. The quantities that were derived from the analysis of these data are the ensemble-averaged mean velocity, the corresponding rms velocity, skewness, and kurtosis, as well as the Reynolds stresses. Skewness and kurtosis, the third and fourth moments of the probability distribution, respectively, provide useful indications on the temporal variation of the instantaneous velocity, flow unsteadiness and the temporal nature of the velocity fluctuations involved in the mixing process.

In Figure 2 the velocity vector map showing the flow field along a plane over half of the vessel is presented. A reference vector representing a velocity of 0.1Vtip is also shown. Each vector represents an interrogation cell. The outline of the impeller swept area and the shaft is also shown in this and subsequent figures. The impeller generates a main vortical motion, which occupies most of the vessel. The maximum velocity is 0.26Vtip and is located just below the impeller. In the region z/T < 0.1 and r/T < 0.13 a small counter-rotating vortex can be noted, which rotates in clockwise direction. The measured rms velocities, not shown for brevity, revealed that, as expected, the most turbulent areas were located along the edge of the impeller stream region under the impeller and were characterized by values of uz′ a little higher than 0.1Vtip. In the center of the discharge region low axial rms values were obtained. The radial turbulence level was analyzed as well and found to also be higher along the edge of the main discharge stream, as well as close to the base of the vessel. The skewness, kurtosis, and the Reynolds stresses were analyzed as well, as they provide useful information on the temporal velocity characteristics and turbulence content of the flow. The main axial discharge stream was found to be characterized by a large positive skewness and a high kurtosis. Large negative values of skewness and high values of kurtosis were found along the edges of the main discharge stream. These findings are in agreement to those reported, using the LDA technique, by Fentiman et al.15 with the same impeller and under the same conditions. To assess the PIV data, the results obtained were compared to those obtained with the same impeller and vessel with the LDA technique by Fentiman et al.14 Parts a and b of Figure 3 show characteristic results from these comparisons. Axial mean and rms velocity profiles in the region below the impeller are shown. The results show good agreement between the data obtained with PIV and LDA. Results for other z/T values and for the radial mean and rms velocities showed similar agreement. The results obtained with PIV and LDA were compared throughout the vessel and the averaged difference was found to be 0.018Vtip. This value is similar to or smaller than both the PIV and the LDA measurement errors. Consequently, the data obtained with PIV measurements are similar to those obtained with the LDA technique, and in most locations the differences are within the measurement errors and provide confidence for the more demanding subsequent determination of from PIV data.

Figure 3. (a) Comparison of u′z/Vtip values in the plane z/T ) 0.18 obtained by Fentiman et al.14 with LDA and in the present work with PIV. (b) Comparison of U h z/Vtip values in the plane z/T ) 0.28 obtained by Fentiman et al.14 with LDA and in the present work with PIV.


3.2. Turbulence Energy Dissipation. 3.2.1. Methodology Employed To Acquire Dissipation Data from 2-D PIV Measurements. The experience gained with the mean and turbulence velocity PIV measurements was then harnessed to measure in the vessel, for which the measurement resolution must be sufficient to resolve the dissipative scales, that are much smaller than those involved in the production of turbulence. PIV techniques can be used to measure the gradients of the Reynolds stresses comprising the main unknowns in the (eq 7) below (Sharp and Adrian11), provided that such gradients can be measured over volumes sufficiently small and comparable to the Kolmogorov scales.

{ (( ) ( ) ( ) ) ( ) ( )

)ν 2

∂ui ∂xi

2

+

2

∂uj ∂xj

+

2

∂uk ∂xk

+

∂ui ∂xj

2

( ) ( ) ( ) ( ) ∂ui ∂xk

2

∂uk + ∂xi 2

2

(

∂uj + ∂xk

2

∂uk + ∂xj

2

2

∂uj ∂xi

+

+ Figure 4. Contour plot of the distribution of the normalized turbulence energy dissipation; spatial resolution 1.25 mm.

+

Where φ indicates the tangential direction. Consequently, the expression for becomes

)}

∂ui ∂uj ∂ui ∂uk ∂uj ∂uk ‚ + ‚ + ‚ ∂xj ∂xi ∂xk ∂xi ∂xk ∂xj

(7)

In the above equation, u is the fluctuating velocity, the subscripts i, j, k denote the three Cartesian directions, and ν is the kinematic viscosity. As the measurements were taken using a two-dimensional PIV system, only two velocity components (denoted hereafter as ur and uz, respectively, in cylindrical coordinates, with uφ being the tangential component), and therefore, five terms in the full twelve-term turbulent dissipation energy equation terms were available. For this reason the unknown terms were assumed to be statistically isotropic and thus derivable from the known ones (see Sharp and Adrian11). Fentiman et al.,15 in their LDA study of the same flow, found the differences between radial and tangential rms velocities in many regions of the vessel to be small, and it may therefore be expected that the assumption of local isotropy is acceptable in the present two-dimensional PIV analysis, appreciating nevertheless that the results are affected by these approximations, especially near the impeller blades. Therefore, to estimate the turbulence energy dissipation, the seven missing terms were evaluated by considering turbulence as statistically isotropic and thus using relations (8a-c) below (see Sharp and Adrian11) derived from the equations for isotropic conditions (Hinze8):

( ) (( ) ( ) ) ( ) ( ) ( ) ( ) (( ) ( ) ) ∂uφ ∂φ

∂ur ∂φ

2

∂uφ ) ∂r

2

2

)

∂uz ) ∂φ

1 ∂ur 2 ∂r 2

2

+

∂uφ ) ∂z

∂uz ∂z

2

)

2

(8a)

1 ∂ur 2 ∂z

2

+

∂uz ∂r

2

(8b)

∂ur ∂uφ ∂uz ∂uφ ‚ ) ‚ ) ∂φ ∂r ∂φ ∂z

[( ( ) ) ( ( ) )] -1/2

∂ur ∂r

2

+ -1/2

∂uz ∂z

2

2

-

(( ) ( ) )

1 ∂ur 4 ∂r

2

+

∂uz ∂z

)

2

(8c)

{( ) ( ) ( ) ( )

)ν 2

∂ur ∂r

2

+2

∂uz ∂z

2

+3

∂ur ∂z

2

+3

∂uz ∂r

2

+

}

∂ur ∂uz 2 ‚ ∂z ∂r

(9)

As with the PIV technique, the velocity data is calculated by averaging over an interrogation cell, the spatial resolution of the data is very important. The experiments were therefore performed paying particular attention to the effects of resolution on the calculated fluctuating velocity gradients and related quantities. 3.2.2. Effect of Spatial Resolution. Figure 4 shows results obtained with 1.25 mm resolution. Results over half of the vessel are displayed. The /N3D2 values range from 0 to 0.02. Two regions with high dissipation are evident below the impeller. The remainder of the impeller stream presents lower values. The bulk of the vessel, as might be expected, is characterized by very low energy dissipation. Nevertheless, as this area covers most of the vessel volume, its contribution to the total energy dissipation is rather large. To validate the results an integration of over the whole vessel was performed, the result of which could be compared with the power consumption of the impeller used. The power number was measured by Fentiman et al.14 and found to be 0.22. From the integration a power number corresponding to about 7% of the measured one was estimated. It should be noted that this value is over two times higher than that obtained with a spatial resolution of 2.5 mm. This is expected, as for example, Piirto et al.12 and Saarenrinne et al.13,16 found that the measured values increased with the spatial resolution of the PIV measurements. Lack of resolution of the dissipative scales was clearly the most likely reason for this discrepancy. Nevertheless, the bulk region with relatively low turbulence energy dissipation, is characterised by larger values of the Kolmogorov length scale; therefore, the values in this region are expected to be closer to the actual ones. On the contrary, in the high turbulence region below the impeller, the Kolmogorov length scales are much smaller and the values are expected to be significantly affected by an increase in spatial resolution. This is discussed in detail subsequently.


Figure 5. Contour plot of the distribution of the normalized turbulence energy dissipation in a region below the impeller; spatial resolution 0.111 mm.

Experiments were then performed with higher spatial resolutions. To obtain a higher resolution, the camera had to be focused on an image area smaller than for the previous cases. With a spatial resolution of 0.285 mm, values of 10 times higher than those obtained with the previous case (1.25 mm resolution) were found. As the results covered only a small area of the vessel, estimation of the integrated value of and the consequent comparison with the power number were not possible. Extensive data analysis showed that, as might be expected, increasing resolution does not produce a similar or proportional increment in the values across the vessel: the highest and the lowest values are affected in different way. Therefore, to assess the results, measurements were performed at even higher resolutions and the variation of with spatial resolution was studied and compared with previously reported studies. Measurements were performed with spatial resolutions of 0.444, 0.222, and 0.111 mm. From an earlier estimation of the average Kolmogorov length scale across the vessel from the power number, a value η ) 37 µm was found; this implies that the resolution of 0.111 mm is about 3 times the average η. Nevertheless, it should be noted that the smallest Kolmogorov length scale, as it can be deduced by its definition (η ) (ν3/)0.25), is located in regions where is highest. Therefore, the smallest η value in the vessel should be less than 37 µm. The results obtained from the finest 0.111 mm spatial resolution measurements are shown in Figure 5. An increase in the measured values of in comparison to the results of Figure 4 (1.25 mm resolution) can be noted, mostly in the region where turbulence is highest. There, the dissipation value is about 40 times greater than in the previous case. Values about 80 and 8 times the mean dissipation value (j ) 0.54 m2/s3) were found in the highest and lowest turbulence regions respectively within the measurement area; very similar values were found by Cutter17 and by Zhou and Kresta18 for a vessel stirred by a flat blade impeller and a pitched blade turbine, respectively. Lee and Yianneskis,9 studying the turbulence properties of the impeller stream of a Rushton turbine, found a maximum value of the nondimensional energy dissipation around 20. This value is about 25 times the maximum value found in this experiment (* ) 0.8), the same ratio as that of the power numbers of the Rushton (Po = 5.5) and the hydrofoil impellers (Po = 0.22). Considering the highest dissipation value from Figure 5, ) 0.75N3D2 ) 39 m2/ s3, a value about 13 µm is found for η. This implies that the present resolution may be about 8.5 times the

smallest turbulence length scale. The Taylor spatial microscales, defined by

λij )

x( )

2(u′i)2 ∂ui 2 ∂xj

(10)

were analyzed as well and values around 0.5 mm were found in the highest turbulence region below the impeller. 3.2.3. Comparison with Dimensionally Based Dissipation Approximations. The energy dissipation has often been estimated using the relation expressed in eq 1 obtained from dimensional analysis. This relation is applicable only to flows or regions where the turbulence is homogeneous and isotropic. Moreover, as the length scale varies substantially according to the turbulence level in different regions of the vessel, a single value for L is not appropriate for such a complex flow field. Nevertheless eq 1 has often been regularly used by various authors (see, for example, Kresta and Wood19 and Zhou and Kresta20) in order to estimate turbulence dissipation in stirred vessels due to the inherent difficulties in measuring directly. These estimates of turbulence energy dissipation do not take in consideration the contribution of rms velocity components other than that in the main flow direction. For comparative purposes, the turbulence level data obtained by the PIV analysis and described in the previous paragraphs were used to make an estimation of turbulence dissipation according to eq 1. The resulting contours of normalized are shown in Figure 6. In the impeller stream region, where the highest resolution measurements are available from this work, it can be noted that the highest dissipation values are rather different. In Figure 6, the highest values are around 0.41 while in Figure 5 values around 0.75 are observed. Moreover, to assess the values obtained with eq 1, they were integrated over the whole vessel volume. The integration produced a value of Po ) 0.265, which is only around 20% higher than the actual power number. It should be noted that, even though this value is not very different to Po, eq 1 involves some approximations, which are pointed out below, that can yield an incorrect local distribution of . Considering Figures 4 and 6 some observations must be made. Both plots show values obtained from 1.25 mm resolution experiments; nevertheless, the distributions are different. The energy dissipation obtained


Figure 7. Variation of the turbulence energy dissipation, averaged over the impeller stream region, with the spatial resolution of the PIV measurements. Figure 6. Contour plot of the distribution of the normalized turbulence energy dissipation, as calculated with eq 1.

with eq 1 (Figure 6) presents high values (* > 0.4) in a large region contained in 0.07 < z/T < 0.25 along the boundary of the main axial stream below the impeller. The energy dissipation calculated with the direct definition (Figure 4) shows a distribution with values continuously decreasing with distance from the impeller, as might be expected. Sheng et al.,10 studying turbulence dissipation in a vessel stirred by a pitched blade turbine, obtained a distribution similar to that in Figure 4. Moreover, estimating over the whole vessel using a large eddy PIV method, they found that the dissipation rate obtained by dimensional analysis (for example, through eq 1), yielded much larger values in the bulk of the vessel than those found by large eddy approach, which is also mainly based on the definition employed in this work. Therefore it may be concluded that the estimation of through eq 1 produces results that are reliable if the average energy dissipation and its integration over the vessel is considered, but can only approximate the local values and their distribution over the vessel. The above finding has important implications for both practical and fundamental aspects of stirred vessel flows and the related mixing processes and is worthy of further consideration. It indicates that approximations of the distribution, estimated through equations such as (1), are likely to overestimate the minimum and, consequently, as the average value obtained is approximately correct, underestimate the maximum levels of in a stirred vessel. This is confirmed by the predictions of with large-eddy simulation (LES) methods by Derksen and van den Akker,21 for a vessel stirred by a Rushton turbine. This work showed that the levels of exhibit a near-logarithmic decrease with distance from the impeller. Consequently, much higher and lower levels are found in regions near and far from the blades, respectively, than the less pronounced variations in distributions obtained with expressions of the form of eq 1. Figure 7 shows the effect of the spatial resolution on the measurements of the turbulence energy dissipation in this work. The data shown represent the spatial average over a small area located just below the impeller in the most turbulent region. Six different spatial resolutions were considered for the energy dissipation: 2.5, 1.25, 0.444, 0.285, 0.222, and 0.111 mm. The influence of spatial resolution is due to the strong

dependence of the velocity gradient estimation on the distance between two adjacent measurement locations. The estimates of energy dissipation show small differences at low spatial resolutions (large dx) while they differ significantly at high resolutions (small dx). In the latter case, a small change in the spatial resolution produces strong variations in the value of the estimated . From Figure 7 it can be noted that the strong change in values occurs for resolutions smaller than about 0.5 mm. In this region, a decrease in interrogation cell size, and thus a reduction of the distance dx between adjacent measurement locations, of 50% from 0.444 mm to 0.222 mm, produces an increase in the estimated value of up to 250%. The Taylor microscales were calculated as well and were found to decrease linearly with increasing spatial resolution. The smallest resolution achieved, dx ) 0.111 mm, is about 3 times the mean Kolmogorov length scale (η = 37 µm). The ratio between dx and η is of great importance for the assessment of the results. Piirto et al.12 and Saarenrinne et al.,13 studying Taylor microscales and the energy dissipation, found trends very similar to those presented in this work. According to their findings, the dissipation that can be calculated with PIV is a fraction of the real dissipation value and depends on the spatial resolution normalised by the Kolmogorov length scale. In particular, they found that in order to enable the measurement of 90% and 65% of the actual energy dissipation, the spatial resolution should be around 2η and 9η, respectively. Therefore in the present work, it is estimated that around 80% of the true value in the less turbulent regions and about 65-70% in the most turbulent ones could be measured with a resolution of 0.111 mm. Figure 7 shows an almost asymptotic variation of the estimated over the smaller dx values. However, this trend is worthy of further consideration. The curve represents the interpolating function found to best fit the data obtained from the experiments. As no experiments were performed for spatial resolutions smaller than 0.111 mm, the behavior of the function for values of dx closer to Kolmogorov length scale cannot be reliably assessed. The curve should eventually reach the maximum value for dx close to 13 µm (i.e., the smallest likely Kolmogorov length scale). Further study is needed to assess this. Nevertheless, an estimation of how different the measured values are from the actual ones can be made. As has already pointed out earlier, according to the findings of Piirto et al.12 and Saaren-


rinne et al.,13 with dx ) 3-8.5η, about 80 to 65%, respectively, of the actual values could be estimated in this work. It should be noted here that the turbulence dissipation, and consequently η, vary considerably across the vessel, from very small and large η in the bulk, to large and small η below the impeller, therefore the spatial resolution reached in the present experiments should be sufficient to estimate reliably the dissipation over most of the vessel volume. This can be shown by estimating the area below the curve in Figure 7, between dx ) 0.111 mm and infinity, and comparing it to the remainder of the (undetermined) area corresponding to smaller dx. According to Figure 7, a value ) 13.26 m2/s3 corresponds to dx ) 0.111 mm. Extrapolating the curve, for dx ) 0.066 mm (about 5 times ηmin ) 13 µm) a value ) 25 m2/s3 is found. As has been previously pointed out, the extrapolation is likely to give an overestimation of because the curve should rapidly tend to the asymptote value in the region considered. Therefore the value ) 25 m2/s3 should be close to the actual value. The area under the curve between dx ) 0.066 mm and dx ) 0.111 mm is 16% of the whole area between dx ) 0.066 mm and dx f ∞. This indicates that about 84% of the actual value should be estimated in the present work. This result is therefore in agreement with the findings of Piirto et al.12 and Saarenrinne et al.13 Saarenrinne et al.,13 extending the work of Piirto et al.,12 concluded that the image size of PIV measurements influenced the flow velocity, while the interrogation cell size did not exhibit a clear effect. In the present work, an only very small dependence of flow velocity over interrogation cell size was found: an increase in the velocity values of only 0.006Vtip in the highest velocity region below the impeller was found when changing resolution from dx ) 2.5 mm to dx ) 0.285 mm, corresponding to an interrogation cell area about 80 times smaller. Such a negligible change in the velocity values compared to a rather large refinement in the interrogation cell size does not support the observation of Saarenrinne et al.13 3.2.4. Effect of Number of Samples. It should be noted that in the estimation of other factors can also affect the results: possible sources of uncertainty include the number of samples employed in the analysis. As it has been described earlier, 500 different pairs of pictures were considered in the ensemble-average estimations in the present work. To assess the magnitude of this uncertainty, some estimations were performed using different sets of images chosen from the total 500 data files obtained. The turbulence energy dissipation was calculated in the manner described earlier. Figure 8 shows the results of these estimations: the maximum value obtained with different numbers of samples is presented. As can be easily noted, there is a scatter in the results, more manifest for small numbers of samples. This emphasizes the importance of the number of samples taken into consideration when aiming to produce ensemble-averaged estimations. If, for example, only 100 samples are considered, an error up to 40% could be present in estimating the turbulence energy dissipation. With an estimation based on 200 samples the error decreases to 15% and to 7% when 300 samples are employed. Clearly a larger number of frames is necessary to establish the statistical independence of the results, but, for numbers of samples above 400, the results are expected to be statistically reliable.

Figure 8. Variation of the maximum dissipation value measured with the number of samples employed in the calculation.

3.3. Phase-Resolved Measurements. As indicated earlier, phase-resolved data can help improve understanding of structures present in the flows, such as trailing vortices, and can help eliminate the inherent broadening of the ensemble-averaged turbulence levels. Such broadening stems from the averaging that is employed when ensemble-averages over 360° of revolution are employed to calculate rms values or turbulence levels, as the variation in the mean flow results in an increase in the rms value determined. When phaseresolved averages over 1° are calculated, such broadening of the rms values is essentially eliminated, as has been shown, for example, by Yianneskis et al.1 Phaseresolved measurements were obtained in the present work by means of an encoder attached to the impeller shaft, that produced 2000 pulses per revolution. It should be noted that the shaft rotation was fairly steady; and shaft eccentricity, albeit small, was established to produce an identical of the shaft/impeller in every revolution, via observations with a stroboscope and a reference laser beam. It should be noted that the aforementioned broadening does not affect quantities such as that are based on the instantaneous values of fluctuating velocity gradients. Experiments were performed in order to determine the mean and rms velocities as well as the trailing vortex structures originating from the blades of the rotating impeller. An image area between r/T ) 0.00.5 and z/T ) 0.0-0.7 was considered. The resolution of the camera was 13 pixels/mm. Four different positions of the blades were considered, corresponding to φ ) 0°, 30°, 60°, and 90°. 500 pairs of pictures were taken for each of the four phases or φ angles. Mean and rms velocities were analyzed first. Parts a and b of Figure 9 show the normalized axial velocities measured in three different points in the vessel plotted over φ. Results obtained in the ensemble-averaged measurements described earlier are plotted as well in order to allow comparison. All of the points considered are located in regions greatly affected by the periodic rotation of the impeller: two of them below the impeller region and one at the same axial position as the blades (z/T ) 0.34), but at r/T ) 0.3. As can be noted in both parts of Figure 9, the results from the phase-resolved measurements are dispersed around the value found with the ensemble-averaged measurements, as expected. This is important for both the determination of the true levels of turbulence in a vessel, as well as for the accurate assessment of CFD predictions of the flows, such as LES, that do not treat the flow in an ensemble-


Figure 9. Comparison of the axial mean (a) and rms (b) velocity values in three different locations of the vessel obtained with ensemble-average and phase-resolved measurements.

averaging manner. The actual values of turbulence level at a particular location can vary from one angle to another, and this variation is more pronounced in the impeller stream. In parts a and b of Figure 10, the contour plots of axial rms velocities normalized by Vtip for φ ) 30° and φ ) 90°, i.e., 30° and 90° after the passage of the blade, are shown. The results obtained for φ ) 0° and φ ) 60° did not reveal any additional features and are not shown for economy of presentation. In Figure 10a, a high turbulence region can be noted just below the impeller in the main axial stream region. Values up to 0.145Vtip are present in this area. The distribution of the turbulence levels is rather different from the case shown in Figure 10b representing the results obtained from measurements taken 90° after the passage of the blade. In this second case, high turbulence regions can be noted

along the internal and external borders of the impeller discharge region. The turbulence is generally lower than the previous case, with the highest value being around 0.11Vtip. The different distribution of turbulence levels is due to the development of the impeller stream and the related motion of the trailing vortex in the axial and radial directions and to its increase in size. According to this interpretation, partly supported by the vorticity results described in the following section, the high turbulence areas clearly defined in Figure 10b correspond to the boundary of the impeller stream with the surrounding flow. As has already being pointed out, the ensemble-averaged turbulence levels in the three coordinate directions were previously found to be approximately locally isotropic in many locations. The u′z and u′r results showed that the phase-resolved turbulence levels are also nearly isotropic and that the degree of isotropy increases with distance from the blades. Fentiman et al.’s15 phase-resolved LDA data also showed very small differences between the radial and axial direction phase-resolved turbulence levels. These small differences in turbulence levels in subsequent blade angles stem mainly from the periodic passage of the blades and the trailing vortices originating from them. 3.4. Trailing Vortex Structure. The trailing vortex structure was considered in some detail to characterize such flow features and obtain additional information not available to date through the earlier, predominantly single-point LDA, studies. The mean velocity in the four different φ angles was studied and the presence of the vortices was best identified through determination of the distribution of vorticity in the vessel in four different phase angles. The vorticity was calculated as

ω)

∂Uz ∂Ur ∂Ur ∂Uz

(11)

characteristic results are shown in Figure 11a,b: results at other angles were similar and are not presented. High positive and negative values of vorticity indicate the presence of the vortex core. The plots show a high vorticity region originating from the blade of the impeller and developing downward, turning outward at about z/T ) 0.15. The centers of the vortices can be identified by the highest vorticity values shown in deep red color.

Figure 10. (a) Contour plot of the axial rms velocity at φ ) 30°. (b) Contour plot of the axial rms velocity at φ ) 90°.


Figure 11. Contour plots of the normalized phase-resolved vorticity across the vessel for two different φ positions of the blades: (a) 60°; (b) 90°.

In Figure 11a,b, a number of centers of different vortices can be noted, originating from different blades of the impeller located at different φ angles. The high vorticity region near the wall is not related to the vortices. Consequently, the vorticity and other phase-resolved quantities can allow a more exact determination of the path followed by each vortex than was possible with other techniques, starting from their origin at a blade tip, travelling toward the bottom of the vessel and higher r/R values until they decay and become indistinguishable from the vorticity in the rest of the vessel. Results indicating the vortex path are shown in Figures 12 and 13, using data from more than one vortex trajectory to illustrate the vortex motion. In Figure 12, the axial and radial locations of the vortices are plotted against the φ value corresponding to the angle after the passage of the blade from which they originate. The relatively limited data obtained by Fentiman et al.15 are also shown for comparison, and as can be observed, there is a good agreement. The vortices could be observed up to about 400° after the passage of a blade. Each vortex originates from a blade and moves steadily downward and outward toward higher r/T values. The resulting structure that is revealed is that of a helical vortex with its center moving slightly outward in the radial direction and descending toward the bottom of the vessel before reaching the cylindrical wall. The kinetic energy of the trailing vortices was analyzed as well. Figure 13 shows the vorticity and kinetic

Figure 12. Variation of the axial and radial location of the trailing vortex center in the plane θ ) 0° with angle after the passage of the blade and comparison with the results of Fentiman et al.15

energy at the center of the trailing vortices plotted against the φ value for the blade from which the vortex originated. The vorticity could be analyzed for values of φ from 180° to 400°, whereas the kinetic energy could be analyzed for values of φ from 120° to 200°; for higher values of φ the kinetic energy of vortex could not be determined reliably due to the relatively low values involved (similar to the background turbulence levels in the vessel bulk). In the figure the best fit curves are shown as well. As can be observed, both the vorticity and kinetic energy decay in a similar manner. It should be noted that if the k and ω axes scales are adjusted, the two variations would be nearly identical.


Figure 13. Variation of the normalized vorticity and kinetic energy of the trailing vortex center with angle after the passage of the blade.

4. Conclusions This work has provided a detailed description of the mean flow, turbulence level, and turbulence energy dissipation rate distribution in a vessel stirred by a hydrofoil impeller. At first, 360° ensemble-averaged measurements were performed. The skewness, kurtosis, Reynolds stresses, and vorticity distributions were analyzed as well. The main features of the mean flow and turbulence structures were analyzed and discussed, and agreement was found with less-detailed earlier studies employing single-point methods such as LDA. The dissipation rate data were obtained through direct determination of the instantaneous fluctuating velocity gradients. Particular attention was paid to the effects of spatial resolution on the estimation of energy dissipation. Experiments were performed at spatial resolutions ranging from 2.5 to 0.111 mm. The latter was calculated to be around 3 times the average Kolmogorov length scale in the vessel. Consideration of the present and previously reported PIV approaches for the measurement of indicated that around 90% and 65-70% of the dissipation in the bulk flow and the impeller stream respectively can be determined with the present approach. Energy dissipation results were also compared with findings obtained through dimensional relations or with LDA. Dimensional relations were found to provide reliable estimates of the average energy dissipation across the vessel, but are likely to overestimate the minimum and underestimate the maximum levels of dissipation in a vessel and are thus not very suitable for the determination of the local values and distribution, which are of paramount importance for the optimization of process vessel design (Tatterson22). The present experimental results support the findings of earlier CFD predictions of the flows with LES methods (Derksen and van den Akker21) that indicated a nearlogarithmic decay of dissipation rate with distance from the impeller blades. Phase-resolved PIV measurements were performed as well to study the trailing vortex structures and phaseresolved levels of turbulence in the flow. The path, vorticity and kinetic energy of the vortex centers across the vessel were estimated and found to be in agreement with the findings by Fentiman et al.,15 while the variation of turbulence level with phase angle was found to be significant near the impeller blades.

Most importantly, the present PIV work has produced not only a detailed assessment of the effects of spatial resolution on the direct measurements of the dissipation rate of the kinetic energy of turbulence in stirred vessels, but has also provided hitherto unavailable data and shows much promise for the accurate determination of in such flows in future. The next stages of such work should involve measurements with a 3-D PIV system to determine directly more terms in the equation. Such measurements present new challenges such as the resolution in the third direction that depends on the laser sheet thickness, but the present investigation provides much confidence for measuring directly and facilitating optimization of the design of mixing-sensitive chemical processes and the assessment and development of related prediction methods. There is clearly a need for further work in the future in terms of improvements in measuring techniques and overcoming resolution limitations present at high Re which can be overcome via measurements in larger vessels, and correspondingly larger average Kolmogorov scales, as in view of the size of the scattering particles necessary for PIV measurements, further increase of PIV resolution is not feasible. Second, determination of all fluctuating velocity gradients would alleviate the need for assumption of local isotropy and these could be achieved through measurements with 3-D PIV, which is possible with the current system, provided the measurements are made in a larger vessel. Third, recent developments in PIV systems have achieved fast acquisition rates that should enable time-resolved dissipation measurements that would facilitate the direct temporal determination of the levels in stirred vessels. Nomenclature Roman Characters A ) constant, eq 1 A′ ) constant, eq 4 B ) caffle width (m) C ) impeller off-bottom clearance (m) D ) diameter of the impeller (m) H ) total liquid depth in vessel (m) k ) turbulent kinetic energy (m2/s2) L ) integral length scale (m) N ) number of samples N ) impeller speed (Hz) Po ) impeller power number q ) Turbulent kinetic energy, eq 3 (m2/s2) r ) radial distance from the axis of the vessel (m) r ) radial coordinate of the measuring volume (m) Re ) impeller Reynolds number T ) diameter of the vessel (m) u′z, u′r, u′φ ) axial, radial, and tangential rms velocities (m/s) uz, ur, uφ ) axial, radial, and tangential instantaneous fluctuating velocities (m/s) Uz, Ur, Uφ ) axial, radial and tangential instantaneous velocities (m/s) U h z, U h r, U h φ ) axial, radial and tangential mean velocities (m/s) Vtip ) velocity of the tip of the impeller blades (m/s) z ) vertical coordinate of the measuring volume (m) Greek Characters j ) turbulence energy dissipation rate averaged over the whole vessel (kJ/kg/s) ) turbulence energy dissipation rate (kJ/kg/s)

7016 Ind. Eng. Chem. Res., Vol. 42, No. 26, 2003 * ) nondimensional turbulence energy dissipation, {}/{(N3D2)} Λx ) integral length scale, eq 6 (m) η ) Kolmogorov length scale (µm) φ ) tangential coordinate (angle from the middle of the span of a reference blade) (deg) κ ) wavenumber, eq 2 (m-1) λ ) Taylor microscale (mm) µ ) dynamic viscosity (kg/m/s) ν ) kinematic viscosity (m2/s) θ ) tangential coordinate of the measuring volume (deg) F ) density (kg/m3) ω ) vorticity (s-1) Abbreviations 2D ) two-dimensional dx ) grid vector spacing, spatial resolution (m) LDA/LDV ) laser doppler anemometry/velocimetry PIV ) particle image velocimetry rms ) root mean square

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Received for review October 21, 2002 Revised manuscript received October 3, 2003 Accepted October 9, 2003 IE0208265

On the Direct Measurement of Turbulence Energy Dissipation in

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