Dynamics of Flow Structures and Transport Phenomena, 2

Ind. Eng. Chem. Res. 2009, 48, 8285–8311

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Dynamics of Flow Structures and Transport Phenomena, 2. Relationship with Design Objectives and Design Optimization Channamallikarjun S. Mathpati, Mandar V. Tabib, Sagar S. Deshpande, and Jyeshtharaj B. Joshi* Department of Chemical Engineering, Institute of Chemical Technology, Matunga, Mumbai-400019, India

There have been several approaches in the literature to identify and characterize flow structures qualitatively as well as quantitatively. In the first part of this review, the methodologies and applications of various experimental fluid dynamics and computational fluid dynamics techniques, as well as mathematical techniques, have been discussed. Their chronological developments, and relative merits and demerits, have been presented to allow readers to make a judgment as to which techniques to adopt. In the present part of the review series, a stepwise procedure is suggested for the design of equipment using flow structure knowledge. An attempt has been 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.). This understanding of flow structures has brought improvements in the formulations of heuristic models of mass and heat transfer. This review makes an effort in developing insights into the views of earlier established analytic and heuristic theories of heat and mass transfer. The recently revealed dynamics of flow structures (as uncovered through the use of various techniques) has helped in furthering the efforts of developing new theories of heat, mass, and momentum transfer. Such an understanding between the structure dynamics and the transport phenomena has helped in the optimization of flow pattern (for instance, maximization of ratios of heat and mass transfer, as well as mixing, with respect to energy input). In this direction, some success stories have been described that have already been implemented in industry and have good potential for implementation. 1. Introduction Chemical processing involves the activities necessary for deciding the best possible hardware and operating protocols to perform desired transformations of raw materials into valueadded products. These processes and protocols are used in many areas, such as the energy sector, the refinery and petrochemical industry, the glass and cement industry, fertilizers, and the synthesis of a variety of materials and specialty chemicals. The chemical transformations of raw materials to desired products requires a sequence of chemical and physical changes that are classified as unit processes and unit operations, respectively. Most of the industrially important processes are predominantly heterogeneous. They require two or more phases to be in contact; the processes can be either organic or inorganic in nature. The successful design requires expertise in thermodynamics, chemistry and catalysis, reaction engineering, fluid dynamics, mixing, and heat and mass transfer. The first three disciplines explain transformations at the molecular level, so the information collected at the laboratory scale applies equally to the industrial scale. However, the equipment-scale-dependent phenomena, such as mass and heat transfer and mixing, must be understood for the reliable and efficient design of equipment for unit operations and processes. To date, design engineers rely on accumulated knowledge (in the form of empirical and semiempirical correlations) and pilot-plant experience with various reactor arrangements. There are almost 400 different types of reactors and multiphase contactors in practice, which are * To whom correspondence should be addressed. Tel.: +91-22-2414 0865. Fax: +91-22-2414 5614. E-mail address: [email protected].

designed by modifications in few conventional hardware configurations (straight and coiled pipes, jets and venturies, ejectors, packed columns (random and structured packings), falling and agitated film contactors, staged column contactors, bubble columns, fluidized beds, stirred tanks, etc.). When such modifications are successful, they are reproduced as such for other production facilities of the same chemical or scaled-up or scaled-down, depending on the requirements. By following the aforementioned procedure, the design may prove unreliable (and/or expensive), because of the lack of understanding of transport phenomena. This is mainly because, as mentioned previously, the present status of the design is still closer to an “art” rather than the desired status of “science”. Thus, in practice, all these plants are always overdesigned for a given production rates. Such empirical practices lead to high capital and operating costs, long start-up times, and expensive ways of solving plant problems. Furthermore, such apparent overdesigns have resulted into underdesigns in several commercial operations. In addition, the selectivity levels may be low and lead to byproducts, which, in turn, adds a burden on the environment. The overdesign also means energy-intensive operation. Hence, it was thought desirable to classify these reactors, then examine their design consideration and implement the knowledge of flow structures to optimize these reactors. 2. Reactor Classification on the Basis of Energy Input For various applications, energy is required and there are various ways by which energy can be supplied to the equipment. In fact, all the equipment can be conveniently classified (see

10.1021/ie900396k CCC: $40.75  2009 American Chemical Society Published on Web 06/04/2009

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Table 1. Reactor Classification, on the Basis of Energy Input Class 1

Class 2

2,3

bubble columns packed bubble columns1,2 internal-loop air-lift reactors2 external-loop air-lift reactors1,2,4 sectionalized bubble columns1,2,5 plate columns6 pipeline and coil reactors1 static mixers7 fixed-bed reactor6 expanded-bed reactor8 fluidized-bed reactor9,10 ion-exchange column/ chromatographic separators11 dryers12 spray columns6 plate extraction columns6 circulating fluidized bed13,14

falling film reactor/evaparator packed column with random and structure packing17,18 agitated film dryer/evaporator/reactor19 trickle-bed rectors20 trickling filters in wastewater treatment21 condensers22 rotating disk contactors6,23 rotating basket (of packing) contactors24 disk and donut contactors solar water/air heaters25,26 wetted wall reactors27 wetted sphere reactors28

stirred reactors (single and multiple impeller)30 gas-inducing reactor31 surface aerators32,33 rotating disk contactor6–23 annular centrifugal extractors34 asymmetric rotating-disk contactor6 jet loop reactors35 mixer-settler6 plunging jet reactors36 Scheibel columns6 centrifuges37 Higee contactors38

spinning disk contactors29

the work of Joshi and Doraiswamy1) based on the mode of energy supply: (1) pressure energy, (2) potential energy, and (3) kinetic energy (see Table 1). The Class 1 equipment is also called column-type equipment. Under this category, there is a variety of multiphase contactors. The gas-liquid contactors include bubble columns, packed bubble columns, internal-loop and external-loop air-lift reactors, sectionalized bubble columns, plate columns, etc. Solid-fluid (liquid or gas) contactors include the static mixers, fixed beds, expanded beds, fluidized beds, transport reactors or contactors, etc. For instance, a fixed-bed geometry is used in unit operations such as ion exchange, adsorptive and chromatographic separations, and drying, as well as in catalytic reactors. The liquid-liquid contactors include spray columns, packed extraction columns, plate extraction columns, static mixers, etc. Similar equipment are widely used in liquid-gas, gas-liquid-solid, gas-liquid-liquid, and gas-liquid-liquid-solid contactors. In all of these multiphase contactors, the dispersed phase consists of bubbles or drops or particles, or two or all three of them. The continuous phase is either liquid or gas. The governing and the characteristic features of the dispersed phase is its size distribution and velocity distribution, and these two features have a major impact on the performance of this equipment. The other governing parameters include the column diameter, the column height, the sparger design, and the design of internals. Furthermore, in Class 1 equipment, the energy is supplied through the introduction of phases. For instance, in the case of bubble columns and gas-fluidized beds, the gas is supplied against the static pressure of multiphase dispersion and the energy input rate is given by the following equation: Ei ) QGFDHDg

Class 3 15,16

(1)

where FD is the average density of the dispersion. Ei is the pressure energy, and, therefore, Class 1 equipment are classified under pressure energy. For the energy balance of different types of Class 1 equipment, the reader is referred to the works of Joshi and co-workers.39–47 In Class 2 equipment, the energy supplied is in the form of potential energy associated with the liquid. For instance, in a conventional packed column (used for either distillation or absorption), the liquid is pumped upward to the top of the column and distributed over the packing. If the gas is introduced at the bottom, the gas phase has the pressure energy; however, it is usually negligible, compared to the potential energy of the liquid. Even if the gas and liquid phases flow co-currently downward, the major contribution to the energy is from the

liquid phase. Because the liquid flows in the form of a film, this equipment (Class 2) may also be termed as film contactors. Other equipment in this category include trickle-bed reactors and falling film reactors/evaporators. In packed columns, a variety of packing shapes (and, of course, sizes and material of construction) are used in practice, including the Raschig rings (in old installations), Berl saddles, partition rings, Intalox saddles, Pall rings, Hy-pak rings, structured packings, etc. Equally important is the uniformity of liquid distribution, liquid redistribution, and packing supports. The Class 3 equipment includes stirred tank reactors. In this case, the mode of energy supply is through the impeller rotation. The flow leaving the impeller has kinetic energy in the form of mean or/and turbulent. The energy is used for a variety of objectives such as blending, heat transfer, liquid-liquid dispersions, gas-liquid dispersions, solid-liquid dispersions, and higher-order dispersions. Stirred reactors, in which one or more impellers are used to generate the desired flow and mixing are among the most widely used reactors in chemical and allied industries. Stirred reactors offer unmatched flexibility and control over various transport processes that occur within the reactor. The performance of a stirred reactor can be optimized via appropriate adjustments of the reactor hardware and the operating parameters. Parameters such as vessel diameter, aspect ratio design, number, type, location and size of impellers, and degree of baffling provide effective handles to control the performance of stirred reactors. However, the availability of such a large number of parameters also makes the job of selecting a suitable configuration of the stirred reactor quite difficult. Stirred tank reactors are extensively used in the chemical process and allied industries to conduct mixing, heat transfer, solid suspension, and a large number of dispersion applications, such as solid-liquid, gas-liquid, liquid-liquid, gas-liquid-solid, gas-liquid-liquid-solid, etc. This equipment is the most sought after, because of their reliability and flexibility toward the process requirements. In the stirred reactor, the primary mode of energy supply is through the impeller. The flow leaving the impeller receives mean and turbulent kinetic energy, as a result of the impeller motion. The introduction of a second/third phase (in multiphase contacting) may also introduce additional energy. The different impeller designs develop different flow patterns and a local degree of turbulence in the vessel. The impeller designs are mainly categorized as open-type or closed-type impellers. The open-type impeller designs are generally classified into two categories: (i) radial flow impellers and (ii) axial

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flow impellers. The Rushton turbine, straight-blade turbines, curved-blade turbines, Brumagin turbines, etc. belong to the radial flow category. These impellers generally produce high shear and, in turn, high turbulence. Axial flow impellers can be operated in either the “up-pumping” mode or “down-pumping” mode. Marine propellers, pitched-blade turbines, hydrofoil impellers, etc. fall under this category. These impellers provide more convection than turbulence, compared to radial flow impellers. If the understanding of transport phenomena is improved to some extent, and the energy can be made to dissipate at the desired locations, then the energy input can be decreased by a factor of at least 5-10. All the previously mentioned equipment is generally operated in the turbulent regime (except highviscosity applications, such as the production and processing of polymers) to achieve the high rates of heat, mass, and momentum transfer. The turbulent flows contain flow structures with a wide range of length and time scales, which control the transport processes. The length scales of these structures can range from column dimensions (highest) to Kolmogorov scales (lowest). However, not all the scales of turbulence contribute equally to the transport rates and mixing. Hence, it is important to understand the basic mechanism of participation of different scales of motions with the rates of different transport processes. The present empirical design practices do not consider these basic mechanisms and conceals detailed local information about the relationship between the turbulence and the equipment performance. The theme of the proposed work is to understand the flow structure-reactor performance relationship for reliable designs. This discussion forms the motivation for addressing the proposed task. 3. Design Consideration As already discussed, there are almost 400 different types of reactors and multiphase contactors in practice. Although the list appears to be seemingly huge, the intended applications and/or design consideration can be classified into the following categories: (i) mixing or blending, (ii) heat transfer, (iii) solid suspension, (iv) gas dispersion and mass transfer, (v) liquid dispersion and mass transfer, (vi) desired mode of gas/liquid/ solid flow, (vii) degree of backmixing in continuous and dispersed phases, (viii) dispersed-phase hold-up profile, (ix) dispersed-phase particle size distribution, (x) drag coefficient, and (xi) quality of interface (extent of slip between phases). The desired contribution of convection to turbulence varies based on the application (see Figure 1). 3.1. Mixing/Blending. If the objective is blending, then we must have a configuration that offers lower mixing time at a lower power per unit volume. The quality of mixing in stirred vessels is dependent on the relative distribution of mean and turbulent kinetic energy. This could be achieved by controlling the turbulence, vis-a`-vis, the mean convective flow patterns. However, the blending process occurs at three levels: molecular, eddy, and bulk (convection). The blending process is controlled by convective patterns than the eddy motion (turbulence).48 A small increase in eddy diffusivity can help to reduce the mixing time, but a larger increase (for example, by more than a factor of 5) does not result in a substantial reduction in mixing time. Thus, as far as blending is concerned, the design should offer scope to optimize the mean convective flow patterns and maintain a low level of turbulence (low eddy diffusivity). However, if the system generates no turbulence (zero eddy diffusivity), then faster mixing at the macro scale will not be achieved. In such a case, the reactant would just move with the

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Figure 1. Relative role of convection and turbulence for various applications.

flow without mixing at smaller scales. Thus, we must ensure that an optimum low level of turbulence exists, along with strong convective currents. From the standpoint of impeller design, the low-power-number (0.1-0.5) impellers (axial flow) generate mean flow, whereas the high-power-number impellers (>3) (radial flow) generate flow with significant turbulent kinetic energy. The hydrofoil (HF) impellers are known to offer high convective currents and low turbulence (Np ) 0.5) and have been used greatly for blending operations. Whereas impellers such as the disk turbine (Np ) 5) produces high turbulence and gives higher mixing times. Using a disk turbine for the mixing operations is a waste of power, because the higher turbulence levels will not be effective in reducing the mixing time above a certain limit. Further design modifications, such as giving 10° to 20° blade twists toward the horizontal plane, improves the hydraulic efficiency and, thus, reduces the mixing time. It has been observed that, at the same level of power consumption, the mixing time is reduces by 13% by simply twisting the impeller blades by 20°. Thus, it has been understood by now that the macromixing applications prefer all the energy in the form of mean kinetic energy and just an appropriate amount of turbulent kinetic energy (even 5% of that available at all locations in the tank). Computation fluid dynamics (CFD) studies have analyzed these problems, and recommendations have been made regarding the optimization.48–51 For macromixing applications, a possible reduction in turbulence means a substantial savings in operating costs. 3.2. Gas-Liquid Dispersion. If the objective is gas-liquid and liquid-liquid dispersions, then we require small bubble and drop size and their size distribution, to achieve high interfacial area (a) and also a high true mass-transfer coefficient (kL). The turbulent energy dissipation rate (ε) and its distribution throughout the reactor is a major parameter that governs the droplet size in dispersions. This requires a very high level of turbulence, and, hence, radial flow impellers suit these applications. Furthermore, there may be circumstances when two or more agitators must be used. For example, a radial impeller that generates high turbulence can be used to break the bubble to the desired size, and then a hydrofoil or axial impeller can be used to ensure that these small bubbles are uniformly dispersed throughout the equipment. 3.3. Heat Transfer. The mechanism of heat transfer at column walls is almost similar to that of single-phase pipe flow. The only variation, in the pipe flow, is turbulence that is

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Figure 2. Parity plot of overall mass-transfer coefficient in the bubble column using various correlations. (Source: Gandhi et al.141)

generated at the walls, which is responsible for heat transfer. However, in the case of other reactors, in addition to friction at the wall, turbulence generation occurs at other locations. If the turbulence generated far from the wall can be made to convect toward the wall and then dissipate, it enhances the heat transfer at a relatively lesser increase in pressure drop and, in turn, lesser operating cost. In the case of heat-transfer applications, stirred tanks are usually provided with either external jackets and/or internal coils. In the case of bubble columns, fluidized beds, either tubes bundles/coils are added and inside or external heat exchangers are used. Heat-transfer applications require both convection and turbulence in appropriate proportion (this subject will be discussed later in section 5.2) for enhanced rates of surface renewal. 3.4. Solid Suspension. In the case of solid-catalyzed reactions, the availability of the entire surface of the catalyst to the liquid is very important. The complete off-bottom suspension ensures a minimum speed of agitation at which all particles are completely suspended (NCS) and, hence, a majority of the solid surface area becomes available for the chemical reaction, heat and mass transport, etc. Knowledge of the critical impeller speed is important, because additional energy beyond that required for NCS may not be useful. The particles will go in the state of suspension when the upward liquid velocities are higher than the settling velocities of the particles. However, there is a risk of particles accumulating in the corners. The solution is to increase the turbulence in the bottom region, because it increases the drag coefficient (decreases the settling velocity) and, hence, makes the particles more amenable for suspension. Furthermore, the turbulent dispersion forces will try to homogenize the suspended particles. Hence, relatively speaking, the turbulent flow structures at the bottom surface are more important than those in the bulk. Computational and experimental studies have helped to quantify the turbulence properties at the bottom, as

well as providing the local solid phase concentration with the reactor, which is otherwise difficult with experimentation. Experimentally,52–54 it has been proven that the axial flow impellers are more efficient for solid suspensions. 4. Suggested Methodology for the Process Equipment Design In the current industrial practice of the process design, much empiricism is involved, in terms of using empirical/semiempirical equations to estimate the design parameters. Figure 2 shows the parity plot of correlations available for the estimation of mass-transfer coefficient in the bubble column. It can be observed that the points are widely scattered on the graph. Such a present status of the available correlations does not provide any confidence in the process design. These involved overestimations/underestimations make the designs unreliable. The possible reasons are (i) not all the variables are taken into consideration, (ii) the range covered is limited, (iii) a complex interaction exists between the variables (which means that the extent of the effect of one variable on kL a can be a function of other variables that affect kL a). Currently, there is some type of polarization between practicing engineers and the researching engineering faculty in the universities. With the practicing engineers, there is a great amount of accumulated knowledge regarding standardized practices for design and scaleup, which, at best, is closer to an art than the desired state of science. In contrast, in the universities, substantial research has been occurring with regard to understanding the mechanisms of transport phenomena. Currently, there is a need to transfer the knowledge generated by the academicians to the practicing engineers and vice versa. The approach for industrial reactor design through combined efforts can be classified into three categories: (i) equipment selection, (ii) research work performed

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Figure 3. Flowchart of the suggested methodology for the process equipment design.

at laboratory scale, and (iii) stepwise procedures for design and optimization. There are almost 400 different types of equipment used in the chemical process industries. The classification of this equipment has already been discussed in section 2. A flowchart for the selection, design, and optimization of process equipment is presented in Figure 3. For the first two objectives, it is desired that these are undertaken via a partnership between academia and industry. Academicians (with the active participation of industrial practitioners) can take the initiative to recommend the guidelines for the selection of equipment for a given objective. The practicing engineers then can contribute to determine the rate-controlling steps. This can be achived using model contactors.1,55,56 The practicing engineers can conduct

experiments for a range of feed composition, flow rates, catalyst loading, excess reactant, particle size, temperature, and pressure on real systems. The real challenge is in the estimation of design parameters, as already shown in Figure 2. This challenge can be addressed by understanding the fluid mechanics in a variety of equipment. In particular, the real issue is to understand the physics of turbulence and the role of turbulence in governing the transport phenomena. The understanding of the role played by mean flow and turbulence, and its implication on transport phenomena, can greatly contribute to reliable and energyefficient designs. Energy is one component that costs money; however, the major issue is the reliability and confidence in the design for the purpose for which the equipment is proposed

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to function. Such confidence can only be brought by understanding the fluid mechanics. Academicians can conduct measurements using advanced techniques57–61 such as laser Doppler velocimetry (LDV), hot film anemometry (HFA), particle image velocimetry (PIV), and holographic particle image velocimetry (HPIV) and computations using direct numerical simulation (DNS) and large eddy simulation (LES) to understand the fluid mechanics and the role of turbulent flow structures. The instantaneous flow structures, sometimes, are very much different than the mean flow patterns. The mean flow patterns can be obtained using Reynolds-averaged numerical structure (RANS) models and the PIV technique. HFA and LDV provide good information at a point in time with a data rate that exceeds 1 kHz. PIV and HPIV provides planar/volumetric information, although at low data rate, but they are advantageous in terms of spatial information. DNS can give point/planar and volumetric data at a sufficiently high data rate, which can be used for the quantification of all the structures. When the dynamics of large scale structures has a marked effect on the transport rates, LES can give vital information for the quantification of these structures. These experimental and computational techniques provide velocity/pressure/temperature and/or concentration data. This data can be analyzed using advanced mathematical techniques such as fast Fourier transform (FFT), eddy isolation method (EIM), wavelet techniques, proper orthogonal decomposition (POD), and hybrid POD-wavelet techniques. The relative merits of these techniques are discussed in Part 1 of this review.140 These techniques give the size, shape, penetration depth, and energy content of every flow structure in the system. This flow structure information can be used to estimate design parameters such as mixing time, as well as heat- and masstransfer coefficients, and construct the energy spectrum. The energy spectrum can provide useful information, such as local turbulent viscosity and local energy dissipation rate. This can help us to identify the regions where energy is getting wasted and where we need to dissipate more energy for a given application. Hardware can be modified accordingly and all the steps from performing experiments and simulations, up to estimation of the design variables, can be repeated until we obtain an optimum design. The third part (i.e., design and overall optimization) can be done through a partnership between academicians and practicing engineers. The optimization involves analysis inside the equipment, within the clusters of equipment and across the plant. The time series obtained from advanced measurement techniques is of combined interest. This data can be used to devise the control strategy for the safe operation of the equipment. 5. Estimation of Design Parameters 5.1. Mixing Time. In the literature, there are several correlations for the estimation of mixing time. These correlations are based on gross parameters (dimensionless mixing time (Nθmix), average circulation velocity, power consumption per unit volume, and geometrical details) that are specific to the equipment. Here, our objective has been to broaden the perspective by suggesting a single unified correlation that can be used for the computation of mixing time in all of the equipment. This has been made possible by considering the view that, in all of the equipment, the flow structures (of different time and length scales) contribute to mixing by advecting the reactants from one place to another at the macrolevel and continue to advect it until they break down or dissipate. Thus, the mixing time can be correlated to the length scale distribution of the flow structures, which can be very well-represented by

Figure 4. Parity plot of mixing time from experiments and correlation obtained from flow structure properties.

statistical parameters, in terms of higher-order moments (skewness, kurtosis, etc.). The length scale distribution has been obtained using the image processing of the spatial POD modes and wavelet transform decomposition scales. The parameters that have been considered for developing the correlation are power input per unit volume (P/V), which indicates the energy input for creating the flow; the Reynolds number Re, which indicates the operating flow conditions; the dimensionless ratio of height of the equipment to the diameter of the equipment (H/T), which represents the impact of the geometry; and the statistics that capture the turbulent length scale distribution. For each piece of equipment operating at different parameters, the length scale distribution obtained has some uniqueness, which is characterized by measuring the statistical parameters related to the distribution, such as the observed mean of the distribution j ), the standard deviation from the mean (σ) of distribution, (L the skewness (γ1) and kurtosis (γ2) of the distribution, and observed dominant length scale (Lmax). The values of mixing time in this equipment are obtained through the use of either available correlations or CFD runs. The following correlation was obtained by Deshpande et al.,62 who related the observed mixing time for the equipment to the length scale distribution within it, the power dissipated per unit volume (P/V), and the Reynolds number (Re): θmix ) 2.70

-0.04

( VP )

(Re)-0.21(γ1)-7.102(γ2)4.966(σ)0.16

( ) Lmax j L

-0.88

(2)

Such an attempt at relating the length scale distribution to the mixing time is quite unique. Although this approach can be quite cumbersome, because one must first use the CFD/EFD to get the flow field, and then apply the data analysis techniques to extract the flow structure and length scales. For instance, Figure 4 shows the comparison of the mixing time between experimental data and predictions from eq 2 for various equipment. The future work in this area may involve developing techniques that can extract flow structure and lengths scales in a simple and effective manner and then attempting to capture the effect of flow structures on the transport phenomena. 5.2. Heat- and Mass-Transfer Coefficient. Solid-fluid and fluid-fluid heat- and mass-transfer coefficients plays an important role in the design of chemical process equipment (for example, heat exchangers, reactors, and separation processes).

Ind. Eng. Chem. Res., Vol. 48, No. 17, 2009 Table 2. Proposed Theories of Heat Transfer by Various Investigators reference

heat-transfer coefficient

theory k δ

Lewis and Whitman64

film theory

h)

Higbie65

penetration theory

h ) 2FCp

Danckwerts66

surface renewal

h ) FCp(RS)1/2

Fortescue and Pearson67

large eddy

h ) 1.46FCp

( ) R πtc

1/2

( ) Ru2,rms lI

1/2

Scott small eddy and co-workers68,69

h ) 0.4FCp

[ υ/
R ]

Luk and Lee70

single eddy

h ) 0.9FCp

( )

Banerjee71

h ) FCp[R(γ2)1/2]1/2 surface divergence model where γ)

(

1/2

(

)1/2

R jl /ujE

∂u
3 ∂u
2 + ∂z ∂z

1/2

)|

int

The existing literature63 shows that most of the study can be listed under two major categories: (i) the analytical approach and (ii) the heuristic approach. The major resistance to heat and mass transfer lies in a region very close to the interface (within a few micrometers). Hence, many heuristic theories (references) are proposed in the literature64–71 (see Table 2). These theories assume that (i) the replacement of the fluid elements at the interface is a stochastic process motivated by the turbulent flow field and (ii) the rate of surface renewal significantly affects the heat- and mass-transfer rates. The surface renewal rate is strongly dependent on the quality of the interface and the turbulence intensity at the interface. The rate of surface renewal is dictated by the structure age distribution. Figure 5 shows the age distribution at the solid-liquid (SL) interface (no slip) and the vapor-liquid (VL) interface (finite slip) in a condensation jet reactor (CJR).72,73 It can be observed that the mean age at the VL interface is considerably lower than the SL interface, because of a higher turbulence intensity at the VL interface. The heat transfer at the VL interface is 2-3 orders of magnitude higher than that at the SL interface. Although surface renewal theories are simple and involve only one or two parameters, their applicability is limited. The accurate estimation of age distribution and rate of surface renewal is not possible, because experimental measurements are very difficult. Resolving such a small region close to the interface at high Re requires a very large number of grids, which is not possible with the computational power available today. Because of these difficulties, indirect methods have been used in the literature74–88 to estimate the age distribution. These techniques use some peculiar turbulent flow features that occur at the time of surface renewal. Most of these techniques use a velocity-time series obtained from LDV/HFA close to the interface, and conditional

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sampling is performed to detect surface renewal events. These techniques have been tested and standardized for flow in turbulent boundary layers. A brief account of these techniques has been presented in Part 1.140 The structure characterization techniques have been applied on to the available HFA/LES dataset of channel flow to identify the age distribution of flow structures. The channel flow has been considered because of its wide acceptance for validating the structure charaterization techniques. These methods then are applied on an industrially relevant piece of single-phase equipment (jet loop reactor (JLR)) and multiphase equipment (bubble column), and the information has been used to suggest a suitable equipment specification, which we call the design of flow patterns. The HFA and LES data have been used for this study. Care has been taken to ensure that the data captures the effect of small eddy sizes. For the HFA dataset, the data acquisition rate was very high (>10 kHz) with an equal time spacing. The low- datarate sample may not show the contribution of very small age eddies and the distribution will always start with a maxima, thereby misinterpreting the exponential age distribution to be the correct observation. The HFA high data rate enables us to capture the resolution until the dissipation range and avoid a biased view, which resulted from a low data rate. For the case of LES, the mesh size in the near-wall region has been on the order of ∼0.1 mm and the time steps of 0.005 s has been used. This helps us to captures the dynamics of smaller scales (∼10 times the Kolmogorov scale) which affect the heat-transfer coefficient (HTC). The accuracy with which the structure characterization technique capture the age distribution ultimately determines the accuracy with which they compute the heat-transfer coefficient. In the literature, HTC is correlated with only one structure propetry (i.e., age distribution function). The literature contains various formulations for age distribution. Figure 5 shows the age distribution for CJR near the vapor/liquid (VL) interface. Figure 6 shows the age distributions computed using these techniques (namely, PRT, QUAD, EIM, MVITA, VITA, YULI, ULV, BT) for the channel flow. In addition to these techniques, age distribution from CWT and DWT are also incorporated for

Figure 5. Comparison of structure age distribution functions, using the eddy isolation method-continuous wavelet transform (EIM-CWT) for a condensation jet reactor (CJR). (Legend: (O) VL interface and (0) SL interface.)

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Figure 6. Age distribution as obtained using different structure characterization techniques: (9) ULV, (2) VLV, (3) QUAD, (×) PRT, (b) YULI, (4) BT, (+) VITA, and (s) MVITA.

comparion. HTC has been estimated using these ages using the following expression: hi ) 2



DA π(∆t)i

∑

1 h hj ) N i)1:N i

(3) (4)

Table 3 shows that the wavelet-based techniques (discrete wavelet transform (DWT) and continuous wavelet transform (CWT)) have been able to provide fairly accurate estimates of the heat-transfer coefficient for the channel flow, with a deviation within 10%-12% from the experimentally known heat-transfer coefficient values. The wavelet-based methods provide the scale-time decomposition of the fluctuating velocity-time series. They preserve the temporal information regarding the occurrence of an eddy, and the criteria for eddy identification is fairly accurate. Among the two wavelet transform techniques, the CWT method gives a finer frequency resolution than the DWT and, thus, captures the age of an eddy more accurately, resulting in more-accurate prediction of the heat-transfer coefficient than of the DWT. These techniques perform well with the noisy data as well. In the case of noisy data, generally, the singularities incorporated by the noise at a higher frequency often gets mistakenly identified as an eddy. With wavelet transform (WT), this noise is alienated in the high-frequency scale and does not affect the overall behavior of the eddies at various scales. Therefore, WT distinguishes different eddies based on their energy, their scale, and their time of occurrence. Because the noise portion can be easily separated, the noise effect is minimal. On the other hand, the EIM shows the presence of large number of small eddies for noisy data. This may result in deviation from the reality and, hence, the distribution; therefore, noise removal at the high-frequency region is essential in the case of the eddy isolation method (EIM). Also, the EIM methodology identifies an eddy based on the zero crossing. There is a high probabliity that the fluctuating part between the two zero crossings could be a resultant of the passage of smaller subeddies within an eddy, rather than a single eddy. Therefore, the EIM cannot identify the subeddies within an eddy that passes at the same time, but the WT can discern if eddies of different length scales

(subeddies within an eddy) are passing over a point at the same time. Thus, the EIM technique may result in the underprediction of HTC because the small-scale eddies are not being captured and the age distribution shows a higher mean age. As a result, the EIM shows higher deviations (>13%) than the WT-based methods. The advantage that lies with the EIM technique, on the other hand, is that it is simple to use. As the heat transfer, shear at the interface and the Reynolds stresses are closely related; the quadrant analysis technique was determined to give very good agreement with the experimental heat-transfer coefficient, because this technique detects the events with the spontaneous movement of fluid close to the wall. The remaining techniques use only streamwise velocity components, compared to the quadrant analysis, which uses both streamwise and wall-normal velocity components. The ULV and VLV techniques have been determined to deviate significantly, as they cannot always detect the bursting process. However, there exists some correlation between the streamwise and wall-normal velocity in the bursting process. These velocities are out of phase in most of the bursting cycle and is one of the reasons for the similarities in the predictions of the heat-transfer coefficients via these two methods. The pattern recognition technique (PRT) was determined to perform very well close to the wall, because it accounts for the acceleration and deceleration features of sweeps and ejections. The other four techniquessnamely, BT, YULI, VITA, and MVITAsuse short time-averaged turbulent signals. Because the bursting process is associated with the high degree of velocity fluctuations, the use of a long time average may remove the pertinent features of the flow. The MVITA technique is based on the detection of an event directly on the large short-timeaveraged energy, which includes the contribution of all frequencies in the turbulence signal. Hence, this technique takes care of the age of the flow structure, as well as the influence of energy of the structure. The VITA technique considers the variance of the streamwise velocity component. The results clearly show the improvement in prediction with the modification in the standard VITA technique. The BT technique detects the large-scale motions with high-frequency large-amplitude fluctuations and the choice of filter becomes important. Therefore, based on the interaction between the signal and filter functions, the results may vary. Thus, a qualitative relationship between the heat-transfer coefficient and age distribution can be obtained from these methods. Probably, we can design equipment such that we obtain the desired age distribution and the desired heat-transfer coefficient. Out of many structure properties, such as age, size, shape, and energy distribution, only age distribution has been considered as a major parameter that affects heat transfer in the literature. Hence, it was thought desirable to study the sensitivity of the HTC calculations on the age distribution function; analysis has been performed using two extreme age distributions (the Higbie and Danckwerts models). The HTC expressions for these theories are as follows:



Higbie Theory: hH ) 2FCp

R πtc

(5)



Danckwerts theory: hD ) FCp√RS ) FCp

R tc

(6)

Note that the percentage change in the prediction of HTC evaluated from these two theories, which is given as

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Table 3. Heat-Transfer Coefficients, as Predicted by Various Structure Characterization Techniques Heat-Transfer Coefficient equipment

expt HTC value

CWT

DWT

channel at Re ) 5600 (Validation: Deviation from expt HTC value)

1644

1464.9 10.8%

1457.4 11.3%

channel at Re ) 18000

4184

3644.6

jet reactor at Re ) 17000

650

727.4

jet reactor at Re ) 33000

1200

bubble column (sieve)

5109

PRT

QUAD

EIM

MVITA

VITA

YULI

ULV

BT

VLV

1448.4 11.8%

1375.6 16.3%

1408.3 13.7%

1403.15 14.6%

1375.3 16.3%

1306.6 20%

1208 26.5%

1142.4 30.5%

1092.043 33.5%

3619.1

3393.4

3250.1

3493.2

3048.0

3020

2990.7

282.37

2608.7

2403.29

728.5

733.07

874.2

750.1

874.2

889.5

960.5

969.1

984.3

1025.8

1354

1364

1376.4

1559.2

1415.7

1641.3

1696.8

1806

1826.7

1851.8

1926

4476.7

4389.25

4349.1

2809.95

4212.83

2733.31

2631.13

2120.2

1877.2

1815.73

1606.7

Experimental HTC Value

Higbie’s Penetration

Surface Renewal

Large Eddy

Small Eddy

1644

643.29

817.72

1925.4

1233

-60.87%

-50.26%

17.12%

-25%

channel at Re ) 5600 (Validation: Deviation from expt HTC value)

hH - hD × 100 (%) ) hD



2FCp

FCp

(



R - FCp πtc



R tc

R tc

× 100

)

2 - 1 × 100 √π ≈ 13% (7) Equation 7 suggests that, for either age distribution, the predictions can vary only within 13% of the mean age. Thus, it is clear that the age distribution cannot be the sole parameter that governs the heat- and mass-transfer behavior at the interfaces. Therefore, a new formulation is required that takes into account age, energy, size, shape of structures, and their penetration distance from the interface. A new formulation has been proposed by Deshpande et al.73 (modified capacitance model) for the evaluation of average heat flux, which takes into account the combined effects of age and size (indirectly energy) distributions. Figures 7A and 7B show the eddy size distributions using CWT for three different eddy ages for the CJR at the VL and SL interfaces, respectively. The study of distribution functions obtained from CWT shows that, for the flow structures of same age, there is a wide distribution of eddy size and the energy content. The degree of spread around unity suggests the presence of a broader distribution of multiple size eddies. At the VL interface, the size distribution is determined to be wide for small eddy ages (i.e., 0.003 s and 0.02 s; see Figure 7A, lines 1 and 2, respectively), whereas for large eddy ages (0.5 s), it is narrower (see Figure 7A, line 3). This suggests that larger eddy ages contribute to a lesser extent, with regard to affecting the HTC. On the other hand, the eddies with a broader size distribution can enhance the HTC value, which is clearly observed at the VL interface. On the other hand, at the SL interface, the distribution shows variation in a narrow band at small ages (i.e., at 0.1 s; see Figure 7B, line 1). It becomes wider for ages near the peak of eddy age distribution (Figure 7B, line 2 for t ) 2 s) and again narrows down toward the larger ages (Figure 7C, line 3 for t ) 7 s). At the SL interface, the area distribution is observed to be narrower, compared to the VL interface, and also it is skewed toward smaller eddy sizes. This results in a small reduction in the HTC. )

Figure 7. Normalized size distribution function, based on area AE [of O(l2)] of flow structures and obtained by EIM-CWT for the CJR: (A) VL interface (line 1, ∆t ) 0.003 s; line 2, ∆t ) 0.02 s; line 3, ∆t ) 0.5 s) and (B) SL interface (line 1, ∆t ) 0.1 s; line 2, ∆t ) 2 s; line 3, ∆t ) 7 s).

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Figure 8. Parity plot at the SL interface for the CJR between the hjexp and predicted hj, using (9) CWT, (2) EIM, (9) small eddy model, and (4) large eddy model. Also shown are data for (×) the surface renewal model and (]) the Higbie model.

Figure 9. Parity plot at the VL interface for the CJR between the hjexp and predicted hj using (9) CWT, (2) EIM, (0) small eddy model, and (4) large eddy model. Also shown are data for (×) the surface renewal model and (]) the Higbie model.

The results obtained from the modified capacitance model have been compared with the experimental HTC obtained from HFA. The values of hjexp have been compared with the calculated hj from the heuristic models presented in Table 3 for SL and VL interfaces as parity plots in Figures 8 and 9. Similar results are observed in the case of the SL interface, as shown in Figure 8. At the SL interface, the hjexp is observed in the range of 200-1300 W/(m2 K). The flow structures developed at the SL interface are mainly governed by the generation of turbulence at the wall as well as turbulence convected from the bulk by large eddies. The latter contribution is determined to be significant, compared to the first. Therefore, the flow structures at the interface can be characterized by the exposure time estimated from bulk turbulence parameters. At the SL interface, the surface renewal and Higbie models are determined to considerably deviate from the hjexp values. Because of the narrow age distribution, the most frequently observed exposure time (∼2.2 s) is very close to the average exposure time (∼2.31 s).

The large eddy model is observed to be in good agreement with the hjexp values, with slight overprediction, and this can be readily explained by length scale distribution, as shown in Figure 8. Unlike the VL interface, large length scale eddies are greater in number at the SL interface. Hence, the effective exposure time (1.15 s), based on integral length scale and rms velocity, is determined to be lesser than that from the Higbie model. The small eddy model underpredicts the experimental values of hjexp by 60%. This may be because the small eddy model considers the Kolomogorov length scale to be the controlling size group, and this range [O(10-6 m)] cannot be monitored by HFA. It is reflected in terms of a lower value of the effective exposure time (1.83 s). In the case of EIM based on the zero crossing and CWT based on the modified capacitance model, the length scale distribution shows a very narrow distribution for smaller ages, whereas the distribution is broadened for larger ones and becomes significant for ages close to the mean exposure time. In this way, the area-based corrections for the large-eddy age elements result in a net reduction of hj. Therefore, the hj value, based on the modified capacitance model, is determined to be lower than that obtained from the large-eddy model but lying close to the experimental values of hjexp. Also, the CWT and EIM methodologies show effective exposure times of 1.26 and 1.34 s, respectively, and hence, the CWT shows higher values of hj than EIM, making it more accurate. In a CJR at the VL interface, hjexp are observed in a range of 0.1-4 MW/(m2 K). All the theories have been compared based on the effective exposure time. This exposure time is defined as tc for Higbie, 1/S for Danckwerts, jl/uj E for single eddy, lI/u2,rms for large eddy models, and (υ/ε)1/2 for small eddy models (see Table 3). Using the experimentally monitored velocity values (u2), the exposure time and hj for the aforementioned models have been calculated (with the formulas presented in Table 3). In all the theories, the constant of proportionality is ∼1 (see Table 3). Therefore, the hj predictions are very sensitive to the calculated values of the effective exposure time of eddies. Figure 9 clearly shows that the Higbie and Danckwerts theories deviate significantly from experimental values, because the skewness of age distributions, toward small eddies at VL interface, is not taken in their formulation. The most frequently encountered exposure time is observed to be 0.018 s, whereas the mean exposure time is 0.031 s at (x1, x2) ) (0.005 m, 0.03 m). The small eddy and large eddy models have different considerations regarding the size group of eddies controlling the heat transfer at the interface. The large eddy model was specifically developed for GL interfaces in open channel flows where major turbulence generation is away from the wall. Therefore, the large eddies from the bulk come closer to the interface and renew the surface without significant breakup into small eddies. Therefore, in this formulation, integral length scale and rms velocity of the bulk are considered. The small eddy model was developed for flows such as falling films, bubble transport through pipelines, etc. This model considered the contribution of small eddies at the interface by considering the dissipation rate at the interface without going into more details of their formation. The dissipation rate was estimated from the bulk turbulence energy spectrum. In the present study, at (x1, x2) ) (0.005 m, 0.03 m), the effective exposure time for small eddy model (∼0.006 s) is much smaller than the large eddy model (∼0.0135 s). However, all these theories could not consider the combined effect of exposure time and total surface area of individual eddies at the interface. This effect has been taken into account by modified capacitance model and the model parameters such as eddy size, eddy energy, and age distributions

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have been estimated using the EIM and CWT methodologies. The area- and energy-weighted effective exposure times were determined to be 0.0023 s using EIM and 0.00192 s by CWT. 5.3. Drag Coefficient. In chemical engineering literature, the bubble, drop, and particle columns are respectively known as (a) gas-liquid bubble columns, (b) liquid-liquid spray columns, and (c) solid-liquid fluidized beds. In all of these cases, liquid forms the continuous phase and bubbles, drops, and particles form the dispersed phase in the respective three types of equipment. This equipment operates in the homogeneous (where dispersed-phase holdup is uniform everywhere) or heterogeneous (dispersed phase radial and axial variation exists) regime. The dispersed entities (bubbles, drops, and particles) have several similarities and differences: (i) the size (and distribution) and the shape (as well as the distribution) of the solid particles can be measured, whereas the measurement of size and shape distribution of bubbles and drops is (particularly under operating conditions) is difficult with the present status of knowledge; (ii) with regard to the boundary conditions at the interface: (a) no-slip boundary conditions prevail at the solid/fluid interface, (b) complete slip prevails at the gas/liquid interface when the interface is clean and devoid of either surface-active impurities or fine particles, and (c) different extents of finite slip at the gas/liquid and liquid/liquid interfaces exist. Henceforth, bubbles, drops, and particles will be collectively termed as particles, with the difference being the interface boundary condition. Three major types of interfacial forces are acting on the particles: (i) drag force, (ii) lift force, and (iii) virtual mass force. The formulation of these forces is given by the following equations:

( )

CD 3 |V - VL |(VG - VL) Drag force ) -
GFL 4 dB G

(8)

Lift force ) CL
GFL(VG - VL) × ∇ × VL

(9)

Virtual mass force ) -Cv
GFL

D(VG - VL) Dt

(10)

The contribution of these forces to the transport phenomena differs for bubbles, drops, and particles. These forces have been studied in the literature over a wide range of Reynolds number and the cases of (i) a single particle in an infinite medium, (ii) a single particle with the wall effect, and (iii) a multiparticle system in the absence and presence of a wall effect. These contributions for drag force have been briefly reviewed below. 5.3.1. Drag Coefficient under Terminal Conditions. The value of the drag coefficient can be derived for particles, bubbles typically when Re , 1. Above this Re value, the drag coefficient values are estimated using empirical correlations. Note that, for solid particles, the CD-Re relationship has a unique behavior, mainly because of no-slip boundary conditions, whereas for bubbles and drops, a large number of correlations are available where their predictions substantially differ from each other. This may be mainly due to the lack of knowledge of the quality of the interface and, hence, the interface boundary conditions. Furthermore, the extent of drag varies substantially from the front stagnation point to the rear stagnation point and it continues to change from point to point on the interface. In the published literature,89–93 DNS has been performed for spherical-shaped particles for the two cases of complete slip and no slip. It is now desired that the finite slip and nonspherical shapes be analyzed in the future work. 5.3.2. Drag Coefficient under the Influence of Wall Effect. The settling and rise of particles and bubbles can be said to be terminal when the container size is 10 times larger

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than the particle/bubble diameter. When the ratio of particle diameter to column diameter is 1, the value of m is 3.75 and 3.26 for clean and contaminated liquid, respectively. For ellipsoidal particles, the value of m is 1.48. For spherical cap particles, the value of m is expected to be zero, because the drag force is independent of Re. For some typical values of m, the velocity-hold-up relationship is shown in Figure 10.94,95 In multiparticle systems, when the motion of particles do not affect the flow field significantly, the slope of dispersed-phase holdup versus dispersed-phase superficial gas velocity is equal to 1/VB∞ and the value of m is 1. However, an increase in the particle population increases the turbulence intensity. This modifies the flow field around the particles and, hence, the drag on each particle. The rise velocity is less than that in an infinite medium and the slope is greater than 1/VB∞ and m approaches a value of 3.4 for Re > 1 (see Figure 11).43 5.3.4. Multiparticle Heterogeneous Dispersion. In the multiparticle heterogeneous regime, a hold-up gradient forms in the system, because of density differences. The bubbles in

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Figure 10. Effect of fractional gas holdup on slip velocity for various quality interfaces.

Figure 11. Fractional gas holdup, as a function of superficial gas velocity, for different Richardson-Zaki indexes (m): m ) 1 (curve 1) and m ) 3.4 (curve 2).

the center rise faster, compared to the homogeneous regime, and, hence, experience lower drag. Because the energy transferred is equal to the drag force multiplied by the slip velocity, the energy transfer rate is high in the heterogeneous regime. The transferred energy is utilized by the liquid phase to develop circulation cells. The bubble size is greater in the central region and coalescence and breakup frequently occur. The liquid flows downward near the wall, where some bubbles may get trapped and move downward in the wall region. Thus, the direction of energy supply is from bubbles to liquid in the central region and liquid to bubbles in the wall region. The actual mechanism of energy transfer from bubble interface to liquid is still a mystery for turbulence researchers. There are many possibilities by which energy transfer can occur. Some of these are listed below: (1) Vortex breaking and dissipation of turbulent kinetic energy on the surface of immersed objects

Figure 12. Profiles of turbulent eddy diffusivity from various approaches (smooth lines show the best fits) at H/T ) 3.2. The values of υt from the k-ε model use modified CD values, which would give a reasonable match with the profiles from EIM.

(2) Wakes and shedding of the vortices behind the immersed objects (3) Enhancement of fluid velocity gradients between two neighboring immersed objects (4) Deformation and vibrations of the surface of the immersed objects In the literature, to understand the interaction between bubbles and its effect on liquid-phase turbulence, efforts are ongoing using a population balance approach.3,96 Population balance is a well-established method for computing the size distribution of the bubbles and accounting for the breakage and coalescence effects in bubbly flows. This approach is concerned with maintaining a record for the number of bubbles, whose presence or occurrence may dictate the behavior of the system under consideration. 5.4. Turbulent Viscosity from Energy Spectra. Eddy diffusivity is the means through which the time and length scale effects of turbulent flows are introduced into the equations of the mean flow. Thus, analogous to the kinetic theory, modeling υt requires specification about the local length and time scales (or, equivalently, the local velocity and length scales). The complexity in a turbulent flow field is mainly due to its sensitivity toward the various events that are happening over a range of wavenumbers/scales. The small scales include the dissipative range responsible for most of the energy dissipation and some tail portion of the energy transfer range (the inertial range, in the case of isotropic turbulence). The transfer range scales are large, compared to dissipative scales, but small compared to the large scales that extract energy from the mean flow. Hence, it is important to understand the interaction between flow structures and their break-up pattern. Various models (such as the P, L, P-L, and β models) are discussed in the previous part of this review.140 The local eddy diffusivity values from these various models at different radial positions (Figure 12) have shown that, in almost all cases, the value of υt remains almost constant in the central region of the column and then decreases toward the wall.97 However, a clear difference can be seen in all of the profiles, indicating that the mode in which intermittency is incorporated certainly makes a difference to the analysis of local turbulence. The comparison of the results from the eddy isolation model with υt from the random β-model

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Figure 13. Bubble column reactor: (A) homogeneous regime and (B) heterogeneous regime.

and the P-model showed good agreement, with a deviation on the order of (3%-8% for higher values (i.e., in the center of the column), which is acceptable. 5.5. Dissipation Rate from Energy Spectra. Furthermore, if the ε distribution is known throughout the equipment, its volume integral gives the total energy dissipation rate (ED) in the tank. Its value can be compared with the energy introduction rate. A close observation of energy spectrum shows the existence of flow instabilities and flow structures. The characteristic flow structures have a signature over the energy spectrum, and one can get the frequency of occurrence of these structures. The knowledge of the ε distribution is very important from the point of view of the design of turbulence. However, ε can only be accurately evaluated from HFA data by constructing a 3D energy spectrum. Measurement of the three-dimensional (3D) velocity and over the entire domain using HFA is time-consuming, as well as costly. Thus, an alternative methodology is used in the literature,98–100 which is based on the structure function approach. This approach can be used for HFA data, provided the minimum possible distance between the two probes would be sufficiently close to lie within the inertial subrange. However, the simultaneous measurement at two neighboring positions is rarely practiced using HFA. In the framework of these limitations of HFA, the PIV emerges as a useful technique, because it gives simultaneous measurements at several locations. In the case of PIV, ε is evaluated on a plane using the structure function based method98,99 by maintaining the spatial resolution to lie within the inertial range. Deshpande et al.100 have developed an algorithm for the estimation of ε from PIV with the support of HFA data in a jet loop reactor (JLR). In their algorithm, they had incorporated an overall energy balance to make the algorithm robust. Initially, they assumed a set of parameters and constructed the 3D energy spectrum and checked for energy balance. This procedure was repeated by tuning the parameters until an energy balance was established. They compared their results with large eddy simulation (LES) and the agreement for ε was within 10%.

6. Design of Flow Pattern for Performance Improvement 6.1. Homogeneous and Heterogeneous Reactors: Bubble Column. Bubble columns are widely used for the absorption of gas with or without chemical reaction. Bubble column reactors are popular because of their simple construction, high gas hold-up and interfacial area, and high heat- and mass-transfer rates. The bubble column, when used as a reactor, consists of a vertical cylindrical vessel with a height-to-diameter ratio in the range of 1-20 (often 3-10). Gas is introduced at the bottom via a sparger. Sparger types include sieve plate, ring, spider, radial sparger, ejector, injector, etc. The bubble column is operated either in a semicontinuous mode (gas, continuous; liquid, batch) or in a continuous mode (gas and liquid both continuous). In the latter case, the liquid phase may flow either co-currently or counter-currently to the gas. In bubble columns, because the gas bubbles are dispersed in the continuous liquid phase, fractional gas holdup (
G) is an important design parameter that affects column performance. The most direct and obvious effect is on the column volume, because a significant fraction of the volume is occupied by the gas. The indirect influences have many implications. For instance, the possible spatial variation (particularly transverse) of
G causes pressure variation, which results in intense motion of the liquid phase. These secondary motions govern the rates of mixing plus heat and mass transfer.
G is strongly dependent on the superficial gas velocity and the nature of the gas-liquid system. Column diameter and height often have relatively less influence on
G. However, the quantitative effect is dependent on the regime of dispersion in bubble columns: homogeneous or heterogeneous (see Figure 13). The homogeneous regime (Figure 13A) is characterized by almost-uniformly sized bubbles. Furthermore, the concentration of bubbles is also uniform, particularly in the transverse direction. If the gas is sparged uniformly at the bottom of the column, it remains uniformly distributed across the column. All bubbles rise virtually vertically with only minor transverse and axial oscillations and practically no coalescence or redispersion. Hence, the size of the bubbles in the homogeneous regime is

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Table 4. Variation of Mixing Parameters for Two Different Column Diameters (T ) 0.2 m and T ) 2 m): (A) Homogeneous Regime and (B) Heterogeneous Regime T ) 0.2 m VG (m/s)

εG

u′ (m/s)

0.01 0.02 0.03 0.04 0.08 0.1 0.12

0.04 0.08 0.12 0.16 0.32 0.4 0.48

0.015 0.03 0.045 0.06 0.12 0.15 0.18

0.01 0.02 0.03 0.04 0.08 0.1 0.12

0.031 0.059 0.083 0.105 0.174 0.200 0.222

0.041 0.055 0.066 0.076 0.105 0.116 0.126

VC (m/s)

Dt (m2/s)

T)2m tmix (s)

a_ (m2/m3)

HTC (W/(m2 K))

VG (m/s)

εG

u′ (m/s)

VC (m/s)

Dt (m2/s)

tmix (s)

a_ (m2/m3)

HTC (W/(m2 K))

48 96 144 192 384 480 576

1042 2085 3127 4170 8340 10425 12510

38 71 100 126 209 240 267

6147 8229 9864 11258 15565 17265 18773

(A) Homogeneous Regime 48 96 144 192 384 480 576

1042 2085 3127 4170 8340 10425 12510

0.01 0.02 0.03 0.04 0.08 0.1 0.12

0.04 0.08 0.12 0.16 0.32 0.4 0.48

0.015 0.03 0.045 0.06 0.12 0.15 0.18

(B) Heterogeneous Regime 0.217 0.290 0.348 0.397 0.549 0.609 0.662

0.014 0.019 0.023 0.026 0.036 0.040 0.044

50 38 31 27 20 18 16

38 71 100 126 209 240 267

2875 3849 4614 5266 7280 8075 8781

governed mainly by the sparger design and the physical properties of the system. In contrast, the heterogeneous regime is characterized by nonuniform bubble concentration, particularly in the transverse direction. The
G profile is usually parabolic (with maxima at the center) and results into pressure profiles with a minima at the center. These profiles cause upward liquid circulation in the central region and downward near the column wall. In such a recirculatory flow, turbulence is much higher, compared to the homogeneous regime, where the recirculation is absent. In the heterogeneous regime (Figure 13 B), the bubbles retain their identity only over a small distance from the sparger, and this region is called the sparger region. In the bulk region, the bubble size is governed by the coalescence and redispersion phenomena. A wide bubble size distribution occurs in the bulk, and the average bubble size (which is called the secondary bubble size, dBS) is decided by the balance between the breaking (viscous and turbulent shear stress), the retaining (surface tension) forces, and the coalescing nature of the liquid phase. The fractional gas holdup is related to the superficial gas velocity (Figure 11) as
¯ G ∝ V Ga

(12)

The value of a is usually equal to 1 in the homogeneous regime, whereas it is 99% can be achieved in a residence time of a few seconds. The schematic diagram of coaxial cylinders [(1) and (2)] is shown in Figure 1834 (the cylinders are identified as “1” and “2” in the figure). The immiscible feed liquids enter at points “3A” and “3B”, into the annular region of the two cylinders. The rotating impeller imparts power (in the range of 1-500 kW/m3), which results in a very fine dispersion of the two immiscible liquids. The dispersion flows downward in the annular region (where the mass transfer occurs) and then flows radially inward in the region below the rotating cylinder (“4A” and “4B” in Figure 18) and finally enters the central opening of the rotating cylinder (“5” in Figure 18). Baffles (marked as “6” in Figure 18) are provided in the bottom region, which are either attached to the base of the outer cylinder (as shown in Figure 18) or attached on the bottom of the rotating cylinder (the role of these baffles will be explained later). The dispersion

Figure 18. Schematic diagram of annular centrifugal extractor. Legend: (1) stationary cylinder; (2) rotating cylinder; (3A) light phase inlet; (3B) heavy phase inlet; (4A and 4B) region below rotating cylinder; (5) central opening for rotating cylinder; (6) radial baffles on the stationary bottom plate; (7) deflecting baffle in the rotor; (8) vertical baffles in the rotor; (9) interface between air and the light phase; (10A and B) overflow weirs for lighter and heavier phase, respectively; (11) clean width for heavy phase; (12) clean width for light phase; (13A and 13B) outlets for light and heavy phases, respectively; and (14) liquid level in the annulus.

entering the central hole gets deflected toward the wall by the horizontal baffle (“7” in Figure 18) provided close to the entrance. Above the level of point “7”, the rotor is provided with vertical baffles (marked as point “8” in the figure), to create several chambers ranging from 4 to 8. The rotating cylinder gives the liquid a practically rigid body rotation, the inner surface of which is almost vertical in shape (“9” in Figure 18), because of high g, except a small parabolic portion at the bottom. The dispersion entering at the bottom gets separated as it moves upward, as shown in Figure 19. The rate of separation is dependent on the drop size distribution, their settling velocities under the centrifugal action (rω2), densities, viscosities, and the coalescing behavior of the two phases. For complete separation (which is considered to be a flagship advantage of ACEs), adequate height must be provided for a given level of (rω2). After complete separation, as shown in Figure 19E, the overflow weirs (“10A” and “10B” in Figure 18) are provided in such a way that only very clean light and heavy phases pass over the weirs. The size and location of the weirs are provided in the hardware, according to the relative flow rates of heavy and light phases and their corresponding clean widths (“11” and “12”) that are shown in Figures 18 and 19. The flow of liquids from points 3A and 3B to 13A and 13B, respectively, passes through the steps of extraction and separation. The ACE can be considered as one perfect theoretical stage. However, to improve its efficiency further, the concentration gradient is desired from the inlet to outlet in the annular zone. This will improve the overall rate of mass transfer. The benefit is enhanced if the extraction is accompanied by chemical reaction. Therefore, it is highly desirable to convert the completely back-mixed behavior of the annular region to almost-

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Figure 19. Separation of dispersion in the rotating cylinder at different levels in annular centrifugal extractors (ACEs).

plug flow behavior. This can be done by incorporating radial baffles and/or helical baffles. The spacing of radial baffles and helix design can be selected by understanding the turbulent structures inside the annular region. In the annular region at higher stirring speeds, turbulent Taylor-vortex flow (TTVF) prevails. Between the two Taylor vortices, mass is transferred by intervortex recirculation. The baffles are added in such a way that the recirculation between the vortices is reduced to a greater extent. Furthermore, in the presence of these baffles, the number of vortices is typically twice the number of baffles. Whereas, in the absence of baffles, the number of vortices decreases with an increase in power consumption and it can be even 1, resulting into completely back-mixed behavior. Thus, the combined effect (of increased number of vortices and reduction in the interstage recirculation) can be designed to obtain almost-plug-flow behavior. 6.6. Gas-Inducing Impellers. Gas-inducing impellers are advantageous in situations where an internal recycle of unreacted gas is desirable. This condition arises in several industrially important reactions, such as hydrogenation, alkylation, ethoxylation, ammonolysis, oxidation with pure oxygen, hydrochlorination, etc. In these cases, the use of a gas-inducing impeller is more beneficial than a recycle gas compressor, for the reasons of safety, economy, and reliability. A large number of designs for gas-inducing impeller have been in operation and reported in the published literature.31 The two widely used designs are (1) stator-rotor and (2) hollow shaft-hollow impeller. The mechanism of gas induction has been described by Mundale and Joshi,107 Joshi et al.,108 and Murthy et al.109–111 When the impeller speed is zero, the level of liquid in the hollow pipe and in the vessel is the same. As the impeller speed increases, at any point on the impeller, pressure reduction occurs due to an increase in the kinetic head, according to the Bernoulli equation. At critical impeller speed (NCG), the reduction in pressure at the outlet of orifice on each impeller blade is sufficient to overcome the static head of liquid and the gas is just induced. The determination of the gas induction rate (QG) is an important step in designing the hollow impeller system. The gas induction rate is dependent on the pressure driving force

(local pressure at the orifice-head space pressure) generated due to impeller rotation. From the aforementioned discussion, it is clear that the local pressure at the orifice is an important parameter that determines the rate of gas induction. When the gas starts to be induced into the liquid, the local pressure field itself is altered. This occurs mainly due to the gas-liquid interactions in the impeller zone. The size of bubbles and turbulent flow structures affect the pressure field and, in turn, the rate of induction. The induction rate can be maximized either by (i) putting holes in the low-pressure region, (ii) minimizing the pressure drop associated with gas flow, and (iii) maximizing the pressure differential by changing the shape and/or design of the impeller. Various impeller designs such as disk turbines (radial flow) or pitched-blade turbines (PBTDs) (axial flow) and their modifications can be used as gas-inducing impellers. The location of holes can be identified from the low-pressure region near the impeller surface. In some cases, where solid catalysts are used in dead-end systems, the impeller has two roles to play: (i) maximize the gas induction rate and (ii) complete suspension of solid catalyst. The subject of optimization through the design of the flow pattern has been discussed by Joshi and co-workers,31,108–128 as well as others.129–132 6.7. Turbulence Promoters in Heat Exchangers. Many times, because of the constraints of space and higher heat duties, design engineers look for alternatives that can provide higher heat-transfer coefficients. Different choices are available for such retrofitting, but energy-efficient designs are expected to involve substantial inputs of the physics of turbulence. To enhance the heat-transfer rates, the design modifications that are practiced in the literature are surface coatings, static mixers, roughening surfaces, extended surfaces, swirl flow devices, the convoluted (twisted) tube, and additives for liquid and gases. These modifications improve turbulence and/or swirl, which, in turn, improve the heat-transfer characteristics. However, in most of the cases, turbulence is increased everywhere in the flow, whereas for heat transfer, a localized increase in turbulence close to the wall is sufficient. The heat transfer in the pipe increases when there is an increase in structure dynamics that encourages the renewal of structures on the heat-transfer surface. It is

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Figure 21. Variation of energy efficiency with Nusselt number (Nu) for various turbulence promoters, Legend of curves: (1) empty smooth tube, (2) V-nozzle (PR ) 7), (3) V-nozzle (PR ) 4), (4) V-nozzle (PR ) 2), (5) conical ring (PR ) 0.1), (6) conical ring (PR ) 0.2), (7) conical ring (PR ) 0.3), (8) helical screw (twist ratio ) 1.5), (9) corrugated tube (h/d ) 0.05, p/d ) 1.02), and (10) microfin.

Figure 20. Turbulence promoters in heat exchangers: (a) V-nozzle (Eiamsaard and Promvonge133), (b) conical rings (Yakut et al.134), (c) microfins (Dong and Kyu134), (d) helical screws (Sivashanmugam and Suresh136), and (e) corrugated tubes (Vicente et al.137).

observed that, with an increase in Re or local turbulence, a greater number of small-scale structures with smaller age exists near the wall. This causes an enhancement in the heat transfer. It has been observed that, with an increase in Re, or localized in the boundary layer region of channel flow, the age distribution approaches a higher skewness value, which suggests a higher number density of smaller eddy ages. The skewness of distribution increases by 10% as the Re value is increased by a factor of 3. Hence, a good design should ensure structure age distribution that has higher skewness near the wall, which should come from the creation of higher shear in the near wall region, thus enabling smaller structures near the wall. However, an increase in localized structure dynamics (turbulence) also results in an excessive increase in pressure drop and therefore leads to energy inefficiency. Hence, the process design is always concentrated toward making a good optimum design, which can be achieved via the manipulation of turbulence close to the wall, which is responsible for heat transfer, and not wasting the input energy in creating turbulence in the bulk. In the present work, only few design modifications such as V-nozzle, conical rings, helical screws, microfins, and corrugated tubes have been discussed133–137 (see Figure 20). An attempt has been made to address the modifications of the flow structures, because of the

design changes and their effect on heat transfer and pressure drop. A comparative study of the heat-transfer efficiency of these inserts against a plain tube has been presented. An increase in heat transfer cannot be obtained without a corresponding increase in pressure drop. For a plane tube, the heat transfer increases as u20.8, whereas the pressure drop increases almost proportional to u21.8. Therefore, to take this fact into account, the heat-transfer efficiency (ηh) has been defined as the ratio of the amount of input power (∆P × Q) required for a given heat duty for a plain tube to the power input (∆P × Q) for a particular design modification. Figure 21 shows the heat-transfer efficiency versus the Nusselt number for different designs. The reference line for a plain pipe is plotted at ηh ) 1. The efficiency of external inserts such as V-nozzle and conical rings is less than that of a plain pipe and continues to decrease as the heat-transfer coefficient requirement continues to increase. Therefore, such inserts are very energy inefficient, because they generate turbulence everywhere. The geometric structure of the conical rings is such that it allows periodic redevelopment of the boundary layers. The V-nozzles also help to reduce the thickness of the boundary layer. The helical screws are the most energy inefficient designs, as shown in Figure 21. These helical screws impart a helical path to the flow of the fluid. These all inserts suffer from a problem of excessively high pressure drop. These strategies represent a simple retrofitting of the heat exchangers available in practice. Spirally corrugated tubes have become important in commercial applications for turbulent single-phase flow, because the pressure drop increment is generally compensated by the heat-transfer augmentation. Corrugation increases the heattransfer coefficient in turbulent flow by mixing the flow in the boundary layer and also by increasing the turbulence level of the fluid flow. Figure 21 shows that the corrugated tubes are energy-efficient, compared to V-nozzles, conical rings, helical screws, and plain smooth pipe. The popularity of the corrugated tubes in the industry can also be attributed to the fact that their fabrication is easy and the material required for fabrication is same as that of plain smooth pipe.

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Another method of heat-transfer augmentation is at the stage of initial design itself by corrugating the inside surface or providing microfins over the surfaces. At very low Re values, their efficiency is low; however, with an increase in Re, the efficiency increases dramatically. This can be attributed to the fact that the size of the roughness elements is bigger than the viscous sublayer at higher Re. The roughness generates a large amount of additional turbulence, which increases the turbulent viscosity and the turbulent diffusivity near the wall, so the flow resistance and the heat-transfer rate are increased. These roughness elements break the sweeps and ejections in the sublayer more frequently, and, hence, surface renewal rates are enhanced dramatically. 6.8. Packed Tower Design. Packed-bed reactors are filmtype contactors, where the energy input is in the form of potential energy. The liquid is pumped at the top and it flows as a thin film over packing; gas is introduced in either a cocurrent or counter-current manner. The major internals that affect the performance of packed columns are liquid distributor and type of packing (shape and wetting characteristics). The packing material can be broadly classified into (i) random packing and (ii) structured packing. Random packings are dumped into the tower during installation and allowed to arrange in a random structure. The history of random packing is a long and interesting one. In the past, the packings were irregularly shaped bodies, such as broken stones, rock, tile pieces, and gravel or lumps of coke. Such readily available materials were often inexpensive, but they had a problem of small surface area per unit volume and poor fluid flow characteristics (larger ∆P per unit height). Furthermore, in the past, coke was extensively used in the tower, which handled SO2 and other acidic gases, as well as in the installation, which removes sulfuric acid mists, because of its relative chemical inertness, inexpensive cost, and availability. The use of coke as a tower packing medium has now been almost entirely abandoned, primarily because of its tendency to channel and disintegrate gradually. The improvement has taken the direction of fabricated packings made of chemical stoneware, porcelain, carbon, and metals. Fabricated tower packing are used extensively in nitric, sulfuric, and hydrochloric acid absorption towers; sulfur dioxide, ammonia, chlorine, and carbon dioxide recovery and entrainment towers; and scrubbers, distillation columns, and many types of gas-drying installations. A large variety of packings is available, and making the correct choice for a specific installation is by no means a simple task; it requires an understanding of flow pattern over the packings. The selection should take into account all of the factors given below, which would influence the operation (from Leva17): (1) High surface area per unit volume of packings; (2) High ratio of surface area to total area; (3) High percentage free space (voidage), which should permit the passage of large volumes of fluid through small tower cross sections without loading or flooding; (4) Irregularity in shape to prevent pattern-like packing; (5) Favorable liquor distribution qualities; (6) Low apparent density and high unit strength; (7) Low cost; (8) Low pressure drop; (9) Corrosion-resistant; (10) Durability; and (11) Furthermore, gas pressure drop should be largely the result of skin friction if possible, because this is more effective than form drag in promoting high values of the mass-transfer coefficients.

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The design of flow patterns for performance improvement has a significant role in satisfying these criteria (point 11). The historic development of random packings is covered by Leva,17 Strigle,18 Coulson and Richardson’s Chemical Engineering,138 and Schultes.139 6.8.1. Random Packings. The most common and oldest tower packing is the Raschig ring. Raschig rings are composed of porcelain, chemical stoneware, special ceramic bodies for alkali services, and carbon, as well as of steel and other metal alloys (see Figure 22A). The bulk of the absorption data presented in the literature pertains to the Raschig ring. These rings are essentially pieces of a pipe where the length equals the outside diameter. However, the inside diameter and, consequently, the wall thickness of the Raschig ring may differ considerably. Furthermore, the apparent density and resulting free space are directly related to the wall thickness. Because it is most desirable to have a high fraction of free space and a low apparent density, it is obvious that the better rings should have a smaller wall thickness. The Lessing ring is a modification of the Raschig ring. A web has merely been added inside Raschig ring (see Figure 22B). This packing is frequently made of metal and, for this reason, very small wall thicknesses are possible, as well as smaller sizes. This modification has provided more surface area without much increase in packing material and pressure drop. The quest in the development of packings is to increase the area of contact (maintaining film flow of liquid, which will enhance mass transfer) and reduce pressure drop. A further modification of the Lessing ring is the Cross partition ring, which has, in its center, two webs that cross each other at right angles (see Figure 22C). This type of ring is called as a four-cell ring, in contrast to another type of partition ring that has only three walls. The refined modifications to the partition ring are the spiral-type rings, which can be furnished with a single, double, or triple spiral. The spiral nature provides greater turbulence in the gas stream without disturbing liquid flow and thereby improves the mass-transfer characteristics in the column. The single spiral rings are almost as effective a contacting medium as ordinary partition rings. In multispiral rings, the gas and liquid flow through the interstitial spaces between the packing, in preference to the inside spiral. The spiral rings of all sizes and types have found wide applications in many largesize packed-tower installations. On another front, the modification consists of opening the sides of the conventional Raschig ring, which is called a Pall ring (see Figure 22D). The Pall ring has a significant advantage over the conventional Raschig ring. This would, of course, be expected, because in the conventional Raschig ring, the inside surface area is not accessible. Furthermore, the openings reduce the size of vortices behind the packings and, hence, result in low energy dissipation rates and a pressure drop. By opening the sides, the inside area of the ring has apparently been made more accessible. Of course, the large sizes are always used in stacked beds and their efficiency should be expected to be close to that of ordinary three- or fourcell partition rings. The rings can also be made of metal/plastic and in this construction material, sizes of