The influence of geometry on hydrodynamic and mass transfer characteristics in an external airlift reactor for the cultivation of filamentous fungi

The influences of geometric configuration, mycelial broth rheology and superficial gas velocity (U_sg) were investigated with respect to the following hydrodynamic parameters: gas holdup (), oxygen transfer coefficient (K_La) and mixing time (t_m). Increases in U_sg and height of gas separator (H_t) caused an increase in and K_La, and a decrease in t_m. Consequently, a diameter ratio (D_d/D_r) of 0.71 and H_t 0.20 m were found to be the best geometry and operation parameters to achieve high aeration and mixing efficiency for the high viscous broth system in the cultivation of filamentous fungi. An external airlift reactor (EALR) was developed and designed for the cultivation of filamentous fungi. The EALR with two spargers excels in reliability and high aeration and mass transfer coefficiency, resulting in a fast mycelial growth and high biomass productivity in the cultivation of the fungus Rhizopus oryzae.     

Oxygen mass transfer and gas holdup in a bubble column bioreactor with biosynthesis liquids

The influence of the rheology of some antibiotic biosynthesis liquids produced by Streptomyces aureofaciens, Nocardia mediterranei and Penicillium chrysogenum on the volumetric liquid phase oxygen transfer coefficient, k_La, and gas holdup, ε_G, together with the influence of superficial gas velocity, were studied in a bubble column bioreactor, using samples of fermentation liquids taken from industrial stirred tank fermenters, at 30-hour intervals during fermentation batch. The results were compared to those of previous studies from literature on non-Newtonian homogeneous fluids, such as CMC-Na, xanthan and starch solutions, respectively. In the heterogeneous broths, ε_G and k_La decreased with increasing apparent viscosity of the broth and increased with increasing superficial velocity. The experimental data were correlated using non-linear regression with correlation coefficients above 0.85.     

A comparative evaluation of oxygen mass transfer and broth viscosity using Cephalosporin-C production as a case strategy

The production of Cephalosporin-C (CPC) a secondary metabolite, using a mold Acremonium chrysogenum was studied in a lab scale Internal loop air lift reactor. Cephalosporin-C production process is a highly aerobic fermentation process. Volumetric gas–liquid mass transfer coefficient (k_La) and viscosity (η) were evaluated, during the growth and production phases of the microbial physiology. An attempt has been made to correlate the broth viscosity, η and volumetric oxygen transfer coefficient, k_La during the Cephalosporin-C production in an air lift reactor. The impact of biomass concentration and mycelial morphology on broth viscosity has been also evaluated. The broth exhibits a typical non-Newtonian fermentation broth. Rheology parameters like consistency index and fluidity index are also studied.     

Characterization and improvement of oxygen transfer in pilot plant external air-lift bioreactor for mycelial biomass production

The oxygen transfer dynamics in a pilot plant external air-lift bioreactor (EALB) during the cultivation of mycelial biomass were characterized with respect to hydrodynamic parameters of gas holdup (), oxygen transfer coefficient (K_La) and superficial gas velocity (U _g), and dissolved oxygen (DO). An increased flow rate of air supply was required to meet the increased oxygen demand with mycelial biomass growth. Consequently, an increase in air flow rate led to an increase in , K_La and the DO level. The enhancement of oxygen transfer rate in the cultivated broth system, however, was limited with highly increased viscosity of the mycelial broth. An increase in air flow rate from 1.25 to 2.00 v/v/m resulted in a low increment of oxygen transfer. The newly designed pilot plant EALB with two air spargers significantly improved processing reliability, aeration rate and K_La. The pilot plant EALB process, operated under a top pressure from 0 to 1.0 bars, also demonstrated a significant improvement of oxygenation efficiency by more than 20% in DO and K_La. The performance of the two sparger EALB process under top pressure demonstrated an efficient and economical aerobic system with fast mycelial growth and high biomass productivity in mycelial biomass production and wastewater treatment.     

Syngas fermentation to biofuel: Evaluation of carbon monoxide mass transfer coefficient (kLa) in different reactor configurations

Lignocellulosic biomass such as agri‐residues, agri‐processing by‐products, and energy crops do not compete with food and feed, and is considered to be the ideal renewable feedstocks for biofuel production. Gasification of biomass produces synthesis gas (syngas), a mixture primarily consisting of CO and H₂. The produced syngas can be converted to ethanol by anaerobic microbial catalysts especially acetogenic bacteria such as various clostridia species.One of the major drawbacks associated with syngas fermentation is the mass transfer limitation of these sparingly soluble gases in the aqueous phase. One way of addressing this issue is the improvement in reactor design to achieve a higher volumetric mass transfer coefficient (k_La). In this study, different reactor configurations such as a column diffuser, a 20‐μm bulb diffuser, gas sparger, gas sparger with mechanical mixing, air‐lift reactor combined with a 20‐μm bulb diffuser, air‐lift reactor combined with a single gas entry point, and a submerged composite hollow fiber membrane (CHFM) module were employed to examine the k_La values. The k_La values reported in this study ranged from 0.4 to 91.08 h^?1. The highest k_La of 91.08 h^?1 was obtained in the air‐lift reactor combined with a 20‐μm bulb diffuser, whereas the reactor with the CHFM showed the lowest k_La of 0.4 h^?1. By considering both the k_La value and the statistical significance of each configuration, the air‐lift reactor combined with a 20‐μm bulb diffuser was found to be the ideal reactor configuration for carbon monoxide mass transfer in an aqueous phase. © 2010 American Institute of Chemical Engineers Biotechnol. Prog., 2011     

Oxygen transfer characteristics of a centrifugal, packed-bed reactor during viscous xanthan fermentation

The oxygen transfer properties of a novel, centrifugal, packed-bed reactor (CPBR) during viscous xanthan fermentation were determined with respect to the effects of the arrangement of the centrifugal, packed bed (CPB) and the recirculation loop (RL). Characterized by the maximum volumetric transfer coefficient (k_La) in xanthan broth, the aeration efficiency of CPBR was compared to those in stirred-tank reactors (STR) equipped with disc turbines (DT) or marine propellers (MP), and to that in a water-in-oil emulsion (WIO). As expected, STR-WIO showed the highest k_La (0.038 s^-1 at 2%) among all systems studied due to reduced broth viscosity; however, practical difficulties exist in product recovery. It was found that, at 3.5% xanthan the k_La in CPBR (0.018 s^-1) was higher than that of STR (0.005 s^-1) and close to that of STR-WIO (0.020 s^-1), indicating improved oxygen transfer at such a xanthan concentration. The exterior baffles along the rotating fibrous matrix offer additional agitation in the viscous broth. A gas-continuous arrangement, in which the CPB was kept above the broth, was able to elevate k_La to 0.023 s^-1, higher than that of STR-WIO. The external RL operated by a peristaltic pump was found to play an important role in CPBR aeration by providing better gas-liquid contact. With the improved oxygen transfer efficiency in CPBR at high xanthan concentrations, the CPBR system is practically the preferred system for xanthan fermentation. The characteristic roles of CPB arrangement and the RL should be considered primarily during scale-up operation.     

The influence of polymeric membrane gas spargers on hydrodynamics and mass transfer in bubble column bioreactors

Gas sparging performances of a flat sheet and tubular polymeric membranes were investigated in 3.1 m bubble column bioreactor operated in a semi batch mode. Air–water and air–CMC (Carboxymethyl cellulose) solutions of 0.5, 0.75 and 1.0 % w/w were used as interacting gas–liquid mediums. CMC solutions were employed in the study to simulate rheological properties of bioreactor broth. Gas holdup, bubble size distribution, interfacial area and gas–liquid mass transfer were studied in the homogeneous bubbly flow hydrodynamic regime with superficial gas velocity (U _G) range of 0.0004–0.0025 m/s. The study indicated that the tubular membrane sparger produced the highest gas holdup and densely populated fine bubbles with narrow size distribution. An increase in liquid viscosity promoted a shift in bubble size distribution to large stable bubbles and smaller specific interfacial area. The tubular membrane sparger achieved greater interfacial area and an enhanced overall mass transfer coefficient (K _La) by a factor of 1.2–1.9 compared to the flat sheet membrane.     

Vergleich von Messungen des kLa-Wertes nach der stationären und der dynamischen Methode bei der Fermentation von L-Lysin und Turimycin

The calculation and scale-up of fermentation processes need k_La as one of the most important engineering data. There are two methods to determine k_La depending on power input, aeration rate and the properties of the fermentation broth: static with a balance between air supply and exit, dynamic gassing out with following the changes of dissolved oxygen concentration during periods of air off and a following air on. Within early intervals of fermentation time the data from both methods agree well, while for later time intervals the dynamic method always gives much lower values for k_La than static. The only explanations for this phenomenon are quick changes in the oxygen metabolism or an enzymatic storage of oxygen. For both gassing out and saturation period it is possible to calculate the same absolute amounts of this additional oxygen.     

Continuous production of acetic acid using immobilized acetobacter aceti in a three-phase fluidized bed bioreactor

Continuous operation of a three-phase fluidized bed bioreactor using immobilized cells showed that both immobilized and suspended cells contributed to the production of acetic acid. Unlike the rapid decrease in the productivity at dilution rates above 0.25 h⁻¹ in the free cell fermentation, the productivity was little affected by the dilution rate in the immobilized cell fermentation. Theoretical models were proposed for the continuous process. The models approximately agreed with the experimental results. Experimental results and/or theoretical calculations based on the kinetic models showed that suspended cells were important in the production of acetic acid if the solid holdup was small or if gel radius was large. Theoretical calculations showed that an optimal solid holdup or gel size existed at higher dilution rates because of the k_La dependence on solid holdup and particle diameter.     

Glutamic acid production in an airlift reactor with net draft tube

Cultivation of Brevibacterium divaricatum for glutamic acid production in an airlift reactor with net draft tube was developed. Cell concentration gave an index for adding penicillin G. On-line estimation of total sugar concentration yielded an identified model which was used for determination of the substrate addition. Fermentation for glutamic acid production requires high oxygen concentration in the broth. The proposed reactor has the capability to provide sufficient oxygen for the fermentation. Since the reactor is suitable for fed-batch culture, the cultivation of B. divaricatum for glutamic acid production in the proposed reactor is successfully carried out.List of Symbols a system parameter - b system parameter - C _c,in mole fraction carbon dioxide in the gas inlet - C _c,out mole fraction carbon dioxide in the gas outlet - C _L mole/dm³ oxygen concentration in liquid phase - C _L ^* mole/dm³ saturated oxygen concentration in liquid phase - C _0,in mole fraction of oxygen in the gas inlet - C _0,out mole fraction of oxygen in the gas outlet - CPR mole/h/dm³ carbon dioxide production rate based on total broth - E(t) error signal - F _in mole/h inlet gas flow rate - k ₁ constant defined by Eq. (4) - k ₂ constant defined by Eq. (5) - k _L a 1/h volumetric mass transfer coefficient of gas-liquid phase - OUR mole/h/dm³ oxygen uptake rate based on total broth - P atm pressure in the reactor - t h time - TS _c g total sugar consumption - TS _s g/dm³ set point of total sugar concentration - TS _* g/dm³ reference value of total sugar concentration - TS(t) g/dm³ total sugar concentration in the broth at timet - u(t) cm³/min feed rate at timet - V dm³ total broth volume - VVM (dm³/min)/dm³ flow rate per unit liquid volume - a negative constant defined by Eq. (7)     

Comparison of the hydrodynamics and mass transfer characteristics in internal-loop airlift bioreactors utilizing either a novel membrane-tube sparger or perforated plate sparger

Two gas spargers, a novel membrane-tube sparger and a perforated plate sparger, were compared in terms of hydrodynamics and mass transfer (or oxygen transfer) performance in an internal-loop airlift bioreactor. The overall gas holdup ε _T, downcomer liquid velocity V _d, and volumetric mass transfer coefficient K _L a were examined depending on superficial gas velocity U _G increased in Newtonian and non-Newtonian fluids for the both spargers. Compared with the perforated plate sparger, the bioreactor with the membrane-tube sparger increased the values of ε _T by 4.9–48.8 % in air–water system when the U _G was from 0.004 to 0.04 m/s, and by 65.1–512.6 % in air–CMC solution system. The V _d value for the membrane-tube sparger was improved by 40.0–86.3 %. The value of K _L a was increased by 52.8–84.4 % in air–water system, and by 63.3–836.3 % in air–CMC solution system. Empirical correlations of ε _T, V _d, and K _L a were proposed, and well corresponding with the experimental data with the deviation of 10 %.     

Saccharomyces cerevisiae fermentations in a pilot scale airlift bioreactor: Comparison of air sparger configurations

Hydrodynamic and oxygen transfer comparisons were made between two ring sparger locations, draft tube and annulus, in a concentric pilot scale airlift reactor with a baker's yeast suspension. Sectional hydrodynamic measurements were made and a mobile DOT probe was used to characterise the oxygen transfer performance through the individual sections of the reactor. The hydrodynamic performance of the reactor was improved by using a draft tube ring sparger rather than the annulus ring sparger. This was due to the influence of the ratio of the cross sectional area of the downcomer and riser (A _D/A_R) in conjunction with the effect of liquid velocity and a parameter,C ₀, describing the distribution of the liquid velocity and gas holdup across the riser on the bubble coalescence rates. The mixing performance of the reactor was dominated by the frequency of the passage of the broth through the end sections of the reactor. An optimum liquid height above the draft tube, for liquid mixing was demonstrated, above which no further improvement in mixing occurred. The liquid velocity and degree of gas entrainment showed little dependency on top section size for both sparger configurations. Extreme dissolved oxygen heterogeneity was demonstrated around the vessel with both sparger configurations and was shown to be detrimental to the oxygen uptake rate of the baker's yeast. Dissolved oxygen tensions below 1% air saturation occurred along the length of the riser and then rose in the downcomer. The greater oxygen transfer rate in the downcomer than in the riser was caused by the combined effects of a larger slip velocity in the downcomer which enhancedk _La and gas residence time, high downcomer gas holdup, and the change in bubble size distribution between the riser and downcomer. The position of greatest oxygen transfer rate in the downcomer was shown to be affected by the reactor from the influence on downcomer liquid linear velocity. UCL is the Biotechnology and Biological Sciences Research Council sponsored Advanced Centre for Biochemical Engineering and the Council's support is greatly acknowledged.     

Microcalorimetric investigations of the metabolism of yeasts. growth in batch and chemostat cultures on ethanol medium

Summary In the presence of protein, Hansenula polymorpha cultivation medium exhibits a maximum volumetric mass transfer coefficient, k_La, as function of the employed antifoam agents (soy oil and Desmophen 3600). With diminishing superficial gas velocity this maximum disappeas.Symbols E_G Relative gas holdup - k_La Volumetric mass transfer coefficient (s^–1) - w_SL Superficial liquid velocity (cm s^–1) - w_SG Superficial gas velocity (cm s^–1)     

Gas holdup and mass transfer characteristics of carboxymethyl cellulose solutions in a bubble column with a radial gas sparger

The gas phase holdup and mass transfer characteristics of carboxymethyl cellulose (CMC) solutions in a bubble column having a radial gas sparger have been determined and a new flow regime map has been proposed. The gas holdup increases with gas velocity in the bubbly flow regime, decreases in the churn-turbulent flow regime, and increases again in the slug flow regime. The volumetric mass transfer coefficient (k _La) significantly decreases with increasing liquid viscosity. The gas holdup and k _La values in the present bubble column of CMC solutions are found to be much higher than those in bubble columns or external-loop airlift columns with a plate-type sparger. The obtained gas phase holdup ( _g) and k _La data have been correlated with pertinent dimensionless groups in both the bubbly and the churn-turbulent flow regimes.List of Symbols a m^–1 specific gas-liquid interfacial area per total volume - A _d m² cross-sectional area of downcomer - A _r m² cross-sectional area of riser - d _b m individual bubble diameter - d _vs m Sauter mean bubble diameter - D _c m column diameter - D _L m²/s oxygen diffusivity in the liquid - Fr Froude number, U _g/(g D_c)^1/2 - g m/s² gravitational acceleration - G _a Galileo number, gD _c ^{3 2}/² _app - H _a m aerated liquid height - H _c m unaerated liquid height - K Pa · sⁿ fluid consistency index - k _L a s^–1 volumetric mass transfer coefficient - n flow behavior index - N _i number of bubbles having diameter d _bi - Sc Schmidt number, _app/( D _L) - Sh Sherwood number, k _L a D _c ² /D_L - U _sg m/s superficial gas velocity - U _gr m/s superficial riser gas velocity - V _a m³ aerated liquid volume - V _c m³ unaerated liquid volume - N/m surface tension of the liquid phase - _g gas holdup - _app Pa · s effective viscosity of non-Newtonian liquid - kg/m³ liquid density - ý s^–1 shear rate - Pa shear stress     

Cephalosporin C production in a stirred tank reactor

Summary Cephalosporin C was produced with the moldCephalosporium acremonium in a 20 1 stirred tank reactor with 100 kg/m³ peanut flour in fed-batch operation. The growth and product formation was followed by on-line analysis of the broth composition. The cell concentration was estimated from the RNA-content of the cells. By optimization of the fed-batch operation and by increasing the phosphate content in the broth, a final cephalosporin C concentration of 12 kg/m³ was attained.Nomenclature CPC cephalosphorin C - DAC deacetylcephalosporin C - DAOC deacetoxycephalosporin C - k _L a volumetric mass transfer coefficient - MMBS 2-Hydroxy-4-methylmercaptobutyric acid - PABAH p-Hydroxybenzoicacidhydrazid - RNA ribonucleic acid - RQ respiratory quotient - oxygen transfer rate - CO₂-production rate - t fermentation time     

Hydrodynamics of non-Newtonian liquids in external-loop airlift bioreactors

Gas holdup investigation was performed in two external-loop airlift bioreactors of laboratory (V _L =1.189·10^?3? 1.880·10^?3 m³; H _R =1.16 ? 1.56 m; H _D = 1.10 m; A _D /A _R = 0.111 ? 1.000) and pilot scale (V _L =0.157?0.170 m³; H _R =4.3?4.7 m; H _D =4.0?4.4 m;A _D /A _R =0.04?0.1225), respectively, using as liquid phase non-Newtonian starch solutions of different concentration with K=0.061?3.518 Pa sⁿ and n=0.86?0.39 and fermentation broths of P. chrysogenum, S. griseus, S. erythreus, B. licheniformis and C. acremonium at different hours since inoculation and from different batches. The influence of bioreactor geometry, liquid properties and the amount of introduced compressed air was investigated. The effect of sparger design on gas holdup was found to be negligible. It was found that gas holdup depends on the flow media index, ?_GR decreasing with the increase of liquid pseudoplasticity, A _D /A _R ratio and H _R /H _D ratio. The experimental data are in agreement with those presented in literature by Popovic and Robinson, which take into account liquid properties, geometric parameters and gas superficial velocity, with a maximum error of ±30%. It was obtained a correlation for gas holdup estimation taking into account the non-Newtonian behaviour of the fermentation broths and the dry weight of the solid phase, as well. The concordance between the experimental data and those calculated with the proposed correlation was good, with a maximum error of ±17%. Also, a dimensionless correlation for gas holdup involving superficial velocities of gas and liquid, cross sectional areas ratio, dispersion height to riser diameter ratio, as well as Froude and Morton numbers, was obtained.     

Desorption of carbon dioxide from fermentation broth

Rates of CO₂ desorption from fermentation broths under actual operating conditions were determined by measuring the CO₂ partial pressure in the exit gas. The concentrations of CO₂ physically dissolved in the broths were measured by the so-called tubing method. Values of k_La for CO₂ desorption calculated from these values agreed well with the k_La values for oxygen absorption corrected for the difference in gas diffusivities. The dissolved CO₂ concentration in the broth, which seems to bean important operating parameter, can easily be estimated from the CO₂ partial pressure in the exit gas, a more easily measurable quantity, if the k_La value is known. For a given value of k_La, assumption of perfect mixing or plug flow in the gas phase made little difference in the calculated values of the dissolved CO₂ concentration, indicating that the gas phase was probably in between perfect mixing and plug flow. In industrial fermentors, the CO₂ partial pressure in the exit gas can practically be assumed to be in equilibrium with the dissolved CO₂ concentration.     

An immobilized cell reactor with simultaneous product separation. II. Experimental reactor performance

The simultaneous separation of volatile fermentation products from product-inhibited fermentations can greatly increase the productivity of a bioreactor by reducing the product concentration in the bioreactor, as well as concentrating the product in an output stream free of cells, substrate, or other feed impurities. The Immobilized Cell Reactor-Separator (ICRS) consists of two column reactors: a cocurrent gas-liquid "enricher" followed by a countercurrent "stripper" The columns are four-phase tubular reactors consisting of (1) an inert gas phase, (2) the liquid fermentation broth, (3) the solid column internal packing, and (4) the immobilized biological catalyst or cells. The application of the ICRS to the ethanol-from-whey-lactose fermentation system has been investigated. Operation in the liquid continuous or bubble flow regime allows a high liquid holdup in the reactor and consequent long and controllable liquid residence time but results in a high gas phase pressure drop over the length of the reactor and low gas flow rates. Operation in the gas continuous regime gives high gas flow rates and low pressure drop but also results in short liquid residence time and incomplete column wetting at low liquid loading rates using conventional gas-liquid column packings. Using cells absorbed to conventional ceramic column packing (0.25-in. Intalox saddles), it was found that a good reaction could be obtained in the liquid continuous mode, but little separation, while in the gas continuous mode there was little reaction but good separation. Using cells sorbed to an absorbant matrix allowed operation in the gas continuous regime with a liquid holdup of up to 30% of the total reactor volume. Good reaction rates and product separation were obtained using this matrix. High reaction rates were obtained due to high density cell loading in the reactor. A dry cell density of up to 92 g/L reactor was obtained in the enricher. The enricher ethanol productivity ranged from 50 to 160 g/L h while the stripper productivity varied from 0 to 32 g/L h at different feed rates and concentrations. A separation efficiency of as high as 98% was obtained from the system.     

Scale-up and application of a cyclone reactor for fermentation processes

A cyclone reactor for microbial fermentation processes was developed with high oxygen transfer capabilities. Three geometrically similar cyclone reactors with 0.5?l, 2.5?l and 15?l liquid volume, respectively, were characterized with respect to oxygen mass transfer, mixing time and residence time distribution. Semi-empirically correlations for prediction of oxygen mass transfer and mixing times were identified for scale-up of cyclone reactors. A volumetric oxygen mass transfer coefficient k _L a of 1.0?s^?1 (available oxygen transfer rate with air: 29?kg?m^?3?h^?1) was achieved with the cyclone reactor at a volumetric power input of 40?kW?m^?3 and an aeration gas flow rate of 0.2?s^?1. Continuous methanol controlled production of formate dehydrogenase (FDH) with Candida boidinii in a 15?l cyclone reactor resulted in more than 100% improvement in dry cell mass concentration (64.5?g?l^?1) and in about 100% improvement in FDH space-time yield (300?U?l^?1?h^?1) compared to steady state results of a continuous stirred tank reactor.     

Viscous media in tower bioreactors: Hydrodynamic characteristics and mass transfer properties

Studies in tower reactors with viscous liquids on flow regime, effective shear rate, liquid mixing, gas holdup and gas/ liquid mass transfer (k _La) are reviewed. Additional new data are reported for solutions of glycerol, CMC, PAA, and xanthan in bubble columns with diameters of 0.06, 0.14 and 0.30 m diameter. The wide variation of the flow behaviour index (1 to 0.18) allows to evaluate the effective shear rate due to the gas flow. New dimensionless correlations are developed based on the own and literature data, applied to predict k _La in fermentation broths, and compared to other reactor types.List of Symbols a(a) m^–1 specific interfacial area referred to reactor (liquid) volume - Bo Bond number (g D _c ² _L/) - c _L(c _L ^* ) kmol m^–3 (equilibrium) liquid phase oxygen concentration - C coefficient characterising the velocity profile in liquid slugs - C _s m^–1 coefficient in Eq. (2) - d _B(d_vs) m bubble diameter (Sauter mean of d _B) - d ₀ m diameter of the openings in the gas distributor plate - D _c m column diameter - D _L m²s^–1 diffusivity - E _L(E_W) m² s^–1 dispersion coefficient (in water) - E ² square relative error - Fr Froude number (u _G/(g D_c)^0.5) - g m s^–2 gravity acceleration - Ga Gallilei number (g D _c ³ _L ² / _eff ² ) - h m height above the gas distributor the gas holdup is characteristic for - k Pasⁿ fluid consistency index (Eq. 1) - k _L m s^–1 liquid side mass transfer coefficient - k _La(k_La) s^–1 volumetric mass transfer coefficient referred to reactor (liquid) volume - L m dispersion height - n flow behaviour index (Eq. 1) - P W power input - Re liquid slug Reynolds number ( _L(u _G +u _L) D _c/_eff) - Sc Schmidt number ( _eff/( _L D _L )) - Sh Sherwood number (k _La D _c ² /D_L) - t s time - u _B(u_sw) m s^–1 bubble (swarm) rise velocity - u _G(u_L) m s^–1 superficial gas (liquid) velocity - V(V_L) m³ reactor (liquid) volume Greec Symbols W m^–2 K^–1 heat transfer coefficient - y(y _eff) s^–1 (effective) shear rate - _G relative gas holdup - s relaxation time of viscoelastic liquid - _L(_eff) Pa s (effective) liquid viscosity (Eq. 1) - _L kg m^–3 liquid density - N/m surface tension