Hydrodynamics of churn turbulent bubble columns: gas–liquid recirculation and mechanistic modeling

A phenomenological (mechanistic) model has been developed for describing the gas and liquid/slurry phase mixing in churn turbulent bubble columns. The gas and liquid phase recirculation rates in the reactor, which are needed as inputs to the mechanistic reactor model are estimated via a sub-model which uses the two-fluid approach in solving the Navier–Stokes equations. This sub-model estimates the effective bubble diameter in the reactor cross-section and provides a consistent basis for the estimation of the volumetric mass transfer coefficients. The strategy for the numerical solution of the sub-model equations is presented along with the simulation results for a few cases. The overall reactor model has been tested against experimental data from radioactive gas tracer experiments conducted at the Alternate Fuels Development Unit (AFDU), La Porte, TX under conditions of methanol synthesis.     

Measurement of local specific interfacial area in bubble columns via a non-isokinetic withdrawal method coupled to electro-optical detector

Precise measurement of gas-liquid interfacial surface area is essential to reactor design and operation. Mass transfer from the gas phase to the liquid phase is often a key feature that controls the overall process. Measurement of gas-liquid interfacial area is often made through a separate measurement of the gas holdup and bubble size with complex and/or sophisticated methods. In this work, an inexpensive method is presented for the simultaneous determination of both local gas holdup and bubble diameter. The method is based on the withdrawal of the air-liquid dispersion under non-isokinetic conditions and on bubble counting via a simple optical device. The method was calibrated in a bubble column with several withdrawal pressures using coalescing and non-coalescing media. During the same calibration experiment, gas holdup was also measured manometrically and individual bubble diameters were measured by a photographic method. With a vacuum pressure of 3 kPa, local interfacial area measured with the withdrawal method produced a relative error below 13%, compared to the manometric/photographic method. The method was then used to characterize local specific interfacial area in a bubble column under several operating conditions with coalescing and non-coalescing media. In coalescing media and with superficial gas velocities (v_g) from 0.25 to 3.5 cm/s, the average interfacial area ranged from 17 to . With non-coalescing media the average interfacial area ranged from 40 to . Under the test condition it was observed that gas holdup is a parameter that has a greater distribution (standard deviation from 30% to 70%) than the volume-mean bubble diameter (standard deviation from 6% to 12%). It is shown that a model previously developed for characterizing gas holdup homogeneity is also suitable for characterizing interfacial area homogeneity.     

Simultaneous measurement of gas hold-up and mass transfer coefficient by tracer dynamic technique in “Emulsair” reactor with an emulsion-venturi distributor

Hydrodynamics and gas-liquid mass transfer have been investigated in an “Emulsair” reactor with cocurrent downflow of gas and liquid. This consists of a cylindrical tank with conical bottom topped by an emulsion-venturi as the gas-liquid distributor in which the gas is self-aspired by action of the kinetic energy of the liquid recirculation. An original tracer dynamic technique using the CO₂-N₂/water system that enables the simultaneous measurement of overall gas hold-up and overall volumetric mass transfer coefficient has been developed and validated using conventional techniques, such as volume expansion for gas hold-up and dynamic oxygenation for mass transfer. It has been shown that gas hold-up and K_La_L can be deduced from the moments of experimental response curves using a CO₂ pulse in the gas feed. Experimental results have proved that hydrodynamics and mass transfer in the Emulsair reactor are strongly influenced by the flow regime in the divergent. Two different regimes have been observed depending on the liquid recirculation flow rate: annular and homogeneous bubbling flows. In both regimes, self-aspired gas flow rate, gas hold-up and K_La_L have been reported to increase with the liquid flow rate. The operation effectiveness, estimated from the gas-to-liquid flow rate ratio, has been shown to pass through a maximum around 0.59 as a function of liquid recirculation. A comparison with the literature proved that this maximum is higher than those observed for other kinds of gas-liquid reactors equipped with a venturi. Correlations for mass transfer estimation have been derived and are in agreement with the literature.     

Multiphase fluidization in large-scale slurry jet loop bubble columns for methanol and or dimethyl ether production

A solution methodology is proposed for the process development and process engineering of a continuously operated jet loop bubble column including integrated external or internal steam generation for, e.g., a high-efficiency large-scale medium pressure methanol and or dimethyl ether production, or other gas to liquid Fischer-Tropsch applications.A jet loop bubble column is defined in the present process development to study the combined integration of a jet-eductor draft tube system with an upper bubble column. The major advantages resulting from the integrated jet-eductor draft tube system in large-scale bubble columns are the guidance and good mixing efficiency of the multiphase flow up to the upper part of the bubble column. Reducing the bubble column operating liquid level at about 2.5-3.0 times of the column diameter above the upper end of the draft tube results in a classical jet-eductor draft tube reactor suitable for small and or medium-scale industrial applications.Methanol synthesis is usually executed catalytically in multistage packed beds at higher pressure, e.g. 26 MPa, and about 350-, resulting in a higher plant installation and operating cost. The successful scale-up of a slurry jet loop bubble column can achieve a higher catalytic selectivity at a lower pressure and temperature , and therefore reduce the overall plant investment and production cost [Toseland, 1999. Three phase flows under extreme conditions of pressure and temperature, Part II: industrial applications, Air products and Chemicals, Inc. Presented at the A.I.Ch.E. Annual Meeting, Dallas, TX; Fan, 1999. Three phase flows under extreme conditions of pressure and temperature, Part I: fundmental characteristics, Department of Chemical Engineering, The Ohio State University. Presented at the A.I.Ch.E. Annuxal Meeting, Dallas, TX]. In addition, the separate slurry production of dimethyl ether, or even coproduction with methanol, can be a more cost-effective process than the classical methanol dehydration process.The new Modified Slurry Process^© for large-scale methanol and or dimethyl ether production is presented including internal or external heat exchanger location for steam production.A process concept is developed of a Large Scale Slurry Jet Loop Bubble Column^© with external separator, auxiliary internal heat exchanger equipment and high-efficiency gas-liquid slurry jet-eductor mixing system including draft tubes and an upper bubble column. In addition, as comparison a simplified concept is discussed for a small-to-medium-scale slurry jet loop reactor including external steam production and bottom nozzle jet-eductor installation without the presence of an upper bubble column.The basic geometrical parameters of the proposed slurry jet loop bubble column and jet loop reactor are discussed. The influence of the selected geometrical parameters on the gas holdup, interfacial area and mixing is analyzed. Information about catalyst type and particle size distribution is also presented.The definition of optimal operating conditions related to the influence of the fluid dynamics and mixing on mass transfer efficiency and also information for the minimum required power input per unit volume for startup or stable reactor operation are discussed.A simplified estimation method is presented for the expected axial temperature difference across the overall length of the jet bubble column, and also the required heat transfer area of a new construction-type internal compact heat exchanger for efficient reactor cooling and operation.Scale-up is possible for large diameter jet loop bubble columns, typically up to 5 m diameter and 60 m height, including continuous three-phase slurry operation at higher power input and interfacial area, for more efficient synthesis gas absorption and reaction than in classical slurry bubble columns. Integration of suitable designed sieve trays can further guarantee an efficient operation of the lower jet loop draft tube system at higher column diameters and also achieve an efficient reactor operation in the upper bubble column section.     

Hydrodynamics in rectangular air-lift reactors: Scale-up and the scale-up and the influence of gas-liquid separator design

Experiments were conducted on air-lift reactors with different liquid volumes, aspect ratios, and gas-liquid separator configurations, with constant downcomer to riser cross-sectional area ratio and riser/downcomer height. Reactor performance was highly dependent on the gas-liquid separator design. By changing the latter, the mode of operation could be changed from that of a concentric tube reactor (with gas recirculation) to an external loop reactor (without gas recirculation). Liquid velocity and overall gas hold-up data were correlated to the power per liquid volume and “Disengagement Ratio”, a geometric parameter describing the separator's configuration. During scale-up, similar gas hold-ups and liquid velocities were found in laboratory and pilot reactors. A modified air-lift reactor with an enlarged channel for a gas-liquid separator also was studied.     

Mass transfer in sparged and stirred reactors: influence of carbon particles and electrolyte

Mass transfer in multiphase systems is one of the most studied topics in chemical engineering. However, in three-phase systems containing small particles, the mechanisms playing a role in the increased rate of mass transfer compared to two-phase systems without particles, are still not clear. Therefore, mass transfer measurements were carried out in a 2D slurry bubble column reactor , a stirred tank reactor with a flat gas-liquid interface, and in a stirred tank reactor with a gas inducing impeller. The rate of mass transfer in these reactors was investigated with various concentrations of active carbon particles (average particle size of ), with electrolyte (sodium gluconate), and with combinations of these. In the bubble column, high-speed video recordings were captured from which the bubble size distribution and the specific bubble area were determined. In this way, the specific mass transfer area a_gl was determined separately from the mass transfer coefficient k_l. Mechanisms proposed in literature to describe mass transfer and mass transfer enhancement in stirred tank reactors and bubble columns are compared. It is shown that the increased rates of mass transfer in the 2D bubble column and in the stirred tank reactor with the gas inducing impeller are completely caused by an increased gas-liquid interfacial area upon addition of carbon particles and electrolyte. It is suggested that an increased level of turbulence at the gas-liquid interface caused by carbon particles accounts for a smaller effective boundary layer thickness and an enhancement of mass transfer in the flat gas-liquid surface stirred tank reactor. However, for the carbon particles used in this study, it is rather unlikely that mass transfer enhancement takes place due to the well-known shuttle or grazing effect.     

Gas holdup, liquid circulating velocity and mass transfer properties in a mini-scale external loop airlift bubble column

The external loop airlift bubble column has been regarded as a promising type of gas-liquid or gas-liquid-solid biooreactor because of the liquid circulating flow between the riser and downcomer. A mini-scale column is useful and efficient in the process research and development for highly specialized materials such as fine chemicals, advanced bioproducts and biocatalysts utilized in two or three phase system. In this work, a mini-scale glass column of in volume was designed and characterized. The gas holdup ε_G in the riser was obtained by measuring the volume expansion through photographs taken with a digital camera. The liquid circulating velocity U_L was measured by observing the time required for a tracer particle to travel a fixed distance in the downcomer through analysis of the images taken by a video camera. The gas-liquid volumetric oxygen transfer coefficient k_La and liquid-solid oxygen transfer coefficient k_S were determined by our previous method in which the air oxidation of glucose was catalysed by the immobilized glucose oxidase gel beads suspended in the column to obtain a pseudo steady state concentration of the dissolved oxygen and the corresponding constant rate of glucose consumption. It was shown that even such a mini-scale external loop bubble column could be characterized in terms of gas holdup, liquid circulating velocity and mass transfer properties according to our previous correlations proposed for the bench to pilot scale column.     

三相鼓泡淤浆床环氧乙烷合成反应器数学模拟及工程分析

建立了加压三相鼓泡淤浆床环氧乙烷合成反应器的数学模型 ,计入了催化剂颗粒在床层中沉降形成沿床高浓度分布对反应的影响以及由于惰性液相载体部分返混对传递的影响 ,进一步利用经实验验证的上述数学模型模拟不同表观气速、床高、反应器直径 (扣除传热元件截面积 )、进口乙烯摩尔分数等参数对床层中催化剂浓度随床高的分布、出口环氧乙烷摩尔分数、环氧乙烷选择率以及单位质量催化剂环氧乙烷年产量的影响 .通过模拟分析预示了工业三相床环氧乙烷反应器的合理尺寸、表观气速、环氧乙烷选择率以及时空产率 ,为工业化提供必要的设计依据     

Highly effective cobalt catalyst for wax production in Fischer-Tropsch synthesis

Fischer-Tropsch synthesis (FTS) was carried out in a fixed bed reactor with a highly effective cobalt catalyst for wax production. The procedure for reducing the inactive cobalt oxide to the active cobalt catalyst was examined by X-ray diffraction (XRD) and temperature-programmed reduction (TPR). The results showed that 300 ml/min H₂ at 350 °C for 16 h was suitable for reducing the inactive Co oxides to active metallic Co sites. In the case of the powder and pellet type cobalt catalysts with a reactant (H₂/CO = 2:1) flow rate of 15 g_cat min L⁻¹, catalyst deactivation occurred as a result of mass transfer limitations of the hydrocarbon and water produced on the catalyst. On the other hand, the pellet type cobalt catalyst with a reactant flow rate of 45 g_cat min L⁻¹ showed activity not only for liquid hydrocarbon (C₅₊) formation but also for gas product (CH₄ and CO₂) formation. In particular, the methane yield reached almost 20% due to heat transfer limitation in the catalyst. Considering the heat and mass transfer limitations in the cobalt catalyst, a Co-foam catalyst with an inner metallic foam frame and an outer cobalt catalyst was developed. SEM-EDS Co-mapping revealed the cobalt atoms to be distributed equally over the surface of the Co-foam catalyst. The Co-foam catalyst was highly selective toward liquid hydrocarbon production and the liquid hydrocarbon productivity at 203 °C was 52.5 ml  h⁻¹, which was higher than that by the Co-pellet. In addition, the chain length probability, α, by the Co-foam catalyst was 0.923 and wax formation was especially favored.     

Investigations in an external-loop airlift photobioreactor with annular light chambers and swirling flow

Photosynthetic microorganisms could serve as valuable compounds, but also for environmental applications. Their production under controlled conditions implies to design specific reactors, named photobioreactors, in which light supply is the main constraint. This paper was devoted to an original external-loop airlift photobioreactor (PBR) with annular light chambers in which a swirling motion was induced. The aim was to characterize this novel geometrical configuration in terms of gas-liquid hydrodynamics, and to test its potentiality for algal cultures. This PBR consisted of two identical columns connected by flanges defining tangential inlets, each column being made of two transparent concentric tubes (6 L in liquid volume, 50 m⁻¹ in specific illuminated area). Firstly, the global flow characteristics (circulation and mixing times) were determined by a tracer method and modelled by an axial dispersed plug flow with complete recirculation (Péclet number). By means of a double optical probe, both local and global time-averaged parameters of the gas phase were measured, namely void fraction, bubble velocity, frequency and size. The gas-liquid mass transfer were also characterized, in tap water and in culture medium, by measuring overall volumetric mass transfer coefficients. In a second time, cultures of the microalga Chlamydomonas reinhardtii were run in batch mode. The variations of biomass concentration and pigment content with time from inoculation were successfully obtained. All these findings highlighted: (i) some significant differences in terms of gas-liquid hydrodynamics between the present PBR and the usual airlift systems, (ii) the interest of this configuration for algal cultures, even if complementary studies and technological improvements are still required for definitively validating its scale-up.     

Dynamic analysis of a trickle bed reactor by moment technique

The performance of a trickle bed reactor is investigated by the moment technique. Residence time distributions of SO₂ tracer in both gas (Helium) and liquid (distilled water) effluents are used to predict zero reduced and first absolute moments and these values are compared with the derived theoretical expressions. Correlations are suggested for gas-liquid mass transfer coefficient, liquid hold up, and extent of axial mixing in liquid phase.True adsorption equilibrium constant of the system is estimated as 0.378 from liquid full bed experiments and contacting efficiency of the trickle bed reactor is found as 0.987.Effect of axial dispersion is not significant on gas-liquid mass transfer coefficient since absorption factor is small, but is found to be quite important on the true estimation of adsorption factor.     

Experimental investigation of a pilot-scale jet bubbling reactor for wet flue gas desulphurisation

In the present work, an experimental parameter study was conducted in a pilot-scale jet bubbling reactor for wet flue gas desulphurisation (FGD). The pilot plant is downscaled from a limestone-based, gypsum producing full-scale wet FGD plant. Important process parameters, such as slurry pH, inlet flue gas concentration of SO₂, reactor temperature, and slurry concentration of Cl⁻ have been varied. The degree of desulphurisation, residual limestone content of the gypsum, liquid phase concentrations, and solids content of the slurry were measured during the experimental series.The SO₂ removal efficiency increased from 66.1% to 71.5% when the reactor slurry pH was changed from 3.5 to 5.5. Addition of Cl⁻ (in the form of CaCl₂·2H₂O) to the slurry increased the degree of desulphurisation to above 99%, due to the onset of extensive foaming, which substantially increased the gas-liquid contact area. An increase in the inlet flue gas SO₂ concentration from 502 to led to a decrease in the SO₂ removal efficiency from 80.1% to 69.4%. A temperature increase from 296 to caused a reduction in the degree of desulphurisation from 69.4% to 68.1%, but this result is almost within the experimental uncertainty. The residual limestone level in the gypsum formed increased with increasing values of reactor slurry pH, inlet flue gas SO₂ concentration, and slurry concentration of Cl⁻.     

Interfacial area and volumetric mass transfer coefficient in a bubble reactor at elevated pressures

The present study deals with the pressure effects on mass transfer parameters within a bubble reactor operating at pressures up to . The gas-liquid systems are N₂/CO₂-aqueous solution of Na₂CO₃-NaHCO₃ and N₂/CO₂-aqueous solution of NaOH. A sintered powder plate is used as a gas distributor. Three parameters characterizing the mass transfer are identified and investigated with respect to pressure: the gas-liquid interfacial area a, the volumetric liquid side mass transfer coefficient k_La and the volumetric gas side mass transfer coefficient k_Ga. The gas-liquid absorption with chemical reaction is used and the mass transfer parameters are determined by using the model reaction between CO₂ and the aqueous solutions of Na₂CO₃-NaHCO₃ and NaOH. For a given gas mass flow rate, the interfacial area as well as the volumetric liquid mass transfer coefficient decrease with increasing operating pressure. However, for a given pressure, a and k_La increase with increasing gas mass flow rates. The mass transfer coefficient k_L is independent of pressure. Furthermore, the pressure increase results in a decrease of k_G and k_Ga for a given gas mass flow rate. The values of the interfacial area, which are obtained from both chemical systems are found to be different. These discrepancies are attributed to the choice of the liquid system in the absorption reaction model.     

CFD simulation of dilute phase gas-solid riser reactors: part II—simultaneous adsorption of SO2-NOx from flue gases

Simultaneous adsorption of SO₂-NO_x in a riser configuration is a novel route for flue gas cleaning. The riser operates at a low flux of small diameter Na-γ-Al₂O₃ sorbent particles. The reaction scheme is adopted from previous work (Ind. Eng. Chem. Res. 40 (2001) 119), without adjusting any of the kinetic parameters. The significant concentration gradient between the gas and solid phase mainly arises from the low solid fraction (typically 5×10⁻⁴) in the riser. Enhancing the fluctuating kinetic motion of gas and solid phase increases the SO₂ adsorption, whereas the NO adsorption is decreased marginally. The solid recirculation in the top section of the riser, induced by the abrupt T outlets, significantly decreases the NO and NO₂ removal, while the SO₂ removal remains mostly unaffected. Therefore, it is desirable to avoid recirculation for a maximum NO_x removal. A comparison of the 3D and a 1D model shows that higher SO₂ and NO removal efficiencies are predicted by the 3D model in the major part of the riser. However, these positive effects are largely neutralized by the negative effects of the outlet-induced recirculation, resulting in similar overall removal efficiencies calculated by the two models. Unlike the 1D model, the 3D simulation shows a considerable axial variation in the solid fraction and slip velocity. The 3D simulation also allows to calculate the effects of outlet geometry on the flow and reaction fields. The reactor efficiency can be improved by modifying the outlet configuration to minimize the recirculation.     

Study of the Characteristics of Methanol Synthesis in a Recirculation Slurry Reactor – A Novel Three‐Phase Synthesis Reactor

A process feasibility analysis on the liquid phase methanol synthesis (LPMeOH^TM) process was performed in a recirculation slurry reactor (RSR). In the three‐phase RSR system, a fine catalyst is slurried in the paraffin and this catalyst slurry is continuously recirculated through the nozzle from the slurry sector to the entrained sector by a pump. The syngas is fed concurrently with the downward flow of slurry to form the methanol product. A laboratory scale mini‐pilot plant version of a recirculation slurry reactor system was successfully designed and built to carry out process engineering research, and in addition, an identical cold model was built to measure the mass transfer coefficient in the recirculation slurry reactor. The effects of operating conditions, including temperature, pressure, gas flow rate and catalyst slurry recirculation flow rate on the productivity of methanol were studied. This experimental data helps the scale‐up and commercialization of the methanol synthesis process in recirculation slurry reactors.     

Mathematical modeling of gas-liquid mass transfer rate in bubble columns operated in the heterogeneous regime

In this paper, a multi-scale approach is followed to study gas-liquid mass transfer in bubble columns. First, a single bubble of equivalent diameter d is considered. Its morphology and its gas to liquid relative velocity are related to the bubble diameter through the use of known correlations. Then, the gas-liquid mass transfer between the bubble and the surrounding liquid is studied theoretically. An equation describing the transport of the transferred species in the viscous boundary layer around the bubble is solved. In a second step, a bubble column of 6-10 m height is studied experimentally. The gas phase in the column is characterized experimentally by means of a gammametric technique. Finally, the two studies are linked, yielding a 1D mathematical model able to predict the gas-liquid mass transfer rate in a bubble column operated in the heterogeneous regime.     

Reduction and oxidation kinetics of Mn3O4/Mg-ZrO2 oxygen carrier particles for chemical-looping combustion

The kinetics of reduction with methane and oxidation with oxygen of Mn₃O₄ supported on Mg-ZrO₂ prepared by freeze granulation has been investigated. The reactivity experiments were performed in a thermogravimetric analyzer (TGA) using different reacting gas concentrations and temperatures in the range of 1073-1223 K. The oxygen carrier particles showed high reactivity during both reduction and oxidation at all investigated temperatures. An empirical reaction model, which assumes a linear relation between time and conversion, was used to determine the kinetic parameters for reduction and oxidation, with chemical reaction being the main resistance to the reaction. The order of reaction found was 1 with respect to CH₄ and 0.65 with respect to O₂. The activation energy for the reduction reaction was 119 and for the oxidation reaction. The reactivity data and kinetic parameters were used to estimate the solid inventory in the air and fuel reactor of a CLC system. The optimum solid inventory obtained was at a value of ΔX_s=0.4. At these conditions, the recirculation rate of oxygen carrier between air and fuel reactor was per MW of fuel, which could be accomplished in an industrial reactor. The high reactivity of the Mn₃O₄/Mg-ZrO₂ with both methane and oxygen showed that this is a very promising oxygen carrier for CLC.     

CFD simulation of effects of the configuration of gas distributors on gas-liquid flow and mixing in a bubble column

Numerical simulations of gas-liquid flow in a cylindrical bubble column of 400 mm in diameter at the superficial gas velocity were conducted to investigate effects of the configuration of gas distributors on hydrodynamic behaviour, gas hold-up and mixing characteristics. Eight different gas distributors were adopted in the simulation. The simulation results clearly show that the configuration of gas distributor have an important impact on liquid velocity and local gas hold-up in the vicinity of the gas distributor. Comparisons of the overall gas holdup and mixing time among different gas distributors have demonstrated that none of the adopted gas distributors was able to produce the highest interfacial area and also yield the shortest mixing time. The CFD modelling results reveal that an increase in the number of gas sparging pipes used in gas distributors is beneficial in improving the gas hold-up but is disadvantageous in reducing bubble size due to a decrease in turbulent kinetic dissipation. It has been demonstrated from the simulations that the appearance of asymmetrical flow patterns in the bubble column and the adoption of smaller gas sparging pipes for gas distributors are effective in improving the mixing characteristics.     

Influence of Pressure on the Hydrodynamics and Mass Transfer Parameters of an Agitated Bubble Reactor

The present study deals with the pressure effects on the hydrodynamic flow and mass transfer within an agitated bubble reactor operated at pressures between 10⁵ and 100 × 10⁵ Pa. In order to clarify the flow behavior within the reactor, liquid phase residence time distributions (RTD) for different operating pressures and gas velocities ranging between 0.005 m/s and 0.03 m/s are determined experimentally by the tracer method for which a KCl solution is used as a tracer. The result of the analysis of the liquid‐phase RTD curves justifies the tank‐in‐series model flow for the operating pressure range. Good agreement is obtained between theoretical and experimental results assuming the reactor is operating as perfectly mixed. Two parameters characterizing the mass transfer are identified and investigated in respect to pressure: the gas‐liquid interfacial area and volumetric liquid‐side mass transfer coefficient. The chemical absorption method is used. For a given gas mass flow rate, the interfacial area as well as the volumetric liquid mass transfer coefficient decrease with increasing operating pressure. However, for a given pressure, a and k_La increase with increasing gas mass flow rates. The mass transfer coefficient k_L is independent of pressure.     

The voidage problem in gas-electrolyte dispersions

Assessing the ohmic interelectrode resistance of electrochemical reactors with gas evolution requires data for the gas void fraction of gas-electrolyte dispersions. A voidage equation is derived taking account of the internal liquid flow in stationary electrolytes and at small liquid superficial velocities. The equation is a general form of available voidage equations.Nomenclature C non-dimensional constant, Equation 8 - n exponent, Equation 5 - S cross-sectional area (m²) - v _G gas velocity (m s^–1) - v _L liquid velocity (m s^–1) - v _s rising velocity of a bubble swarm (m s^–1) - v _l terminal rising velocity of a single bubble (m s^–1) - V_G volume flow rate of gas (m³ s^–1) - V_L volume flow rate of liquid (m³ s^–1) - fraction of cross-sectional area - volume (void) fraction of gas - _m geometric maximum of void fraction - maximum of void fraction in infinite gas flow Indices i internal - t total