Calculation, Measurement and Validation for Estimating the Biomass of the Biofilm on Microcarriers

Abstract

Traditional carriers play a major role in wastewater treatment worldwide due to their reliability, ease of production, well-established analytical methods, and strong treatment performance. Recent studies indicate that polyvinyl-alcohol-based microcarriers may surpass conventional media, as their smaller size, higher porosity, and increased specific surface area enable them to retain substantially more biomass within reactors. However, their practical application remains limited because fewer analytical methods and studies exist for these materials, largely due to their small dimensions and heat sensitivity, and their behaviour under industrial conditions—including their kinetics—has yet to be fully characterised and validated. This study aims to address these gaps by reviewing existing biomass measurement standards and highlighting their limitations when applied to microcarriers and by proposing alternative experimental approaches better suited for evaluating biomass on such sensitive yet high-capacity carriers. We present a set of experimental methods (still subject to further refinement) that demonstrate reliable performance with these materials, and to validate our approach, we quantified biomass in both in vitro systems and containerised-scale technologies, reaching up to 14 kg/m³ during winter and 8.7 kg/m³ in spring. Laboratory-scale experiments showed that both heterotrophic and autotrophic cultures can achieve high biomass levels of up to 21 kg/m³ and 16 kg/m³, respectively. Heterotrophs exhibited lower growth inhibition under shear stress, while autotrophs displayed a distinct shear-force niche around 0.09 µN within the reactor.

1. Introduction

Nowadays, the conventional activated sludge (CAS) process remains the most widely applied biological treatment method for municipal wastewater [1]. However, increasingly strict discharge regulations and the need for more compact treatment technologies, especially in situations where minimising the treatment footprint is a priority, have intensified interest in more efficient alternatives [1]. Moving bed biofilm reactors (MBBRs) represent one such option, offering higher pollutant removal rates than CAS systems under laboratory and pilot-scale conditions [2]. Traditional MBBR carriers, typically composed of high-density polyethylene (HDPE) and ranging from a few millimetres to several centimetres in size, are well established and extensively documented in the literature (e.g., [3,4,5]). In contrast, polyvinyl alcohol (PVA) based microcarriers provide a potentially significant improvement due to their higher specific surface area, yet published experience with PVA carriers on the millimetre-to-micrometre scale remains scarce.

In this work, we summarise the most prominent methods used to measure key biofilm characteristics—biofilm thickness, volumetric biofilm density, surface coverage, and biomass—and evaluate their applicability to PVA-based microcarriers. Biofilm thickness is a fundamental parameter because it provides insight into microbial activity and mass-transfer behaviour. Light microscopy offers a simple and inexpensive means of estimating thickness, although it suffers from low resolution and a dependence on image analysis techniques [3]. More advanced methods, including fluorescence microscopy and confocal laser scanning microscopy (CLSM), provide significantly higher resolution but require costly fluorophores, which are not only more expensive than dyes used in light microscopy but may also be more toxic to microorganisms [3]. Optical coherence tomography (OCT) provides similar or better non-invasive imaging capabilities without the need for fluorophores, but its high cost and limited availability restrict broader use [6]. Scanning electron microscopy (SEM) can deliver exceptional surface detail, yet it offers no penetration depth and requires destructive, labour-intensive sample preparation [3]. Magnetic resonance imaging (MRI), conversely, provides excellent penetration depth but is limited by poor spatial resolution and high operational costs [5].

Surface coverage is another important design- and performance-related parameter, as it indicates how effectively a given carrier geometry supports biofilm attachment. When geometry permits, light microscopy can be used to assess coverage [7]. A more sophisticated alternative is micro-computed tomography (µCT), which enables high-precision 3D coverage measurement but requires expensive equipment and non-trivial data processing [8]. Volumetric biofilm density, in turn, is most commonly determined using total adhered solids (TAS) [9] or volatile adhered solids (VAS) [4], which are simple, effective, and inexpensive. However, these methods rely on thermal steps and therefore cannot be applied to heat-sensitive PVA microcarriers. Biomass quantification traditionally relies on TAS, VAS, or total organic carbon (TOC) analysis. Unlike TAS and VAS, TOC avoids thermal degradation but requires specialised and costly instrumentation while providing only indirect biomass information [9].

Because PVA microcarriers are both heat-sensitive and considerably smaller than conventional HDPE carriers, several widely used biofilm measurement techniques cannot be applied directly. Consequently, we implemented and optimised alternative methods tailored specifically to PVA media. For volumetric biofilm density, a combined approach based on settling velocity measurements and acidic charring was employed. Although determining drag coefficients introduces a degree of uncertainty, treating the quasi-spherical carriers as spheres allowed empirical correlations to be used for drag estimation. For biomass quantification, instead of relying on TAS/VAS or TOC, we calculated biomass indirectly from biofilm thickness, surface coverage, volumetric density, and the quantified mass or volume of carrier material present in the reactor. Tracking the amount of media was necessary because PVA carriers can undergo significant amortisation due to pumping and biodegradation [10]. Overall, our aim was to provide a set of measurement techniques that remain accurate, reproducible, and broadly accessible, while being fully compatible with the constraints of heat-sensitive PVA microcarriers.

2. Literature Summary

In this chapter we will summarise the most prominent methods for measuring certain parameters of the biofilm.

2.1. Biofilm Thickness

Measuring biofilm thickness is an important step in modelling and calculating biomass and understanding the behaviour of microbes. Numerous methods were developed for this purpose, each with its advantages and disadvantages.

Light microscopy is the simplest method for this measurement. It is cost-effective and widely accessible. It can be used to determine biofilm thickness with image processing. The sample preparation it requires is simpler than with other methods described in this article, especially for conventional carriers, which do not require staining [3]. The samples are washed, a stain (e.g., 2,3,5-Triphenyl-tetrazolium chloride, TTC) is used to differentiate between empty or dead biofilm-covered and active biofilm-covered areas of the media (only required for microcarriers, but can be used for any media), the samples are washed again to remove any excess stain and, finally, pictures are taken. An image analysis software (e.g., Image Pro Plus (version: 6.3) or ImageJ (version: 1.54p) can be used to simplify the calculations [7]. Its limited resolution and poor depth penetration can be drawbacks if the structural analysis of the biofilm is also an aim. All in all, light microscopy is an easy-to-learn, widely available, cheap and reliable method that, however, gives no information about the structure of the biofilm [3]. While stains are needed for certain measurements, these are cheaper and less toxic than most dyes needed for more specialised methods (e.g., fluorophores).

Fluorescence microscopy relies on stains (fluorophores) to dye specific parts of microorganisms, which enables identification of specific parts of the biofilm. It offers better contrast compared to light microscopy, but it still suffers from poor penetration depth [3]. CLSM (confocal laser scanning microscopy) systems are more precise versions of fluorescence microscopy. Regarding biofilms, it is mostly used for structural analysis thanks to the fact that it can take images from the inside of the biofilm but can also be used for quantitative analysis with image processing. When compared to light microscopy, these methods are more precise and can give more information. However, these benefits are reflected in their cost as well; their operation and the required sample preparation are also of higher complexity [11]. In addition, fluorophores can prove toxic to certain microbes, which can be detrimental [12].

Like CLSM, OCT also gives 3D structural information. Without using any traditional stains, the sample preparation is less complex, and the dye’s potential toxicity will not cause problems. It has a far greater penetration depth than CLSM. Its biggest drawbacks are its high price and, since it is not commonly used for studying biofilms, its narrow availability [6,13].

Out of the mentioned methods described above, SEM has the highest resolution but has no penetration depth, giving no direct information on biofilm thickness, so it must be acquired through image processing. It is very expensive and requires extensive sample preparation [3].

MRI is the exact opposite of SEM in regard to biofilm thickness measurements. It has the highest penetration but has the lowest resolution. It gives direct information on biofilm thickness and can be used for structural analysis but thanks to the low resolution compared to the optical methods, it is less precise. It is very expensive and not commonly used for studying biofilms, making it narrowly available [5].

2.2. Surface Coverage

Surface coverage is an important parameter of the media, since it tells us that at certain environmental parameters what percentage of the media’s surface is utilised by the biofilm. The methods for measuring this parameter are not as numerous for biofilm thickness.

The light microscopy method and image analysis described above can also be used to determine the coverage of the carrier. Like in biofilm thickness measurements, conventional carrier’s surface coverage can be measured without using stains [6], but for microcarriers, staining is required. The only difference is in the calculation, so the same measurement can be used to determine both parameters [7].

The surface coverage can also be measured with µCT (micro-computed tomography). The samples are washed and treated with a contrast agent (e.g., Lugol’s iodide solution); the excess contrast agent is removed from the sample with paper towels, and it is fixed in a sample container and then measured in a µCT apparatus. With image analysis, the surface area of the carrier to which the biofilm could adhere is part of the image, and the area on which the biofilm connects to the carrier can be determined, the fraction of which gives the coverage. While this method is very precise, the reliance on expensive specialised equipment reduces its accessibility [8].

2.3. Volumetric Density of the Biofilm

The simplest method is to calculate dry biomass density from the TAS measurement and the biofilm’s volume. TAS is measured by drying the carriers in an oven at a constant temperature (usually from 104 °C to 110 °C) until a constant mass is achieved [9]. The carrier’s mass is measured, then the biofilm is removed, and the carrier’s mass is measured again. The difference of the two measurements gives the TAS. The volume of the biofilm can be calculated from the biofilm thickness and the surface coverage, with regard to the carrier’s geometry. While this method is cheap, simple and widely accessible, it has numerous drawbacks that limit its applicability. It cannot be used on media that is sensitive to heat or if the biofilm is not easily removable from the carrier [9]. Another version of this measurement is that the biofilm is scraped off from a known area of the carrier, dried and its mass is measured. Together with the biofilm thickness, this can be used to calculate the biofilm’s density [14].

A similar method is VAS measurement. VAS is measured by drying the carriers in an oven at a constant temperature (usually from 104 °C to 110 °C) until a constant mass is achieved. The carrier’s mass is measured, then the carrier is put back in an oven, this time around 600 °C which makes the attached organic matter burn away, and the carrier’s mass is measured again. The difference of the two measurements gives the VAS. The volume of the biofilm can be calculated from the biofilm thickness and the surface coverage, with regard to the carrier’s geometry. This method has fewer problems dealing with complex geometries compared to TAS; however, it is suitable for even fewer materials [4].

2.4. Biomass

We can calculate biomass from TAS measurement and the sampling volume. Take all the carriers as samples from a known volume of the reactor. TAS is measured by drying the carriers in an oven at a constant temperature (usually from 104 °C to 110 °C) until a constant mass is achieved. The carrier’s mass is measured, then the biofilm is removed, and the carrier’s mass is measured again. The difference of the two measurements gives the TAS. Dividing this value with the sampling volume gives us the biofilm present on the carriers in the reactor [15]. While this method is cheap, simple and widely accessible, it has numerous drawbacks that limit its applicability. It cannot be used on media that is sensitive to heat or if the biofilm is not easily removable from the carrier [9].

A similar method is VAS measurement. VAS is measured by drying the carriers in an oven at a constant temperature (usually from 104 °C to 110 °C) until a constant mass is achieved. The carrier’s mass is measured, then the carrier is put back in an oven, this time around 600 °C which makes the attached organic matter burn away, and the carrier’s mass is measured again. The difference of the two measurements gives the VAS. This method has fewer problems dealing with complex geometries compared to TAS; however, it is suitable for even fewer materials [4,15].

TOC can serve as a good indirect way to estimate the biomass present. It is usually not measured outright but calculated from total carbon (TC) and inorganic carbon (IC) (TOC = TC − IC). While it correlates well with biomass, the exact way to estimate biomass from TOC is dependent on numerous factors like the measurement methods used or the health and dominant species of the microbes in the biofilm. Thus, the correlation must be determined using a direct measurement method, and this needs to be repeated each time at least one parameter changes. The other drawback of this method is the need for expensive specialised equipment [9].

Hereby, in

Table 1. Biofilm measurement methods.

Measurement	Method	Advantages	Disadvantages
Biofilm thickness	Light microscopy	Cheap and easy sample preparation, Imaging larger areas of the sample	Limited resolution, Using stains might be necessary
	Fluorescence microscopy/CLSM	High resolution, Decent penetration depth, 3D imaging	Expensive fluorophores are required, Interference from the biofilm, Expensive equipment
	OCT	High resolution, Good penetration depth, 3D imaging, No staining required	Expensive equipment
	SEM	High resolution, Ability to image complex samples	Sample preparation is time consuming, Cannot scan large sample area, Low penetration depth, Expensive equipment
	MRI	High penetration depth, No staining required	Low resolution, Expensive equipment
Surface coverage	Light microscopy	Cheap and easy sample preparation, Imaging larger areas of the sample	Limited resolution, Using stains might be necessary
	µCT	High precision	Using a contrast agent is necessary, Expensive specialised equipment
Volumetric density	TAS	Cheap and easy sample preparation and measurement	Not applicable to heat-sensitive materials
	VAS	Same as TAS but can deal with more complex geometries	bit more expensive than TAS
Biomass	TAS	Cheap and easy sample preparation and measurement	Not applicable to heat-sensitive materials
	VAS	Same as TAS but can deal with more complex geometries	bit more expensive than TAS
	TOC	Faster, more automated	Indirect measurement, Expensive specialised equipment

, we summarised available and suitable methods for each measurement parameter. For biofilm thickness and surface coverage measurements, our choice was light microscopy. While alternatives offer higher resolution and/or information on the biofilm’s structure, these are unnecessary for us and would result in our methods being less widely available, which would be contrary to our goals. The methods available in the literature for biofilm thickness and biomass are ill suited for PVA microcarriers because of the heat sensitivity of the media and because our carriers’ size makes precisely removing the biofilm from the carriers challenging at best. For measuring the volumetric density of the biofilm, we developed an experimental method based on settling speed and acidic charring, described in Section 3. We decided to use a mathematical formula to calculate the biomass instead of measuring it and made another experimental method which is based on pore volume for certain spheres with added size similar to our microcarriers.

2.5. Summarising the Available Techniques and Their Applicability

To summarise and evaluate our work, we made a table (

Table 2. Advances and applications of analytic methods.

Measurement Method	Experience Traditional Media	Experience PVA-Based Media	Advantages PVA	Disadvantages PVA
Biofilm thickness
Light microscopy	Extensive; widely documented.	Very limited.	Simple, cheap, non-destructive.	Insufficient resolution; difficult to stabilise microcarriers.
Fluorescence microscopy	Well-established.	Scarce; no standard protocols.	High contrast; improved visualisation.	Dye toxicity; diffusion into PVA matrix.
CLSM	Extensive for 3D biofilm mapping.	Exploratory only.	3D structure possible if immobilised.	Hard to stabilise; requires fluorophores.
OCT	Used in multiple studies.	No reports.	Non-destructive; no staining needed.	Resolution too low for microcarriers.
SEM	Common for morphology.	Minimal due to deformation.	High-resolution dry imaging.	PVA collapses during drying; destructive.
MRI	Limited; low resolution.	None.	Non-invasive (theoretical).	Resolution far too low for micro-scale.
Surface coverage
Light microscopy	Frequently used.	Very limited.	Cheap, accessible.	Clumping prevents accurate coverage.
µCT	Demonstrated for complex carriers.	No reports.	Potential 3D quantification.	Voxel size too large; expensive.
Volumetric density
TAS/VAS	Gold-standard.	Not usable.	None.	Heat destroys PVA; incompatible.
Settling velocity + acidic charring	Rarely used.	Developed specifically for PVA.	Non-thermal; compatible with heat-sensitive media.	Requires modelling; shape assumptions.
Biomass in reactor
TAS/VAS	Gold-standard.	Not usable.	None.	Heat destroys PVA; incompatible.
TOC	Occasionally used.	Limited.	Non-thermal; avoids PVA degradation.	Expensive; indirect biomass proxy.
Computed biomass	Rarely necessary.	Increasingly used.	Non-destructive; integrates multiple parameters.	Error propagation across inputs.
Amount of media in reactor
Media amount tracking	Typically unnecessary.	Required due to abrasion.	Ensures biomass accuracy.	Frequent measurement needed.

) that summarises the key differences in both well-known and freshly developed techniques for analysing various parameters of any type of microorganisms carrying media.

From Table 2, it becomes clear that many traditional biofilm characterisation methods are supported by extensive scientific experience and standardised protocols. However, these techniques are often unsuitable for PVA-based microcarriers because their small size and heat-sensitive polymeric nature fall outside the operational constraints of most classical analytical approaches. Even in cases where a method may be technically applicable to PVA-based systems, its practical use is frequently limited by the high cost and limited availability of the necessary advanced instrumentation. Consequently, emerging PVA microcarriers—which hold considerable promise for enhancing biofilm stability and improving pollutant-removal performance—remain without a robust set of validated, accessible measurement tools. This methodological gap underscores not only the need for targeted analytical development in this field but also the novelty and relevance of the work presented in this article.

3. Materials, Methods and Mathematical Formulas

This chapter outlines the experimental methods applied in our study. Laboratory experiments and measuremnts were carried out at the Headquarter of Inno-Water Inc. (1028 Budapest, Hungary). Six in vitro reactors, each with a working volume of 5 L, were operated throughout the laboratory campaign. All reactors were supplied with artificial wastewater prepared from laboratory-grade chemicals and supplemented with treated effluent from a local wastewater treatment plant (2083 Solymár, Hungary). At the same facility, a containerised treatment unit capable of processing up to 32 m³ of municipal wastewater per day was also monitored. Two on-site monitoring campaigns were conducted at this installation, each lasting approximately two and a half months—one during spring and the other during autumn and winter.

Heterotrophic cultures were supplemented with 0.263 g potassium dihydrogen phosphate, 5 g sodium acetate, and 1 teaspoon of calcium carbonate. Autotrophic cultures received 50 g potassium dihydrogen phosphate, 7 g ammonium chloride, a reduced dose of sodium acetate (1.3 g), sodium carbonate, and 1 teaspoon of sodium hydrogen carbonate, all added to a 30-litre plastic barrel. The reactors underwent a 50% water exchange every 6 h, consisting of 10 min of settling followed by 5 min of water replacement using peristaltic pumps. Each heterotrophic reactor was operated under distinct mixing conditions, analogous to those applied to the autotrophic reactors.

The experiment ran for one month, during which samples were collected every 2–3 days to monitor biofilm development on the carrier surfaces. At the containerised scale, two separate monitoring campaigns were conducted: one during the autumn–winter period and another in spring. Both campaigns relied on microscopic analysis for biofilm characterisation.

Three in vitro control reactors were also run, which was necessary to determine the core diameter of media. These ran with similar conditions (mixing, aeration); however, these reactors did not receive supply water with substances.

Determining biofilm thickness in our system is feasible; however, additional analyses are required to fully characterise the biofilm. To estimate biofilm density, several experiments were conducted to replace the values typically taken from the literature. This was necessary because most published data refer to biofilms grown on traditional carrier materials, which differ substantially from ours in structure, available surface area, and the influence of shear forces.

As we transitioned from traditional assumptions to our own experimental measurements, we selected a statistically representative carrier type as the basis for our calculations and needed to determine its quantity per litre of reactor volume. This posed several challenges: the media operate at a microscopic scale, making it impossible to count individual carriers or determine the mass of a single unit and scale it up. Consequently, we developed and validated a new experimental formula to determine this parameter accurately.

In this chapter we also describe the mathematical formulas used by our methods described in Section 3. This includes the biofilm thickness and surface coverage calculations already described in one of our previous publications, experimental formulas for volumetric density of the biofilm and amount of carriers in the reactor, and two formulas that build on top of these without requiring additional measurements, dry volumetric biofilm density and biomass.

3.1. Biofilm Thickness

Because our aim was to measure biofilm thickness rather than examine its structural characteristics, and to ensure that the method remained cost-effective and widely accessible, light microscopy was selected for imaging. To distinguish living from dead cells, a 1% TTC (2,3,5-triphenyl-tetrazolium chloride) solution at a 1:20 volumetric ratio relative to the sample (Figure 1) was applied, following the untreated control shown in Figure 2. After staining, 30–50 images were captured for each sample within 8 h using a light microscope (Zeiss Lab A1, Carl Zeiss, Göttingen, Germany). Samples were placed on glass plates, excess water was removed, and the plates were immediately transferred to the microscope. Imaging was performed at 5× magnification under bright-field illumination. To ensure adequate coverage, photographs were taken (via Zeiss AxioCam ERc 5s (Carl Zeiss, Göttingen, Germany)) across the entire plate to obtain at least 30 usable images. Only areas of the medium that were clearly separated from neighbouring media and flocs were imaged, ensuring well-defined boundaries. Image analysis was conducted using Image Pro Plus software (Version: 6.3) (Figure 3), employing RGB colour-based quantification. Additional methodological details can be found in the references.

Several geometric properties were quantified using the software, including the mean, minimum, and maximum diameters of each medium. The maximum and minimum diameters were defined as the largest and smallest distances, respectively, between two boundary points passing through the centre point. The mean diameter was calculated as the average of all boundary-to-boundary distances intersecting the centre. Across both the laboratory-scale and container-scale experiments, more than 4900 data rows containing usable measurements for these parameters were obtained.

When calculating the thickness of the biofilm, the diameter of the carrier Equation (2) is needed. If the carrier in question does not contain biodegradable parts, the $d_c$ parameter is equal to 0. Otherwise, because of the loss of volume, thanks to the microbes consuming parts of the carriers, it should be determined for the given carriers and microorganism culture by monitoring the change in minimum diameter of fresh carriers with surface coverages below 20% using Equation (1). Since $d_{20}$ changes as the microorganisms degrade parts of the carrier, this value should be calculated every time when calculating biofilm thickness until it reaches a constant value, from which point it can be treated as such until adding fresh media to the reactor [16].

$$d_c=d_b-d_{20}$$

(1)

$$h={\displaystyle \frac{d_{50}-(d_b-d_c)}{2}}$$

(2)

where $d_c$ is the correctional value of the base diameter [µm], $d_b$ is the average diameter of the media prior to it being added to the reactor [µm], $d_{20}$ is the minimum diameter of media with surface coverage below 20% [µm], $h$ is the thickness of the biofilm [µm], and $d_{50}$ is the average diameter of media with a surface coverage above 50% [µm].

3.2. Surface Coverage

As with the biofilm thickness measurements, Image Pro software was used to perform RGB analysis on light-microscope images of TTC-stained samples. The staining and imaging parameters were identical to those used for thickness determination, ensuring that each image could be analysed for both purposes. In this analysis, two area-based parameters were extracted: the total area of the medium and, in a separate measurement, the area of the TTC-stained region. The ratio of stained area to total area provided an approximate estimate of the colonised surface of the medium. These values were then used to calculate the colonised area of the carriers as described in [17].

The following Equation (3) can be used to calculate the surface coverage from 2D imaging.

$$B={\displaystyle \frac{A_n}{A_w}}·100$$

(3)

where $B$ is the surface coverage [%], $A_w$ is the total area of the carrier [pixel], and $A_n$ is the area of the carrier covered in biofilm [pixel].

3.3. Volumetric Density of the Biofilm

While the literature shows that several established methods can accurately quantify biofilm density and mass on traditional carrier materials—typically by removing the biofilm from the surface or by applying mass-based measurements involving heating—these approaches are not suitable for our system. The carriers used in our study are heat-sensitive; the crosslinked PVA matrix degrades at elevated temperatures, preventing the use of thermal mass-determination techniques. In addition, the biofilm cannot be removed from the carrier surface with sufficient precision to allow reliable gravimetric analysis. Therefore, an alternative method was required that is compatible with our carrier material and capable of estimating biofilm density. Our approach consists of two complementary measurements:

• One is measuring the sedimentation velocity. A 1000 mL measuring cylinder was used. Two marks were put on the cylinder, leaving enough space at the top for the carriers to reach terminal velocity, and measured the time it takes for the carriers to sink from the upper mark to the lower. Only a few were added each time, so they would not interfere with each other while settling, and these were added around the axis of the cylinder to minimise the effect of the friction caused by the wall of the cylinder (Figure 4). Fourteen attempts were made to determine their settling speed both on colonised and freshly produced, noncolonised media.
• The second measurement involved determining the proportion of carrier volume occupied by biofilm. This was achieved through acidic charring. Carriers were placed into 100-mL graduated cylinders; water was added to a total volume of approximately 99 mL; and the suspension was allowed to settle before recording the volume displaced by the carriers. Subsequently, 1 mL of concentrated sulfuric acid was added (after testing several volumes, this amount proved optimal); the mixture was gently homogenised, allowed to settle again, and the carrier volume was re-measured. Both measurements were performed on biofilm-covered carriers and on clean, unused carriers for comparison (Figure 5). To ensure accuracy and scientific reliability, two independent experiments were conducted, each with three replicates.

To calculate the density of the biofilm, we need to determine the density of carriers both covered in biofilm and empty, as well as calculate what portion of the biofilm-covered carrier’s volume the biofilm represents. The first step is to calculate the Reynolds number using Equation (4) with the measured sedimentation velocity.

$$Re={\displaystyle \frac{d{v}_s}{\nu }}$$

(4)

where $Re$ is the Reynolds number [-], $v_s$ is the sedimentation velocity [m/s], and $\nu$ is the kinematic viscosity of the fluid [m²/s].

If the Reynolds number is below 1 (Re < 1), the Stokes equation can be used to calculate the particle density Equation (5) [16]. The Stokes law can be rearranged for ease of use into Equation (6) for the purposes of these calculations.

$$v_s={\displaystyle \frac{g({\rho }_{p }-{\rho }_f)d^2}{18\nu }}$$

(5)

$${\rho }_p={\rho }_f+{\displaystyle \frac{18\nu v_s}{g{d}^2}}$$

(6)

where $v_s$ is the sedimentation velocity [m/s], $g$ is the gravitational acceleration [m/s²], ${\rho }_p$ is the density of the particle [kg/m³], ${\rho }_f$ is the density of the fluid [kg/m³], $d$ is the diameter of the particle [m], and $\nu$ is the kinematic viscosity of the fluid [m²/s].

If the Reynolds number is greater than 1 (Re > 1), we need to use Newton’s law and calculate particle density based on equality of forces at terminal velocity using Equation (8), which requires the drag coefficient ($C_d$). $C_d$ can be calculated with empirical Equation (7) [18]:

$$C_d={\displaystyle \frac{24}{Re}}+{\displaystyle \frac{3}{\sqrt{Re}}}+0.34$$

(7)

$$\left({\rho }_p-{\rho }_f\right)g{\displaystyle \frac{\pi }{6}}{d}^3={\displaystyle \frac{1}{2}}{C}_d{\rho }_f{\displaystyle \frac{\pi }{4}}{d}^2{v}_s^2$$

(8)

Equation (8) shows the balance of forces affecting the settling media, which can be rearranged into Equation (9) for ease of use.

$${\rho }_p={\rho }_f+{\displaystyle \frac{3{\rho }_f{C}_d{v}_s^2}{4gd}}$$

(9)

where $C_d$ is the drag coefficient [-], $Re$ is the Reynolds number [-], ${\rho }_p$ is the volumetric density of the particle [kg/m³], ${\rho }_f$ is the volumetric density of the fluid [kg/m³], $g$ is the gravitational acceleration [m/s²], $d$ is the diameter of the particle [m], and $v_s$ is the sedimentation velocity [m/s].

To calculate what portion of the biofilm covered carrier’s volume does the biofilm represents from the acidic charring measurements described in Section 3 the following Equation (10) can be used.

$$C_v={\displaystyle \frac{V_{cp}/V_{ca}-V_{np}/V_{na}}{V_{cp}/V_{ca}}}$$

(10)

where $C_v$ is the portion of the biofilm [-], $V_{cp}$ is the volume of carriers covered with biofilm prior to acidic charring [mL], $V_{ca}$ is the volume of carriers covered with biofilm after acidic charring [mL], $V_{np}$ is the volume of carriers not covered with biofilm prior to acidic charring [mL], and $V_{na}$ is the volume of carriers not covered with biofilm after acidic charring [mL].

Finally, the volumetric density of the biofilm can be calculated with the following Equation (11). For this formula, both ${\rho }_c$ and ${\rho }_n$ can be calculated as ${\rho }_p$ from previous equations (Equation (6) or Equation (9) depending on the Reynolds number).

$${\rho }_b={\displaystyle \frac{({\rho }_c-{\rho }_n\left(1-C_v\right))}{C_v}}$$

(11)

where ${\rho }_b$ is the volumetric density of the biofilm [kg/m³], ${\rho }_c$ is the volumetric density of carriers covered in biofilm [kg/m³], ${\rho }_n$ is the volumetric density of carriers not covered in biofilm [kg/m³], and $C_v$ is the portion of the media constituted by the biofilm [-].

3.4. Amount of Carriers in the Reactor

For traditional carrier materials, parameters such as filling ratio, size, and weight can be measured easily and accurately. As a result, measurements taken from a small number of carriers can be reliably scaled up to represent the entire reactor. In contrast, for microscopic-scale PVA-based media, even if the average biofilm mass or thickness is known from a statistically representative subset, this information alone is insufficient for accurate reactor-scale extrapolation. Furthermore, previous studies have shown that although PVA-based media are promising, their degradation rate can vary over longer experimental periods [16]. This variability further highlights the need for analytical techniques specifically adapted to these newer carrier types.

This method is based on determining the mass of water obtained after filtering a known volume of settled media, which allows us to calculate the void-to-solid ratio of the carriers. First, measuring cylinders were filled with carriers and water, and the carriers were allowed to settle (Figure 6). The volume occupied by the media and the total volume were then recorded. The contents were subsequently poured through a sieve with a mesh size small enough to retain the carriers, after which the mass of the filtered water was measured. Using this value, together with the average carrier diameter, the total number of carriers present in the reactor was calculated.

The following Equations (12) and (13) can be used to calculate the number of carriers in the reactor.

$$r={\displaystyle \frac{\frac{m_t-m_0}{{\rho }_f}-\left(V_t-V_c\right)}{V_c}}·100$$

(12)

$$a={\displaystyle \frac{f}{100}}·{\displaystyle \frac{\left(1-\frac{r}{100}\right)}{\frac{4\pi {\left(d_a/2\right)}^3}{3}}}$$

(13)

where $r$ is the void-to-solid ratio of carriers in the reactor [%], $m_0$ is the empty mass of vessel used for measuring water mass [kg], $m_t$ is the total mass of vessel used for measuring water mass [kg], ${\rho }_f$ is the density of the fluid [kg/m³], $V_t$ is the total volume measured in the vessel used for settling the carriers [m³], $V_c$ is the volume of the settled carriers [m³], $a$ is the amount of carriers in the reactor [1/m³], $V$ is the reactor volume [m³], $f$ is the filling ratio of the reactor [%], and $d_a$ is the diameter of the carriers [m].

3.5. Dry Biofilm Density

The following Equation (14) can be used to estimate dry biofilm density from biofilm density.

$${\rho }_d={\rho }_b\left(1-{\displaystyle \frac{C_w}{100}}\right)$$

(14)

where ${\rho }_d$ is the density of the dried biofilm (no change in volume) [kg/m³], ${\rho }_b$ is the volumetric density of the biofilm [kg/m³], and $C_w$ is the water content of the biofilm [wt%].

3.6. Biomass

In one of our previous publications, we used a method to calculate biomass in the reactor; however, we have made three significant improvements to it since Equation (15):

• The first is that we no longer estimate the biofilm density from relative and absolute biofilm thickness based on the proportions other articles found but instead have a measurement well suited for our media.
• The second is that a more reliable method to measure the amount of carriers in the reactor was developed.
• The third is an update made to the calculation to fit the geometry of the biofilm (biofilm developed on a quasi-spherical carrier) better by encapsulating the curvature of the surface more accurately:

$$M={\displaystyle \frac{4\pi }{3}}·\left({\left(h+{\displaystyle \frac{d_b-d_c}{2}}\right)}^3-{\left({\displaystyle \frac{d_b-d_c}{2}}\right)}^3\right)·{\displaystyle \frac{B}{100}}·{\rho }_b·a$$

(15)

where $M$ is the amount of biofilm present on carriers in a given volume of the reactor [kg/m³], $h$ is the thickness of the biofilm [m], $d_c$ is the correctional value of the base diameter [m], $d_b$ is the average diameter of the media prior to it being added to the reactor [m], $B$ is the surface coverage [%], ${\rho }_b$ is the volumetric density of the biofilm [kg/m³], and $a$ is the amount of carriers in the reactor [1/m³].

4. Experimental Results

The primary objective of our experiment is to quantify the biomass that developed on the surface of the growth media. This section presents the complete set of experimental results together with the numerical outcomes obtained from our calculations. Its purpose is to demonstrate the capabilities of the proposed technology and to illustrate its performance using real-world data. Because several calculations required the precise density of water—and to ensure that our methodology remains reproducible for the scientific community—we measured it directly. The resulting value was 997.442 kg/m³.

To determine the biomass density, we combined settling velocity measurements with the observed volume loss following acidic charring. This approach relies on two key simplifications:

• We treat our carriers (both colonised and non-colonised) as spheres.
• The neglection of the media’s non-vertical movement during settling.

We measured settling times for both colonised and non-colonised media (

Table 3. Settling time of carriers.

[s]	Colonised	Not Colonised
t1	34	44
t2	45	38
t3	39	27
t4	37	27
t5	48	42
t6	38	34
t7	35	28
t8	38	31
t9	36	32
t10	46	37
t11	42	36
t12	48	40
t13	35	27
t14	33	35
Mean	39.56	34.05
Std. Deviation	5.02	5.48

) over a 21 cm distance, leaving space at the top of the cylinder for the carriers to reach terminal velocity.

These give an average settling time of 39.56 s for colonised and 34.05 s for non-colonised media, meaning an average settling velocity of 0.0054 m/s and 0.0063 m/s, respectively.

Next, the acidic charring experiments (

Table 4. Volume of media before and after charring.

[mL]	Colonised	Not Colonised
first sample before acidic charring	90	88
second sample before acidic charring	72	95
third sample before acidic charring	80	87
Mean	80.67	90.00
Std. Deviation	7.36	3.56
first sample after acidic charring	74	81
second sample after acidic charring	54	87
third sample after acidic charring	64	79
Mean	64.00	82.33
Std. Deviation	8.16	3.40

) were carried out to determine what percentage of the colonised media the biofilm constitutes. In order to correct with the volume loss of the media during the process, the experiment was also carried out with non-colonised media. It is worth noting that the non-colonised media showed a lower standard deviation, indicating fewer experimental errors. The variation remains within 10%, which is reasonable given the heterogeneity of the media’s surface and shape caused by biological activity during colonisation. Furthermore, the mean values indicate that approximately 8.5% of the volume of non-colonised media was lost during the acidic charring procedure. This apparent degradation may be attributed to partial breakdown of the PVA matrix under acidic conditions. However, due to the lack of analysis on the molecular structure or chemical integrity of the carriers, the observed loss could also stem from other, unidentified mechanisms. Importantly, these potential mechanisms do not influence the objectives of our study, as the primary purpose of the acidic charring step is to compare relative changes rather than to establish the exact chemical pathways responsible for material loss.

From this point on, the calculations described in Section 3 were followed. Our Reynolds number was greater than 1, so the Stokes formula could not be used. Our drag coefficients were 6.1 and 7.2, which, using Newton’s law, gives us 1011.0 kg/m³ and 1028.9 kg/m³ for the density of the colonised and non-colonised carriers. Based on the acidic charring experiment, the portion of the carrier’s volume, which the biofilm constitutes, was calculated to be 13.6%. According to these results and measurements, the density of the biofilm is 897.9 kg/m³. It is difficult to compare this result with the literature since most studies that deal with the density of the biofilm measured dry biomass. A solution to this could be if the water content of the biofilm could be determined; however, since our carriers are sensitive to heat and the biofilm is not easily removable from them, our best option for this is to use an estimate based on the literature. Hu et al. [19] found the water content of the biofilm to change between 75.34% and 93.3%. Saini et al. [20] gives a water content of 97% for the biofilm. Based on those using a rough estimate of 90% water content for our biofilm, we obtain a dry biofilm density of 89.8 kg/m³ (0.0898 g/cm³), which is well within the literature parameters [21,22,23,24,25,26].

Next, we quantified the number of carriers present in the reactor. Using the procedure described above, we recorded the total volume, media volume, vessel mass, total mass, and the resulting void ratio. These measurements, which form the basis for calculating the carrier count, are summarised in

Table 5. Void-to-solid ratio measurements.

Sample	Total Volume	Media Volume	Vessel Mass	Total Mass	Void Ratio
	[mL]		[g]		[-]
1	50	9	77.2940	121.1557	0.330
2	50	33	80.0581	109.8078	0.389
3	25	6.5	77.3891	98.0065	0.334
4	25	9	77.8872	97.1944	0.373
5	25	8	76.1014	96.2474	0.400
Mean					0.365
Std. Deviation					0.032

.

The 64% stacking efficiency of randomly arranged spheres is described by Jaeger and Nagel [27]. Based on these results, the biomass for some of our previous measurements was recalculated with the new density and amount values and with the new formula shown in

Table 6. Metadata for the lab measurements.

Reactor	Shear Stress [µN]	Culture
1	0.16	autotroph
2	0.11	autotroph
3	0.09	autotroph
4	0.16	heterotroph
5	0.11	heterotroph
6	0.09	heterotroph

,

Table 7. Margin of error of diameter values used for biomass estimation.

Reactor	Days After Experiment Start [-]	Number of Media [-]	Std. Diameter [μm]	Margin of Error [µm]	Relative Margin of Error [%]
4	3	63	241.75	23.24	4.07
	10	46	274.02	25.34	4.10
	14	29	181.69	15.27	2.04
	17	44	283.20	23.61	3.11
	21	33	201.91	16.52	2.10
	23	35	207.35	16.72	2.06
5	3	46	276.76	23.09	3.05
	10	36	295.43	22.95	2.63
	14	32	209.74	17.16	2.18
	17	41	276.29	22.34	2.77
	21	35	268.94	20.16	2.15
	23	20	207.23	16.47	1.97
6	3	40	268.75	23.12	3.24
	10	45	295.34	26.58	4.08
	14	42	194.44	17.88	2.87
	17	52	250.37	22.70	3.54
	21	60	186.55	15.66	2.09
	23	42	170.47	14.51	1.99
1	3	46	166.15	15.85	2.73
	10	40	269.61	23.29	3.29
	14	48	206.31	17.91	2.56
	17	40	261.90	22.88	3.31
	21	35	199.60	15.33	1.71
	23	30	180.53	13.62	1.47
2	3	55	324.61	30.03	4.87
	10	44	277.59	24.61	3.67
	14	37	229.60	19.96	2.86
	17	43	273.86	22.15	2.74
	21	34	215.93	16.39	1.79
	23	36	267.58	21.05	2.47
3	3	33	396.43	28.89	2.91
	10	37	337.58	28.41	3.81
	14	37	236.62	19.39	2.47
	17	40	247.59	20.95	2.84
	21	35	255.92	19.48	2.14
	23	32	218.85	16.05	1.64
Container	3	68	265.49	24.31	3.86
	8	53	243.20	22.96	3.88
	10	54	280.65	23.60	3.16
	13	49	244.45	21.01	2.94
	17	47	240.64	21.50	3.25
	28	91	161.77	19.14	5.08
	34	50	229.99	22.64	4.16
	38	61	223.05	23.24	4.78
	41	60	189.07	20.09	4.30
	44	65	266.11	27.72	5.70
	51	49	246.37	24.77	4.75
	57	46	313.49	30.09	5.25
	59	54	286.41	27.07	4.58
	62	45	279.49	25.83	4.18
	66	41	388.77	35.50	5.61
	3	23	312.37	28.99	4.73
	15	27	190.31	19.47	3.86
	17	30	261.81	26.23	4.99
	22	33	218.25	21.59	4.01
	36	24	307.03	30.65	5.79
	64	11	219.25	21.23	3.77

and

Table 8. Laboratory measurements.

Reactor	Days Since Experiment Start	Biomass	Reactor	Days Since Experiment Start	Biomass
	[d]	[kg/m3]		[d]	[kg/m3]
4	3	0.0	1	3	0.1
	10	2.9		10	6.5
	14	7.9		14	5.4
	17	11.3		17	9.5
	21	8.5		21	16.6
	23	12.3		23	16.6
5	3	2.3	2	3	1.2
	10	8.0		10	9.0
	14	11.8		14	14.4
	17	14.1		17	13.4
	21	16.3		21	23.5
	23	16.8		23	15.9
6	3	1.9	3	3	1.6
	10	3.9		10	12.8
	14	3.5		14	18.9
	17	5.6		17	17.3
	21	9.4		21	21.0
	23	10.3		23	21.7

. A biofilm density of 897.9 kg/m³ was used as described above. The starting filling ratio for the lab experiments was 10%, and for the pilot plant 7%.

Because our results are derived from mean values and depend on previously defined variables, it was not possible to compute error bars for the final calculated quantities directly. The underlying formulas operate on aggregated means rather than on individual measurements. The mean diameter was selected as the basis for our calculations because it most accurately represents the biomass characteristics relevant to the governing equations. To express the uncertainty of the calculated values, their relative margin of error was quantified by determining the margin of error using the following expression:

$$MOE=t^{*}\times {\displaystyle \frac{s}{\sqrt{n}}}$$

(16)

Here, MOE denotes the margin of error, t^∗ is the critical value of the t-distribution corresponding to the chosen confidence level (95%), $s$ is the standard deviation, and n is the sample size. The relative margin of error was obtained by dividing $MOE$ by the mean diameter of each sample’s carriers and multiplied by 100. Table 7 presents the resulting margin of errors for both the laboratory and the containerised experiments.

Our statistical analysis shows that although the standard deviation is relatively high—reaching nearly 50% of the corresponding mean diameter values—the calculated margin of error remains consistently below 6% when evaluated using 95% confidence intervals. This high level of precision is achieved because each mean diameter is based on a sufficiently large number of media particles, with an average of approximately 43 particles measured per sample, which stabilises the estimates despite the inherent variability.

Based on these results, shear stress significantly affects the amount of biomass present in the reactor for both heterotroph (Figure 7) and autotroph (Figure 8) organisms in the parameters of our experiment. For heterotrophs, the optimum seems to be below 0.09 µN, while for autotrophs, the optimum seems to be between 0.16 and 0.09 µN. All reactors show a steady, gradual increase in biomass during the experiment. It can also be observed that the biomass of the autotrophs is lower than that of the autotrophs, which is to be expected based on experience.

Our biomass measurements (

Table 9. Pilot plant measurements.

Season	Days Since Experiment Start	Biomass
	[d]	[kg/m3]
autumn	3	6.9
	8	9.5
	10	11.1
	13	11.7
	17	11.2
	28	6.8
	34	13.0
	38	7.7
	41	9.1
	44	10.8
	51	10.3
	57	12.4
winter	59	14.2
	62	12.8
	66	12.7
spring	3	3.0
	15	6.3
	17	4.4
	22	7.0
	36	8.7
	64	7.9

) in the pilot plant in autumn and winter show a clear increasing trend which starts quickly and slows down over time eventually turning into stagnation (Figure 9 and Figure 10). This fits well onto the Monod equation generally used to model microbial growth. Due to the measurements in spring being less numerous, the same trend is less certain but still observable. The biomass measured in late autumn, early winter was higher, reaching stagnation around 13 kg/m³, while in spring it stayed around 8 kg/m³.

Our biomass measurements in the pilot plant in autumn and winter show a clear increasing trend which starts quickly and slows down over time eventually turning into stagnation. This fits well onto the Monod equation generally used to model microbial growth. Due to the measurements in spring being less numerous, the same trend is less certain but still observable. The biomass measured in late autumn, early winter was higher, reaching stagnation around 13 kg/m³, while in spring it stayed around 8 kg/m³.

5. Conclusions

We compared the methods available in the literature for the measurement of different parameters of MBBR carriers with the aim of selecting the ones best suited for use with PVA microcarriers that are still widely accessible. For biofilm thickness and surface coverage, light microscopy was used. For the volumetric density of the biofilm and the amount of media present, experimental methods were made, since the available methods for the former either involve heating or specialised equipment, and for the latter, they are not needed for conventional carriers. For biomass present in the reactor, a mathematical formula was used that calculates it from carrier size and geometry, biofilm thickness and density, surface coverage, filling ratio, and amount of carriers present. Our measurements and methods gave back values in correlation with values expected based on the literature. While there is still room for improvements with our experimental methods, the set of methods described in Section 3 are well suited to determine many parameters of quasi-spherical carriers that have a size interval range from up to the lower end of the millimetre scale down to the higher end of the micrometre scale. Since the methods do not involve heating, they are perfectly useable on heat-sensitive media.

Our laboratory measurements showcase the effect of shear stress on autotroph and heterotroph organisms. While the biomass in all reactors shows a steady gradual increase, the speed of this increase was heavily influenced by shear stress in both autotroph and heterotroph cultures. The heterotrophs reached a higher biomass (21.7 kg/m³) compared to the autotrophs (16.8 kg/m³). Within the parameters of our experiment, heterotrophs seem to prefer the lowest shear stress, while autotrophs seem to have an optimum between 0.16 and 0.09 µN.

In the results from our pilot plant, an intensive increase can be observed at the beginning of the measurements, which slows down over time, eventually reaching stagnation. These results fit well on the curve of the Monod equation. During the late autumn and early winter period, a higher peak biomass was measured (14.2 kg/m³) compared to the spring period (8.7 kg/m³).