Review of Bubble Column Reactors with Vibration - Industrial

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Review

Review of Bubble Column Reactors with Vibration Brian R. Elbing, Adam L. Still, and Afshin Ghajar Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.5b02535 • Publication Date (Web): 22 Dec 2015 Downloaded from http://pubs.acs.org on December 26, 2015

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Industrial & Engineering Chemistry Research

Review of Bubble Column Reactors with Vibration Brian R. Elbing*, Adam L. Still† & Afshin J. Ghajar Mechanical & Aerospace Engineering, Oklahoma State University, 218 Engineering North, Stillwater, OK, 74078, United States of America

Abstract Vibrating a bubble column reactor can increase the gas holdup (void fraction) as well the mass transfer rate. Since the seminal work in the 1960’s, there has been minimal effort focused on this topic until the early 2000’s. Currently there are several groups studying this problem making advancements in our fundamental understanding of the process with detailed experiments, theoretical analyses and physics based models. However, throughout the literature there are inconsistencies with both experimental results and proposed scaling of the fundamental properties as well as minimal data spanning the parameter space. This review serves as an overview of key works from the 1960’s and the 2000’s as well as to identify these inconsistencies between key studies. Recommendations for how to proceed with future work is provided with an emphasis on defining the parameter space in terms of the Reynolds number and Froude number. Keywords: bubble column, multiphase flow, fluid mechanics, void fraction, mass transfer

* †

Author to whom correspondence should be addressed: [email protected] A.L. Still is currently at Sandia National Laboratory, Albuquerque, NM, USA.

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Review of Vibrating Bubble Column Reactor

Page 2 of 59 Elbing, Still and Ghajar

1 Introduction 1.1 Motivation Numerous real-world systems involve multiphase flow, which produce complex flow patterns that are dependent on the balance of forces and phase distribution. The control of these flows for engineered systems, such as piping networks for the petroleum industry, require a complete understanding of the phase interactions with the boundary conditions and the other phase(s). However, for many of these flow-fields there is relative velocity between phases and nonhomogeneous distributions that prevent the governing equations from being solved as a mixture with average fluid properties. Consequently, each phase has to be solved individually, which requires an understanding of the mass, force and energy interaction terms between the phases. Currently these relationships are unknown, which forces researchers to form heuristic models for a specific flow pattern.1 For this reason, the study of multiphase flows are typically confined to specific flow configurations and rely heavily on experimental data and empirical modeling. The current review focuses on a bubble column reactor (BCR) with vibration, which involves two phases interacting with each other to dissolve the dispersed phase (gas) into the continuous phase (liquid mixture). BCRs are widely used throughout the chemical, biochemical and petrochemical industries due to their simple design, low cost, compactness, ease of operation and high heat/mass transfer rates. Some BCR applications include waste-water treatment, aeration of organic organisms in bio-reactors, solvent gasification for chemical reactions and hydrogenation of coal-slurries to produce synthetic fuels. The last example has gained significant attention recently due to several military and civilian aircraft being qualified to fly with synthetic fuel

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Industrial & Engineering Chemistry Research Review of Vibrating Bubble Column Reactor

Elbing, Still and Ghajar

blends, which are produced with the Fischer-Tropsch process. United Airlines ran the first flight demonstration of a commercial airliner in the United States using synthetic fuel blends with an Airbus A319 in April of 2010.2 Currently there are several United States Air Force fighters and aircraft (e.g. A-10, C-17, KC-135 and F-22) approved to fly with synthetic fuel.3-6 This example, as well as numerous others related to chemical processing, highlight the ever increasing use and application of BCRs in our society. Thus there is also an increasing interest in the ability to control the physical processes within the reactor. While the primary limiting factor in FischerTropsch process example is not the mass transfer rate,7 mass transfer rate and gas holdup are critical parameters for any BCR application. It was discovered in the early 1960’s that vibrating a BCR could increase the mass transfer rate and gas holdup. While some additional research expanded the theory, minimal research effort focused on this phenomena until the early 2000’s. A summary of the primary studies spanning this range are provided in Table 1 along with the primary measurements from each study.8-23 Recent research has gone so far as to develop theoretical, physics based models to predict mass transfer and void fraction in these systems. While there is a nontrivial body of work on vibrating BCRs, to date the parameter space has been explored in a rather haphazard fashion. With numerous studies available there is a need for a systematic, dimensionally reasoned analysis of the available data to identify specific flow regimes and the flow physics that dominate the given range of conditions. The available models have been tested against limited data, but there are inconsistencies with both experimental results as well as the proposed scaling of the fundamental properties. In addition, the large gap in time between studies provide motivation to reexamine these past studies due to the significant advancements in experimental and computational tools. This review will provide a broad overview of previous studies, identify



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key findings that have established our current knowledge and note inconsistencies in the literature that need further examination.

1.2 Background It is beneficial to first review some basic physical properties and common definitions used studying vibrating BCRs. From an industrial point of view, the chemical reaction rate is typically the most important property. It is often assumed to be proportional to the volumetric mass transfer coefficient (kLa), which is the product of the mass transfer coefficient (kL) and the interfacial surface area per unit mixture volume (a). For simplicity, the volumetric mass transfer coefficient (kLa) is generally referred to as the mass transfer rate. For BCR’s as well as other aerated systems, kLa is used to relate the time rate of change of the concentration (dC/dt) to a concentration potential,

(

)

dC = kLa C * − C , dt

(1)

where C is the concentration of gas dissolved in the liquid and C* is the saturation concentration. If spherical bubbles are assumed, then the interfacial bubble area per unit volume (a) can be determined from the mean bubble diameter (db) and the void fraction or gas holdup (ε), a = 6ε d b . Here void fraction is defined as the ratio of gas phase volume (Vg) to the total

volume (Vg + Vl) in a multiphase system, ε = V g (V l + V g ) .

It should be noted that this estimate of a is an approximation used by some to simplify large reactor systems or for the purpose of modeling.20,22,24 The validity of this approximation is primarily dependent on the bubble size, which when db < 2 mm it is general assumed valid.25 At this size capillary effects typically dominate the governing dynamics. Experimental observations 4 ACS Paragon Plus Environment

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of bubbles in vibrating BCR show that typically db < 5 mm due to hydrodynamic breakup,8 which is smaller than that observed without vibration. While the average diameter indicates that the bubbles are nearly spherical, in most instances the bubble surface geometry constantly changes as it is formed and rises.26-28 These surface oscillations affect both the bubble volume and shape, but it is generally noted that the shape oscillations primarily influence mass transfer at the interface.29 While it is desirable to use a single characteristic length scale for the bubble diameter, there are inherent variations in the distribution of sizes and shapes that prevent a universally accepted length scale. One method measures the bubble chord length probability distribution and use the resulting mean diameter as the characteristic length.30-32 The advantage of this method is that it does not rely on the spherical bubble assumption, and it produces an accurate measure of the bubble size variation. However, many researchers16,22,33 prefer to use the Sauter mean diameter (d32), which is the ratio of the representative bubble volume to the bubble surface area, n

∑n d

3 b ,i

∑n d

2 b ,i

i

d 32 =

i =1 n

i

.

(2)

i =1

Here ni is the number of bubbles with size db,i. The Sauter mean diameter is commonly used when bubble size is measured from 2D bubble images. Here the projected bubble area (Aproj) in each 2D image is used to estimate an equivalent bubble diameter (deq) assuming a spherical bubble,16 d eq = 4 Aproj π . The equivalent diameter is used as the estimated bubble diameter (db,i) in Eq. (2) to calculate the mean Sauter bubble diameter for a given condition. The gas phase velocity is also a critical measurement, as this determines the bubble residence time. It is typically reported as the bubble velocity (Ub), gas phase velocity (Ug) and/or 5 ACS Paragon Plus Environment


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superficial gas velocity (Usg). While there is some variation in the use of each term in the literature, Ub and Ug refer to the actual measured phase velocity and Usg is a representation of the area or volume averaged phased velocity. Thus the superficial gas velocity is equal to the volumetric gas flux (Qg) divided by the column cross-sectional area (Acs), which can be related to the average gas phase velocity using control volume analysis, U sg = Q g Acs = ε U g . Here the brackets denote averaged quantities. It should be noted that Ub is often used in the literature as a specific description of a particular condition in which individual bubbles or clouds are measured. In stationary bubble columns with no flow Ub is simply the rise velocity. However, when there is flow in/out of the column, or when the fluid column is moving, Ub can be related to the slip velocity and the void fraction typically through a correlation.34 Specifically in the case of an oscillating BCR, Ub is an important indication of the forces acting on the bubble as will be discussed subsequently. Therefore, a fundamental understanding of the multiphase flow properties such as void fraction and bubble size distribution as well as the related mass transfer properties are crucial to understanding and thereby improving the operation of BCRs. A review of the relevant literature pertaining to these measurements in a vibrating or oscillating BCR is provided here. This review is divided into four sections. First, a review of the most pertinent work with an emphasis on the fundamental physics governing the observed phenomena. Second, a review of common experimental methods including testing procedures, facilities and instrumentation. The emphasis on experimental techniques is required since, as previously stated, this research area is heavily dependent on experimental data due to the need for heuristic models specific to a given flow regime. Third, there is a brief review of other relevant literature as well as computational efforts, which provide additional perspective as well as direction for future work. Finally, the manuscript


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concludes with an overview of the topic with an emphasis on what is the required knowledge to advance and implement vibrating bubble columns.

2 History of Vibrating BCR Research 2.1 Houghton and Harbaum (1960-1963) Harbaum & Houghton8 used Minnaert26 and Smith35 to reason that vibrating a bubbly flow could impact mass and heat transfer. This was confirmed by measuring CO2 absorption rate in a water column and its dependence on vibration amplitude and frequency. Subsequently, Harbaum & Houghton8 showed that the mass transfer rate was not necessarily linked to the amplitude, but attributed the increase to “resonance effects” that were only frequency (f) dependent. Then they attempted to differentiate the frequency effects on kL and a individually. The results (shown in Figure 1) indicate that an increase in mass transfer was primarily due to an increase in a associate with an increase in ε at specific frequencies rather than an increase in kL, which showed a marked dip at the same frequency. Experimental observations indicated that vibrations cause the bubbles to become smaller and more spherical, though elongated in the direction of motion. However, no quantitative discussion of bubble size was offered, only bubble count per volume.



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Figure 1. Data from Harbaum & Houghton8 showing the effect of vibration frequency on interfacial area per unit volume (a), bubble number density (n), bubble superficial gas velocity (Usg) and mass transfer (kL) for a water column with CO2. Adapted with permission from Journal of Chemical Technology & Biotechnology, vol. 12, K.L. Harbaum and G. Houghton, “Effects of sonic vibrations on the rate of absorption of carbon dioxide in gas bubble-beds,” 234-240. Copyright 1962 John Wiley and Sons. Houghton36 studied particles suspended in an oscillating velocity field by applying a force balance to a particle (gas phase in bubbly flow) in a velocity field u = Aω cos (ωt ) . Here A is the amplitude, ω is the radian frequency and more details about this field are provided in Still37. This analysis decomposes to a linearized steady state solution similar to a Mathieu equation when the transient drag effects approach zero. Solutions of the Mathieu equation can be readily interpreted using a Mathieu stability diagram (see Houghton36), with regions of the parameter space having stable solutions. The important implication of this analysis is that careful selection of the frequency and amplitude will result in a motionless bubble (with respect to a fixed reference). Houghton36 concluded that the particle behavior within an oscillating flow should be predictable based on drag coefficients obtained from terminal velocity measurements of the particles.


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2.2 Buchanan et al. (1962) During the same period, Buchanan et al.9 explored the effect of vibration on bubble migration in a liquid column. This was the first report of unified bubble migration, which acts against buoyancy at specific frequencies. Using electromagnetic-hydrodynamic analogies,38 a frequency was derived at which the bubbles are stable (stationary). Imagine a bubble placed at a location h below the free surface of a fluid body undergoing oscillatory motion z = A sin(ωt ) , where z is the vertical position relative to a fixed coordinate system as illustrated in Figure 2.

Figure 2. Diagram of a single bubble in an oscillating BCR (prime denotes varying quantity). The following analysis makes several assumptions; (1) the bubble’s resonant pulsation frequency is greater than the vibration frequency, (2) the bubble is sufficiently large that surface tension can be neglected, (3) the bubble expands and contracts isothermally, (4) the bubble internal pressure follows Boyle’s law and (5) spherical momentum effects caused by the bubble surface oscillation are negligible. Here the instantaneous total pressure can be divided into three components pT = pe + ps + pv , where pe is the ambient pressure, ps is the hydrostatic pressure (= ρgh), g is the gravitational acceleration and pv is the pressure due to vibration. The pressure field can be solved for by using the unsteady, incompressible Navier-Stokes equation and applying the boundary

condition

that

pT = p e

when

h

=


0.

From

this

solution,


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pT = pe + ρgh − ρhω 2 A sin(ωt ) , it is apparent that the contributions due to the column vibrating is pv = − ρhω 2 A sin ωt . The instantaneous bubble volume can be written as V = Vo + ∆V , where Vo is the initial bubble volume (i.e. when pv = 0) and ∆V is the deviation from Vo. Inserting the total pressure and instantaneous bubble volume into Boyle’s law ( p oVo = pV ) and rearranging produces

ρhω 2 A sin (ωt ) ∆V = . Vo + ∆V po

(3)

Here p0 is the total pressure without oscillation (or at the initial condition) p 0 = p e + ρgh . The maximum volume displacement is achieved at the top of the stroke when sin(ωt) = 1. Therefore Eq. (3) can be used to establish a relationship for the maximum volume displacement,

∆Vmax ρhω 2 A = . V0 p0 − ρhω 2 A

(4)

Bjerknes38 states that a body in translational motion within a fluid is subject to a buoyancy force that is the product of the body’s acceleration and the mass of the fluid displaced by the body. Applying this principle to a force balance on the bubble results in a temporally evolving force,

(

)

F (t ) = ρ (V0 + ∆V ) g − ω 2 A sin(ωt ) . It is important to note that Buchanan et al.9 erroneously neglected the liquid density term in their derivation. Addition of the term is required to properly account for the mass of the fluid. Integration of the instantaneous force over a period of oscillation (T) gives the average force on the bubble, T

Favg =

1 1   F (t )dt = ρ V0 g − ω 2 A∆Vmax  ∫ T0 2   10 ACS Paragon Plus Environment

(5)

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When the average force is zero, the bubble motion relative to the fluid is zero. Thus, setting the average force to zero and applying Eq. (4) produces a quadratic equation that can be solved for the stabilization frequency (ωs) required for a stationary bubble,

ω s 2 A = − g ± 3g 2 +

2 gpe . ρh

It is interesting to note that this frequency is independent of the volume, which is due to the fact that in this analysis gravitational acceleration is balanced with the unsteady acceleration. Buchanan et al.9 performed this analysis and compared the predicted stabilization frequency to that of an experimentally determined “cut-out” frequency (ωc). The cut-out (or critical) frequency was defined as the minimum frequency where bubble cyclic migration was observed, which corresponds to when bubbles move against gravity. If the assumptions used in the analytical analysis were valid then the stabilization frequency should be nearly equal to the cutout frequency, since the zero bubble velocity condition would be the boundary between ascending and descending bubbles. The cut-out frequency results9 are plotted versus the stabilization frequency in Figure 3, which shows excellent agreement noting that the solid line corresponds to ωc = ω s . 2

2



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Figure 3. Data for different column inner diameters (ID) from Buchanan et al.9 of the cut-out frequency (ωc) plotted versus the stabilization frequency (ωs) for a water column with air injection. Data was acquired at atmospheric pressure and room temperature. Adapted with permission from Industrial & Engineering Chemistry Fundamentals, vol. 1, R. H. Buchanan, G. Jameson and D. Oedjoe, “Cyclic migration of bubbles in vertically vibrating liquid columns,” 82-86. Copyright 1962 American Chemical Society.

2.3 Baird et al. (1962-1963) Around this same period, Baird & Davidson39 investigated gas absorption by singularly rising CO2 bubbles in a stationary BCR. This work provided two key insights into gas absorption (mass transfer) in stationary liquid columns, which are critical to understanding the more complex condition with vibrations. First, the mass transfer coefficient (kL) of smaller bubbles (deq < 25 mm) rising at a steady rate within a stationary water column is not (or weakly) time dependent. This is demonstrated in Figure 4 with mass transfer coefficients for various bubbles sizes39 plotted as a function of time. Here the smallest bubbles have negligible variation in the mass transfer coefficient over time while the larger bubbles decay proportional to t-0.5.


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Figure 4. Mass transfer coefficient as a function of time for CO2 bubbles in a quiescent water column. Data and curves are from Baird & Davidson.39 Adapted from Chemical Engineering Science, 17, M.H.I. Baird & J.F. Davidson, “Gas absorption by large rising bubbles,” 87-93, Copyright 1962, with permission from Elsevier.

Figure 5. Surface tension effects on mass transfer coefficient of CO2 in solution. Figure has been adapted from Baird & Davidson39 with each data point representing the average of several single bubble measurements. Adapted from Chemical Engineering Science, 17, M.H.I. Baird & J.F. Davidson, “Gas absorption by large rising bubbles,” 87-93, Copyright 1962, with permission from Elsevier. The second key insight from this work is a corollary to the first, which proposed that circulation and renewal effects from the bubble wake might be responsible for establishing a steady-state condition. It was noted that a stagnation layer was established for bubbles rising in 13 ACS Paragon Plus Environment


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n-hexanol and n-butanol solutions, where surface tension between CO2-solution was lower than CO2-water. In essence, the reduction of the turbulent wake behind the spherical bubble due to surface tension “smoothing” the interface reduces the concentration gradient. Thus the gas phase is insulated from the liquid by a thin boundary layer of partially dissolved gas-solution, which would be mixed with the unsaturated liquid if there were more turbulent mixing. This was demonstrated in Baird & Davidson39 (and shown in Figure 5) noting the decrease in the wake profile and subsequent decrease in kL with the addition of surfactant (Lissapol). This theory could explain the increase in kLa with vibration, the oscillating shear forces cause the bubble boundary layer to detach, which prevents the buildup of the high-concentrated layer around small bubbles. Baird11 acquired measurements of the resonant frequency of stationary singular bubbles and slugs within a vibrating liquid column. These results were compared with established theoretical estimates of bubble resonant frequency (fr).26 The theoretical estimates matched experiments between 5-85%. Consequently, Baird11 proposed a modified relationship that included secondary effects (e.g. confinement effects and bubble depth),

1 fr = 2π ravg

 3 γ po     ρ 

1/ 2

  ravg 1 +    R  

 4h    − 1     R

−1 / 2

,

which matched observations to within ±20%. Here ravg is the mean radius of the bubble, γ is the specific heat ratio of gas (= 1.4 for air), ρ is the fluid density and R is the column radius. In addition, stroboscopic photography was used to confirm that bubbles do reach a maximum volume at the top of the stroke, which supports the analysis that produced Eq. (5). In addition, it was observed that the larger and more visible slugs experienced larger expansion amplitudes than contraction amplitudes, which leads to an amplitude disparity. This disparity in slug motion could be applied to smaller bubbles, and can be explained by Bjerknes38 who notes that “the


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light…body [bubble] will in the two extreme positions be in different masses of [liquid], and if these have not exactly the same motion, it will be subject in these two positions to kinetic buoyancies not exactly equal and not exactly opposite in direction.” Since these pioneering efforts, Baird and coworkers have continued advancing these concepts and related applications. Additional details from this group’s work will be discussed in the experimental methods section, but the interested reader is directed to Ni et al.44 for an excellent review of recent activity with an emphasis on baffled columns and their application to producing enhancements for the chemical and process industries.

2.4 Krishna, Ellenberger and coworkers (2000-2007) There was minimal research activity on vibrating and pulsing BCRs between the 1960’s and the start of the 21st century. Krishna et al.33 provided data on bubble break-up within a bubble column vibrating at 100-200 Hz and the associated energy requirements. Krishna & Ellenberger17 provided a more detailed investigation reexamining measurements of void fraction and mass transfer in a vibrating BCR. Here a marked improvement in void fraction was observed with vibration (see Figure 6), which had a local maxima ε at specific “critical” frequencies (Figure 7). Here Ho is the water column height measured above the injection location with no air injection and H is the column height for the given condition.



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Figure 6. Images illustrating typical improvement in the void fraction (gas holdup) with vibrations from Krishna & Ellenberger;17 (a) no gas injection, (b) gas injection with no vibration and (c) gas injection and vibration. Reprinted from International Journal of Multiphase Flow, 28, R. Krishna & J. Ellenberger, “Improving gas-liquid contacting in bubble columns by vibration excitement,” 1223-1234, Copyright 2002, with permission from Elsevier. Optimum combinations of frequency and amplitude were found to double the void fraction as the gas superficial velocity was increased. The results showed that kLa increased 1.5-2 times more than the increase in void fraction at higher frequencies (see Figure 7). However, contrary to Harbaum & Houghton,8 Krishna & Ellenberger17 proposed that the mass transfer rate increase was not solely due to an increase in interfacial area per unit volume (a). They claimed that due to an increase in turbulence the mass transfer coefficient (kL) could have also increased. However, Krishna & Ellenberger17 did not independently measure kL and a as Harbaum & Houghton8 did to base their conclusions upon. It is possible for both assessments to be valid if two mechanisms are active with one or the other dominating for a given flow regime.


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Figure 7. Vibration induced improvement of volumetric mass transfer coefficient and void fraction (gas holdup) with Usg = 1.0 cm/s and A = 0.5 mm. Mass transfer and void fraction data from Krishna & Ellenberger17 scaled versus the mass transfer coefficient (kLa)o and void fraction εo with no vibration, respectively. Adapted from International Journal of Multiphase Flow, 28, R. Krishna & J. Ellenberger, “Improving gas-liquid contacting in bubble columns by vibration excitement,” 1223-1234, Copyright 2002, with permission from Elsevier. Ellenberger & Krishna34 also attempted to distinguish between the effects of vibration frequency and the amplitude on void fraction. Their results show increased void fraction for both increasing frequency and amplitude, but the general trends differ in shape, as illustrated in Figure 8. Here increasing the vibration amplitude has a more abrupt impact on the void fraction compared to increasing the frequency. For example, examining the condition of 1 cm/s there is a 17% increase in void fraction between 40 and 60 Hz whereas a 50% increase in amplitude (0.5 mm to 0.75 mm) increases the void fraction 36%. This example is illustrated in Figure 8 with the overlaid dashed lines. Since a rise in ε is expected to directly increase a it suggests a greater increase in kLa can be achieved through higher amplitude as well.



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Figure 8. Data from Ellenberger & Krishna34 showing the influence of (a) vibration frequency and (b) vibration amplitude on void fraction for air injected into a vibrating water column. Dashed lines are used for reference to compare behavior at Usg = 1 cm/s. Adapted from Chemical Engineering Science, 58, J. Ellenberger & R. Krishna, “Shaken, not stirred, bubble column reactors: Enhancement of mass transfer by vibration excitement,” 705-710, Copyright 2003, with permission from Elsevier. While the experimental setup of Ellenberger & Krishna34 was limited to smaller amplitudes (A ≤ 1.2 mm), these results are plotted in Figure 9 and provides the relationships between the vibration amplitude and the void fraction (ε) or the single bubble rise velocity (Ub). These have inverse trends with the void fraction increasing with amplitude while the bubble velocity decreases. Thus the increase in void fraction was attributed to the decrease in rise velocity, which was suspected of being due to the generation of standing waves in the column. In addition, measurements of bubbles produced from a single capillary tube showed a 40-50% reduction in size due to the vibration.


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Figure 9. Dependence of the rise velocity of a single bubble (Ub) and the void fraction (gas holdup) at regime transition (εtrans) on the vibration amplitude. Data from Ellenberger & Krishna.34 Adapted from Chemical Engineering Science, 58, J. Ellenberger & R. Krishna, “Shaken, not stirred, bubble column reactors: Enhancement of mass transfer by vibration excitement,” 705-710, Copyright 2003, with permission from Elsevier. Ellenberger et al.18 produced a theoretical estimate of the local void fraction and bubble velocity as a function of liquid column height using the Rayleigh-Plesset equation, 3  ∂2r 3 ∂r   σ  r0  σ µ ∂r ,   ( ) = − + p + − − − p − p z t 2 2 4 ,   e e  r0  r  r r ∂t 2 r ∂t   ∂t 2 

and a force balance on a bubble,

ρ gV

∂U b dp ( z , t ) 1 =− V + (ρ − ρ g )V g − C D ρ U b U b πr 2 . ∂t dz 2

Here r is the bubble radius, ro is the initial bubble radius, ρg is the gas density, p(z,t) is the local pressure, σ is surface tension, µ is the liquid phase viscosity and CD is the coefficient of drag that includes viscous and pressure effects. The Rayleigh-Plesset equation describes the change in bubble radius (r) as a function of time and vertical height (z). The force balance on a bubble includes the Bjerknes force as a pressure gradient. It should be noted that bubble force balance above is not exactly the equation presented in Ellenberger et al.,18 which omitted the mass term 19 ACS Paragon Plus Environment


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on the left hand side. The correction has been made here to make the equation dimensionally consistent. It appears that this error was merely a misprint and that subsequent calculations were not impacted, but the omission would only impact the magnitudes not the trends. The solution to this set of equations, illustrated in Figure 10, suggest that local void fraction increases at harmonic wavelengths, which is most prevalent at the pressure antinodes and higher harmonic modes. Thus supports observations of increased mass transfer at specific frequencies.

Figure 10. Results from the analytical analysis of Ellenberger et al.,18 which shows the influence of harmonic modes (HM) on local void fraction within a water column with air injection. Reprinted from Chemical Engineering Science, 60(22), J. Ellenberger, J.M. van Baten, & R. Krishna, “Exploiting the Bjerknes force in bubble column reactors,” 5962-5970, Copyright 2005, with permission from Elsevier. The validity of these theoretical predictions was assessed with local void fraction measurements at various column heights with an electrical conductivity meter.18 Measurements of the local void fraction allows for the examination of spatial effects such as the impact of phase on the bubble distribution, which cannot be assessed with global measurements. The results from these measurements are provided in Figure 11, which show that the local void fraction (gas holdup) peaks at corresponding nodes for each harmonic mode (HM). The identification of peaks corresponding to the number of nodes for each condition is striking. However, the slope of the 20 ACS Paragon Plus Environment

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axis of symmetry (dashed lines) typically has either a positive or negative slope that is not predicted from the analysis. This is suspected to be the product of an unaccounted force in the analytical analysis or experimental technique, but further investigation is required to deduce the primary cause.

Figure 11. Local void fraction (gas holdup) experimental results from Ellenberger et al.18 plotted versus the location within the column height for various harmonic modes (HM). All the conditions were acquired at Usg = 0.01 m/s and A = 0.5 mm, except for HM-5 that was tested at A = 0.4 mm. The Ho = 0.8 and 1.1 m for the top and bottom row, respectively. Dashed lines are the linear fit to the data in order to show trends. Adapted from Chemical Engineering Science, 60(22), J. Ellenberger, J.M. van Baten, & R. Krishna, “Exploiting the Bjerknes force in bubble column reactors,” 5962-5970, Copyright 2005, with permission from Elsevier. This group’s most recent work45,46 provides experimental data demonstrating that single gas bubbles and slugs can be forced to levitate (i.e. held stationary) within a vibrating bubble column. The columns were oscillated at frequencies between 50 and 400 Hz in order to levitate the bubbles over a range of column diameters, fill heights, liquid densities and viscosities, vibration amplitudes and operating pressure. The experimentally determined conditions for levitation were in good agreement with the theoretical model of Baird.11



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2.5 Knopf and coworkers (2006-2009) This group’s first main contribution to oscillating bubble column performance focused on low gas rates20 and jetting gas rates.21 They expanded upon previous research, which focused primarily on the effect of frequency, in three key areas. First, a unified model to predict void fraction and mass transfer coefficient based on correlations and theory was proposed, which compared well with experimental results.47 Second, they provided experimental results showing the effect of viscosity on kLa and a theoretical relationship that predicts the viscosity dependence.22 Finally, a model to predict the bubble size distribution at varying column heights and vibration frequencies based on population-balance-modeling, which was compared with experimental results with some success.48 Each of these key results are discussed in greater detail here, but the interested reader is directed to Waghmare49 as well as the individual papers for more details. Waghmare et al.47 proposed a model to predict void fraction that builds upon the analysis of Buchanan et al.9 and is based primarily on bubble breakage scaling with the power input. Here it is assumed that the column static pressure is large compared to the sum of the hydrostatic pressure and vibration pressure, which allows Eq. (4) to be rewritten as

∆Vmax ρhω 2 A = . V0 pe This assumption obviously limits the universality of the analysis due to the fact that for many test conditions the magnitude of the vibrational pressure is on the same order as the external (atmospheric) pressure. For example, it is not uncommon to oscillate a 78 cm column of water with air injection at atmospheric pressure with a frequency of 40 Hz and an amplitude of 2.5 mm. This condition results in a magnitude equal to ρ gH − ρ H ω 2 A = 115,000 Pa, which is


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comparable to atmospheric pressure and violates the requirement p e >> ρ gH − ρ H ω 2 A . While this assumption might limit the applicability of the analysis, it offers physical insights into this flow regime. This analysis produces a relationship for void fraction based on input parameters,

(

)

H U gU + 0.5ω 3 A 2 2 / 5  1 sg sg E ( Bj ) , ε = ∫ ε (h)dh = 2.25 2/3 H 0  (σ ρ )3 / 5 g ν   

(

)

(6)

where the bracket on ε indicates that it is the average value, ν is the kinematic viscosity of the

[

]

liquid phase, E ( Bj) = 3 1 − (1 − Bj)1 / 3 Bj and Bj is the Bjerknes number ( = ρH ω 4 A 2 (2 gp e ) ). Waghmare et al.47 couples this relationship with a modified penetration theory,

k L a = 4D πt c (6ε d 32 ) , where D is the molecular-diffusion coefficient of the species (units of length2/time) and tc is the contact time for mass transfer estimated as d 32 U b . This combined relationship produces an equation for the average mass transfer as a function of input parameters,

(

3 2 U sg D gU sg + 0.5ω A k L a = 4.58 1/ 3  (σ ρ )6 / 5 g ν 

[

where G( Bj) = 3 1 − (1 − Bj) 2 / 3

(

)

)

4/5

 G ( Bj ) ,  

(7)

] (2Bj) . The derivation of Eqs. (6) and (7) was based upon several

additional assumptions, including (i) gas concentration is dilute ( ε

Review of Bubble Column Reactors with Vibration - Industrial

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