Estimation of oxygen effective diffusion coefficient in a ... - Springer Link

Environmental Science and Pollution Research https://doi.org/10.1007/s11356-018-1227-8

RESEARCH ARTICLE

Estimation of oxygen effective diffusion coefficient in a non-steady-state biofilm based on response time Jian-Hui Wang 1 & Hai-Yan Li 1 & You-Peng Chen 1,2 & Shao-Yang Liu 3 & Peng Yan 1 & Yu Shen 4 & Jin-Song Guo 1,2 & Fang Fang 1 Received: 19 September 2017 / Accepted: 4 January 2018 # Springer-Verlag GmbH Germany, part of Springer Nature 2018

Abstract In wastewater treatment, oxygen effective diffusion coefficient (Deff) is a key parameter in the study of oxygen diffusion-reaction process and mechanism in biofilms. Almost all the reported methods for estimating the Deff rely on other biokinetic parameters, such as substrate consumption rate and reaction rate constant. Then, the estimation was complex. In this study, a method independent of other biokinetic parameters was proposed for estimating the dissolved oxygen (DO) Deff in biofilms. It was based on the dynamic DO microdistribution in a non-steady-state inactive biofilm, which was measured by the oxygen transfer modeling device (OTMD) combining with an oxygen microelectrode system. A pure DO diffusion model was employed, and the expression of the DO Deff was obtained by applying the analytical solution of the model to a selected critical DO concentration. DO Deff in the biofilm from the bioreactor was calculated as (1.054 ± 0.041) × 10−9 m2/s, and it was in the same order of magnitude with the reported results. Therefore, the method proposed in this study was effective and feasible. Without measurement of any other biokinetic parameters, this method was convenient and will benefit the study of oxygen transport-reaction process in biofilms and other biofouling deposits.

Keywords Oxygen microelectrode . Wastewater treatment . Dynamic oxygen profiles . Diffusion distance . Response time . Dynamic oxygen microdistribution

Introduction Jian-Hui Wang and Hai-Yan Li contributed equally to this work. Responsible editor: Marcus Schulz * You-Peng Chen [email protected] * Jin-Song Guo [email protected] 1

Key Laboratory of the Three Gorges Reservoir Region’s Eco-Environments of MOE, Chongqing University, Chongqing 400045, China

2

Key Laboratory of Reservoir Aquatic Environment of CAS, Chongqing Institute of Green and Intelligent Technology, Chinese Academy of Sciences, Chongqing 400714, China

3

Department of Chemistry and Physics, Troy University, Troy, AL 36082, USA

4

National Base of International Science and Technology Cooperation for Intelligent Manufacturing Service, Chongqing Technology and Business University, Chongqing 400067, China

In recent years, water treatments based on biological methods (Daims et al. 2015; Moya et al. 2015; Qi et al. 2015), physical methods (Flavigny and Cord-Ruwisch 2015; Zhang et al. 2017; Zhang et al. 2016), and chemical methods (Zhang et al. 2015) have been widely reported. The studies of waste treatment microprocess and mechanism have developed to microcosmic directions. In these studies, microprocess and mechanism in biofilms become focus during wastewater treatment. Oxygen transport-reaction process is an important aspect when studying wastewater treatment by aerobic biofilms. Understanding this process will benefit the study of the dynamic dissolved oxygen (DO) microdistribution and the microbial metabolic mechanism inside biofilms, and will contribute to the operation strategy of biological reactors. In biofilms, mass is transferred by advection and diffusion processes (Debeer et al. 1994), while in most biofilms, diffusion controls the mass transfer process (Lewandowski and Beyenal 2013; Phoenix et al. 2008; Xu et al. 1998). It can be considered that the oxygen is mainly transported by diffusion and

Environ Sci Pollut Res

consumed by biochemical reactions (Eberl et al. 2000). Oxygen diffusion coefficient is a key parameter in the study of oxygen diffusion-reaction process and mechanism. However, due to the heterogeneous biofilm structure, the diffusion coefficients and relevant biokinetic parameters change along with biofilm depth (Beyenal and Lewandowski 2002), and they are difficult to be measured. Effective diffusion coefficient (Deff) was usually used to lump together transport properties (Zhang et al. 1998), and it was widely used in models for description and calculation of mass transport (Beyenal et al. 1997; Hille et al. 2009; Ning et al. 2012; Ning et al. 2014; Zhou et al. 2013; Zhao et al. 2016; Wang et al. 2017b). In general, Deff in steady-state biofilms and non-steadystate biofilms is estimated by combining diffusion-reaction models and relevant measurements. Early in the 1960s, Bungay et al. determined the diffusivities and respiration rates in microbial slime systems (i.e., biofilms) by microprobe techniques and diffusion-reaction models (Bungay et al. 1969). According to the model, the diffusivities were calculated by substrate utilization rates and the numerical derivation of substrate profiles. After that, with the development of computer technology, numerical solutions of diffusion-reaction models were used. Beyenal et al. evaluated the substrates Deff in biofilms by a diffusion-reaction model, which was built at pseudo-steady state (Beyenal and Tanyolac 1996). The model was solved numerically, and the values of substrate consumption rates were necessary. In recent year, analytic solutions of diffusion-reaction models were used. Ning et al. calculated oxygen Deff in a steady-state biofilm by in situ oxygen profile measurement using a microelectrode (Ning et al. 2012). Combining with the analytic solution of the built diffusionreaction model, the Deff can be calculated by the oxygen uptake rate (OUR) and the first-order reaction rate constant. Khlebnikov et al. estimated oxygen diffusion coefficient in non-steady-state biofilms based on sample OUR measurements and a transient mathematical model (Khlebnikov et al. 1998a, b). Chen et al. applied the unsteady-state diffusion model in a gel bead to estimate the Deff and the partition coefficients of azo dye (Chen et al. 2003). The model was solved numerically, and the values of maximum specific reactive rates, and the Michaelis-Menten constant were necessary. Guimerà et al. characterized dynamic external and internal mass transport in heterotrophic biofilms and estimated the oxygen effective diffusivity from oxygenation profiles (Guimerà et al. 2015; Guimerà et al. 2016). Although the studies are abundant, almost all the reported methods for estimating Deff in biofilms rely on other biokinetic parameters, such as substrate consumption rate and reaction rate constant. It is necessary to estimate the relevant biokinetic parameters at first, which complicated the estimation of oxygen diffusion coefficient. Moreover, the estimation accuracy will also be affected by relevant biokinetic parameters.

A method for estimating the oxygen Deff in biofilms was proposed in this study. It was based on the dynamic DO microdistribution during the oxygen-infuse process in a nonsteady-state inactive biofilm. The best advantage of this method is that it was independent of other biokinetic parameters. Comparing with the reported methods, this method was more convenient. Due to the advantage of being independent of other biokinetic parameters, the estimate accuracy was also improved. Besides, with the support of the oxygen transfer model device (OTMD) reported in our previous study (Wang et al. 2017a,b), the external environment can be simulated accurately. Application of this method will benefit the study of oxygen diffusion-reaction process in biofilms and other biofouling deposits.

Materials and methods Mathematical principle of the experimental method Basic assumptions of DO diffusion-reaction process The following assumptions were made in the theoretical analysis of DO diffusion-reaction microprocess inside biofilms (Chen et al. 2016; Zhou et al. 2013): 1. The oxygen in biofilm was only transported by diffusion, and its concentration change and microdistribution could be described by the Fick’s first law and second law. The diffusivity in the biofilm was represented by the average value (effective diffusion coefficient, Deff). 2. The properties of the biofilm change only in the direction perpendicular to the biofilm-support interface, and DO was diffused only in the perpendicular direction.

DO diffusion-reaction model inside biofilms Based on the mass balance of DO inside biofilms, the threedimensional diffusion-reaction model can be obtained (Wanner et al. 2006): ∂C ∂J x ∂ J y ∂ J z − − −r ¼− ∂x ∂y ∂z ∂t

ð1Þ

where t is time (T); x, y, and z are spatial coordinates (L); z is the distance from the surface of biofilms (L); C is the oxygen concentration (ML−3); Jx, Jy, and Jz are the components of the mass flux J (ML−2 T−1) along the coordinates; and r is the consumption rate (ML−3) of oxygen. Considering the assumption (1), the convective flux of DO could be neglected. According to the assumption (2), the Jx and Jy could also be neglected. Therefore, C and r are both the


function of t and z. Based on the Fick’s first law, Eq. (1) can be simplified to

be independent to the r(t, z). The following experimental method was based on this mathematical principle.

∂C ðt; zÞ ∂C 2 ðt; zÞ ¼ Deff −rðt; zÞ ∂t ∂z2

Bioreactor and biofilm

ð2Þ

where Deff is the effective diffusion coefficient (L2 T−1), and it represents the effective diffusion coefficients in the entire biofilm and reflects the macro mass transfer; C(t, z) is the oxygen concentration at position z and time t (ML−3); r(t, z) is the consumption rate (ML−3) of oxygen. The other parameters were as described above.

Theoretical analysis for estimating Deff Equation (2) is a widely used expression of DO diffusionreaction model in steady-state and non-steady-state biofilms. The schematic of the DO microdistribution and change in steady-state biofilms, non-steady-state biofilms, and inactivated biofilms is shown in Fig. 1. In non-steady-state biofilms, it can be seen in Eq. (2) and Fig. 1b that the Deff and r(t, z) are closely connected with each other. It means that when calculating or measuring the Deff by this model, the r(t, z) must also be calculated or measured. In steady-state biofilms, shown in Fig. 1a, the DO profile did not change with time (3):

(∂C∂tðt;zÞ

0 ¼ Deff

¼ 0 ), and then, Eq. (2) can be simplified to Eq.

∂C 2 ðt; zÞ −rðt; zÞ ∂z2

ð3Þ

That is the DO diffusion-reaction model in steady-state biofilms, and the Deff and r(t, z) are also closely connected with each other. So, the Deff can not be obtained without calculating r(t, z). However, if the r(t, z) in Eq. (2) decreased to zero, the DO diffusion-reaction model will degenerate into a pure DO diffusion model, shown in Fig. 1c. Then, the Deff will Fig. 1 The schematic of the DO microdistribution and change in a steady-state biofilms, b nonsteady-state biofilms, and c nonsteady-state inactivated biofilms

The bioreactor used in this study was a laboratory-scale rotating biological cage, which is a modified conventional rotating biological contactor (RBC). It is composed of rotating cages and reaction tank, and the cages were filled by biocarriers with high specific surface area (500 m2/m3). Biofilms firmly attached to and grew on the biocarrier. This bioreactor was used for synthetic refractory high-salt wastewater treatment. Before the experiment, it was continuously operated for 25 months, and it was consistently operated with influent flow rate of 6.8 L/day for 6 months. Additional details of the bioreactor and synthetic wastewater were reported in our previous study (Wang et al. 2017b). The biofilm used in this study was produced by this bioreactor, and it was taken out with the biocarrier which it attached to.

Experimental solution pretreatment The experimental solution used in this study was taken at the same site with the biofilm in the RBC. Then, the solution was deoxygenated by bubbling N2 about 20 min to decrease the DO concentration in it to near zero (Guimerà et al. 2016). In order to make sure DO concentration in the solution was kept very low, N2 was bubbled when the biofilm was exposed to air. Before the biofilm was immerged in the solution, the bubble was stopped. In order to avoid the effect of bioactivity on DO transfer, microorganism respiration in the biofilm should be inhibited before the test is performed. As reported, sodium azide (NaN3) showed better respiration inhibition to microorganism (Fu et al. 1994). Before the experiments, 200 mg/L NaN3 was added into the experimental solution to inactivate the biofilm


(Horn and Morgenroth 2006). Because the biofilm was deactivated by chemical methods, it can be considered that the biofilm structure was not changed. Then, it was assumed that the DO Deff remained constant with the value in active biofilms before inactivation.

Dynamic DO microdistribution measurement Dynamic DO microdistributions in the biofilm were measured by the OTMD combining with an oxygen microelectrode system, which was reported in our previous study (Wang et al. 2017b). Experimental solution was loaded in a beaker, and the beaker was left on the OTMD. As reported, the OTMD can provide vertical movement of the beaker, while the biofilm was fixed on a stationary carrier above the beaker. With the up-and-down movement of the beaker, the biofilm was exposed to the experimental solution and air alternately. The oxygen microelectrode system also remained stationary, and the oxygen microelectrode was placed upon the carrier and biofilm. The oxygen microelectrode (Unisense, OX25 fast, Denmark) with a response time of less than 0.3 s was controlled by a three-dimensional microelectrode propeller (WN101TA150M, Jindaqingchuang Ltd., Beijing, China) with an accuracy of 10 μm. The schematic of the oxygen measurement system is depicted in Fig. 2. Additional details about the OTMD and oxygen measurement system can be found in our previous study (Wang et al. 2017b). The experimental solution in the beaker was lifted and dropped to expose the biofilm to solution and air alternately. Following a 1-min immersion, the DO concentrations in the biofilm closed to 0 mg/L. After 1 min of exposure in air, the increase of DO concentrations in biofilm was very slow. According to these results, the biofilm was periodically exposed to the solution and air for both 1 min in the measurements. In order to ensure that the DO microdistribution in the

biofilm was periodically stable, the OTMD was run for about 30 min before performing the measurement. Then, the DO concentration varying with time at each depth was measured, and a series of depths at intervals of 30 μm were measured. Moreover, in order to protect the microelectrode, the maximum measuring depth should be less than the biofilm thickness. The biofilm thickness was measured by the propeller and a lathy glass tube (tip diameter 10–20 μm) as described in our previous study (Ning et al. 2012). Because the DO microdistribution in the biofilm was periodically stable, the DO concentrations at a series of depths were considered to be measured at the same time. The measured data at equally spaced depths and independent time were aligned at the beginning of the exposure process. Then, the dynamic DO microdistribution could be obtained. The measurement was repeated for three times at one point, and all experiments were performed at room temperature, about 20 °C.

DO pure diffusion model in inactive biofilms The following assumptions were made in the theoretical analysis of DO diffusion and microdistribution in biofilms (Wanner and Gujer 1986; Zhou et al. 2013): 1. The biofilm was deactivated as mentioned above, so DO transfer microprocess in it can be considered as a pure diffusion process without biological reaction (consumption). 2. The DO in biofilms was only transported by diffusion, and its concentration change and microdistribution could be described by the Fick’s first law and second law. The diffusivity in the biofilm was represented by the average value. 3. The properties of the biofilm change only in the direction perpendicular to the biofilm-support interface, and DO was diffused only in the perpendicular direction. Based on the Fick’s first law and second law, the mass balance of an infinitesimal layer inside the biofilm can be obtained as follows (Guimerà et al. 2015; Taherzadeh et al. 2012; Wang et al. 2017b): ∂C ðt; zÞ ∂C 2 ðt; zÞ ¼ Deff ∂t ∂z2

Fig. 2 The schematic of the dynamic DO microdistribution measurement system

ð4Þ

where z is the distance from the surface of biofilms (L), and the biofilm is considered in the positive direction; t is time (T). The other parameters were as described above. The analytical solution of this pure diffusion model relied on the initial condition and boundary condition. In this study, the two conditions were obtained by the following experiments.


Results and discussion

Dynamic DO microdistributions inside the biofilm

Dynamic DO microdistribution

The dynamic DO microdistributions during exposure process inside the biofilm were monitored. The results of three measurements at one typical point are shown in Fig. 4. During the exposure process, the DO concentration increased with time and decreased with biofilm depth, in accord with the results reported in active biofilms (Guimerà et al. 2016; Wang et al. 2017b). In the top layer of the biofilm (0 μm < z < 100 μm), DO concentration could reach over 8 mg/L during the exposure (60 s). The maximum DO concentration at the biofilm surface should be close to the saturated oxygen concentration in the experimental solution (C A , ML −3 ), which can be calculated by Henry’s law and oxygen concentration in air. At room temperature under one atmospheric pressure, the calculated CA is 9.317 mg/L, while the measured CA is about 9.02 mg/L. To characterize the beginning of oxygen-infuse process, 1% of CA (~ 0.09 mg/L) was selected as the critical DO concentration. The time needed to reach the critical value at different biofilm depths was indicated on the gray planes in Fig. 4. It took less than 10 s for DO to penetrate 300-μm-thick biofilm. Due to the inhibition of NaN3 in the experimental solution, it can be considered that there were no biochemical reactions in the biofilm. The DO was diffused from air to biofilm during the exposure process, and it is a pure diffusion process. In theory, the DO concentrations in the biofilm will eventually get close to that at the biofilm surface after a very long time.

Dynamic DO concentrations outside of the biofilm The dynamic DO concentrations outside of the biofilm were measured at four positions with distances of − 150, − 100, − 50, and 0 μm to the biofilm surface. This experiment was designed and implemented to provide bases of the boundary conditions for the diffusion model. The dynamic DO concentrations during the whole period (120 s) at the distances are illustrated in Fig. 3a, and the DO concentrations at the beginning of exposure and immersion processes at the four positions outside of the biofilm are shown in Fig. 3b. The results indicate that the DO concentrations outside of the biofilm rapidly increased to over 7 mg/L at the beginning of the biofilm exposure process, and quickly decreased to 0 mg/L when the biofilm was immersed in the anaerobic experimental solution again. At the distance of more than − 100 μm from the biofilm surface, the DO concentration increased to 7 mg/L within 0.7 s. And at the distance of less than − 100 μm from the biofilm surface (including the surface), the DO concentration increased to 7 mg/L in about 1.5 s. At the beginning of immersion process, the DO concentration at the biofilm surface decreased to 0 mg/L in about 1.0 s. The rapid increase and decrease of DO concentration near the biofilm surface defined the boundary and initial conditions of the diffusion model during the exposure process.

Fig. 3 a Dynamic DO concentrations during the whole period (120 s) at the sites closing to the biofilm surface. b Dynamic DO concentrations at the beginning of exposure and immersion processes at the four positions

(the negative sign in biofilm depth shows that the locations were outside of the biofilm)


Fig. 4 Dynamic DO microdistributions inside the biofilm and the critical values: a, b, and c are the results of the triplicate measurements, respectively

Solution of the DO diffusion model during exposure process Initial condition and boundary condition According to the experimental results, the initial and boundary conditions of the DO diffusion model of the inactive biofilms during exposure process can be obtained as follows: Initial condition: In order to simplify the analysis and the model solving, the biofilm was dipped into the anaerobic experimental solution for 1 min, and the DO concentration inside it was close to 0 mg/L. So, the initial condition of the exposure process can be simplified as C(t, z)|t = 0 = 0. Boundary condition: According to Fig. 3, the DO concentration at the biofilm surface increased to a higher value (7 mg/ L) in 1.5 s. It is fairly quick when comparing with the time of the whole exposure process (60 s). In this study, the microorganism respiration was inhibited by NaN3, and the bioreaction in the liquid layer and biofilm can be ignored. For this reason, the oxygen concentration gradient in the liquid layer was much smaller. Moreover, there was no air flux produced by a fan, and the air speed was 0 m/s. That is because in our laboratory reactor, the rotating rate of cages was 3 rpm, and the diameter was 15 cm, the relative motion between the biofilm and the air during exposure was only 0.02356 m/s. In our previous work, when the air speed rate increased from 0 to 0.5 m/s, the DO concentration at 150 μm increased from 5.6 to 5.8 mg/L, and the effect was small. In comparison with this air speed, the effect of 0.02356 m/s air speed can be ignored. Therefore, the Dirichlet boundary condition C(z = 0) = CA = constant is a feasible approximate processing method. Therefore, the boundary condition during exposure process can be assumed that the DO concentration at the biofilm

surface was constant and equal to the saturated oxygen concentration in the experimental solution: C(t, z)|t = 0 = CA. Then, the model with the initial and boundary conditions can be rewritten as follows: 8 > ∂C ðt; zÞ ∂2 C ðt; zÞ > < ¼ Deff ∂t ∂z2 ð5Þ Initial condition : C ðt; zÞjt¼0 ¼ 0 > > : Boundary condition : C ðt; zÞjz¼0 ¼ C A

Analytical solution of the DO diffusion model Based on the initial and boundary conditions during the exposure process, the analytical solution of the DO diffusion model can be obtained by the method of Laplace transform (Crank 1975): ! z C ðt; zÞ ¼ C A erfc pffiffiffiffiffiffiffiffiffiffi ð6Þ 2 Deff t where erfc is referred to as the error-function complement and Deff, t, z, and CA were defined previously.

Theoretical derivation of the DO Deff As mentioned above, 1% of CA was selected as the critical concentration to characterize the beginning of oxygen-infuse process. The time needed to reach the critical DO concentration, which was defined as response time in this study, increased with the biofilm depth. The relationship between diffusion distance (z) and response time (t) to reach the critical DO concentration is shown in Fig. 5. In fact, due to the lack of


Then, Eq. (8) can be rewritten as Eq. (9): z2 ¼ 13:264Deff t

ð9Þ

The z and t at the beginning of oxygen-infuse process at different biofilm depths were obtained in the experiment. Therefore, the DO Deff can be calculated by fitting the experimental data to Eq. (9).

Estimation of DO Deff by data fitting Fig. 5 The relationship between diffusion distance (z) and response time (t) to reach the critical DO concentration

bioreaction, when the biofilm was exposed to air, the DO concentration in the biofilm will increase to a constant (shown in Fig. 1c) which equals to that at the biofilm surface. The scope of this method based on the specified condition was just in the initial phase of the oxygen-infuse process. At the critical DO concentration, the relationship between t, z, and DO concentration could be expressed as Eq. (7): z C ðt; zÞ ¼ C A erfc pffiffiffiffiffiffiffiffiffiffi 2 Deff t

! ¼ 0:01C A

ð7Þ

zffiffiffiffiffiffiffi p So, erfc ¼ 0:01, and it can be calculated by 2

Deff t

MATLAB that erfc(1.821) ≈ 0.010. Therefore, Eq. (8) can be obtained: 2

z pffiffiffiffiffiffiffiffiffiffi ¼ 1:821 Deff t

ð8Þ

According to the theoretical derivation above, the DO Deff was calculated based on the experimental diffusion distance (z) and response time (t) at the critical DO concentration. When the DO concentration of a biofilm depth increased to the critical value, the diffusion distance (z) and response time (t) were extracted, as shown on the gray planes in Fig. 4. The red balls in Fig. 4 showed when the critical DO concentration was reached at different depths. Based on the extracted data, the square of diffusion distance (z2) and response time (t) are presented in Fig. 6 and linear fittings were performed. The square of diffusion distance (z2) and response time (t) were simulated by linear fitting to Eq. (9), as shown in Fig. 6. The R2 (coefficient of correlation) was 0.990, 0.984, and 0.976, respectively, indicating that the experimental data were well fitted to the DO concentration changes described by the pure diffusion model. Moreover, the DO Deff can be considered as a property that depends on the biofilm structure, so this parameter can be considered as constant under the condition in this study and under normal operating condition. The values of DO Deff in the three measurements were calculated to be 1.010 × 10−9, 1.060 × 10−9, and 1.092 × 10−9 m2/s, respectively. The average value was

Fig. 6 Linear fit of the square of diffusion distance and response time: a, b, and c were the three measurements

Environ Sci Pollut Res Table 1 DO Deff inside biofilms from literatures 1 2 3 4 5

Literatures

Deff values (m2/s)

Biofilm source

Fu et al. (1994) Beyenal and Tanyolac (1996) Horn and Morgenroth (2006) Kumar et al. (2012) Ning et al. (2012)

0.500 × 10−9–2.430 × 10−9 2.107 × 10−9 0.996 × 10−9–2.489 × 10−9

Completely mixed biodrum reactor Differential fluidized bed biofilm reactor Biofilm tubular reactor Membrane biofilm reactor Sequencing batch biofilm reactor

(1.054 ± 0.041) × 10−9 m2/s, and the standard deviation was 3.961%. Table 1 lists DO Deff values inside biofilms reported in previous studies. Comparing with the reported results, the DO Deff in this work was in the same order of magnitude, indicating that the method proposed in this study was effective. In fact, the response time at each biofilm depth can also be seen in some previous studies, such as the Fig. 3 in the paper reported by Fu et al. (1994), the Fig. 6a in the paper reported by Guimerà et al. (2016), and the Figs. 3 and 4 in our previous study (Wang et al. 2017a). Although the biofilms used in some papers were not deactivated, the response time still existed, which was affected by the microbial activity in the biofilm. However, these response times were not used for estimating the DO Deff inside biofilms independent of other biokinetic parameters.

Conclusion In this study, the dynamic DO microdistributions in the inactive biofilm were measured, and a method for estimating DO Deff in biofilms was developed. The following observations were made: An oxygen microelectrode system combining with the OTMD was used to measure the dynamic DO microdistribution in an inactive biofilm. A method was proposed for estimating DO Deff in biofilms. Comparing with the reported methods, the virtue of this method is that the DO Deff can be estimated independent of other biokinetic parameters in biofilms. No assumptions and measurements regarding reaction kinetics were needed in this method. The DO Deff in the biofilm in this study was calculated as (1.054 ± 0.041) × 10−9 m2/s. It was in the same order of magnitude with the reported results. Therefore, the method proposed in this study was effective and feasible. Funding information The authors gratefully acknowledge the financial support of the Chongqing Science and Technology Commission (cstc2014yykfC20001, cstc2015shms-ztzx20001) and the National Key Project of China (2015ZX07103-007).Nomenclature C, The oxygen concentration in biofilms; CA, The saturated oxygen concentration in the experimental solution; C(t, z), The DO concentration at position z and time t (ML–3); DO, Dissolved oxygen; Deff, The effective diffusion

1.125 × 10−9 0.799 × 10−9

coefficient (L2T–1); erfc, The error-function complement; Jx, Jy, and Jz, The components of the mass flux J (ML–2T–1) along the coordinates; OUR, The oxygen uptake rate (ML3T–1); r(t, z), The oxygen uptake rate at z site and t time (ML–3); t, Time (T); RBC, Rotating biological contactor; z, The distance from the surface of biofilms (L)

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