Identification and characterization of flow structures in chemical process equipment using multiresolution techniques

Chemical Engineering Science 63 (2008) 5330 -- 5346

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Identification and characterization of flow structures in chemical process equipment using multiresolution techniques Sagar S. Deshpande a , Jyeshtharaj B. Joshi a,∗ , V. Ravi Kumar b , B.D. Kulkarni b,∗ a b

Institute of Chemical Technology, University of Mumbai, Matunga, Mumbai 400019, India Chemical Engineering and Process Development Division, National Chemical Laboratory (NCL), Dr. Homi Bhabha Road, Pune 411008, India

A R T I C L E

I N F O

Article history: Received 5 February 2008 Received in revised form 1 June 2008 Accepted 10 June 2008 Available online 22 June 2008 Keywords: Chemical reactors Length scale distribution Hydrodynamics Mixing Model reduction Turbulence Wavelet transform

A B S T R A C T

Planar information of velocity from 2D particle image velocimetry (PIV) and large eddy simulation (LES) data have been studied using multiresolution wavelet transform (WT) formalisms, i.e., discrete and continuous WT. Identification of dominant energy containing structures with their characterization in terms of fractal spectra have been carried out for industrially important equipment exhibiting turbulent behavior. These include annular centrifugal contactor, jet loop reactor, ultrasound reactor, channel flow, stirred tank and bubble column reactor. The characterization of their dynamics based on denoising the data and studying the local energy along the WT scales show sensitive variation and this helps in identifying the size and shape of structures. A dependency is seen between mixing time and the higher order moments of length scale distribution, viz., skewness and kurtosis and a generalized correlation has been built up for important types of equipment and associated flow parameters. The correlation is not only based on the knowledge of reactor geometry and operating conditions but also on the flow structures via their statistical parameters. Wavelet transform modulus maxima (WTMM) methodology has been used to study the evolution of structures and their interaction in a reduced dimensionality by evaluating the fractal spectra. Classification studies have been carried out using principal component analysis (PCA) of the fractal spectra. The results obtained show clear classes for the six types of equipments and delineate regimes to obtain benchmark patterns of flow hydrodynamics based on PCA co-ordinates. This methodology offers a generalized way for the optimal design and operation of different types of reactors. © 2008 Published by Elsevier Ltd.

1. Introduction Widely used industrial equipment, jet reactors, bubble columns, stirred tanks and annular centrifugal contactors (ACC) must be designed to operate at high efficiencies to meet the economical and environmental constraints of the industry. Unfortunately, most of the equipments are over-designed because of the high degree of empiricism involved. Considerable improvements in their design are possible by studying in detail the structures that form and evolve in the flow. For this purpose, a number of techniques have been developed for identification and characterization of flow structures. These include: conditional sampling (vorticity based methods); pattern recognition analysis; proper orthogonal decomposition; stochastic estimation; topological concept-based methods, etc. Reviews and applications of these methods can be found in literature (Joshi and Sharma, 1979; Antonia, 1981; Hussain, 1983; Aubry et al., 1988;

∗ Corresponding authors. Tel.: +91 22 24145616, fax: +91 22 24145614 (J.B. Joshi), Tel.: +91 20 25902150; fax:+91 20 25902162 (B.D. Kulkarni). E-mail addresses: [email protected] (J.B. Joshi), [email protected] (B.D. Kulkarni).

0009-2509/$ - see front matter © 2008 Published by Elsevier Ltd. doi:10.1016/j.ces.2008.06.010

Adrian and Moin, 1988; Glauser and George, 1992; Berkooz et al., 1993; Kevlahan et al., 1993; Bonnet et al., 1998; Joshi, 2001; Joshi et al., 2002). However, there is still a need to find novel methodologies that could identify and study the dynamical behavior of flow structures that are present in the hydrodynamics of turbulent systems. In particular, wavelet transform (WT) has gained attention and a wide range of applications is seen in recent times because it studies the energy decomposition of non-stationary flows in a localized way. This it does by convolution of data with suitable wavelet basis kernel functions that have finite energy with well-defined properties. The wavelet coefficients obtained by the WT systematically span different time and spatial scales in the flow. Therefore, convolution of wavelet functions with data can identify different types of structures that are simultaneously present in the flow dynamics. Studies have shown that discrete wavelet transform (DWT) is useful for scalewise decomposition of non-stationary data in an orthogonal multiresolution framework (Farge, 1992; Camussi and Guj, 1997). In fact, DWT has been used to develop strategies that are useful for denoising data corrupted with noise arising due to measurement and systemic errors (Roy et al., 1999; Goring and Nikora, 2002; Ganesam et al., 2004). Alternatively, a continuous wavelet transform (CWT)

S.S. Deshpande et al. / Chemical Engineering Science 63 (2008) 5330 -- 5346

approach may be used to study the evolution and interaction of structures. This may be carried out by using the wavelet transform modulus maxima (WTMM)methodology for evaluating the multifractal behavior of complex systems (Meneveau and Sreenivasan, 1987; Muzy et al., 1991; Arrault et al., 1997). Studies using DWT and CWT have been extended to the identification of flow structures in planar data sets (Kailas and Narasimha, 1999; Camussi, 2002; Srinivas et al., 2007) and the corresponding fractal spectra (Decoster et al., 2000). In the present work, we study the particle image velocimetry (PIV) and large eddy simulations (LES) data using WT to identify the energy containing structures and their associated multifractal spectra for systems of industrial importance. This is desirable because characterization of various equipment on a common ground may be extremely useful for the selection of the right one for a specific purpose. Therefore, we undertake a comprehensive study of six industrially important chemical process equipments and study them on a common framework that includes: (1) Comparison of length scale distribution of flow structures. (2) Correlation for mixing time as a function of length scale of dominant structures and statistical parameters associated with length scale distribution. (3) Study of fractal spectra behavior of data in terms of quantifiers that study the nature and strength of the singularities. (4) Regime classification by principal component analysis (PCA) of a large database of fractal spectra for varying system properties including reactor geometry, operating conditions and statistical properties of flow structures. 2. Literature There have been several notable applications of WT to study flow structures in turbulent flow. Most studies use data sets obtained from a point in the spatial domain (1D time series). Farge (1992) used wavelets to study flow structures by analyzing scales specific to regions of high vorticity concentration in direct numerical simulations (DNSs). The reconstructions obtained by inverse wavelet transform (IWT) were used to study the dynamics of these structures. Kailas and Narasimha (1999) applied WT on planar images to identify flow behavior in the turbulent mixing layer in a free jet. On application of the threshold for the wavelet coefficients in each scale, the flow structures of various types could be brought out in a scalewise fashion. Three levels of scale-specific structures were revealed, namely, the nearly homogeneously distributed small scale structure, an intermediate regime where scale organization of the vortex boundaries (with some internal sub-structures) occur, and the large scale coherence of each vortex itself. The WT was advantageously able to extract out the hidden small scale homogeneous structures, which were not perceptible in the flow images. Siddhartha et al. (2000) observed that the nature of the turbulent motion is most strongly evident in the azimuthal component of the vorticity, and which is organized in the form of a toroidal base supporting a relatively thin conical sheath. It was observed that the interior of the structure is nearly devoid of azimuthal vorticity with radial and stream wise components giving evidence of secondary structures in the form of helically organized vortex pairs. Camussi (2002) devised a methodology for the identification of flow structures based on 1D transform over 2D PIV data such that radial direction was used for convolution with axial velocity component and vice versa. The results then become equivalent to studying the vorticity. Criteria for cutoff based on local intermittency measure (LIM) was studied. In contrast to the conventional vorticity methods that showed diffused structures, this methodology, brought out the presence of well-defined structures in the flow.

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Saxena et al. (2002) studied the wavelet analysis of planar laserinduced fluorescence (PLIF) images of diametral sections of a jet. At larger scales, they observed a five lobbed sinuous ring-like structures of the unheated plume with a void inside and a green outer band. The lobes observed were similar to the `cells' obtained by DNS vorticity field by Basu and Narasimha (1999). Srinivas et al. (2007) depicted the life cycle of a flow structure through temporal animation of the wavelet transformation on PLIF images for diametric sections of a turbulent jet. Thus, by analyzing the amount of ambient fluid in the core of the jet, a quasi-cyclic behavior with an average period t given by U⊂/ t/y1/2 of about 6 was observed, where U⊂/ is the center-line velocity in the jet and y1/2 is its half-width. The period t was also seen to be the average time taken by a flow structure to pass the observation section. The present paper studies both single and multiphase systems with a view to present a generalized methodology that identifies and characterizes their flow behavior on a common basis. The application of the developed methodology for important classes of chemical process equipment aids making similarities and dissimilarities in the hydrodynamic behavior of the system. This paper is organized as follows: In Section 3, the details of the equipment and data generation are outlined. The multiresolution framework employed for denoising the data, evaluating the length scale distribution and obtaining the multifractal properties of the hydrodynamics of these systems is described in Section 4. The advantages of using both DWT and CWT for these purposes becomes apparent. The results of analysis of the individual equipment are presented in Section 5. A comparison of length scale distributions leading to a mixing time correlation and multifractal singularity spectra is discussed in detail. Section 5 also presents the results of classification of flow hydrodynamics using PCA of the singularity spectra database. The basics of PCA methodology has been described in the literature (Diamantaras and Kung, 1989; Hyvarien et al., 2001; Hastie et al., 2001; Kuriakose et al., 2004). 3. Equipment description Six equipments have been studied for the hydrodynamic analysis, viz., ACC, jet loop reactor (JLR), ultrasound reactor (USR), channel (CHA), stirred tank reactor (STR), and bubble column reactor (BCR). The geometrical description, methodology used for obtaining the data as well as flow details for each of these equipment are summarized in Table 1. For the first three equipment (ACC, JLR and USR) experimental data that have been obtained by PIV is used. The PIV apparatus consists of a pulsed dual Nd:YAG laser from TSI Inc., USA, and having a pulse duration of 6 ns was synchronized with a camera. The optics included a combination of cylindrical and spherical lenses that created a thin laser sheet of 1 mm thickness. A high resolution 4M CCD camera (Powerview plus, TSI) of 15 Hz frequency was positioned at right angle to the laser sheet. Lens of Nikon make was used for capturing a larger size window (150 mm × 150 mm). For covering smaller areas (up to 40 mm × 40 mm), a zoom lens (AF micro Nikkor, 60 mm) from Nikon was used. The images were recorded with a resolution of 2048 × 2048 pixels. The flow was seeded with hollow glass–silver beads of 20 m diameter and 1050 kg/m3 making sure that the Stokes number is small enough for the particles to follow the path of the fluid. The recorded images were divided into interrogation area of 64 × 64 pixels with a 50% overlap, resulting in approximately 4000 vectors for the image. The time difference between the two laser pulses was optimized based on the Nyquist criterion. A nonlinear calibration analysis was carried out in order to obtain accurate values of the velocities. The personal computer functioned as a central data acquisition and also processing unit. The details about the PIV measurements and error minimization can be obtained in the literature (Adrian, 1991; Camussi, 2002).

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Table 1 Flow and geometrical details of equipment Equipment

Data type

Snapshot details

Tank details (m)

Pin (W/m3 )

Variation in geometry boundary conditions

Re

Case

ACC

PIV

512 snapshots sampling rate 1 Hz

H = 0.06 Din = 0.038 Dout = 0.05

700 7000

N = 10 rps N = 20 rps

14, 3000 314, 140

ACC-1 ACC-2

JLR

LES

512 snapshots sampling rate 20 Hz 512 snapshots sampling rate 1 Hz

D = 0.3 H = 0.4 HN = 0.3

4 30 12

d = 0.02 m; Q = 0.286 kg/s d = 0.02m; Q = 0.542 kg/s d = 0.015 m; Q = 0.286 kg/s

17, 000 35, 000 33, 000

JLR-1 JLR-2 JLR-3

V = 0.002 m3 V = 0.002 m3

PIV

USR

PIV

512 snapshots sampling rate 1 Hz

D = 0.15 H = 0.14

15 35

Channel

LES

1000 snapshots sampling rate 50 Hz

W = 1.0 L = 3.0

1867 2160

u2 = 15 m/s u2 = 18 m/s

5000 18, 000

CHA-1 CHA-2

STR

LES

400 snapshots sampling rate 10 Hz

D = 0.3 H = 0.3 HI = 0.1

13 90 225

Hydrofoil PBTD45◦ Disc turbine

140, 000 140, 000 140, 000

STR-1 STR-2 STR-3

BCR

LES

400 snapshots sampling rate 10 Hz

D = 0.15 H=1

Single hole sparger Sieve plate sparger Sintered plate sparger

37, 500 31, 500 12, 000

BCR-1 BCR-2 BCR-3

0.19 0.19 0.19

USR-1 USR-2

Table 2 LES details for meshing, prediction and comparison S.N.

Equipment

Simulation details

Remarks

1

CHA

Grid size: 0.4 million, near wall grid: 4 nodes up to y+ = 5 so that viscous region is fully resolved up to Kolmogorov scale (Mathpati and Joshi, 2007) Sub-grid scale (SGS) model: one equation k model

Mean velocity and turbulence parameters within 5% as compared to DNS simulations

2

STR

Grid size: 1.1 million; SGS model: one equation k model (Mathpati and Joshi, 2007)

Mean velocity, turbulence parameters and power numbers were found in close agreement with the experimental LDV data (within 10%)

3

BCR

Grid size: 0.2 million; grid size is set such that the bubbles are not resolved. Fine grid is maintained near the wall while coarse grid is kept in the center. SGS model: Smagorinsky model (Tabib et al., 2008)

Mean velocity and turbulence parameters were found in good agreement with the experimental LDV data (within 10%)

In the event when dynamic 2D experimental data are not available, carrying out LES studies helps in easing the situation. For further studies, the LES predictions have been used as data sets for the remaining three equipment, viz., CHA, STR and BCR. The corresponding reference equations for LES predictions have been listed in Appendix A. In the present study, the LES simulations were initially validated by matching for the local mean velocities (namely, axial, radial and tangential velocity components) and turbulence parameters (turbulent kinetic energy and energy dissipation rate) and the energy balance either using DNS simulations for CHA or using experimental LDV time series for STR and BCR available with us. The details of the LES prediction for the three types of equipment are presented in Table 2. After validating the LES predictions, a collection of planar cuts of the data have been used as snapshots for the transient studies. The multiresolution wavelet methodology proposed here for identification and characterization of flow structures is then applied to both PIV and LES data for all the six types of equipments. ACC: Fig. 1A shows the schematic of the ACC. Two rotation speeds have been considered namely10 and 20 rps. The measurements have been performed in the annular gap of 6 mm. A vertical plane has been captured for the study. More details about the measurements can be seen in Deshmukh et al. (2007). JLR: Fig. 1B shows the schematic of the JLR. The Re has been varied from 16,000 to 35,000 for different combinations of nozzle diameter and flow rate. The window size used is 150 mm × 150 mm. Thus, the maximum scale captured is that of the radius of the tank.

USR: Fig. 1C shows the schematic of the USR. The energy is supplied through the ultrasonic horn and for its variation the flow patterns have been studied. Three variations of power input per unit volume are studied, namely, 15, 35 and 50 kW/m3 . For the first two cases, the power supplied was 30 and 70 W, respectively, with volume of tank, V = 0.002 m3 . In the third case, the power input, Pin = 50 W with V = 0.001 m3 was employed. CHA: Fig. 1D shows the schematic of a CHA. The channel flow has been simulated for the case of Re = 5000 and 18,000. The flow behavior near the boundary layer within y+ =50 has been considered to identify the large scale structures. This system has been considered mainly to verify the multiresolution algorithm by comparing with known results. STR: Fig. 1E shows the schematic of the STR. The data have been obtained for three impellers at identical rotation speeds, namely, disc turbine (DT), pitched blade turbine (PBT) and hydrofoil (HF). The impeller diameter was kept the same. BCR: Fig. 1F shows the schematic of the BCR. Three sparger types, namely, single hole, sieve plate and sintered plate have been studied for the chosen reactor geometry and constant gas flow rate. 4. Multiresolution methodologies In the present work, both the DWT and CWT have been used for the identification of flow structures and their characterization. The

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5333

Data plane Data plane Flow Direction

H

H W L

Data plane

Din Dout Data plane

Nozzle

H

HI

DI H D

HN

D

D

Ultrasonic horn Data plane H

Data plane

H

Gas Sparger D

Gas In Fig. 1. Schematic diagrams of the six equipments. (A) ACC; (B) JLR; (C) USR; (D) CHA; (E) STR; (F)BCR.

DWT has been used for data denoising and structure identification, while, the CWT is used to study the scalewise interactions by obtaining the fractal spectra for estimating the strength of the singularities. The description of the overall algorithm used for the studies is described below.

4.1. 2D discrete wavelet transform (DWT) The DWT separates the information content in the data from a fine scale to a coarser scale, systematically, by isolating the fine scale variability in terms of wavelet coefficients representing the

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details from the corresponding coarser scale coefficients depicting the smoothness. This procedure can be repeated iteratively over smooth scales to obtain scalewise detailed decomposition in a multiresolution framework. The wavelet coefficient matrix W a,b (x) may be obtained by the convolution of the 2D snapshot data, say u(x), with a suitably chosen mother wavelet function a,b (x) as W a,b (x) =



u(x)a,b (x) dx

(1)

where (2)

In this case, x represents the spatial domain of the 2D planar data with a and b defining the scaling and translation parameters. When the wavelet functions a,b (x) are chosen to be orthogonal to their dyadic dilations by 2−a and their translations by discrete steps 2−a b, for b = 1, . . . , 2a , it allows a multiresolution analysis in wavelet scales a = Ji − 1, . . . , 2, 1, 0 that can be carried out in the 2D domain. Here, Ji refers to the number of scales in the directions of x and depends upon the data resolution. For clarity in notation, we refer to the scale S1 as representing the smooth scale for a = 0 and D1, D2, . . ., representing the detailed scales for a = 1, 2, . . ., etc, with the higher scales studying the high wavenumber information in the data. An analysis of the length scale distributions in the data is then carried out as follows: (1) Obtain wavelet coefficients Wia,b (x) as shown in Eq. (1) from the two components of velocity ui (x) where i = 1, 2 represent radial and axial components. (2) The Wia,b (x) may be used to obtain scalewise reconstructions uai , by carrying out the IWT. This is carried out scalewise by choosing the WT coefficients corresponding to a particular scale for an IWT and setting the coefficients at all other WT scales equal to zero. Note that the reconstructions uai lie in the spatial domain x and this is therefore assumed in the notation for u. (3) A denoising algorithm, is then applied to uai by studying the energy spectra at the WT scales. The procedure employed is similar to the one used by Roy et al. (1999), except that it has been appropriately formulated here for application to 2D planar data, rather than for the 1D situation carried out in Roy et al. (1999). The denoising method uses the derivative of the scalewise data uai for the WT with a chosen mother wavelet function (say, Daub  (a,b)

(x) in the derivative 12) in Eq. (2). We thus obtain the Wi domain. By doing this, the deterministic component (true data) gets shifted to lower scales and the noise component to higher scales Roy et al. (1999). An automatic threshold for identifying scales with noise component can then be obtained by studying (a) the power spectra P  i at different scales. (a)

=



(a,b)

(Wi

d,(a,b)

(x). matrix Wi (5) The structures can be identified by setting a threshold cutoff to d,(a,b)

the Wi (x), obtained in step (4), by analyzing the local energy distribution in a scalewise fashion defined as:  ⎡⎛ ⎞⎤   d,(a,b) d,(a,b) W1 (x)W2 (x)   a ⎠⎦ (4) E(x) = ⎣⎝  d,(a,b) d,(a,b)   maxa W1 (x)W2 (x)b E(x)a > [E(x)a − E(x)a ]2 1/2 .

a,b (x) = 2a/2 (2a (x) − b)

Pi

subjected to convolution in x2 direction, while u2 has been subjected to convolution in x1 direction to obtain the coefficient

(x))2

(5)

a

Here, E(x) represents the local energy value at WT scale a and · the spatial average. As shown in Eq. (4), energy E(x)a has been normalized using the maximum of the average energy values obtained at all WT scales. The normalization suppresses the effects of local high frequency perturbations. The right-hand side of Eq. (5) brings out the spatial energy dominance at every scale and is therefore useful as a suitable criteria for identification of structures in a local region. This local energy measure (LEM) methodology is similar to evaluating LIM (Farge, 1992; Kulkarni et al., 2001; Camussi, 2002). However, the present analysis takes care to contain the effects of LIM shooting to high values in the higher scales by the improved normalization over all scales rather than within scales. (6) By studying the vorticity patterns in the data, it is possible to obtain the length scale distribution of structures. For this purpose, we show the advantage of using scalewise vorticity a (x) d,(a)

obtained by taking the curl of the velocity components u1

and

d,(a) as: u2

d,(a) a (x) = |ud,(a) × u2 | 1

(6)

and evaluating LEM of this data, i.e., steps (4) and (5). The scalewise LEM values from vorticity data [a (x)] may be used to identify the location, magnitude and shape of the structures from contour maps. It may be noted that while LEM of velocity data identifies structures, the LEM of vorticity data help in quantifying the properties of these structures. The area of the contours may be used as an indirect measure to evaluate the equivalent diameter and length scale distribution characteristics of the structures in the data. (7) The percentage energy content scalewise may be calculated by

d,(a) 2 ) x (ui × 100 Eˆ a = d,(a) 2 (u ) a x i

(7)

The above methodology has been used for identification and length scale distribution of velocity and vorticity information for each of the equipment.

(3)

b

4.2. Fractal spectrum of planar data using WTMM based method  (a)

vs. a. The scales and locating an inflection in the plot of Pi higher than the inflection scale has the noisy component. This thresholding scale may be identified in an automated fashion for denoising purposes. Denoised 2D data may therefore be obtained by carrying out an IWT after setting WT coefficients in the scales higher than the threshold scale to zero. Integrating back from d,(a)

the derivative domain then gives noise-reduced data ui that advantageously retains the true dynamical features of the system which would otherwise have been obscured. d,(a) (4) Denoised ui , obtained in the above manner, are collated in matrices for i = 1, 2 and are subjected to 1D WT similar to the methodology suggested by Camussi (2002), i.e., u1 has been

To estimate the multifractal scaling properties of heterogeneous spatiotemporal data, a methodology based on the WTMM method obtained by CWT has been used here. The WTMM method has the particular advantage that it does not require stationarity in data while simultaneously preserving hierarchial organization of singularity information during analysis. The methodology employed in the 2D spatial domain is as follows: First, a continuous wavelet basis function  is chosen to decompose the planar data (x) into elementary space-scale components by translations b and dilations a.    x−b 1 T a,b (x) = dx (8) (x) a a

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where  can be replaced by ui in case of velocity or by  for vorticity. The mother wavelet function  can be selected as a nth derivative of the Gaussian: dn −(1/2)x2 e . dxn

D1

42.4 %

41.5 %

5.8 %

D2

D1

17.5 %

2.1 %

S1

25

(11)

20 15 10 5 0 0 6 x1, (mm) Data

10.4 % D4

D3

S1

50 40 x2, (mm)

d(q) dq

D2

30

For certain values of q, exponent (q) has a special significance. For q > 0, Z(a, q) represents scaling of large fluctuations and strong singularities, while for q < 0, Z(a, q) reflects scaling of small fluctuations and weak singularities. Monofractal signals exhibit a linear (q) behavior, whereas, nonlinear behavior suggests that the signal is multifractal. Thus, from the (q), the fractal dimension spectrum [f ()] may be obtained by a Legendre transform as

30

(12)

20

where  is defined as the standard singularity strength or otherwise called as the local Holder exponent. The maximum value of the Holder exponent is the Hurst exponent that determines the type of multifractality. In the case when fmax is high, the 2D pattern exhibits large scale correlations, while if fmax is low, the features in the data are anti-correlated. Therefore, the availability of multifractal spectra f – for a variety of single and multiphase systems and operating under different conditions can be beneficially studied for comparing their flow characteristics.

10

=

D3

(9)

Using the Gaussian function, the deterministic trends up to (n − 1)th order may be eliminated by a local polynomial fit, thereby, extracting the masked singularity in the data. Further, only the local maxima of |T (a,b) (x)| are chosen and grouped to obtain the loci of maxima as a function of x across the scales a. These maxima lines represent the locations of singularities. A partition function Z(a, q) may be calculated scalewise as: q    ,b a Z(a, q) = (x)| ≈ a(q) (10) sup |T a  a

f () = min(q − (q))

D4

35

x2, (mm)

(n) (x) ≡

Data

5335

0 0 6 x1, (mm)

31.0 %

49.4 %

Fig. 2. Scalewise vorticity WT reconstructions for ACC based on 2D DWT (presented in Section 4). Data: vorticity data; S1: for smooth scale representing the average behavior; and D1, D2, D3, D4: for detailed scales; % numbers represent the relative energy content Eˆ a at the respective scales; (A) ACC-1; (B) ACC-2.

5. Results and discussion We first present the studies that focus on identifying the structures in the flow using DWT for different equipments. Subsequently, we discuss the statistical properties and length scale distributions of these structures. The WTMM methodology, using CWT, characterizes the multifractal behavior of different equipments and their usefulness in classification studies by PCA is then shown. 5.1. Flow structure characterization by DWT The results obtained by DWT for the individual equipment and the variety seen in their flow structures using the methodology presented in Section 4 is discussed here. 5.1.1. Annular centrifugal contactor (ACC) In the case of ACC, a 6 mm×60 mm contactor has been considered for the analysis of flow structures. WT of the snapshots of PIV data could be resolved in five wavelet scales in both axial and radial directions. Fig. 2 shows the scalewise reconstructed plots of the vorticity  after carrying out the DWT (Eq. (6)). Figs. 2A and B correspond to the scalewise reconstruction of the ACC operating at Re = 143, 000 and 314, 140, respectively. For Re = 143, 000 (Fig. 2A), scales D2 and D3 have most of the energy, Eˆ a (calculated by Eq. (7)), namely, 42.4% and 41.5%, respectively. Panels D2 and D3 are the respective scalewise reconstruction. For Re = 314, 140 (Fig. 2B), scales D3 and D4

are the most dominant with Eˆ a equal to 49.4% and 31%, respectively. Thus, it is observed that based on the relative energy content, the average dominant length scales are distributed in the range 6 × 10−4 to 1 × 10−3 m and 4 × 10−4 to 8 × 10−4 m, respectively, for the two situations. Note that the total energy content for the higher Re case is approximately four times higher. It is interesting to note that the lowest scale (i.e., smooth scale, S1) shows circulation cells having width and height nearly equal to that of the annular width (Fig. 2A). Also, at D4 we observe irregularly shaped, slender vortices near the wall and in the central region in the axial direction. Scales D2 and D3 are found to have the dominant energy content with smaller vortices lying in the periphery of larger structures identified in D1. We interestingly observe that the motion of smaller structures follow that of the larger. The shape of the structures in D2 and D3 are elongated and kidney like, while at D1, the shape is seen to be circular. Thus, for ACC, higher WT scales contribute in a larger measure for the turbulencegeneration. Snapshot studies of planar data over time clearly showed the motion of these structures in the radial direction followed by axial direction and lying along the periphery of mean circulation cells. These snapshot studies corroborate the observation discussed above on the formation of structures at the inner wall. The analysis of these patterns will therefore be useful in estimating the characteristics of structures in terms of length scale distribution and scalewise interactions.

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Data

D5

D4

42.6 %

38.6 %

34.0 %

41.6 %

D3

D2

D1

3.1 %

1.1 %

18.4 %

4.5 %

1.5 %

15.8 %

4.8 %

1.7 %

x2, (m)

0.15 0.1 0.05

0.05 0.1 x1, (m)

0.15

14.5 %

x2, (m)

0.15 0.1 0.05

0.05 0.1 x1, (m)

0.15

x2, (m)

0.15 0.1 0.05

0.05 0.1 x1, (m)

0.15

37.6 %

40.0 %

Fig. 3. Scalewise vorticity WT reconstructions for JLR based on 2D DWT (presented in Section 4). Data: vorticity data; S1: for smooth scale representing the average behavior; and D1, D2, D3, D4: for detailed scales; % numbers represent the relative energy content Eˆ a at the respective scales; (A) JLR-1; (B) JLR-2; (C) JLR-3.

5.1.2. Jet loop reactor (JLR) In the case of the JLR, a planar section of size 0.15 m × 0.15 m has been considered and studies have been made over two sections that cover the region below the jet. Thus, the maximum size of length scale considered is that of the radius of the tank, i.e., 0.15 m while, the minimum size was set to be 0.0025 m for all the flow conditions. Figs. 3A–C correspond to Re = 17, 000, 26,000 and 33,000, respectively. In the case of Re = 17, 000 (Fig. 3A), scale D5 is the most energetic (42.6%) and this reduces at lower scales. On the other hand, for Re = 26, 000 (Fig. 3B), D5 captures 21.5% of the energy, while D4 shows maximum energy content, i.e., 41.6%. The energy content reduces to 18.4% and 6.8% for D2 and D1, respectively. Similarly, for the case of Re = 33, 000, the maximum energy gets shifted to higher scales, i.e., D3 and D4. Thus, for a variation in flow rate and nozzle diameter, smaller size structures are seen to become more energetic. When the scalewise shapes of the structures are compared, at D1, large vortex tubes are seen along the periphery of the shear region with large circulation loops visible. At D2, leading edge vortices are seen in the shear region. Co-existence of smaller structures is seen in D2–D4. For variations in the flow rate and using a nozzle diameter of 20 mm (Figs. 3A and B), turbulence in the bulk region gets affected and this is clearly seen by the presence of different types of structures. On reducing the nozzle diameter to 15 mm (Fig. 3C), a further increase in the size of structures is seen. In fact, the effects of increasing the power supplied to the system is seen by the structures penetrating deeper into the bulk region. Similar qualitative characteristics have been observed in the near wall region (x1 = 0.12–0.15 m) in our analysis. The properties of the structures in the shear region and bulk would be important for studying fast

homogeneous reactions as well as mixing sensitive systems. On the other hand, the surface renewal and age distribution of flow structures at the wall becomes important for the understanding the behavior of heat exchange from the system to the wall and vice-versa. 5.1.3. Ultrasound reactor (USR) For the case of USR, planar section of size 0.15 m × 0.08 m has been considered for the study of flow structures based on the power input conditions. Therefore, the maximum size of the length scale considered is of the diameter of tank, viz., 0.15 m. The minimum size was set to be 2.5 × 10−3 m for all the conditions of power input. Figs. 4A and B correspond to the variation in the power supplied to the system, namely, 15 and 35 W/m3 , respectively. In the case of 15 W/m3 (Fig. 4A), scales D4 and D5 contain 40% and 38% of the energy (Eˆ a ), respectively, and these correspond to high energy content. On the other hand, scales D2 and D3 contribute only 12% and 8%, respectively. For power of 15 W/m3 (Fig. 4B), a similar distribution is observed except that D2 now contributes 20% of the energy. The overall scalewise behavior in both the cases (Figs. 4A and B) is similar except for the variation in the shape of structures. At scale D2, two high intensity counter-rotating vortices near the source of ultrasound are observed. For a higher power input, the length of these vortices become elongated. Similarly at scale D3, the use of higher power is seen to distribute the structures more in the bulk region. Sharp energy structures of small sizes are observed in scales D4 and D5 below the horn. The sharp gradients observed in this region are due to high frequency contraction and rarefaction of energy packets. Further, small scale irregular structures are also observed in the near wall region and they are seen to follow an upward motion.

x2, (m)

x2, (m)

S.S. Deshpande et al. / Chemical Engineering Science 63 (2008) 5330 -- 5346

Data

D5

D4

D3

0.05 0.1 x1, (m)

40.0 %

38.3 %

Data

D5

D4

D3

30.1 %

18.0 %

5337

D2

D1

0.14 0.12 0.1 0.08 0.06 0.04 0.02 12.4 %

8.5 % D2

0.8 % D1

0.14 0.12 0.1 0.08 0.06 0.04 0.02 0.05 0.1 x1, (m)

30.1 %

20.9 %

0.9 %

Fig. 4. Scalewise vorticity WT reconstructions for USR based on 2D DWT (presented in Section 4). Data: vorticity data; S1: for smooth scale representing the average behavior; and D1, D2, D3, D4: for detailed scales; % numbers represent the relative energy content Eˆ a at the respective scales; (A) USR-1; (B) USR-2.

In Figs. 4A and B, the dominant structures are seen to have asymmetric patterns due to the oscillations present in the plume. Thus, intricate dynamical features in the data can be identified and the effects of parameters of operation in changing the hydrodynamical behavior are studied quantitatively. 5.1.4. Channel flow (CHA) In the case of CHA, the behavior in the boundary layer region has been studied up to 0.25 m (y+ < 50) in the wall normal direction, and 2.9 m in the longitudinal direction. The scalewise reconstruction of velocity and vorticity patterns have been studied. The WT of the snapshots of LES predictions could be analyzed in five wavelet scales in both axial and wall normal directions for the data resolution. Thus, the range of scales investigated have been in the range from 0.008 to 0.25 m in wall normal direction and 0.09–2.9 m in longitudinal direction. Fig. 5 shows the scalewise reconstructed plots of vorticity  as obtained by Eq. (6). Figs. 5A and B correspond to Re = 5000 and 18,000, respectively. In the case of Re = 5000 (Fig. 5A), scale D1 is the most energy containing scale (50.7%) and the energy content reduces toward D5. On the other hand, for Re = 18, 000 (Fig. 5B), D1 shows 21.5% of the energy, while D2 shows maximum energy content corresponding to 43.6%. Towards higher WT scales, the energy content falls to 28.6% and 6.8% for D3 and D4, respectively. Thus, the corresponding average length scales for the two Re conditions are 0.12 and 0.06 m, respectively. Fig. 5A for scale D1 suggests that the strong vorticity areas are away from the wall. A snapshot study in time of the data also showed that the structures at this scale do not reach the wall. On the other hand, D2 onwards, high vorticity regions are observed even in the near vicinity of the wall. Thus, it may be concluded that as the Re increases, the velocity gradient increases and hence the possibility of structures penetrating into the film region also increases. The corresponding snapshot study showed the formation of structures at various scales because of the velocity patterns that are developed in the boundary layer. Streaks of various length scales are seen ascending at an inclination angle of 5◦ –15◦ . Most of the streaks follow each other, and the interactions between structures are seen to be minimal. Also, coalescence of consecutive structures is not observed. Most of the smaller streaks end their life

within the turbulent boundary layer itself. Few of the larger streaks were seen either moving out of the boundary layer or breaking down within the y+ < 50 region. Such phenomena are in-line with the observations in the literature. Thus, it is verified that the identification and characterization of structures from data are systematically possible using the WT methodology presented here and it can be extended to study the behavior of other complex chemical engineering systems. 5.1.5. Stirred tank reactor (STR) In case of the STR, a planar section of size 0.3 m × 0.3 m has been considered for the study of flow structures. Thus, the maximum size of length scale considered is of the size of diameter of tank, i.e., 0.3 m. The minimum size is set to be 0.0025 m for all the geometrical conditions considered. Figs. 6A–C correspond to the variation in the impeller type, namely, DT, PBT with inclination of 45◦ (PBTD45) and HF, respectively. In the case of DT (Fig. 6A), scales D1 and D2 contain 24% and 14% energy while scale D3 has the maximum energy with 33%. For PBTD45, scale D2 shows maximum of the energy, Eˆ a ≈ 35% (Fig. 6B). Similarly, for HF (Fig. 6C), energy is distributed prominently in scales D2 ≈ 30% and D3 ≈ 25%. The spatial distribution of dominant structures is also found to vary for each of the cases based on the type of the impeller. In the case of DT, trailing vortices are observed at scale D1 and D2. Also, the structures are seen to collapse toward the side wall in the impeller plane. These structures lose energy and move upward and downward. At the shaft of the impeller, a small structure is seen at scales D2–D4. In the case of PBTD45 (Fig. 6B), a broad jet is formed inclined at an angle ≈ 35◦ from the vertical axis and in the lower part of the tank because of the thrust from the impeller. These structures are found to deviate in the range 30◦ –50◦ when snapshots are studied in time. The structures are carried upward in the near wall region and form a large circulation loop. Small reverse loops are formed below the impeller with formation of small scale structures. Additionally, at scale D2, trailing vortices are observed in the impeller blade region. In the case of HF (Fig. 6C), at scale D2 a similar structure is observed but with an angle of approximately 25◦ . The energetic structures are mainly observed in the portion below the impeller and these are carried upward in

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Data x2, (m)

x2, (m)

Data 0.2 0.1 0 0

0.2 0.1 0

2.9

0

2.9375 x1, (m)

x1, (m) D4

11.4 %

D4

6.8 %

D3

18.1 %

D3

28.1 %

D2

19.8 %

D2

43.6 %

D1

50.7 %

D1

21.5 %

S1

S1

Fig. 5. Scalewise vorticity WT reconstructions for CHA based on 2D DWT (presented in Section 4). Data: vorticity data; S1: for smooth scale representing the average behavior; and D1, D2, D3, D4: for detailed scales; % numbers represent the relative energy content Eˆ a at the respective scales; (A) CHA-1; (B) CHA-2.

Data

D6

0.2 x1, (m)

9.4 %

D5

D4

D3

15.6 %

33.0 %

14.5 %

D2

D1

21.5 %

6.0 %

x2, (m)

0.3 0.2 0.1 0 0

x2, (m)

0.3

Sc1 12%

Sc2 14.8%

Sc3 19.2%

SD4 35.1%

Sc5 15%

Sc6 3.9%

12.0 %

14.8 %

19.2 %

35.1 %

15.0 %

6.3 %

Sc1 12.7%

Sc2 14.6%

Sc3 25.4%

Sc4 30.5%

Sc5 12.4%

Sc6 4.4%

12.7 %

14.6 %

25.4 %

30.5 %

12.4 %

4.4 %

0.2 0.1 0 0

0.2 x1, (m)

x2, (m)

0.3 0.2 0.1 0 0

0.2 x1, (m)

Fig. 6. Scalewise vorticity WT reconstructions for STR based on 2D DWT (presented in Section 4). Data: vorticity data; S1: for smooth scale representing the average behavior; and D1, D2, D3, D4: for detailed scales; % numbers represent the relative energy content Eˆ a at the respective scales; (A) STR-1; (B) STR-2; (C) STR-3.

S.S. Deshpande et al. / Chemical Engineering Science 63 (2008) 5330 -- 5346

the near wall region. Unlike DT and PBTD, no dominant structure is observed at the top surface near the shaft in HF.

D5

D4

D3

D2

D1

0.15 0 x1, (m)

8.5 %

28.9 %

44.2 %

17.8 %

0.5 %

0.15 x1, (m)

10.7 %

43.1 %

31.3 %

12.1 %

2.8 %

0.15 0 x1, (m)

11.8 %

42.3 %

31.5 %

12.7 %

1.8 %

1

0.8

x2, (m)

0.6

0.4

0.2

0

1

0.8

0.6 x2, (m)

5.1.6. Bubble column reactor (BCR) In the case of BCR, a planar section of size 0.15 m × 1 m has been considered for the study of flow structures. WT of the snapshots of LES data have been analyzed in five wavelet scales in both the axial and radial directions. Thus, the length scales of structures lie in the range from 0.003 to 0.15 m corresponding to the radial direction, while in the axial direction it lies between 0.031 and 1 m. Figs. 7A–C, correspond to the variation in the spargers type, namely, single hole, sieve plate and sintered plate, respectively. In the case of single hole sparger (Fig. 7A), scale D3 contains 44.2% of the energy while scales D2 and D4 contain 17.8% and 28.9% energy, respectively. For sieve plate (Fig. 7B) and sintered plate sparger (Fig. 7C), scale D4 shows maximum, i.e., 43% of the energy while scales D2 and D3 contribute Eˆ a ≈ 12% and 31%. Though the relative energy content is similar for sieve and sintered plate spargers, the spatial distribution of dominant structures is found to vary in each of the cases. For the single hole sparger (Fig. 7A), elongated counter-rotating trailing vortices are observed in scales D1 and D2 just above the sparger because of the plume formation at the gas entry. In scales D3 and D4, plume developing regions corresponding to the gas passage are observed in the snapshots. These structures were found to oscillate in time. In the case of sieve plate sparger (Fig. 7B) structures are seen over the entire tank in scales D2 and D3. The nature of these vortices is found to be elongated with irregularity in shape. Interaction of structures is observed at various locations and are seen in a snapshot animation over time. Small structures found in scale D5 are mainly found to dominant near the wall. In the case of sintered plate sparger (Fig. 7C), symmetric structures of larger size are observed at regular intervals in the axial direction. The structures are elongated and the dominant structures are observed at the near wall region for scale D3. This is mainly because of the unidirectional motion of bubbles of equal size where significant gradients occur in the near wall region.

Data

5339

0.4

0.2

0 0

5.2. Comparison of length scale distribution 1

0.8

0.6 x2, (m)

The length scale distribution has been characterized by statistical parameters such as the averaged length scale corresponding towards dominant energy containing scales (lequi ), a mean value of length scale based on over all length scale distribution (lmean ), the standard deviation from the mean (l ), the skewness (Sl ) and the kurtosis (Kl ). The statistical parameters calculated for the different equipment are given in Table 3. The skewness (Sl ) suggests the presence of a symmetry in the length scale distribution. A higher positive value of Sl suggests that the length scale distribution has higher number density of small scale structures. The Kl value measures the nonuniformity in the length scale distribution whereby a positive Kl indicates a peaked distribution while negative Kl indicates a flat distribution. As the operating conditions and geometric conditions are changed, the flow pattern and the length scale distribution varies and this can be captured by lequi , lmean , l , Sl and Kl . In the case of ACC, large changes have been observed in length scale statistics by varying the rotation speed from 10 (ACC-1) to 20 rps (ACC-2). The lequi /Dequi has been reduced from 0.167 to 0.1 and the corresponding dominant energy scale contributes 49% energy as compared to 42% in ACC-1 (Table 3). Further, an increase in l values by 110% occurs while, Sl and Kl also shows an increase by 50% and 70%, respectively. In the case of jet, from JLR-1 through JLR-3 flow has been raised by 90%. This is clearly reflected in terms of reduction in lequi /Dequi by approximately half. However, the statistical behavior of normalized length scale distribution is observed to be similar for both cases. This suggests that the length scale distribution patterns are similar for both JLR and they lie in the same regime of operation. USR-1

0.4

0.2

0

Fig. 7. Scalewise vorticity WT reconstructions for BCR based on 2D DWT (presented in Section 4). Data: vorticity data; S1: for smooth scale representing the average behavior; and D1, D2, D3, D4: for detailed scales; % numbers represent the relative energy content Eˆ a at the respective scales; (A) BCR-1; (B) BCR-2; (C) BCR-3.

and USR-2 show different trends in the statistical parameters on varying the power input. The lequi /Dequi is observed to remain same while, both Sl as well as Kl are reduced by 40% and 15%,respectively.

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Table 3 Statistical parameters evaluated based on length scale distribution Equipment

mix,expt

Re

Pin

lequi

Dequi

lequi /Dequi

max Eˆ a

Rl

lmean

l

Sl

Kl

ACC-1 ACC-2 JLR-1 JLR-2 JLR-3 USR-1 USR-2 CHA-1 CHA-2 STR-1 STR-2 STR-3 BCR-1 BCR-2 BCR-3

9 7 32 20 13 – 17 – – 24 18 15 22 32 40

143,000 314,140 17,000 26,000 35,000 5700 16,000 5000 18,000 140,000 140,000 140,000 37,500 31,500 12,000

700 7000 4 12 30 15000 35000 1867 2160 225 90 13 0.19 0.19 0.19

0.001 0.0006 0.017 0.0098 0.003 0.008 0.01 0.1 0.05 0.015 0.03 0.023 0.015 0.009 0.02

0.006 0.006 0.3 0.3 0.3 0.14 0.14 0.25 0.25 0.3 0.3 0.3 0.15 0.15 0.15

0.17 0.1 0.06 0.03 0.01 0.06 0.07 0.4 0.2 0.05 0.1 0.08 0.1 0.06 0.13

0.42 0.49 0.4 0.38 0.38 0.39 0.3 0.5 0.43 0.33 0.35 0.3 0.44 0.43 0.42

0.44 0.72 0.95 0.94 0.95 0.87 0.65 0.31 0.44 0.84 0.85 0.86 0.39 0.42 0.45

0.16 0.12 0.04 0.04 0.04 0.08 0.12 0.32 0.24 0.08 0.08 0.08 0.24 0.2 0.16

0.01 0.02 0.036 0.04 0.04 0.03 0.02 0.0051 0.0087 0.027 0.03 0.03 0.01 0.01 0.01

3.28 4.37 4.69 4.68 4.69 4.65 4.11 2.51 3.52 3.93 4.26 4.65 2.76 2.9 3.29

13.15 20.92 23.01 22.99 23.01 22.75 19.02 9.41 15.47 18.60 20.82 22.78 11.26 12.5 13.1

50

15 %

40 Predicted Mixing Time (θmix), s

The CHA shows significant deviation in the length scale distribution pattern for an increase in Re of 3.5 times (i.e., from Re = 5000 to 18,000), where the lequi /Dequi is decreased by half and both Sl , Kl increase by approximately 66%. In the case of STR, for all the three impeller designs (STR-1, STR-2 and STR-3), a significant deviation is observed in Sl and Kl with higher values observed for HF (STR-3). It is seen that the axial flow impellers, because of the lower values of turbulence intensity, have more number of small scale structures than the radial flow impellers. In bubble column, the changes in flow pattern occurring due to the type of sparger design can be nicely captured by means of the statistics in terms of their length scale distribution. For example, as compared to the BCR-1 (single hole sparger), both Sl and Kl values obtained for BCR-2 and BCR-3 with sintered plate are higher by ≈ 10% and 15%, respectively. The increase in the value of Sl and Kl is due to the higher number density of small scale structures in a sintered sparger as compared to the sieve and single hole spargers. Thus, the statistics of length scale distribution acts as a signature for each equipment and may be used to judge the role of transport phenomena in the equipment.

-15 %

30

20

10

0 0

5.3. Mixing time correlation based on MRA statistics Mixing time is significantly dependent on the hydrodynamics of each equipment and the structures developed within it. The literature shows various correlations for mixing time that have been developed for the individual equipment (Yianneskis, 1991; Patwardhan, 2002). But, these correlations are mainly based on the Re, P/V and geometrical ratios involving parameters such as H, D, HN , etc., and are specific to each type of equipment. It is now possible to develop a unified relationship characterizing the extent of mixing in different types of chemical process equipment based on the patterns in the flow structures that are formed. In developing the relationship, we include the knowledge of reactor geometry and operating conditions, and also information with respect to the flow structures that are developed in the system via the statistical parameters (Section 5.2) l , Sl , Kl , dominant energy quantifiers lequi /Dequi and scalewise energy ratio max(Eˆa ). A generalized correlation for mixing times can therefore be developed and is given by the following equation:

mix = 181.25(Re)−0.31 (l )−0.095 (Sl )0.17 (Kl )1.13 ×

(Pin )−0.095



 lequi 0.37 E [max(Eˆa )]−0.17 l Dequi Ep

(13)

For developing the above correlation, the mixing time in the equipment have been obtained from available experimental data or verified through CFD predictions (Table 3) (Kumar et al., 2007;

10

20 30 40 Experimental Mixing Time (θmix), s

50

Fig. 8. Parity plot of predicted vs. experimental mixing times using the developed correlation (Eq. (13)).

Patwardhan, 2002; Tabib et al., 2008; Mathpati and Joshi, 2007). The above correlation could fit the data within ±10% accuracy as shown in the parity plot in Fig. 8. This correlation does not include the channel flow as it is a continuous system. It suggests that the mixing time strongly depends also on the statistical parameters obtained from the length scale distribution using the procedure employed in Section 4. Note that the single correlation (Eq. (13)) is valid for all the five types of equipments and flow parameters studied. Additionally, its basis depends on the flow structures and the dominant energy content identified using the experimentally monitored velocity data ui . There, however, exists a requirement for improving the calculation of mixing time from data and this issue needs to be separately addressed. The correlation derived here opens up the possibility of improving the calculations of mixing times by incorporating multiresolution processing techniques as is used here. 5.4. Evaluation of fractal spectrum Before describing the multifractal behavior of all the equipment, we discuss it for an illustrative case of the JLR. Fig. 9 (JLR-1–JLR-3) compare the results obtained for three different cases discussed

S.S. Deshpande et al. / Chemical Engineering Science 63 (2008) 5330 -- 5346

5341

Fig. 9. The WTMM and fractal spectrum analysis for jet loop reactor. (JLR-1) Re = 17, 000; (JLR-2) Re = 35, 000; (JLR-3) Re = 33, 000; (A) Vorticity data; (B) The CWT at scale indices a1 = 2, a2 = 12, a3 = 22 and a4 = 32; (C) WTMM loci lines overlapped with the CWT slices at the scale indices a1 − a4, (D) Singularity fractal spectra

earlier in Section 5.1.2. The 2D planar data for the three situations is shown in column A (Fig. 9; JLR-1,A–JLR-3,A), while the corresponding scalewise variation of flow structures in the snapshot are shown in Fig. 9 (JLR-1,B–JLR-3,B). The relative comparison among the three cases show the formation of strong singularity patterns in the jet shear region. The strength of the singularity is observed to be highest for case JLR-2 and is followed by cases JLR-3 and JLR-1. Cases JLR1 and JLR-3 show smoother patterns in comparison to JLR-2, which exhibits high wavenumber scale patterns across a broader area of the image. This result clearly shows the effects of increase in Re on instantaneous flow behavior. More information, in terms of strong and weak singularities and their relative characterization may be obtained by evaluating the WTMM of the data (Fig. 9; JLR-1,C–JLR3,C). The maxima lines depict the loci of local discontinuities due to flow structures occurring at various scales. These loci lines have been overlapped with the scalewise CWT coefficients and show the locations of maxima intersections on respective contour slices. These lines show the singularity behavior, scalewise at each location and bring out the presence of small structures within larger ones. The nature of their evolution and interactions in terms of space filling nature is also seen. In the shear region, two parallel loci lines are observed that passes through the centroid of the flow structures. Both the lines are parallel to each other and indicate that two similarly sized structures follow the mean flow directions. The curvature of the lines are such that the higher wavenumber scales are upstream in the flow direction. Also, the multiple branching of singularity loci lines toward higher scales show the complex break-up phenomena. Therefore, it is suggested that the origin of loci lines and their span across the scales may be beneficially used to study the space-scale correlations and flow structure evolution. Quantification may be carried out by obtaining the singularity fractal spectrum of the data (see Fig. 9; JLR-1,D–JLR-3,D). This is important because by dimensionality reduction, relevant information is extracted from the 2D planar data.

In Fig. 9 (JLR-1,D–JLR-3,D), the corresponding fractal spectra for three cases show characteristic multifractal hump shaped patterns with maxima at fmax . The value of fmax (corresponding to the abscissa at fmax ) is related to the Hurst exponent of the 2D data. This maximum value shows the most probable self-similar pattern, with the reduced magnitude indicating the variation in the break-up distribution of large and small scale structures. The complete set of  values are the Holder exponents and their spread brings out the multifractal dimensions in the flow. It is observed that the flow becomes correlated and persistent toward the higher  values, while it becomes anti-persistent for  < 0.5. Thus, the abscissa of the fractal spectrum can be directly related to the length scale distribution present in instantaneous 2D flow data. From the instantaneous multifractal spectra Fig. 9 (JLR-1,D–JLR-3,D), it can be observed that for a change in flow rate by a factor of 2, the fmax is reduced from 1.5 to 1.1. On the other hand, in case of JLR-3 where the nozzle diameter is reduced by 25%, the value of fmax remains around 1.5, while fmax is reduced considerably from 1.75 to 1.45. The value of fmax is found to be lowest in JLR-2, i.e., 1.35. The singularity multifractal spectrum varies over a finite range of  with corresponding  and max values. These values suggest, respectively, the limits of small and large energy containing structures. It indicates that the large scale structures breakdown in a particular manner as seen by the trend across the wavelet scales. The minimum value, min characterizes finer structures, while maximum value, max represents coarser structures. The relative comparison of Fig. 9 (JLR-1,D–JLR-3,D) show that min values lie in the range of 0.9 for all the three cases. On the other hand, wide variation is observed in the max values, viz, 1.9, 2.2 and 2.9, respectively. Thus, it can be seen that the spread of the fractal spectrum is broadened for higher Re and suggests an increase in the complexity in the flow behavior. Fractal spectrum can also be analyzed based on other factors such as area under the multifractal curve (Af ). A monofractal signal shows

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Table 4 Parameters evaluated from fractal spectrum Equipment

Rf

mean



S

K

Af

min

max

fmax

fmax

ACC-1 ACC-2 JLR-1 JLR-2 JLR-3 USR-1 USR-2 CHA-1 CHA-2 STR-1 STR-2 STR-3 BCR-1 BCR-2 BCR-3

0.65 0.54 1.21 1.14 1.25 1.17 1.10 1.03 0.85 1.16 1.16 1.16 0.68 0.70 0.65

0.82 1.85 1.08 0.74 0.97 0.43 0.46 2.51 2.17 1.60 1.78 1.80 2.12 1.62 1.97

0.045 0.030 0.143 0.112 0.150 0.120 0.111 0.091 0.053 0.101 0.122 0.120 0.041 0.042 0.038

−0.36 −0.78 −0.25 −0.72 −0.37 −0.51 −0.48 −0.94 −1.50 −0.98 −0.65 −0.70 −0.56 −0.76 −0.69

1.96 2.43 1.82 2.46 1.91 2.16 2.15 2.86 4.23 3.04 2.21 2.29 2.16 2.41 2.27

4889.4 709.4 1184.5 4577.9 1113.8 2224.7 6824.0 2263.4 4475.5 5374.1 2451.9 2463.8 1089.9 2342.3 1611.4

0.37 0.46 0.06 0.07 0.06 0.34 0.17 1.20 1.55 0.81 0.63 0.51 0.90 0.88 1.08

0.91 3.08 2.43 1.00 2.03 0.63 0.67 2.81 2.33 1.70 1.91 1.92 2.48 1.74 2.06

0.81 0.70 1.47 1.29 1.52 1.32 1.29 1.16 0.97 1.30 1.29 1.30 0.80 0.82 0.77

0.72 1.09 0.25 0.45 0.23 0.17 0.24 2.18 1.97 1.53 1.60 1.62 1.65 1.44 1.79

a single point and this means that the pattern of structures in the signal is self-similar at all the scales. However, in case of real systems, a distribution in points in the f – curve is observed and this is related to the evolution and break-up of structures. A higher value in the Af indicates a wider distribution of scalewise interaction in flow structures. A comparison of Af show higher values for JLR-2 compared to JLR-1. Case JLR-3 shows stronger variation when compared to JLR-1 and this brings out the role of the reactor diameter in the observed dynamics. A comparison of the mean of the fractal spectrum (mean ) and fmax may be investigated to infer about the dominance of strong and weak singularities. It can be observed that both the wings of the spectrum are equally spaced for JLR-1, while the spread in the case of JLR-2 and JLR-3 is found to be higher. Skewness (Sf ) is a quantification of bias of the fractal spectrum toward a particular side of mean . Thus, Sf shows higher values when the spectrum is inclined towards the higher values of  and vice-versa. The flatness factor (Kf ) shows the presence of spiky behavior. Therefore, Kf shows higher values when the multifractal spectrum becomes sharper in distribution. In the case of JLR, both Sf and Kf show a small increase in magnitude for an increase in flow rate, but change sharply for a reduction in diameter. From studying the snapshots over time, it is observed that the fractal spectra show variation depending on the dominant structures and their interactions. When low energy structures are present in the snapshot WT coefficients change sharply for negative q, while high energy structures show strength at positive q values. The range of q within which the fractal spectra show positive magnitude depicts the nonlinear characteristics of the flow. The comparison of the fractal spectra variations for three JLR cases over time clearly show the appearance of various features and the role of different size structures as discussed above. The mean fractal spectra based on the temporal variation has been found to be in accordance with the length scale distribution. This may be verified by comparing the statistical parameters given in Tables 3 and 4 for JLR. Thus, for a change in operating conditions and geometrical conditions, the flow structure characteristics show a change in the fractal spectrum, which in turn may be captured by evaluating the statistical parameters, namely, fmax , fmax , max , min , mean , Sf and Kf . Based on the instantaneous behavior, the trends in the fractal spectra each of the equipment have been studied. The fractal spectra behavior over the snapshots has been averaged to find the overall behavior of singularities in the flow. In the case of ACC, Fig. 10A shows the fractal spectra obtained from vorticity data (). The CWT has been obtained for the snapshot studied earlier and shown in Fig. 2. The corresponding fractal spectra have been studied for both ACC-1 and ACC-2 (Fig. 10A). The instan-

Fig. 10. Fractal spectra for the six equipments. (A) ACC; —:ACC-1, − −:ACC-2; (B) JLR; —:JLR-1, − −:JLR-2,− − −:JLR-3; (C) USR; —:USR-1, − −:USR-2; (D) CHA; —:CHA-1, − −:CHA-2; (E) STR; —:STR-1, − −:STR-2, − − −:STR-3; (F) BCR —:BCR-1, − −:BCR-2, − − −:BCR-3.

taneous spectra show fmax equal to 1.1 and 0.95, with the corresponding values of fmax equal to 1 and 0.75, respectively. Thus, with an increase in the flow, fmax has been shifted to lower values indicating the presence of increased instability. The other parameters,

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namely, min , max , Af show that the finer structures have high energy content in comparison to the coarser. In Fig. 10A, the right side of the curve is broader for ACC-1, while for the case of ACC-2, it is sharper. There is a corresponding increase in the negative magnitude of skewness (Sf ) and flatness factor (Kf ). Fig. 10C shows the fractal spectrum for the variation in input power for the USR. The instantaneous spectra show the maxima, fmax equal to 1.1 in both the cases, with value of fmax being 1.55 and 1.7, respectively. Further, other parameters such as min , max and Af show lower values in case of USR-1 operating at lower power. Also, the trend shows that fractal spectrum for USR-1 is broad in the left wing, while opposite behavior is observed in USR-2 with right wing becoming broad. Thus, the higher values of  in USR-2 suggest the presence of coarser structures by the convective forces as against USR-1. In the case of CHA, Fig. 10D show the fractal spectra in CHA-1 and CHA-2 (Table 4). The instantaneous spectra show the fmax = 1.9 in both the cases. For CHA-1, the right side of the curve is broader while for CHA-2 it is sharper.This variation has been reflected in statistical properties Sf and Kf of the curve. In the case of STR, Fig. 10E shows the plot of fractal spectrum for DT (STR-1), PBTD (STR-2) and HF (STR-3). The maxima values, fmax of 1.5, 1.7 and 1.8 are seen based on the instantaneous fractal spectra. The behavior of spectrum is inclined towards higher values of  for HF showing relatively uniform behavior. The Sf shows highest negative value in DT indicating the instability to be highest. Also, Kf exhibits the highest value in the case of DT. In the case of BCR, the fractal spectra have been plotted in Fig. 10F. The instantaneous fractal spectra show the maxima fmax equal to 1.5, 1.3 and 1.7 for cases BCR-1, BCR-2 and BCR-3, respectively. The magnitude of fmax was found to be nearly 0.8 in all the three cases. The min and max values showed variations each of the cases as seen in Fig. 10F. Further, BCR-1 and BCR-2 show similar spread in both the wings, while in the case of BCR-3 the left side of the curve is broader. The variation is also seen from the higher values of Sf and Kf for curve BCR-2. 5.5. Regime map study The analysis has been further extended toward identifying classes among the equipment and to interpret their behavior with respect to the hydrodynamic parameters. In order to visualize the sensitivity seen in the fractal spectra, a methodology based on the PCA has been used. In the previous section, CWT based methodology has been applied to reduce the dimensionality of the system for analysis and interpretation. In this section, we analyze the complete database of the f vs.  curves and search for correlated behavior and vice versa. For this purpose, the fractal spectra have been interpolated suitably between min and max to obtain f at regularly spaced  values for the entire database. PCA uses a correlation matrix C of multifractal spectrum data f as

Clm =

n 1  f (i, l)f (i, m), M

l, m = 1, . . . , M.

(14)

i=1

The entire data set have been subjected to the PCA using singular value decomposition (SVD), namely,  = CW SW

(15)

 being the required set of eigenfunctions and  where, W S the corresponding eigenvalues. The principal component vectors zlm ∈ Z can be obtained by √ zlm = wlm sm ,

l, m = 1, . . . , M

(16)

Fig. 11. Regime map based on PCA from the fractal spectra database for the six equipments and operating conditions as shown in Table 1. Two hundred 2D snapshots from each equipment were collated to form the database for PCA; (A) zl2 vs. zl1 ; (B) zl3 vs. zl2 ; ×:ACC; ◦:JLR; +:USR; :CHA (channel); :STR; :BCR.

 and sm ∈  with wlm ∈ W S. The number of principal components, say p, required for no redundant information is obtained by considering p columns of Z such that a cost function EPCA shows minimal meansquare error while maximizing the variance for each component. Carrying out the PCA for the multifractal spectra database obtained for the cases considered in Table 1 reveals that three eigenmodes, i.e., p = 3 were sufficient to capture 96% of the total information in the data. The results of PCA have been plotted in Fig. 11. Fig. 11A shows the variation of first principal component (zl1 ) vs. second principal component (zl2 ) using snapshots of each equipment at regular time intervals. Similarly, Fig. 11B shows the variation of the second principal component (zl2 ) vs. the third (zl3 ). A clear separation in the regions for each equipment is observed, although, overlapping regions are also seen. A classification can be made based on magnitudes of the principal component co-ordinates, namely, (zl1 ), (zl2 ) and (zl3 ) (Table 5). ACC shows high value of zl1 , while value of zl2 and zl3 is low. The position of the co-ordinate is therefore at extreme right in Fig 11A, while in Fig. 11B, it is shifted to a region near the origin. In the case of STR, it is observed that the co-ordinates lie

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Table 5 Principal component analysis: a qualitative description

can help in obtaining conditions for operating different equipment in desirable regimes with characteristic flow behavior.

Case

zl1

zl2

zl3

ACC-1 ACC-2 JLR-1 JLR-2 JLR-3 USR-1 USR-2 CHA-1 CHA-2 STR-1 STR-2 STR-3 BCR-1 BCR-2 BCR-3

High High Medium Medium High Low Low Medium Medium High High High High High High

Low Medium High High High High High Medium Medium Low Low Low Medium Medium Low

Low Low Low Low Low Low Medium High High High High Medium Low Low High

near the ACC in Fig. 11A, while they are separated in Fig. 11B due to a high value of zl3 . For both BCR and STR zl1 is found to be on the higher side. However, there is a marked difference in the values of zl2 with zl2 being higher for BCR (Fig. 11A). A more detailed comparison taking Figs. 11A and B together shows that the BCR-2 (sieve plate sparger) is found to overlap with the STR, while BCR-1 (single hole sparger) and BCR-3 (sintered sparger) shows a degree of overlap in the PCA co-ordinates with channel and JLR. JLR shows medium zl1 value and high zl2 value (Fig. 11A). With increase in flow rate its PCA co-ordinates move toward the co-ordinates of the BCR. The regime of BCR is found to lie in between ACC and JLR. Co-ordinates of ACC-2 are found to overlap in the regime of JLR in Fig. 11A. On the other hand, JLR shows a distinct regime lying in the right lower corner (Fig. 11B). Also, it can be observed that, zl1 of STR and JLR are found to be close to each other (Fig. 11A). On the contrary, zl2 and zl3 are found to lie in diagonally opposite regimes. In the case of CHA, Fig. 11A shows its co-ordinates to lie in the diagonal region below the zone of JLR and beside BCR. However, Fig. 11B shows that JLR and BCR regimes are separated from each other. In the case of USR, a distinct regime is seen when compared to the other equipment (Fig. 11A). Similarly, in Fig. 11B USR shows the regime formation along the diagonal line with zl2 and zl3 values equal. In the case of higher power (USR-2), it is also seen that there is some overlap between JLR and USR. Thus, it can be concluded that, the dynamic study in reduced dimension, i.e., in the fractal spectra domain, gives valuable information for the classification of equipment flow behavior based on pattern of structures and their scalewise interactions by PCA. The first component, zl1 clearly classifies the behavior with respect to dominance of large scale structures. Therefore, CHA, BCR, STR and JLR show similar values while USR shows very low values of zl1 . Another observation is that ACC and STR have same value of zl1 . This may be due to similar formation of large scale structures and arising because of dominant tangential motion in both cases. These observations are corroborate the findings discussed in Section 4.1. On the other hand, BCR, JLR, CHA and USR, the major reason for similar structure formation is mainly because of the secondary vortex stretching in the shear region by the unidirectional motion of flow. In fact, the behavior of zl3 shows another interesting feature. Both BCR and JLR are seen to lie in opposite regions with low values of zl3 . This suggests that instabilities occurring in the flow have different nature. The dominance of these instabilities is higher in STR, as compared to that in JLR and BCR. Interestingly, BCR-2 (sieve plater sparger) show relatively higher flow instabilities as compared to BCR-1 and BCR-3 and therefore its co-ordinates lie closer to STR in Fig. 11B. This phenomena is also observed in the form of variations in length scale distribution as discussed in Section 5.2. Thus, it may be concluded that this methodology using fractal spectra and PCA

6. Conclusions In the present study, an attempt has been made to study the particle image velocimetry (PIV) as well as large eddy simulation (LES) data using multiresolution analysis to identify the dominant energy containing structures and evaluating their characteristics in terms of quantifiers based on their multifractal properties. The conclusions are: (1) The characterization based on the energy content of the structures across the WT scales show variation in dominant length scales and their behavior in time. The comparison of length scale distribution for six equipments reveal the gross effects of flow and mixing behavior. (2) A dependency has been seen between mixing time and the higher order statistical quantities of length scale distribution, namely, Sl and Kl . A generalized correlation has been built for a selected and important class of equipment with wide distribution of mixing times and flow behavior. The improvement in correlation is possible because of not only considering the flow characteristics but also the statistical properties characterizing the length scale distribution and evaluated by scalewise WT studies. These statistical parameters are l , Sl , Kl , dominant energy quantifiers lequi /Dequi and scalewise energy ratio max(Eˆa ). (3) Fractal spectra have been evaluated to understand the evolution and interactions of structures along the scales in a reduced dimensionality by WTMM methodology for the 2D data. The nature of fractal spectra behavior is found to depend on the hydrodynamic properties in each equipment. This has been quantified on the basis of the multifractal quantifiers associated with the varying fractal dimension f and singularity strength . The behavior of parameters identified from the multifractal spectra, namely, fmax , fmax , mean , max , min , Af , Sf and Kf are sensitive to the geometrical parameters and flow conditions. (4) Classification has been carried out by PCA of the fractal spectra database obtained from the selected equipment. Similarities and dissimilarities in the operation of different equipment can be diagnosed by this methodology. Thus, the study of dynamics in reduced dimensionality using fractal spectrum and PCA identifies the regimes in terms of PCA co-ordinates for setting the desired benchmark pattern of formation, and breakup of flow structures in the equipment. By including new types of equipments and repeating this study it may now be possible to arrive closer to a generalized methodology that may be used for the selection and optimum design of reactors.

Notation a Af b Clm CS d D Dequi DI Din Dout

wavelet scaling parameter area under the curve of multifractal spectra, m2 wavelet translation parameter correlation matrix in PCA formalism for multifractal spectra database Smagorinsky constant nozzle diameter, m tank diameter, m equivalent diameter of the tank, m impeller diameter for STR, m inner diameter of annular section in ACC, m outer diameter of annular section in ACC, m

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E(x)a Eˆ a fmax f () H HI HN Ji k K lequi lmean L N p Pin P (a) P q R Re sm S  S Sij t T a,b (x) ui ui uai ui  U⊂/ V wlm W  W W a,b (x) Wia,b (x) xi x y1/2 y+ zlm Z(a, q)

local energy content at selected scale a and co-ordinates x, m2 /s2 percentage energy content for DWT scalewise decomposition, % maximum value of fractal dimension in the fractal spectra local fractal dimension for singularity strength  tank height, m impeller clearance from the bottom for STR, m nozzle clearance from bottom for JLR, m number of WT scales in the direction i turbulent kinetic energy per unit mass, m2 /s2 flatness factor or kurtosis, ( − )4 /( − )2 2 equivalent length scale dominant energy content, m mean lengthscale of distribution, m length of the equipment (channel), m rotation speed, rps number of modes of principal component matrix considered for reconstruction power input to the equipment per unit volume, W/m3 power spectral density at DWT scale a, m3 /s2 LES filtered pressure, N/m2 exponent of wavelet coefficients in Eq. (10), selected range −10 to 10. range Reynolds number eigenvalue for mode mth in matrix  S skewness, ( − )3 /( − )3 eigenvalues of the data set in SVD filtered rate of strain, s−1 time, s continuous wavelet transform coefficients instantaneous velocity component in i direction, m/s LES filtered velocity component in i direction, m/s scalewise reconstruction of velocity ini direction at scale a, m/s time average velocity of component i, m/s center-line velocity in the jet, m/s volume of reactor, m3 eigenfunction value for lth set and mth mode in matrix  W width of vessel (e.g. channel width, annular gap in ACC, etc.), m eigenfunction of the data set in SVD DWT coefficient matrix DWT coefficient matrix for ith component of velocity co-ordinate distance in i direction, m 2D co-ordinate direction vector (x1 , x2 ), m half-width of jet, m dimensionless boundary layer distance in channel principal component vectors from PCA for l number of data sets and mth mode partition function in Eq. (10)

Greek letters

 mean min max fmax

mix

standard singularity strength mean value of standard singularity strength minimum value of standard singularity strength maximum value of standard singularity strength standard singularity strength corresponding to fmax filter length for LES resolution, m turbulent kinetic energy dissipation rate per unit mass, m2 /s3 mixing time, s

 SGS  (q) ij (x) a,b (x)

5345

kinematic viscosity, m2 /s sub-grid scale viscosity, m2 /s variance scaling exponent sub-grid scale stress, m2 /s2 generalized variable for planar data wavelet basis function (applicable to both DWT and CWT) vorticity magnitude, s−1



Subscripts and superscripts 

derivative data denoised data related to fractal spectrum co-ordinate direction, i = 1, 2, 3 correspond to radial, axial and tangential component, respectively related to length scale distribution

d f i

l

Acknowledgements V.R.K. and B.D.K. gratefully acknowledge Department of Science and Technology, New Delhi, India for the funding of project SR/S3/CE/054/2003-SERC-Engg. S.S.D. gratefully acknowledges University Grant Commission for a fellowship. We gratefully acknowledge C.S. Mathpati, B.N. Murthy and M.V. Tabib for the help rendered in obtaining LES data used in the present study. Appendix A. LES model equations The LES model equations have been derived and obtained as follows:   jij jui j j j 1 jP (u ) + (u u ) =  − − jt i jxj i j jxj jxj jxi jxj

ij = ui uj − ui uj 1 3

ij − kk ij = −2SGS Sij Sij =

1 2



jui juj + jxj jxi



The closure term SGS can be obtained by Smagorinsky model as

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