Characterization of hydromechanical stress in aerated stirred tanks up to 40 m3 scale by measurement of maximum stable drop size
© Daub et al.; licensee BioMed Central Ltd. 2014
Received: 8 January 2014
Accepted: 30 May 2014
Published: 7 July 2014
Turbulence intensity, or hydromechanical stress, is a parameter that influences a broad range of processes in the fields of chemical engineering and biotechnology. Fermentation processes are often characterized by high agitation and aeration intensity resulting in high gas void fractions of up to 20% in large scale reactors. Very little experimental data on hydromechanical stress for such operating conditions exists because of the problems associated with measuring hydromechanical stress under aeration and intense agitation.
An indirect method to quantify hydromechanical stress for aerated operating conditions by the measurement of maximum stable drop size in a break-up controlled dispersion was applied to characterize hydromechanical stress in reactor scales of 50 L, 3 m3 and 40 m3 volume with a broad range of operating conditions and impeller geometries (Rushton turbines). Results for impellers within each scale for the ratio of maximum to specific energy dissipation rate ϕ based on measured values of maximum stable drop size for aerated operating conditions are qualitatively in agreement with results from literature correlations for unaerated operating conditions. Comparison of data in the different scales shows that there is a scale effect that results in higher values for ϕ in larger reactors. This behavior is not covered by the classic theory of turbulent drop dispersion but is in good agreement with the theory of turbulence intermittency. The data for all impeller configurations and all aeration rates for the three scales can be correlated within ±20% when calculated values for ϕ based on the measured values for dmax are used to calculate the maximum local energy dissipation rate. A correlation of the data for all scales and all impeller configurations in the form ϕ = 2.3∙(ϕunaerated)0.34∙(DR)0.543 is suggested that successfully models the influence of scale and impeller geometry on ϕ for aerated operating conditions.
The results show that besides the impeller geometry, also aeration and scale strongly influence hydromechanical stress. Incorporating these effects is beneficial for a successful scale up or scale down of this parameter. This can be done by applying the suggested correlation or by measuring hydromechanical stress with the experimental method used in this study.
Turbulence intensity, or hydromechanical stress, is a parameter with an important impact on many different processes in the fields of chemical engineering and biotechnology. Since it governs the break-up of bubbles and drops in a turbulent flow field  it is very important in processes where interfacial area for mass transfer can become rate limiting . Furthermore, turbulence intensity has been discussed for a long time to have a large influence on biotechnological processes by a direct action on the biological phase . E.g. cell viability in microcarrier based cell culture processes can be correlated with turbulence intensity . In submerged fungal fermentations turbulence intensity may interact with morphological behaviour of the fungus . It was shown that volumetric power input in shake flasks is comparable to volumetric power input in stirred fermenters [6, 7] for typical operating conditions. Nevertheless, even at comparable levels of volumetric power input pellets grow much larger in shake flask culture compared to cultures in stirred tank reactors because turbulence intensity is much lower than in stirred reactors [8–10]. There is a close interaction between morphology, broth rheology and agitation intensity in submerged fungal fermentations in stirred tank reactors that can impact process performance . Production scale for fermentation processes can be as large as several hundred m3. The development of these large scale processes is conducted in lab or pilot scale fermenters. This poses the question how to scale down the conditions prevalent in large production scale to the development scale and vice versa. Typical parameters of interest may be mass transfer coefficient kLa, volumetric power input P/VL, impeller tip speed utip, turbulence intensity in the form of maximum local energy dissipation rate ϵmax, circulation frequency tc, or a combination of these parameters like the energy dissipation to circulation function EDCF (first introduced by ) that is basically the ratio of ϵmax/tc. Based on an estimation of the values of these parameters in large scale, small scale operating conditions (agitation and aeration rates) can be estimated that resemble the respective values of these different parameters in large scale. Testing these operating conditions in small scale may reveal which of these parameters (if at all) is a suitable proxy for scaling up or scaling down a particular process. The work of Jüsten [11–13], e.g., showed that scale up of mycelial flocs can be correlated with hydromechanical stress and circulation frequency in the form of the energy dissipation to circulation function EDCF. The morphology of the fungus in this case depends on the break up in the impeller region that is governed by hydromechanical stress and the aggregation of mycelial flocs in regions where turbulence intensity is low. Circulation frequency is decisive for the time for aggregation and for the frequency the mycelial flocs pass the high intensity region close to the impeller where they break up.
The whole procedure of testing different parameter candidates strongly depends on the validity of the correlations used to calculate these parameters. Although turbulence intensity is often discussed to influence biological processes only few data exists on the influence of geometry, scale and aeration on this parameter. There are two circumstances that strongly complicate the intention of establishing comparable levels of turbulence intensity in large and small scale: first, in industrial practice, geometrical similarity throughout the scales is hardly found [14, 15]. Therefore, the influence of geometry on turbulence intensity must be known for a successful scale up or scale-down of this parameter. Second, in aerobic fermentations the working medium is a multiphase gas-liquid dispersion which is characterized by a volumetric gas hold-up of up to 20% in production scale reactors . Of course, this is accompanied by much higher gas hold-ups in the vicinity of the turbulence inducing agitators. Very little is known on the influence of such high gas hold-ups on turbulence characteristics in stirred tanks because it is very hard to measure this parameter under these conditions. Particle image velocimetry (PIV) and laser Doppler anemometry (LDA) are often used to measure maximum local energy dissipation rates in small scale, single-phase reactors [16–23]. However, these methods cannot be applied in high gas hold-up conditions. Most available data was measured in lab scale with low agitation intensity. A review on existing data for maximum local energy dissipation was presented in .
It would be desirable to have a simple and practical method at hand to experimentally investigate turbulence intensity in large-scale multiphase reactors. This would enable a more rational approach to the scale-up and scale-down of this parameter for existing process equipment and it would help to throw light onto this range of operating conditions that are important for many processes yet extremely hard to access experimentally.
Therefore, a method was established to measure hydromechanical stress that can be applied in large scale equipment at intense agitation and aeration. Details on the development of the measurement method are specified in Daub et al. . The method is based on the theory of turbulent drop dispersion and uses the well established correlation of the maximum stable drop size dmax with maximum local energy dissipation rate ϵmax for break-up controlled dispersions:
with K1 = 0.23  and K2 = 2.5 . If dmax and ϵØ are known, ϕ can be calculated iteratively from Eqs. 1 and 2. Since this equation models an effect of microturbulence, the constants should be independent of geometry and scale .
This theory is valid if the flow field is fully turbulent and the drop size is much smaller than the macroscale of turbulence Λ = 0.4 h (Λ/dmax > 10) and much larger than the microscale of turbulence λ = (ν3/ϵ)1/4 (dmax/λ > 10). The flow field is fully turbulent when the Reynolds number is Re > 5∙103 and Λ/λ > 150 .
Geometrical details of the reactors used, geometrical details of impeller configurations and ranges of turbulence parameters
Geometrical details of reactors
Geometrical details of impellers
Nominal reactor size
Number of impellers
4.0 – 11.7
4.3 – 12∙104
16 – 32
286 – 571
14 – 28
11 – 39
3.6 – 8.0
8.1 – 18∙104
13 – 23
400 – 688
17 – 21
19 – 40
3.3 – 7.2
9.6 – 21∙104
14 – 23
402 – 674
17 – 21
19 – 39
2.8 – 6.1
1.0 – 2.2∙105
13 – 23
396 – 695
17 – 21
19 – 42
2.1 - 5.8
2.5 – 9.7∙105
11 – 23
1.5 – 3.2∙103
14 – 23
71 – 211
1.9 – 5.1
3.7 – 10∙105
11 – 23
1.5 – 3.2∙103
15 – 26
64 – 185
1.3 – 4.2
2.6 – 11∙105
11 – 25
1.9 – 4.4∙103
15 – 24
82 – 261
0.8 – 2.3
1.1 – 3.3∙106
11 – 22
3.9 – 8.3∙103
13 – 24
163 – 652
The same method will be applied in the current report with 8 different reactor configurations with Rushton type impellers in reactors of scales 50 L, 3 m3 and 40 m3 in a wide range of operating conditions (geometrical details are given in Table 2). It is the goal of this study to provide a broad data basis to demonstrate the applicability of this method to real life equipment of different scales, to characterize different reactor configurations with respect to hydromechanical stress as a basis for successful scale-up or scale-down of this parameter under aerated operating conditions and to test whether established literature correlations for the estimation of hydromechanical stress for different impellers can be applied with acceptable accuracy.
Results and discussion
Parameter values for the turbulence parameters for the range of operating conditions used in the present study are presented in Table 2. The microscale of turbulence was calculated with maximum local energy dissipation rates based on Eq. 1 with values for ϕ also given in Table 2. All parameters are within the validity ranges for the application of the theory of turbulent drop dispersion (Re > 5∙103, Λ/dmax > 10, dmax/λ > 10 and Λ/λ > 150).
Drop size distributions in different scales
The shape of a drop size distribution reveals important insight into the nature of the processes that formed the distribution. It was argued that a similarity of drop size distributions for different operating conditions or different equipment is a strong indication that the microprocesses involved in forming the drop size distribution are comparable. Brown and Pitt , Chen and Middleman , Konno et al. , and Peter et al. , e.g., showed that a plot of a normalized drop diameter versus the cumulative volume distribution reveals invariant drop size distributions with respect to agitation rate and dispersion time. Normalization was done either by the Sauter mean diameter d32 or the maximum stable drop size dmax. Invariance of normalized drop size distributions is referred to as self-similarity. Pacek et al.  point out that the observation of self-similarity might partly be due to a smoothing of fine differences by the cumulative distribution that is typically used for these plots. They argue that differences in drop size distributions might be clearer recognized in volume density distributions. Therefore, these will be used in this work.
Comparison of maximum stable drop sizes d max and Sauter mean diameters d 32 for all three scales
Maximum stable drop size dmax
Sauter mean diameter d32
Maximum stable drop sizes calculated from these distributions are indicators for hydromechanical stress. To perform a scale-up or scale-down of hydromechanical stress it is necessary to correlate the maximum stable drop size with operating conditions. This is the focus of the following paragraphs.
Correlation of maximum stable drop size with impeller tip speed ut
Especially in industrial practice impeller tip speed is frequently applied as a correlator for hydromechanical stress. Margaritis and Zajic  estimate that 20% of the fermentation processes in industry are scaled up based on this rule. The results shown here, in accordance with earlier analyses on the value of impeller tip speed for scale-up of processes [13, 15, 40, 41], clearly show that impeller tip speed is not well suited to correlate hydromechanical stress in fully turbulent aerated stirred tanks.
Correlation of maximum stable drop size with volumetric power input
For cell culture processes, not only the effect of maximum energy dissipation induced by the flow field is relevant for cell damage [42, 43]. More importantly, bubble formation at the sparger and bubble rupture at the liquid surface are known to be the major cause for cell death by hydrodynamic forces in these processes [44–46]. Attachment of cells to bubbles plays an important role in the lethal effects of bursting bubbles in cell culture processes [47, 48]. There is no evidence that these effects may have an influence on the maximum stable drop size. Measurements were conducted with aeration rates varied from 0.1 vvm to 1 vvm. If bubble rupture had a considerable influence on the maximum stable drop size a correlation of maximum stable drop size with aeration intensity migh be expected. Additionally, when unaerated and aerated data were compared in a previous publication , maximum stable drop size was smaller without aeration than with aeration (with the same volumetric power input). Hence, it can be concluded that the experimental results shown in this study are not influenced by the effects of bursting bubbles.
The measurements clearly discriminate between the different impeller configurations in the 50 L reactor. The larger the impeller, the larger the maximum stable drop size at a given volumetric power input. This is equivalent with a decrease in the ratio of maximum to specific energy dissipation rate with increasing impeller size and consistent with existing literature data for unaerated operating conditions .
The characteristics of the impellers used in the 3 m3 reactor are relatively similar in relation to the measurement accuracy for the maximum stable drop size and the impellers can hardly be distinguished. Nevertheless, a sequence of the three impeller configurations is apparent that is in accordance with the results for the 50 L reactor. The smallest impeller B-1 produces the smallest maximum stable drop sizes at a given volumetric power input. The larger impeller B-2 with the same impeller blades as B-1 produces larger drops and the largest impeller with larger impeller blades B-3 results in the largest maximum stable drop sizes at a given volumetric power input.
Quality of the fit for all 8 reactor configurations
Standard deviation [%]
Turbulence intermittency might explain an increased slope for dmax. This extension of the classic theory of drop dispersion takes into account the intermittent character of fine-scale turbulence as laid out by Baldyga and Podgorska  and Baldyga et al. . In this concept, ϵmax is not taken as a constant but as a stochastic variable that fluctuates about its mean value. The theory predicts that rare but strong bursts of high energy become more and more important for the evolution of the maximum stable drop size with increasing dispersion time. This results in a long-term drift of maximum stable drop sizes towards smaller drops. As a consequence, the exponent on maximum energy dissipation rate becomes time-dependent with values up to -0.62 for very long dispersion times . Additionally, intermittency is stronger for higher Reynolds numbers  and, therefore, the effect becomes increasingly important with increasing scale.
All data presented in this study were measured between 100 and 180 min of dispersion time. The maximum stable drop sizes measured were essentially constant during this time span in all reactors. Examples for this were shown in Daub et al. . Extending the experimental time to up to 9 h yielded a further decrease of the maximum stable drop size in the range of 10% compared to the value at 3 h dispersion time in the 3 m3 reactor. This is qualitatively in agreement with the prediction of the effect of intermittent turbulence, but the extend of the effect is relatively small and comparable to the reproducibility of the measured values in independent experiments. The data for the 3 m3 reactor follows equally well the classic theory of drop dispersion as the data for the 50 L reactor. It must be assumed that the scale-effect on the slope cannot be resolved with the applied measurement method because it is below the reproducibilty of the experimental method. If the measurement accuracy in the 40 m3 reactor is similar to that in the other scales and taking into account that the effect of turbulence intermittency on the slope was not strong enough to be detected for the other reactor configurations it seems justified to assume that the differences between measurement and model for the 40 m3 reactor rather reflect measurement inaccuracy than the effect of intermittency on the slope. We, therefore, simplify the analysis and restrict the value of the slope to that of the classic theory of drop dispersion.
The data for the largest impeller in the 50 L reactor (A-4) exhibits a higher slope than expected for all aeration rates. This impeller has an extreme geometry with very large d/DR-ratio of 0.65 and large impeller blades. The distance between the impeller tip and the baffles is only 0.02 m. This probably gives rise to a nonstandard flow-field which may result in a modification of the turbulence characteristics. Additionally, the calculated values for power input based on Eq. 8 might possess a larger error than for the other impellers that are closer to standard geometry. The impellers for A-2, A-3 and A-4 have the same impeller blades but different impeller diameters. A-3 generates larger maximum stable drop sizes at the same power per unit volume than A-2. If this trend is extrapolated to A-4 than larger maximum stable drop sizes may be expected for A-4 than for A-3 at the same volumetric power input. The values for the lower range of power inputs for A-4 up to 4 kW/m3 are in agreement with this expectation but the data for the higher power inputs tend towards smaller maximum stable drop sizes than expected. By interpreting the data on the basis of Eqs. 1 and 2 the value of ϕ for this impeller might be overestimated in comparison to the other impellers.
Estimation of maximum local energy dissipation rate ϵmax and the ratio of maximum to specific energy dissipation rate ϕ
The literature correlations for the ratio of maximum to specific energy dissipation rate ϕ given in Table 1 can be used to calculate the maximum local energy dissipation rate ϵmax for different operating conditions and reactor geometries. It must be emphasized that these correlations were derived for single-phase, unaerated operating conditions and not for aerated operating conditions. However, up to now the only practical way to estimate ϕ for different reactor configurations for aerated operating conditions was to assume that these correlations can be applied also in the presence of aeration. This was first supported by the data of Bourne  that is based on a chemical method to measure micromixing efficiency. Bourne  came to the conclusion that ϕ is not influenced by aeration. Fort et al.  report on a roughly 20% reduction of turbulence intensity in the presence of aeration. However, their data is based on the measurement of pressure fluctuations at constant agitation rate. There is no clear conclusion with regard to the influence of aeration on ϕ. The data presented in Daub et al.  clearly shows that ϕ is reduced by aeration based on the same measurement technique that is used in this study. The relation of these results to the findings of Bourne  and newer literature data incorporating similar methods are discussed in Daub et al. . As an example for the value of the correlations from Table 1 for aerated operating conditions, Figure 6 compares all data obtained in this work with ϵmax calculated with the correlation of McManamey . This equation was used because it might be the most common correlation to estimate ϵmax. Given the simplicity of this type of correlation, the broad spectrum of impeller geometries used in the experiments and the broad range of scales applied, the results are in reasonable agreement. The prediction from this simple correlation is probably accurate enough for crude estimations in industrial practice.
Comparison of experimental values for ϕ and values from literature correlations
Results from experiments with aeration
Correlations for single-phase, unaerated operation
The absolute values for ϕ that were derived from the measurements are small compared to the values from literature correlations. It was shown in Daub et al.  that a comparison of maximum stable drop sizes under aerated and unaerated operating conditions for reactor configurations B-1 and B-3 reveals a strong attenuation of turbulence intensity by the presence of air. ϕ was reduced by aeration by 64% for B-1 and by 52% for B-3 when compared with unaerated operating conditions on the basis of equal volumetric power input. The low values for ϕ found for the data presented in this study fit well into this pattern and support this finding.
If different scales are compared with each other the measured values for ϕ suggest a scale-effect with higher values for ϕ in larger scales. A-1 and B-1 for example are close to geometric similarity and the literature correlations predict similar values for ϕ for the two impeller configurations. McManamey’s  correlation, e.g., predicts that the ratio of the values of ϕ for B-1 compared to A-1 should be 1.1. However, the ratio of the values of ϕ derived from the measurements is 1.7. That means that hydromechanical stress at equal volumetric power input is higher in the 3 m3 reactor than in the 50 L reactor although the impeller geometries are close to geometric similarity. A comparison of the data for the 40 m3 reactor with the 3 m3 reactor also shows this scale-effect. C-1 has a larger diameter impeller than B-1 with similar sized impeller blades in relation to the reactor diameter. Within one scale this combination results in a lower value for ϕ for the larger impeller (e.g. A-2 in comparison with A-1). This is also predicted by the correlation of McManamey  that predicts a ratio of the values of ϕ for C-1 compared to B-1 of 0.6. The experimental data however gives a ratio of 1.7 for C-1 compared to B-1, i.e. hydromechanical stress is higher at the same volumetric power input in C-1 than in B-1. The classic theory of drop dispersion as expressed in Eqs. 1 and 2 does not predict a scale-dependence of maximum stable drop size for geometrically similar reactor configurations for aerated operating conditions. However, Baldyga et al.  show for inviscid drops in unaerated dispersions that turbulence intermittency can explain a scale-dependence of dmax that leads to smaller drops in larger scales. The extent of the scale-dependence is related to dispersion time through the “multifractal scaling exponent”. A parameter that is not readily available for practical applications. For long dispersion times the theory predicts a dependence in the form dmax ~ DR-0.543. If this is the case and the data is still interpreted on the basis of the classic theory of drop dispersion this will result in an apparently higher value of ϕ for large reactors. If the case of inviscid drops is considered with dmax ~ ϵmax-0.4 then ϕ ~ DR0.543 would result following the theory of Baldyga. The proportionality ϕ ~ DR0.543 can be used to calculate the theoretical ratios of ϕ for the different reactor scales used in this study assuming geometric similarity: the ratio of ϕ for the 3 m3 reactor compared to the 50 L reactor is 2.2. The calculated ratio of ϕ for the 40 m3 reactor compared to the 50 L reactor is 3.4 and the ratio of ϕ for the 40 m3 reactor compared to the 3 m3 reactor is 1.6. These differences are in reasonable agreement with the differences seen in the experimental results for ϕ. Although the effect of intermittency on the exponent on energy dissipation rate could not be resolved with the measurement method applied in the experiments, the effect of intermittency on ϕ is strong and must be incorporated in the analysis.
For the first time, results from drop dispersion experiments in aerated stirred tanks were presented that cover a broad range of operating conditions, impeller geometries (Rushton impellers) and reactor scales of 50 L, 3 m3 and 40 m3 volume. A comparison of the volume density distributions in the three different scales show that the drop size distributions are self-similar and that d32/dmax for the aerated dispersions are in the same range as reported by other groups for single-phase, unaerated dispersions e.g. . It was shown that the influence of aeration and scale on hydromechanical stress is not considered correctly when using impeller tip speed as the correlator. The influence of aeration for each impeller type is well reflected by correlating the data with aerated volumetric power input. This is in accordance with the classic theory of break-up controlled drop dispersion if the ratio of maximum to volume averaged energy dissipation rate ϕ is independent of the operating conditions. Absolute values for ϕ that were calculated for each impeller based on literature correlations for unaerated operating conditions differ strongly for the different correlations and in comparison to the values derived from the measurements with aerated operating conditions. The relative order of the impellers within each scale is the same for all correlations for unaerated operating conditions as for the values that are based on the drop size measurements for aerated operating conditions. Hence, the behavior of the impellers relative to each other within each scale is qualitatively well predicted by the correlations even though they are strictly valid only for unaerated operating conditions. The low values for ϕ found for the data presented in this study support the finding reported in Daub et al.  that hydromechanical stress is strongly reduced (ϕ is reduced by approx. 60%) for aerated operating conditions compared to unaerated operating conditions at the same volumetric power input. Comparison of data in the different scales shows that there is a scale effect that results in higher values for ϕ in larger reactors. This behavior is not covered by the classic theory of turbulent drop dispersion but is in good agreement with the theory of turbulence intermittency that predicts an up to 3.4 times larger value for ϕ in the 40 m3 reactor than in the 50 L reactor. The data for all impeller configurations and all aeration rates for the three scales correlate very well when calculated values for ϕ based on the measured values for dmax are used to calculate the maximum local energy dissipation rate. Most of the data lies within 20% around the theoretical prediction from the classic theory of drop dispersion when these values for ϕ are used. A correlation of the data for all scales and all impeller configurations in the form ϕ = 2.3∙(ϕunaerated)0.34∙(DR)0.543 is suggested that successfully models the influence of impeller geometry, aeration and scale on ϕ for aerated operating conditions. Incorporating the effects of aeration and scale on hydromechanical stress is beneficial for a successful scale up or scale down of this parameter. This can be done by applying the suggested correlation or by measuring hydromechanical stress with the experimental method used in this study.
Materials and methods
Reactor and impeller configurations
Measurement of drop size distributions and maximum stable drop size
The development of the experimental procedure to measure drop size distributions and maximum stable drop size including the rationale for the dispersed and continuous phases used for the dispersion experiments were reported in Daub et al. . Details on the preparation of the 50 L reactor were also given there. Experimental details specific for the 3 m3 reactor were presented in Daub et al. . The experiments in the 40 m3 reactor were conducted in the same way as explained for the 3 m3 reactor in Daub et al. . All experiments were conducted with the same production batch of paraffin oil (Weissöl Ph Eur., Brenntag, Germany).
Power input and power number Po
where nbl is the number of impeller blades and h is the impeller blade height, except for impeller configurations B-1 and B-3 where the power number was measured based on experiments without aeration. The reliability of this correlation can be tested by comparing the measured values for B-1 and B-3 with the calculated power numbers for these impeller configurations. The calculated power number for configuration B-1 is 5.5 vs. the measured value of 4.9 (+12%) and for configuration B-3 5.8 vs. the measured value of 5.9 (-2%). Both values are in reasonable agreement with the measured values. This correlation for power number can be considered very helpful and reliable within engineering accuracy and within the accuracy needed for the power data for the analyses conducted in this work. It resembles correctly the relative influence of blade height and impeller diameter.
Power input in the 3 m3 and in the 40 m3 reactors were measured through the electrical power draw of the engine corrected for friction and other losses. For the 3 m3 reactor the power input to the liquid was calculated from the raw value by a linear correction function as described in detail in Daub et al. . Power losses were evaluated by an instationary temperature method that is completely independent of the electrical power measurement. The correlation for power losses was tested against electrical power measurement in the empty reactor and both measurements of power loss were in good agreement. Reproducibility of the electrical power measurement was very good with a standard deviation of 5%. For the 40 m3 reactor a linear correlation of the power data with n3 showed a good correlation of the data with an R2 of 0.99 when an offset of 25.2 kW was used. The two lowest agitation rates had very low power inputs (0.13 kW/m3 for n = 0.78 1/s and 0.36 kW/m3 for n = 1.0 1/s). It was decided to use calculated values for power input for these operating conditions instead of the measured values to avoid large measurement errors in this low range of operating conditions.
a: Constant [-]
C: Bottom clearance of first impeller [m]
ΔC: Impeller spacing [m]
d: Impeller diameter [m]
dmax: Maximum stable drop size [μm]
DR: Reactor diameter [m]
d32: Sauter mean diameter
Fl: Impeller flow number [-]
Fr: Impeller Froude number [-]
h: Impeller blade height [m]
H: Unaerated liquid height [m]
K1: Constant in Eq. 2 [-]
K2: Constant in Eq. 2 [-]
n: Agitation rate [1/s]
nbl: Number of impeller blades [-]
P: (Aerated) power input [W]
P0: Unaerated power input [W]
Po: Power number [-]
ut: Impeller tip speed [m/s]
VL: Total liquid volume [m3]
w: Impeller blade width [m]
ϵØ: Volume averaged energy dissipation rate [W/kg]
ϵmax: Maximum local energy dissipation rate [W/kg]
ρc: Continuous phase density [kg/m3]
ϕ: ϵmax/ϵØ [-]
σ: Interfacial tension between dispersed and continuous phase [N/m]
ηD: Dynamic viscosity of dispersed phase [Pa∙s]
The authors would like to acknowledge financial support for the project from Sandoz GmbH, Kundl, Austria.
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