Error propagation in open respirometric assays

BIOPROCESS ENGINEERING

Error propagation in open respirometric assays

C. C. Lobo^I; N. C. Bertola^I; E. M. Contreras^II,*

^ICentro de Investigación y Desarrollo en Criotecnología de Alimentos, (CIDCA), CONICET, Facultad de Ciencias Exactas, UNLP, 47 y 116 (B1900AJJ), La Plata, Argentina

^IIInstituto de Investigaciones en Ciencia y Tecnología de Materiales, (INTEMA), CONICET, Phone/Fax: + 54 223 4816600, Mar del Plata, Av. Juan B. Justo 4302 (B7608FDQ), Mar del Plata, Argentina. E-mail: [e-mail]

ABSTRACT

This work deals with the calculation of the uncertainty of the exogenous respiration rate (Rex) and the total oxygen consumed (OCT) derived from a single open respirometric profile. Uncertainties were evaluated by applying a linear error propagation method. Results show that standard deviations (SD) of Rex and OCT depend not only on the SD of the dissolved oxygen (σ_C) and kLa (σkLa), but also on the SD of the derivative term (dC/dt) of the oxygen mass balance equation (σ_b). A Monte Carlo technique was employed to assess σ_b; a power law expression for the dependence of σ_b as a function of σ_C, the time window (t_w) and the sampling rate (Δt) was proposed. The equations obtained in the present work are useful to calculate suitable conditions (e.g., biomass concentration, kLa) that minimize the coefficient of variation corresponding to Rex and OCT.

Keywords: Respirometry; Error propagation; Respiration rate; Oxygen consumption.

INTRODUCTION

Respirometry is a well-established procedure in the field of biological wastewater treatment. Respirometric methods have been used to determine parameters of kinetic models that describe the aerobic biodegradation of wastewaters, single compounds, or mixtures of chemicals (Cokgor et al., 2009; Orhon et al., 2009). These techniques were also employed to evaluate the biodegradable fraction of wastewaters (Lagarde et al., 2005), and to monitor the acclimation process of an activated sludge to a new environment (Carvalho et al., 2004; Aktas, 2012). Additionally, several on-line respirometer-based biosensors in wastewater treatment plants have been developed. Based on the comparison bet_ween the actual respiration rate of the activated sludge with a given set point value (e.g., the historical respiration rate), these biosensors allow the early detection of a toxic wastewater (Sollfrank and Gujer, 1990; Vanrolleghem et al., 1994). However, in order to avoid false alarms, the evaluation of the error associated with the measurement of the actual respiration rate is crucial to distinguish bet_ween a real event (e.g., a toxic wastewater) and aleatory fluctuations of the indicator used for toxicity (e.g., the decrease of the respiration rate).

The literature regarding respirometry has given a special emphasis to parameter identifiability, precision, and sensitivity (Dochain et al., 1995; Vanrolleghem et al., 1995, Petersen et al., 2001; Guisasola et al., 2006); conversely, less focus has been given with regard to the error associated with the estimation of some relevant respirometric parameters, such as the respiration rate (R), and the amount of oxygen consumed (OC) during the biodegradation process. Considering that not only R and OC, but also their uncertainties are essential for the appropriate design of aeration devices of wastewater treatment processes, the aim of this work was to obtain the expressions to estimate the error of the measurements of R and OC derived from a respirometric profile.

MATERIALS AND METHODS

Principle of the Open Respirometer

Batch respirometric tests are a powerful tool for the measurement of the respiration rate of both pure and mixture cultures, such as activated sludge. Although several respirometric principles are available (Vanrolleghem and Spangers, 1998), nowadays the open (flowing gas/static liquid) respirometer (Vanrolleghem, 2002) is one of the most used for research applications (Contreras et al., 2008a,b; Lobo et al., 2013).

Figure 1 shows an example of the dissolved oxygen profile (Fig. 1(a)) and the respiration rate (Fig. 1(b)) as a function of time during a typical respirometry. When a pulse of an oxidizable substrate is added (t = 0.05 h), C decreases to reach a minimum value C_min = 4.7 mgO₂ L^-1 due to the increase of the respiration rate associated with the oxidation of the external substrate. After this transient period, the respiration rate remains almost constant up to the substrate depletion. In some cases, experimental conditions are selected to achieve a constant biomass concentration (X) throughout the experiment, simplifying the interpretation of the obtained respirometric profile. For example, the use of a culture medium devoid of the nitrogen source, or an initial substrate concentration (S₀) much lower than X, both prevent a significative increase of X during the experiment. Additionally, these conditions also lead to short assay times, minimizing the possible decay of biomass. If X remains constant throughout the experiment, the dissolved oxygen concentration after the substrate depletion will be equal to the dissolved oxygen concentration before the substrate addition.

The change of the dissolved oxygen concentration (C) as a function of time (Fig. 1(a)) is determined by t_wo competing processes, namely, the oxygen supply and the microbial respiration. Generally, aeration conditions (e.g., air flow rate, agitation) are constants during a typical experiment; however, the respiration rate depends on the availability of an oxidizable substrate. t_wo processes are responsible for the observed respiration rate. In the absence of substrates, the microbial oxygen uptake rate reflects the endogenous respiration rate (R_en) associated with maintenance processes; in this case, the biomass itself serves as the oxidizable substrate.

Conversely, the exogenous respiration rate (R_ex) is defined as the oxygen uptake rate related to the oxidation of an external substrate (Contreras et al., 2008a,b; Lobo et al., 2013). Considering these processes, the dissolved oxygen mass balance in the respirometer is:

where k_La is the volumetric oxygen mass transfer coefficient, and C_S is saturation dissolved oxygen concentration under the experimental conditions. In the absence of substrate R_ex = 0; thus, C reaches a constant value (C_e) due to the equilibrium bet_ween oxygen transfer and endogenous respiration:

Combination of Eqs. (1) and (2) yields the following expression:

In the absence of an external substrate (R_ex = 0), the actual values of k_La, C_S, and C_e in an open respirometer can be obtained by several methods. For example, by stopping the oxygen supply, the first term on the right hand side of Eq. (1) becomes zero. Then, at a certain t = t₀ the air supply is recommenced and C increases according to Eq. (1) (Ros, 1993; Vanrolleghem, 2002). Considering the t_wo above mentioned aeration conditions and assuming that k_La, C_S, and R_en are constants within the tested time interval, integration of Eq. (1) leads to the following expression:

Thus, k_La, C_S, and R_en values can be obtained by fitting Eq. (4) to the data corresponding to dissolved oxygen concentration as a function of t before and after t₀; then, from these parameters, C_e can be calculated using Eq. (2). To determine R_ex by means of Eq. (3), a numerical differentiation, such as a moving regression window, is required:

where R_exn is the exogenous respiration rate evaluated around the point n, which corresponds to the process time t = (n-1)Δt, Δt is the interval bet_ween t_wo successive data (e.g., the sampling frequency), t_w is the time window, and bt_w, _Δt is the slope of the regression line within t_w.

Another important parameter that can be derived from a typical respirometric profile is the total oxygen consumed (OC_T) to oxidize a given amount of the tested substrate (Lobo et al., 2013). By definition, the oxygen consumed (OC) as a function of time during the degradation of the exogenous substrate is

Combining Eqs. (3) and (6), and considering that at t = 0, C₁ = C_e, the following equation was obtained:

Taking into account t = (n-1)Δt, and applying the trapezoidal rule (Euler method) to the integral in Eq. (7):

where OC_n corresponds to OC evaluated at the point n. When the exogenous substrate is depleted, R_ex is zero, thus, C_n = C_e. Based on these considerations, the total oxygen consumed (OC_T) can be calculated as follows:

where N >> 2 is the total number of data. For example, taking into account that the sampling interval (Δt) used to obtain the respirometric profile shown in Figure 1(a) was 1 s, and that the total assay time was 2 h, this assay consisted of N = 7200 data. Figure 1(c) shows that OC increased as a function of time up to the substrate depletion, and then it remained constant; in this example, the total oxygen consumed (OCT) was 15.8 mgO₂ L^-1.

RESULTS AND DISCUSSION

Uncertainty of Respirometric Measurements

Equations (5) and (9) demonstrate that sources of error in the calculation of R_ex and OC_T are the error associated with the measurements of C, k_La, and bt_wΔt . Uncertainty of R_ex, and OC_T can be evaluated by applying the linear error propagation method to Eqs. (5), and (9), respectively. Uncertainties were extrapolated from first-order local sensitivity functions using the following expression (Lucas and Prinn, 2005; Pasternack et al., 2006):

where is the first-order sensitivity of some variable V (e.g., R_ex, OC_T) to the parameter θ_i (e.g., C, k_La), σ²_V and σ²_θi are the variances of V and θ_i respectively, and the summatory is over the M parameters of the equation used to calculate V (Saltelli et al., 2005).

In order to use Eq. (10) to evaluate the variances corresponding to R_ex (σ²R_exn) and OC_T (σ²OC_T), an estimation of the variances associated with the measurements of C (σ²_C), K_La (K_La), and bt_w, Δt (σ²_b) is necessary. With regard to the measurement of the dissolved oxygen concentration (C), we must distinguish bet_ween signal noise and accuracy of the dissolved oxygen probe. While signal noise can be reduced using several smoothing techniques (Hardle, 1991), accuracy depends on the probe characteristics. According to the specifications provided by several manufacturers, the accuracy of a typical optical dissolved oxygen probe within the range 0 -10 mgO₂ L^-1 is ±0.1 mgO₂ L^-1 (YSI, 2013). Besides, the literature has given less attention to the error associated with the volumetric oxygen mass transfer coefficient. Gogate and Pandit (1999) studied several of the methods that are commonly used for the measurement of the mass transfer coefficient; these authors concluded that, within the range of the operating conditions encountered in fermentor operation, errors less than 10% of k_La can be expected. Similar results were reported by Puskeiler and Weuster-Botz (2005). However, at the laboratory scale, k_La errors are usually lower. Moreover, the error associated with kLa can be assumed to be proportional to its value, the proportionality constant being the coefficient of variation of k_La (CV_kLa). For example, Figure 2 shows that CV_kLa measured in our respirometer (500 mL working volume) was approximately 6% of the measured value. Based on the above mentioned considerations, the following standard deviations were assumed in this work: σ_C = 0.1 mgO₂ L^-1, σ_kLa = CV_kLa k_La, where CV_kLa = 0.06.

Evaluation of the Standard Deviation of the Derivative Term (bt_w, Δ t ) by a Monte Carlo Technique

The estimation of the standard deviation of the derivative term bt_w, Δ(σ_b) was assessed using a Monte Carlo technique. For a given set of values corresponding to the time window (t_w), and the interval bet_ween t_wo successive data (Δt), a noiseless profile of the dissolved oxygen concentration (C) was calculated assuming a linear decrease of C as a function of t with a constant slope (b) within t_w. Then, for each C value, a Gaussian noise with σ_C = 0.1 mgO₂ L^-1 was added. Finally, from the linear regression of the dissolved oxygen concentration with the added noise as a function of time, the slope bt_w,_Δt was calculated (Figure 3). This procedure was repeated 10000 times to obtain the distribution of bt_w,_Δt, statistical relevant estimations of the mean, and the standard deviation of bt_w,_Δt within the tested conditions (the routine was implemented in Sigma Plot 10.0 and is available on request).

Figure 4 shows that a Gaussian distribution of the standard deviation of bt_w,_Δt (σ_b) was obtained by the Monte Carlo method.

Simulations show that σ_b did not depend on the dissolved oxygen concentration (C), or on the actual slope (b). Conversely, Figure 5(a) shows a linear increase of σ_b as a function of σ_C. Moreover, Figures 5(b) and 5(c) show that, in the cases when t_w/Δt was higher than 10 (e.g., more than 10 points were used to calculate the slope bt_w,_Δt of the regression line), σ_b has a power law dependence on t_w, and Δ_t. Based on these considerations, the following empirical equation was proposed to calculate σ_b as a function of σ_C, t_w, and Δt:

where σ_b, σ_C, t_w, and Δt were expressed in mgO₂ L^-1 h^-1, mgO₂ L^-1, and s, respectively. Equation (11) was fitted to the data generated using the Monte Carlo method and the following coefficients were obtained: α = 9543, β = -1.44, γ = 0.45 (r² = 0.9950). This equation is useful for calculating the standard deviation (σ_b) of the slope bt_w,_Δt obtained from the regression line of the dissolved oxygen (C) as a function of the standard deviation of C (σ_C), the time window (t_w) and the sampling rate (1/Δt).

While σ_C is an intrinsic feature of the DO probe, t_w, and Δt can be selected to minimize σ_b. Because of the aleatory noise of C, if the time window is small, then large variations on the derivative (therefore, on the respiration rate) are obtained. In most cases, this problem may be overcome by using digital filters or larger time windows (Savitzky and Golay, 1964; Bromba and Zlegler, 1981). Although there are several methods to select the optimal window size of digital filters (Vivo-Truyols and Schoenmakers, 2006), the selection of very large values of t_w may produce a flattened effect on R_ex curves, masking the actual behavior of the respiration rate as a function of time. Moreover, large t_w values in combination with a high sampling rate (e.g., a very low Δt value) produce very large data files, which are difficult to process due to computer limitations. The optimal selection of t_w and Δt to evaluate the derivative arises from a compromise bet_ween data noise characteristics, size of the data file, and accuracy in the estimation of the respiration rate (Marsili-Libelli and Tabani, 2002). It must be emphasized that, once t_w, and Δt are selected, σ_b only depends on σ_C; thus, because the accuracy of a typical optical dissolved oxygen probe is constant when operating in the range 0 - 10 mgO₂ L^-1, σ_b is also a constant.

Evaluation of the Variances Corresponding to R_exn, and OC_T

Once the standard deviations of C (σ_C,), k_La ( σ_kLa), and b_tw,Δt (σ_b) are known, variances corresponding to R_exn, and OC_T were calculated by applying Eq. (10) to Eqs. (5), and (9), respectively:

The first term in Eq. (12) represents the contribution of the error in k_La to the total variance of R_exn. This term changes throughout the experiment from zero (C_n = C_e) to a given maximum (at C_n = C_min). The second term arises from the imprecision of the DO. The third term in Eq. (12) represents the inaccuracy resulting from the estimation of the derivative and it depends on t_w and Δt, as was discussed previously. Equation (13) shows that the variance of the total oxygen consumed (OC_T) is comprised of t_wo terms. The first one depends on the variance of k_La and is proportional to OC_T; the second term depends on the variance of C and is proportional to k_La and to the product Δt N, which represents the total time of the respirogram. For example, considering the experimental conditions used in the respirogram shown in Figure 1 (k_La = 6.8 h^-1, C_e = 7.9 mgO₂ L^-1, t_w = 60 s, Δt = 1 s), according to Eq. (12), σ_b = 2.6 mgO₂ L^-1 h^-1; thus, σR_exn ranges from 2.8 (at C_n = C_e) to 3.1 (at C_n = C_min) mgO₂ L^-1 h^-1. Figure 1(c) shows that OC reaches a maximum value of 15.8 mgO₂ L^-1 at t = 1 h due to the substrate depletion and then it remains constant; at this time, the product ΔtN = 1 h, resulting in σ_OCT = 1.4 mgO₂ L^-1 (Eq. (13)).

From a practical standpoint, the parameters that can be selected by the operator to obtain the most suitable results using an open respirometer are the initial substrate concentration (S₀), the biomass concentration (X), the oxygen transfer coefficient (k_La), and the dissolved oxygen sampling interval (Δt). Note that not all parameters are independent of each other. Equation (13) shows that, among others, the variance of OC_T depends on the value of OC_T, and on the product Δt N, which represents the total time of the experiment. Because both OC_T and the total time are proportional to S₀, σ_OCT increases as a function of S₀. As an example, the following case study is used to derive the relation bet_ween these parameters.

Case Study: the Monod Model

The most used equation to represent the dependence of the substrate consumption rate as a function of the initial exogenous substrate concentration is the Monod equation. This equation can be employed to interpret the dissolved oxygen profile obtained from an open respirometric assay. In some cases, such as the example shown in Figure 1, it can be assumed that KS << S for most of the assay time. If the biomass concentration (X) can be considered constant throughout the experiment (e.g., a culture medium devoid of the nitrogen source, or initial substrate concentration much lower than X), then the dynamics of the respirometer can be described by the following equations:

where S is the substrate concentration that limits the respiration rate, q_Sm is the specific maximum substrate consumption rate, q_en is the specific endogenous respiration rate, and Y_O/S is the substrate oxidation coefficient. Before the addition of S, the dissolved oxygen concentration in the absence of substrate (C_e) is:

After the susbtrate addition, the respiration rate increases up to a maximum value (R_ex-max)

In most respirometric assays, experimental conditions are usually selected to maintain the dissolved oxygen above a certain critical concentration (e.g., C_C = 2 mgO₂ L^-1) to avoid the oxygen limitation of the respiration rate. Thus, Eq. (19) imposes a severe restriction on the suitable values of X and k_La.

The total time necessary to achieve the susbtrate depletion (Δt N) can be calculated by integration of Eq. (14):

where S₀ is the initial substrate concentration. Finally, assuming that the substrate oxidation coefficient (Y_O/S) is a constant, the total oxygen consumed (OC_T) is:

Equations (16) to (20) can be combined with Eqs. (12) and (13) to obtain the following:

Equation (21) shows that σ²_Rex-max is a function of the biomass concentration (X), and of the oxygen transfer coefficient (k_La); moreover, σ²_OCT (Eq. 22) also depends on the initial substrate concentration (S₀). Although equations (21) and (22) are a function of Y_O/S and q_Sm, these coefficients can be easily obtained from a respirogram:

For example, according to Figure 1, R_ex-max = 22 mgO₂ L^-1 h^-1, and OC_T = 15.8 mgO₂ L^-1; considering that S₀ = 20 mgS L^-1, and X = 1.8 gTSS L^-1, according to Eqs. (23) and (24), values corresponding to Y_O/S and q_Sm are 0.79 mgO₂ mgS^-1, and 15.5 mgS gTSS^-1 h^-1, respectively.

Combining Eqs. (21) to (24), the expressions for the coefficients of variation corresponding to R_ex-max, and OC_T (CV_R, CV_OC) can be obtained:

Equations (25) and (26) demonstrate that, in order to minimize CV_R and CV_OC, high values of X and low values of kLa are necessary. However, these conditions may also lead to dissolved oxygen values lower than the critical value (Eq. 18) (Fig. 6). Thus, the equations obtained in the present work are useful to obtain the suitable operative conditions (X, k_La) that minimize CV values corresponding to R_ex-max, and OC_T.

CONCLUSIONS

In this work, expressions were obtained for the evaluation of the standard deviation (SD) corresponding to the exogenous respiration rate (R_ex) and the total oxygen consumed (OC_T) derived from a single open respirometric profile.

Standard deviations of R_ex and OC_T not only depend on the SD of the dissolved oxygen (σ_C) and kLa (σ_kLa), but also on the SD corresponding to the derivative term (σ_b) of the oxygen mass balance equation of the respirometer.

The implemented Monte Carlo technique allowed calculating σ_b; an empirical power law expression for the dependence of σ_b as a function of σ_C, the time window (t_w) and the sampling rate (Δt) was obtained.

The expressions obtained in the present work demonstrate that high values of X and low values of k_La are necessary to minimize the coefficient of variation (CV) of R_ex-max and OC_T. However, these experimental conditions may also lead to dissolved oxygen concentrations that could limit the substrate oxidation rate. For this reason, the equations obtained are useful to achieve the suitable operative conditions that minimize the CV of R_ex-max, and OC_T.

ACKNOWLEDGEMENTS

This work was supported by the Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), by the Universidad Nacional de la Plata (UNLP), and by the Agencia Nacional de Promoción Científica y Tecnológica (ANPCyT), Argentina.

NOMENCLATURE

Latin Letters b_tw,Δ_t slope of the regression line mgO₂L^-1h^-1 C dissolved oxygen concentration mgO₂ L^-1 C_C critical dissolved oxygen concentration mgO₂ L^-1 C_e equilibrium dissolved oxygen concentration mgO₂ L^-1 C_min minimum dissolved oxygen concentration mgO₂ L^-1 C_S saturation dissolved oxygen concentration mgO₂ L^-1 CV_kLa coefficient of variation of k_La CV_OC coefficient of variation of OC_T CV_R coefficient of variation of R_ex-max k_La volumetric oxygen mass transfer coefficient h^-1 N total number of data OC oxygen consumed mgO₂ L^-1 OC_T total oxygen consumed

mgO₂ L^-1

q_en specific endogenous respiration rate mgO₂ L^-1 h^-1 q_Sm maximum specific substrate consumption rate mgS gTSS^-1 h^-1 R_en endogenous respiration rate

mgO₂ L^-1 h^-1

R_ex exogenous respiration rate mgO₂ L^-1 h^-1 S₀ initial substrate concentration mgS L^-1 t time h t_w time window used for the regression line s X biomass concentration

gTSS L^-1

Y_O/S substrate oxidation coefficient

mgO₂ mgS^-1

Greek Letters

α, β, γ

empirical coefficients of Eq. (11)

Δt time interval between two data s σ_b standard deviation of the slope of the regression line mgO₂ L^-1 h^-1 σ_C standard deviation of C mgO₂ L^-1 σ_kLa standard deviation of k_La h^-1 σ_OCT standard deviation of OC_T mgO₂ L^-1 σ_Rexn standard deviation of R_ex at the point n mgO₂ L^-1 h^-1

(Submitted: April 16, 2013 ; Revised: June 18, 2013 ; Accepted: July 23, 2013)

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Publication Dates

• Publication in this collection
  07 July 2014
• Date of issue
  June 2014

History

• Received
  16 Apr 2013
• Accepted
  23 July 2013
• Reviewed
  18 June 2013