Introduction

Uses of measurements for bubbles and pellets

Accurate and representative bubble and pellet size distributions have been used to characterize biochemical processes. Fermentation of industrially important fermentation products involves multiphase dispersion [110], focusing primarily on gas in liquid systems. Gas bubble size depends on (1) the type of sparger configuration (i.e., point, ring, or frit) and its position relative to the impeller, (2) bioreactor operating conditions such as shear (affected by agitator speed/velocity and type), volumetric air flow rate, and temperature, and (3) gas/liquid properties such as the type of media (affected by density and viscosity) and the presence of surface-active components (i.e., surface tension) [44, 67, 92]. Bubble sizes range considerably with some reported to be up to around 5–10 mm in bubble columns containing viscous solutions [59]. Bubble size distributions vary widely within a stirred tank based on the distance from the impeller [6]. In addition, bubbles change in size over the course of the fermentation. Quantification of bubble sizes is important to establish mass transfer characteristics (based on gas–liquid interfacial area) when oxygen transport to cells across gas–liquid interfaces becomes a limiting factor. In these situations, there is a direct influence of bioreactor parameters on culture yields [110], and thus it is useful to reliably quantify bubble size.

Several industrially important cultures grow as multi-cell pellets (vs. filamentous mycelia) for maximum productivity and lower viscosity. Pellet sizes range from 40 to 1,000 μm for Penicillium chrysogenum [34, 35, 58] and 400 to 2,500 μm for Streptomyces tendae [109]. Pellet sizes depend not only on the agitator shear, but also on several other factors such as culture and media [40, 73]. Optimal pellet size avoids nutrient limitations associated with larger pellets which cause cell death and reduced productivity [82]. Thus, quantification of pellet sizes is important to identify transport restrictions (based on pellet diameter and density) when diffusion in the presence of nutrient uptake limits radial penetration of nutrients including oxygen [20]. Interestingly, the surrounding turbulence improves nutrient transport in pellets [113].

In situ versus sampling; on-line versus off-line

On-line measurements are continuous in nature and can be based either on continuous sampling from the bioreactor system or direct sensing. The sample size for on-line analysis is larger; thus more representative data are generated in more timely intervals and greater quantities of data accumulate faster with little manual intervention [19]. On-line analysis should reduce and not increase the laboratory workload [19]. There has been a perceivable trend towards computer-controlled chemical operating plants [47], and most recently process analytical technology (PAT), in part based on on-line sensors, is being applied to many processes [93].

In situ sensing avoids the instrument being limited by removal of a sample as it can be difficult to obtain representative samples due to operator variability and sampling inconsistencies [19]. In situ analysis also avoids sample alteration caused by removal from the processing environment [19]. It omits error caused by sample preparation. Specifically for mycelia undergoing image analysis, sample preparation errors can be  ±  8% [71], and dilution of particles inappropriately can alter particle shape [19]. Furthermore, biomass dilution factors must be optimized; samples of mycelia typically are diluted 100-fold prior to image analysis [57]. Over-dilution increases the number of objects detected, while under-dilution causes overlaid objects; optimal dilution is the lowest dilution that enabled the maximum percentage of recognition [5]. As the overall variance of analysis is composed of the sum of individual variances of each step, a 10% sampling error is large compared with a 1% standard error in size analysis [47]. Errors also exist when the sensor is placed next to the glass window of a tank (applicable primarily to small-scale laboratory apparatus) which can affect measurements due to its curvature, although this error is smaller than the sampling error. In situ methods in which the probe is inserted into the process avoid this problem, but an inserted probe can potentially obstruct bubble position or modify bubble shape [91].

Background

Historical and currently available measurement methods

General

The present state of the art in particle size analysis is characterized by the use of optical methods [47]. Particle size has been measured using several methods such as low-angle laser light scattering, ultrasound, optical image analysis, and direct mechanical measurement; each method results in accurate particle size data within its intended set of parameters [19]. There are four common methods for bubble analysis: photographic, electrical conductivity, electro-optical, and light scattering [10]. The focus of this review is on-line photographic methods using optical image analysis for bubbles size analysis. However, there are far more published studies using image analysis for pellet and morphology than bubbles and applying image analysis to off-line rather than on-line samples. In fact, image analysis is well established for quantifying and characterizing mycelia from off-line samples of fermentation systems [77]. Thus, surveys of past and present image analysis techniques for both bubbles and pellets (morphology) were conducted and evaluated to assess issues and trends relevant to the development of new instrumentation devices.

Early methods of pellet size analysis utilized manual sieves with various mesh sizes [34, 35, 109]. Morphological characterization of filamentous organisms (free cells, mycelia, and pellets) in submerged culture has been significantly enhanced by image analysis technology developments [24]; some of these developments have also been applied to bubble size quantification. Consequently, bubble quantification also has progressed significantly from initial methods that used simple manual measurements of photographs [88]. Many cell imaging systems are stagnant and most bubble systems are dynamic, however. All systems need the ability to focus automatically or manually to obtain a clear image [70]. Costs for fully automated image analysis can be too high for certain applications, however, and additional manual steps can be required that are time-consuming [36].

Early photographic methods

Compared with conductivity, light scattering, and electro-optical methods, the photographic method is the most tedious, but it handles the broadest bubble size distribution and is the most reliable for viscous media [76]. It is one of the best methods for obtaining gas bubble surface area despite the tedious effort required [101]. Overall, photography is more sensitive to smaller bubbles than larger ones, requires relatively clear media, and possesses issues with occlusion and depth of field [76]. Cameras have been attached to specially adapted microscopes located externally to the process apparatus or placed directly adjacent to the apparatus wall with resulting images printed and analyzed after viewing through a microscope [13].

Manual counting and analysis techniques for photographs or videos are laborious, tedious, imprecise, and time-consuming which make them hard to be reliably quantitative [5, 110]. Early photographic methods were manual in nature, performing measurements directly from the microscope stage with the use of a micrometer or indirectly using photographs [77]. Initial investigations of mycelial morphology relied upon these inaccurate and time-consuming manual measurements from photographs [105]. An electronic digitizer increases speed [77], but digitizing tablets are operator-dependent and slow [105]. Digitizing methods also have the disadvantage of being labor-intensive and time-consuming [1] and are not particularly precise or accurate [70].

Image acquisition, processing, and analysis methodology

An on-line, automated system should produce data within a process-relevant timeframe. This time is evaluated through quantification of the (1) data acquisition time (DAT) or time to acquire a single piece of data and (2) measurement acquisition time (MAT) or time to acquire sufficient data to produce an accurate result and reset the system for the next measurement. Ideally the MAT is equal to the DAT [19]. The absolute value of the MAT is most relevant. If the MAT is greater than or equal to the process time constant, the instrument is unable to resolve process temporal changes. If the MAT is less than the process time constant, then only non-quantitative process trending can occur. If the MAT is much less than the process time constant, the instrument can observe process behavior and upsets [19] and thus be useful for real-time monitoring and control applications. Specifically for particle size counters, the MAT is much greater than the DAT [19]. For microbial secondary metabolite or animal cell processes, MATs of no faster than 1–2 per hour are likely to be sufficient; for faster metabolizing Escherichia coli or yeast fermentations, higher MATs of up to 4–6 per hour might be required.

Measurement systems utilize a television camera mounted on a microscope with a video signal of the field of view sent to a computer capable of image processing and analysis. Tube cameras offer high sensitivity especially in low light levels [77]. The image is digitized in both space and tone to produce pixels (picture elements), each of which is assigned a grayness level. Further image processing is done to improve quality, and then images are analyzed to obtain measurements [70]. Conventional video cameras can be synchronized with the flashing of a strobe light to speed up acquisition to avoid blurred images [99], and this technique avoids use of an expensive fast acquisition digital or analog video camera [110]. Sharp images are obtained using either stroboscopic imaging or shutter speeds less than 0.001 s, but these fast shutter speeds require a lot of light [89]. Alternatively, high-speed video cameras may be utilized [99].

Typical hardware consists of high-quality cameras (cine-photographic equipment). The quality of most cameras has a specification of a 49 dB signal-to-noise ratio which translates into 7 bits of real information and 1 bit of noise [89]. Owing to noise and degradation with older cameras, previously there were only about 64 gray levels actually distinguishable in data, compared to about 30 levels in the human eye [89]. Images can be divided into pixels (e.g., 640  ×  480 pixels) with each pixel having 256 brightness levels [89], and current cameras have 256 gray levels as well when used in the 8 bit mode. A calibration factor is required depending on the magnification to convert the inter-pixel distance to microns [70]. There is a “real-time” DAT, for example 30 frames/s (fps) [89], which can be increased if the number of pixels per frame (i.e., resolution) decreases. For high-resolution imaging, an analog signal path for data is avoided; a digital signal path reduces degradation and improves resolution [89].

Basic flow charts are similar for various image analysis systems. They consist of the following common stages: initialization/set up/autofocus, image capture/detection, image optimization/enhancement, segmentation, image processing (sometimes with manual editing), measurement/calculation, archive/file storage, and evaluation/analysis [1, 23, 77]. There is a need to set up (1) the hardware’s focus, brightness, and calibration parameters and (2) the software’s image processing parameters [77]. Segmentation, separation of the image into objects of interest to be measured and background, is an important step between image processing and image analysis; it distinguishes using relative brightness [23, 89]. The threshold value delineates the objects from the background [89], with all pixels brighter (i.e., grayer) than a preset value of interest [16, 70]. Real-time gray level differences between two successive frames are employed to detect moving objects and subtract out stationary objects; the length of the delay between two frames is chosen to avoid image overlapping [33]. Erosion removes pixels from an image that should not be there, and dilation adds pixels to an image [89], respectively, decreasing and increasing an object’s size along its boundaries. A masking binary (i.e., black and white) image is defined which shows which objects have been selected for further processing, and subsequent processing is based on this binary image [1, 70]. Sometimes the criterion that there are no holes in circular objects (i.e., regions of background) is applied when selecting objects of interest [70]. Several other types of image processing filters can also be applied to raw images prior to analysis.

Automation of image analysis makes it independent of the operator and faster [71]; automated quantification also avoids bias by the observer [77]. Image analysis is more precise than the digitizing tablet method [1] and has replaced digitizing tablets in many applications. Digital image processing has greater speed and better size resolution and avoids manual steps, so throughput is improved [82]. Input devices (such as a microscope or macroviewer with video camera, video recorder, and scanner) are interfaced with a high-performance PC with image processing microprocessors designed for speed [77]. The PC can store raw images for later recall, but storage requirements need to be reasonable [36]. Storage limitations have been partially overcome by today’s more advanced, high-capacity hard disks. Technological advances and decreased computational costs have permitted quantitative image analysis to be used for process monitoring [5]. The use of manual editors in automated image analysis systems slows down processing, however [70].

Literature examples of bubbles size distribution methodologies

Table 1 shows characteristics of selected published photographic techniques used for obtaining bubble measurements and of selected non-photographic techniques which were included for comparison purposes. A review of these techniques reveals trends and progressions which when summarized clearly indicate key preferences in technology. All dispersions are gas (discontinuous phase)/liquid (continuous phase) unless otherwise stated. Bubble measurements began as fully manual in the mid-1950s and progressed through various levels of automation, frequently incorporating substantial manual steps to assure objects were selected appropriately. These measurement systems were located externally to the process. Most of them involved measuring only bubbles visible at the outside wall of a transparent vessel. Illumination often was by external flash using shutter speeds ranging from 0.0005 to 0.002 s or an internal strobe light synchronized to 50–100 fps. In a few cases, in situ microscopy was used. Magnification ranged from 3 to over 100-fold, with higher values used to discern liquid drops and solid particles. Curvature effects were quantified using internal standards, and in some cases they were minimized by placing cylindrical vessels and pipes into rectangular boxes filled with the dispersion’s continuous phase. Some authors have identified measurement limitations at higher gas hold-ups and interferences due to additional light absorbing/scattering components of the dispersion. No one camera or image analysis software system has emerged as the standard; rather a wide variety of commercial hardware and somewhat customized software has been utilized.

Table 1 Selected photographic and non-photographic bubble measurement systems from the literature
Full size table

Literature examples of cell, pellet, and morphological distribution methodologies

Table 2 shows characteristics of selected published photographic techniques used for obtaining pellet size and mycelial morphology measurements. A review also reveals trends and progressions, as well as similarities and differences to the task of bubble measurement. Techniques range from manual to digitized to semi-automated and then fully automated analysis. In some cases, manual interaction was initially used to classify images. Dilution from 2 to 800-fold was necessary to obtain distinct objects without overlap. Illumination and shutter speed generally were not reported as broth samples primarily were fixed and still shots from a microscope stage were most common. Magnification ranged from 2 to 200-fold for pellets which was about the same as for bubbles and up to 2,000-fold for mycelial hyphae measurements. No single type of image analysis software was used, but commercially available software often was highly customized.

Table 2 Selected photographic cell, pellet, and morphological measurement systems from the literature
Full size table

Measurement assumptions and trade-offs

Key parameters of past and current photographic measurements systems are summarized in Tables 3 and 4 for bubbles and pellets, respectively, Ranges of these parameters bracket expected minimum requirements for future novel measurement devices. The size and number of objects used for each measurement vary considerably depending on the research application. For bubble measurements (Table 3), the size of the objects ranged from 40 μm to 20 mm, with most studies covering the range from 40 μm to 2 mm. The number of objects measured ranged from 50 to 4,000, with over half of the studies using between 480 and 1,000 objects. Previously, between 300 and 1,000 objects have been found to result in a stable bubble distribution in bubble column measurements [18]. There was a large variation (2–400) in the number of frames required for a single measurement. Measurement errors ranged from  <  2 up to 10–20%, and they were not quantified for half of the studies. The best resolution reported was  ±  0.05 mm [92]. MATs for automated acquisition and analysis were also not quantified often, but available reports ranged from 5 to 60 min.

Table 3 Summary of bubble measurement parameters from the literature in ascending order of objects per measurement
Full size table
Table 4 Summary of pellet and morphological measurement parameters from the literature in ascending order of objects per measurement
Full size table

For pellet and mycelial morphology measurements (Table 4), the size of the pellets ranged from 140 μm to 6.7 mm, a range similar to bubble measurements. The size of the hyphae typically measured ranged up to 1,800 μm, also similar to the bubble size measurement range. The number of objects per measurement ranged from 10 to 24,000 with over half of the studies using between 100 and 500 objects. The number of frames measured ranged from 5 to 236. Measurement errors varied widely, from 2 to  <  27%. Resolution ranged from 1–2 μm/pixel [105] when used for hyphae length to 10–40 μm/pixel [24, 25]. MATs ranged from 10 to 170 min with shorter times of 3.6–10 s for measurements of particle flow velocity.

For small particle size measurements, the typical image analyzer acquired 250,000–500,000 images for one particle size distribution which required up to 10–20 min on some systems [19]. These values were much higher when compared to bubble size or pellet distribution instrumentation for which only hundreds of objects were measured to obtain statistically valid results [77]. Few researchers provide data in which the measurement sample size has been extended until differences in the size distribution parameters diminished into the noise/error range. The required number of objects per measurement becomes important if all images need to be reviewed/saved to evaluate accuracy of data or if it extends the MAT.

In obtaining a satisfactory number of bubbles in a single frame, a compromise existed as too many bubbles caused overlap and too few bubbles required counting many frames [13]. The number of frames that required counting for accurate data analysis decreased with lower mean bubble size [55]. Some measurements have been limited to void fractions  <  10% to avoid large amounts of overlapping bubbles [53] or solid volume fractions  <  0.08 to avoid particles covering too much of the bubble area at high solids loadings [42]. Similarly pellets in broth have been diluted so that the number of pellets in the area of analysis filled only 10–15% of the total image area [85, 86]. In addition, the area of picture with the lowest interference has been selected manually and clumping pellets (P. chrysogenum) were divided manually prior to further processing [82].

There was a trade-off between the elapsed time required for the system to reach steady state after operating conditions were changed and the number of different operating conditions that may be analyzed in a given time period. This time to reach steady state has ranged from 2–5 min in agitated gas–liquid systems [42, 55] to 30 min for oil–water–gas systems [83]. Another trade-off was the size range selected for bubble measurement since bubbles smaller than 1,000 μm generally were spherical and bubbles above 2,000 μm began to become non-spherical [55]; even larger diameter bubbles were highly irregular in shape. Larger bubbles required a greater number of frames to attain the statistically desirable number of items to be measured.

Measurement interferences

Common measurement interferences have been identified by prior researchers. One of the main problems monitoring multiphases in a bioreactor is acquiring clear images in motion [99]. The ability to distinguish bubbles from the background, analyze contiguous bubbles (bubbles touching, in front of, or overlapping other bubbles) either by exclusion or deconvolution, and omit large, irregularly shaped bubbles are also key factors. Size analysis under high gas hold-up conditions was complicated by bubble overlap and inability to clearly distinguish individual bubbles [45]. In a few specific instances, bubbles positioned near a huge bubble swarm were incorrectly included within the swarm by the imaging software. An estimate of gas hold-up is obtainable by quantifying the clear areas of an image comprised of bubbles and bubble swarms.

The effect of broth turbidity on the depth of field and interference by cell solids has been another limitation of optically based methods. The presence of protein decreases image contrast [83] by blurring object edges. In addition, particles do not transmit light as bubbles do. In one application, acceptable bubble images (i.e., objects with dark edges and a shiny middle) and an indication of dispersed biomass between bubbles were obtained only up to 5 g/L biomass dry cell weight [27]. Reflection from stainless steel tank internals (i.e., impeller, agitator shaft, sparger) can also interfere, resulting in bright blotches which are reduced in the presence of medium and cells.

Distribution calculations from data

Specific mathematical equations used for bubbles sizes

Size measurements are obtainable based on direct measurements of diameter, area, or volume, or using back-calculations to obtain an equivalent diameter assuming a spherical shape. There are several expressions used to describe diameter. The sample (arithmetic) mean bubble diameter, d a, is given by [6]:

$${d}_{\rm a} = \left(\sum {d}_{i}\right)/{n},$$
(1)

where n is the total number of bubbles measured and d i the diameter of bubble i. The Sauter mean diameter, d 32 [9, 69, 97], links the area, n i d 2 i , and volume, n i d 3 i , of the dispersed phase (the number of bubbles, n i , of diameter d i ), as shown by:

$${d}_{32} = \sum{n}_{i}d_{i}^{3}/\sum{n}_{i}d_{i}^{2}.$$
(2)

This diameter is important for quantifying mass transfer effects [110].

The log-geometric mean diameter, d g, is calculated using [75]:

$$d_{\rm g} = \left(\sum n_{i} \hbox{log}\,d_{i}\right)/ \sum n_{i}.$$
(3)

It characterizes the log-normal distribution curve, one type of distribution commonly associated with bubble size distributions in agitated systems. Air–water bubble diameter distributions, plotted as the percent relative frequency versus the equivalent spherical bubble diameter, d, also fit the Weibull distribution. An exponential distribution was approximated for air–electrolyte bubble distributions in a 0.15 M NaCl solution [6, 30]. Thus, more than one distribution function may be used to fit bubble size distributions from gas–liquid dispersions.

Another diameter expression, the volumetric mean diameter, d 30, is obtained when the equivalent diameter is back-calculated from total volume measurements by assuming a spherical shape [59] according to:

$$d_{30} = \left(\sum 6{V}_{\rm b}/\pi\right)^{1/3}/n,$$
(4)

where V b is the total volume of bubbles. Volume also can be estimated from the cross-sectional area, A (assuming a spherical shape), where the circularity, C, shape, SF, or form factor [25, 56, 66, 89], a quantitative description of non-sphericity based on the perimeter, P, are given by:

$$\hbox{SF} = 1/{C} = 4\pi {A}/({P}^{2}),$$
(5)

where SF = 1 for a perfect circle and SF approaches 0 for a line [56]. The Feret diameter, d F, or equivalent circular diameter [56], is the equivalent diameter of a circular object with the same area as the irregular object being measured according to:

$$d_{\rm F} = (4A/\pi)^{1/2}.$$
(6)

Other applicable shape descriptors [89] include the roundness, R, given by:

$${R} = 4 {A} /[\pi d_{\rm long}]$$
(7)

and the aspect ratio, AR, given by:

$$\hbox{AR} = d_{\rm long}/d_{\rm short},$$
(8)

where d long and d short are the longest and shortest diameters of the bubble, respectively. Quantification of non-sphericity assists in distinguishing between bubbles originating from spargers and those from other sources (such as vortex entrainment).

Regardless of how the diameter is obtained (direct measurement or back-calculation from area or volume measurements assuming a circular or spherical shape), a “higher moment” approach can be used to obtain the exact description of the bubble size distribution curve [42] for any distribution type. The standard deviation from the arithmetic mean diameter, σa, may be calculated according to [42]:

$$\sigma_{\rm a} = \left[\left\{\sum {n}_{i}(d_{i}-d_{\rm a})^{2}\right\}/{n}\right]^{1/2}$$
(9)

or [6, 7]

$$\sigma_{\rm a} = \left[\left\{\sum(d_{i}-d_{\rm a})^{2}\right\}/\{ {n} - 1\}\right]^{1/2}.$$
(10)

From this quantity, the coefficient of variation, C v [6, 7], may be obtained using:

$${C}_{\rm v} = \sigma_{\rm a} /d_{\rm a},$$
(11)

where C v is the distribution spread relative to its mean.

Two other useful descriptions of a distribution are effective in describing its difference from a normal distribution. The skewness, A 3, is the third moment about d a, divided by σ 3a to make the measurements unitless [42] and is given by:

$${A}_{3} = \left\{\sum {n}_{i}(d_{i}-d_{\rm a})^{3}\right\}/\{\sigma_{\rm a}^{3}{n}\}.$$
(12)

The kurtosis, A 4, is the fourth moment about d a divided by σ 4a to make the measurements unitless [42] according to:

$${A}_{4} = \left[\left\{\sum {n}_{i}(d_{i}-d_{\rm a})^{4}\right\}/\{\sigma_{\rm a}^{4}{n}\}\right] - 3,$$
(13)

where the kurtosis of the normal distribution is 3.

For the log-normal distribution, the standard deviation, σg, is given by [75]:

$$\sigma_{\rm g} = \left[\left\{\sum {n}_{i}(\hbox{log}\,d_{i}-\hbox{log}\,d_{\rm g})^{2}\right\}/{n}\right]^{1/2}$$
(14)

and the characteristics of the log-normal distribution curve may be used to calculate d 32 according to:

$$\hbox{log}\,d_{32} = \hbox{log}\,d_{\rm g} +5.7565\,\hbox{log}^{2} \sigma_{\rm g}.$$
(15)

In Eqs. 14 and 15, the logarithm is the common (base 10) logarithm.

Presentation and analysis of bubble size distribution data

There are several methods in the literature for presenting bubble size distribution data.

Common plots are (1) the percentage number frequency (either incremental or cumulative) or number probability density (Y axis) versus (2) the bubble diameter in a specified range [27] or versus the number of bubbles less than the stated bubble size (X axis) [59]. Selection of size ranges or “bins” directly affects the accuracy of the distribution’s calculated parameters and obviously cannot be less than the established incremental size measurement range of the instrument. Previously, increments of 80 μm [75] up to 0.25 mm [6, 7] were used for bubble size data. Smaller bins result in a more accurate smoothing of bubble size histograms or “stepped” cumulative distribution curves into probability density functions or cumulative distribution functions, respectively. Specifically, cumulative bubble volume distributions have been smoothed by three passes through a triangular digital filter to remove data discontinuities [76].

Specific shapes of bubble size distribution plots can be expected. Plots of the normalized cumulative bubble volume distribution (Y axis) versus the log of bubble diameter (X axis) are sigmoidal in nature [83]. The log of the cumulative volume percent of bubbles of that diameter (Y axis) versus the log of bubble diameter (X axis) results in a nearly linear graph [29, 75]. Probability density versus bubble diameter results in a skewed distribution to the lower or upper bubble diameters depending upon the system [29].

Bubble distributions have been found to be non-normal [6], and specifically number frequency distributions were not symmetrical but showed positive skewness [101]. In some cases, size distributions, such as those obtained for bubbles from perforated plates, have been assumed to follow a logarithmic normal probability distribution [59]. Log-normal distributions, using geometrically increasing bin sizes to accommodate sizes ranging over a few orders of magnitude, have been used for agitated gas–liquid systems, along with calculating d g as well as d 32 [38].

Presentation and analysis of pellet size distribution data

Similar methods have been used to display pellet size distribution data with the exception that a somewhat broader range of possible quantities can be calculated. Key quantities are the percentage of pellets [109] or the number frequency of pellets as function of size (e.g., radius) [86, 103]. Other obtainable quantities include the pellet concentration, pellet volume (sum of individual particle volumes), average diameter (e.g., d a), volume concentration (volume of pellets per liter of sample volume), cumulative volume concentration curve (addition of individual volume concentrations from 0 to d i ), normalized cumulative volume concentration curve (d max  =  1), and median value of diameter, d 50 (diameter for which normalized cumulative volume curve is 0.5) [28].

Pellet size distribution data have also been analyzed over the progression of a fermentation. The pellet fraction (pellet number of a certain size/total number) greater than a set size has been evaluated as a function of fermentation time [34], and the projected area of pellets has been explored for various bioreactor operating parameters [41]. A 3D graph has been constructed using the percentage pellet frequency (Y axis) as a function of fermentation time (X axis) and pellet radius (Z axis) [82]. The measurement procedure often is repeated for several samples, then the size distribution is calculated by averaging [82]. Furthermore, size distribution data have been linked with off-line data to plot the percent dry biomass versus pellet diameter [58] or to obtain the pellet density (division of dry cell weight by pellet volume), ρp [28].

Relationship of bubble size to operating parameters

A summary of the influence of operating parameters and liquid phase properties on bubble size in a gas/liquid dispersion is given in Table 5. Several theoretical and experimental relationships have been established to quantify these influences. These relationships are presented below, then used to quantify expected bubble size changes from expected operating parameter changes to estimate bubble measurement sensitivity requirements. In the future, a similar exercise can be conducted for pellet size changes, although the relationships are more complex owing to the varied nature of pellets relative to bubbles [48, 74, 107].

Table 5 Factors affecting bubble size in stirred tank bioreactors
Full size table

The interfacial area, a 32, is calculated according to [9, 97]:

$${a}_{32} = 6 \Phi/d_{32},$$
(16)

where Φ is the void fraction of the dispersed phase or hold up. For typical Φ of 5–15% and d 32 of 0.5–1.0 mm, a 32 ranges from 0.3 to 3.0 mm−1, and there are 50–150 bubbles/cm3.

Many relationships have been established to relate bubble size distribution characteristics to operating parameters. Often experimental data are required to determine constants in these relationships [55]. Their accuracy depends on the precision of the bubble size measurement technique, and thus relationships can vary when measurement techniques differ among various researchers [55]. Similar statements apply to pellets relationships to operating parameters.

Both the maximum bubble diameter, d max, and d 32 correlate with the power input per unit mass, ɛT, surface tension, σT, and continuous phase density, ρc, according to [27, 101]:

$$d_{32}\;\hbox{or}\; d_{\rm max} \propto \varepsilon_{\rm T}^{-0.4}\sigma_{\rm T}^{0.6} \rho_{\rm c}^{-0.6},$$
(17)

where d max∼ d 99, the diameter that is larger than 99% of all diameters in the cumulative number bubble distribution [75]. The proportional relationship between d 32 and d max was determined experimentally for bubbles produced by fine pore spargers [75]:

$$d_{32}/d_{\rm max} = 0.63.$$
(18)

It was related to the parameters of the log-normal distribution by [75]:

$$d_{32}/d_{\rm max} = \hbox{exp} (2.5\,\hbox{ln}^{2} \sigma_{\rm g} - 2.33\,\hbox{ln}\,\sigma_{\rm g}).$$
(19)

A similar approach is expected to apply to bubble size distributions produced by open pipe or ring spargers.

Another established correlation is the Calderbank equation [55] for gas/liquid and liquid/liquid systems:

$$d_{32} = {A}_{\rm c} [\sigma_{\rm T}^{0.6}/\{({P}/{V}_{\rm L})^{0.4}\rho_{\rm c}^{0.2}\}]\Phi^{\beta} (\mu_{\rm G}/\mu_{\rm L})^{0.25},$$
(20)

where A c and β are determined experimentally, P/V L is the gassed power input per unit volume, and μG and μL are the viscosities of the gas (air) and liquid (water/electrolyte) phases, respectively. This equation has been simplified by various researchers:

  • Gas/liquid mixtures [55]

    $$d_{32} \propto {N}^{-1.2} \sigma_{\rm T}^{0.6},$$
    (20a)

  • Gas/liquid mixtures [111]

    $$d \propto {N}^{-1.5},$$
    (20b)

  • Oil/liquid mixtures [111]

    $$d \propto {N}^{-1.2},$$
    (20c)

  • Gas/liquid mixtures [101]

    $$d_{\rm max} \propto {N}^{-1.2}.$$
    (20d)

Clearly diameters decrease with increasing impeller speed but not as much as implied by Kolomogoroff’s theory of isotropic turbulence [101].

Increasing gas flow rates at constant agitation speed shift distributions towards slightly larger bubbles as bubble density increases with higher gas hold-up leading to greater bubble collision and coalescence rates [6, 30]. Increased superficial gas velocity increases bubble collision frequency leading to higher coalescence rates and greater stable bubble diameters in bubble columns [92]. The effect of gas flow rate on bubble size, for bubbles generated from an orifice, has been quantified by [61, 108]:

$$d \propto \hbox{Q}^{\gamma},$$
(21)

where γ = 0.2–1.0 for gas flow rates, Q, for Newtonian fluids.

The influence of impeller speed, N, on bubble size cannot be properly quantified without considering it together with Q, according to [101]:

$$d_{32} \propto {N}^{-0.50}{Q}^{0.10}$$
(22)

for the impeller region. Increasing N shifts distributions towards smaller bubbles particularly at lower Q; at higher Q, this effect is less pronounced as bubble coalescence is higher [6]. Increasing Q causes a reduction in the turbulence level as impeller gas cavities grow and velocity fluctuations are dampened [30].

Gas bubbles tend to become smaller with lower σT, higher ρG, greater gas molecular weight [101], and decreasing μL [92]. Bubble size distribution shifts to smaller sizes as protein concentration increases (resulting in higher volumetric gas–liquid mass transfer coefficients, KLa) due to a drop in σT [83]. Bubble sizes observed with dispersed mycelia are smaller than those observed with pellets due to segregation occurring at biomass concentrations (> 1.5 g/L) for the dispersed mycelia [52].

For a floatation model system, higher pH results in increased bubble size and higher ionic strength decreases bubble size, but the effect is less than that of pH [67]. An ionic solution (0.15 M NaCl) retards bubble coalescence substantially which causes bubble sizes to drop for similar conditions [6]. Higher temperature results in lower σT and thus decreased bubble size [67].

Larger bubbles are produced at higher μL because as μL decreases possibly liquid films form faster and trap less air in each bubble [67]. The effect of μL on bubble sizes in gas/liquid mixtures is quantified by [111, 112]:

$$d \propto \mu_{\rm L}^{\delta},$$
(23)

where δ is 0.1 for air–aqueous dispersions. Specifically, for a change in μL of 1 cP (water) to 6 cP (50 vol.% glycerol), bubble size is expected to increase 20%. The validity of the correlation needs to be considered relative to the manual size analysis conducted from photographs by these researchers based on available technology. In the case of non-Newtonian fluids where μ L changes as a function of shear, a suitable equation for the apparent viscosity is necessary to relate it to operating parameters such as agitation speed [3, 43, 64].

Estimates of the relative impact of a twofold change in these parameters on bubble diameter are given in Table 5. The suitability of novel bubble size distribution measurement devices can be assessed by the instrument’s ability to reproduce these trends both qualitatively and quantitatively. Specifically, requirements for accuracy and precision may be obtained from these relationships.

Summary

Past and current image analysis technologies for bubble and pellet size measurements have evolved based on available measurement, data acquisition, manipulation, and storage technologies. Substantially more effort has been placed on cell, pellet, and morphological measurements than on bubble measurements as off-line sampling errors were lower although not non-existent for cellular materials. New instrumentation technologies are desired to perform on-line, in situ measurements on a time scale relevant to analysis and control for PAT applications. Sensitivity of these techniques needs to be sufficiently high, and measurement variability sufficiently low, so that the expected effects on bubble size distribution caused by changes in process conditions and/or broth composition are clearly characterized.