Ind. Eng. Chem. Res. 2010, 49, 4471–4473
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Reply to “Comments on ‘Dynamics of Flow Structures and Transport PhenomenasPart I: Experimental and Numerical Techniques for Identification and Energy Content of Flow Structures’” Channamallikarjun S. Mathpati, Mayur J. Sathe, and Jyeshtharaj B. Joshi* Department of Chemical Engineering, Institute of Chemical Technology, Matunga, Mumbai - 400 019, India Sir: We thank Professor Forˇt for reading and complimenting our work. In his comment,1 he has provided few additional experimental techniques to study fluid mechanics in agitated vessels. He has presented mainly two methods, based on visual observation of the circulation of small tracer particles to quantify primary and secondary (induced) flow and, in turn, estimation of gross circulation time and measurement of dynamic pressures using a set of pressure transducers to estimate distribution of axial force and the data were analyzed using proper orthogonal decomposition (POD). This analysis has given valuable information regarding coherent flow structures (macro-instabilities). The methods provided by Professor Forˇt1 in his comment refer to the gross flow parameters such as mean circulation time and presence of secondary cells within agitated tanks. This describes only the momentum transfer, which can be quantified in terms of flow number, power number, and circulation cells. It is known that knowledge of the mean flow is necessary; however, it is not sufficient for understanding the momentum, mass, and heat transfer.2 The flow pattern in the majority of chemical process equipment is turbulent. Therefore, it is inevitable to understand the physics of turbulence to the extent possible with the existing status of experimental and numerical techniques. We must identify the structures (distribution of sizes, shapes, energy, velocities, length scales, an energy spectra) and also discern the dynamics. There are numerous techniques described in the available literature besides the three given in the comment. Detailed description of such techniques is outside the scope of the current response, and these have been discussed by Joshi et al.3,4 To maintain brevity, we have given examples of three additional techniques. However, the major objective of the published review4 and relevant articles from our group5–13 was to describe the detailed nature of turbulent flow inside the vessel, which constitutes the formation mechanism of such secondary cells and the magnitude of overall circulation consolidating the understanding of transport phenomena in stirred vessels and the other chemical processing equipment. The review4 deals with understanding and quantifying the turbulent fluctuations, which are the result of passage of the deterministic organized flow structures and the random disorganized irrotational motions. The review was focused on obtaining all the flow structures and not just gross flow patterns in the system. An attempt was made to relate the structure properties such as age, penetration depth, size, shape, and energy content distribution to the design parameters such as mixing time, heat- and mass-transfer coefficient, drag coefficient, dissipation rate, etc. Because of complex flow patterns in stirred vessels, information obtaining about all the flow structures using experimental techniques is not possible and computational fluid dynamics (CFD) using direct numerical simulation/large eddy simulation (DNS/LES) is more appropriate. Hence, in the review,4 we have described very few experimental techniques * Author to whom correspondence should be addressed. E-mail addresses:
[email protected],
[email protected].
(mainly using laser Doppler anemometer (LDA) and particle image velocimetry (PIV)).14–19 If we take any advanced experimental tool available today, it cannot provide all of the information about flow structures. Hence, every technique is unique in its own sense and must be used in combination with other techniques. For example, the pressure probe technique suggested by Professor Forˇt, in combination with the POD technique, has provided the frequency information regarding the macro-instabilities.1 The following paragraphs provide our view regarding the techniques suggested by Professor Forˇt.1 The first technique described by Professor Forˇt deals with measurement of the mean circulation and, in turn, mixing characteristics. Forˇt et al.20 estimated the presence of overall circulation or two circulation cells in a two-impeller system based on impeller positions. This technique was based on visual observation of tracer particles. It should be pointed out that a simple stimulus response technique employing the impulse input of conductive tracer and two conductivity probes at extreme locations in the tank, viz. near the top and bottom part of the tank, should be sufficient to detect the presence of single and multiple circulation cells, based on the number of peaks obtained in the concentration-time series.21,22 Here, we would like to suggest two more techniques to study the circulation pattern and transient mixing characteristics: (a) liquid crystal thermographic technique23 and (b) pressure probe measurement.8 The liquid crystal thermographic technique makes use of temperature as a passive scalar. This utilizes the change in color of the thermochromic liquid crystals when they are subjected to different temperatures; liquid crystals exhibit a rapid and reversible response to dynamic temperature changes over a wide range of temperatures. The method developed by Kulkarni and Joshi8 was used in bubble columns, but it can be extended for measurement in stirred vessels. Pressure fluctuations can be measured at various locations in the tank. Because of the complex nature of flow and the existence of various time scales and length scales propagating at various speeds, it becomes difficult to estimate the propagation time of various patterns detected in time-history records. Hence, convolution of the two signals obtained from the simultaneous measurements of spatially separated sensors offers an advantage. The crosscorrelation function between the pressure measurements at two locations is given by Rxy(τ) )
1 Tb
∫
Tb
0
Px(t)Py(t - τ) dt
(1)
Upon normalization with the square root of the product of the autocorrelation functions (Rxx and Ryy) at lag zero of the respective series, Rn(τ) )
10.1021/ie100129t 2010 American Chemical Society Published on Web 04/13/2010
Rxy(τ)
√Rxx(0)Ryy(0)
(2)
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From the graph of Rn(τ) versus t (Figure 3 in the earlier work of Kulkarni and Joshi8), the velocity (V) and the length (l) scales are given by the following equations: V)
d τmax
l ) V(τmax - τz)
(3)
(4)
The second technique described by Professor Forˇt1 involves measurement of the time-averaged dynamic pressure profile along the bottom of the tank to quantify the total downflow and the flow reversal radius of the axial flow impellers. However, it should be pointed out that the technique will perform satisfactorily only for the impeller clearances sufficient to form a liquid jet with negligible radial and tangential momentum. At lower bottom clearances, the flow reversals start to affect the flow pattern in the near-impeller region and the radial pumping action becomes significant. In such cases, the radial thrust becomes comparable to the axial thrust and the dynamic pressure measurement along the bottom is not sufficient to estimate the flow velocity, because it does not consider the impact pressure, because of radial or tangential velocities; hence, the velocity estimated from dynamic pressure measurement along the bottom will be significantly underpredicted. The technique becomes highly specific to axial flow impellers and, hence, less attractive to develop reliable and generalized design procedure, as discussed in Part II of the review.13 The third technique deals with measurement of the tangential thrust on the baffles at different axial locations. The technique measures the tangential impact pressure using a small trailing target attached to the baffle at different vertical locations. The target is balanced by a spring and, hence, the deflection has linear dependence on the pressure. The deflection is detected by a photoelectric device. Although the response times of photoelectric devices are much less, the response time of a spring mass system limits the overall response time of the proposed technique and, hence, the dynamic pressure time series may be attenuated by the delayed response. The technique is also biased for tangential velocity fluctuations and, hence, is of limited interest to an axial impeller system, where tangential stresses are less (compared to axial stresses). The pressure probe technique described by Professor Forˇt1 is more suitable supplemental information to the literature presented in the review4 than the first technique. Here, we take the opportunity to give a few more techniques for the quantification of flow structures in agitated vessels. However, we would like to reiterate that these techniques deal with only characteristic flow structures (macro-instabilities). The methods presented here do not attempt to find each and every structure present in the system. Roussinova et al.24 used a velocity decomposition technique to study the macro-instabilities, using a single-component LDV technique. They have explained briefly how LDV measurements have been used to study macroinstability by various investigators. They performed measurements using four different impellers (a pitched-blade turbine, A310, HE3, and a disk turbine), two impeller diameters (D ) T/2 and T/4), two clearance ratios (C/D ) 1.0 and 0.5). The measurements were performed at two locations in the tank (impeller stream and top corner). The impeller stream velocity signal can capture all frequencies present in the flow. It contains the blade passages, the high-frequency turbulent components, and the macro-instabilities.
They have decomposed an instantaneous velocity signal as Ures′ ) UBP′ + UMI′ + Urand′
(5)
Here, the subscripts “BP” and “MI” denote the blade passage and low-frequency components, and the subscript “rand” denotes the pure random or turbulent component. The Ures′ term represents the root-mean-square (rms) velocity component after resampling. With selection of a suitable window length, relative to the blade passage frequency, data smoothing filters out the random (turbulent) and periodic (due to the blade passages) velocity components. The low-frequency rms velocity (UMI′) then can be calculated from the smoothed velocity profile as UMI )
1 n
n
∑ (U
2 MI,i′)
(6)
i)1
Nikiforaki et al.25 used LDV and PIV to measure flow structures in stirred vessels. They studied the effects of operational parameters such as vessel size, impeller design, diameter and blade number, Reynolds number (Re), and baffle number. They have also used velocity decomposition techniques, such as that used by Roussinova et al.25 Guillard et al.26 performed PLIF measurement in a stirred tank with two Rushton turbines. They used a pattern recognition algorithm to quantify the large-scale structures in the flow. In each experimental run, they had acquired 400 images. From image analysis, they have quantified the mixing process at length scales where large fluid elements were deformed and broken up due to the fluid motion. They have defined two conditional functions: (i) the local concentration was compared with a discriminator parameter and the function was set to a value of 1 or 0, and (ii) the local concentration at two distinct locations separated by a distance X were compared to the discriminator parameter and the function was set 1 or 0. These two functions were used to find the two-dimensional topology of the mixing structure around position X. Hence, we would like to conclude that multipoint measurement, planar, and volumetric measurement techniques, in combination with advanced data analysis tools, would contribute significantly to the field of turbulent flow structures. Notation d ) distance between two pressure probes (m) D ) impeller diameter (m) l ) characteristic length scale (m) Px, Py ) pressure at two different locations (Pa) T ) tank diameter (m) Tb ) total time duration of series (s) C ) clearance of the impeller (m) Rxx, Ryy) autocorrelation functions Rxy ) cross-correlation function Rn ) normalized cross-correlation function UMI ) smoothed velocity (m/s) UBP′ ) component of velocity due to blade passage (m/s) UMI′ ) component of velocity due to macroinstability (m/s) Urand′ ) component of velocity due to random fluctuations (m/s) Ures′ ) instantaneous velocity signal (m/s) V ) characteristic velocity (m/s) X ) position vector (m)
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