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In this research, three existing dynamic mathematical models for aerated lagoons were applied to an aerated facultative lagoon (AFL) plant for municipal wastewater treatment. The models’ ability to describe the behavior of the primary lagoon was evaluated, and the advantages and limitations of the three models were compared. For this purpose, 7-year operational data from the municipal WWTP were collected and processed; other necessary data—like dynamic temperature values— were estimated. A 2-year period was used for model calibration, while the remaining 5 years served as validation period. One of the models showed poor calibration fit values in the effluent concentration description (R
^{2} of 0.242 and RMSE of 16.8 mg/L); however, with some modifications the adjust was enhanced (R
^{2} of 0.409 and RMSE of 14.0 mg/L); a second model displayed a poor to moderate adjust (0.489 and 13.0 mg/L, respectively), and the third model achieved a moderate fit (0.528 and 11.9 mg/L), though it provided an overestimation of effluent concentration, especially in periods of heavy and frequent rain; this model with some adaptations the adjust was enhanced (R
^{2} of 0.575 and RMSE of 11.4 mg/L). The validation fits are even lower, illustrating the inability of these models to properly describe the AFL behavior. The possible causes of the models’ inadequacy are discussed. Finally, it is established that in terms of AFL behavior description more research and model development are needed.

Many different systems and methods of treating municipal wastewater exist; the choice mainly depends on the initial investment, operating and maintenance cost, and the water quality required at the effluent. Within these systems, lagoon processes are widely used in developing countries, and can be classified into stabilization ponds and aerated lagoons; among the latter, aerated facultative lagoons (AFL) are the most widely used. These are partially aerated, partially mixed lagoons combining aerobic and anaerobic degradation of organic matter. AFL treatment trains can have high BOD removal efficiencies (70% - 90%) [

Aerated facultative lagoons (AFL) are a good economic option for wastewater treatment, especially for cities/ towns with intermediate land availability, because AFL are more efficient than facultative ponds and, therefore, require significantly lower surfaces; furthermore, their operation and maintenance are simpler than activated sludge systems [

A tool increasingly used to design, analyze, optimize and control systems of wastewater treatment is mathematical modeling [

Mathematical modeling strategies for AFL depend on the mathematical complexity the researcher is willing to adopt, on the quantity and quality of meteorological data, on the available characterization of influent and effluent, and on the available plant operating data. This article discusses the full-scale implementation and evaluation of three AFL dynamic models existing in the literature and involving a limited number of variables (aggregated organic matter/oxygen demand or volatile solids) and processes.

The municipal wastewater treatment plant (WWTP) North of San Luis Potosí, SLP, Mexico, (Planta Norte) is located at the northeast of the city of the same name, at 22˚12'24" of north latitude and 100˚56'49" of west longitude, and 1848 m of elevation. It is a train of four aerated lagoons in series, with an average operation flowrate of 23,126 m^{3}/d for the studied period (2004-2010).

Regular sampling and data recording was performed by Proagua by obtaining grab samples of the WWTP influent and effluent every 4 h, starting at 00:00, recording the date, time, point of monitoring and the instant flowrate. Daily composite samples were prepared from the grab samples, in accordance with the NMX-AA-003- 1980 standard [_{5}) in accordance with the NMX- AA-034-SCFI-2001, NMX-AA-030/1-SCFI-2012, NMX-AA-028-SCFI-2001 standards [

In the scope of AFL dynamic modeling, there are very few studies published on the scientific literature, most focused primarily on hydrodynamic aspects [

water characterization as well as specialized software [

In the next subsections, three AFL dynamic models (in non-steady state) from the literature are presented [

The model consists of two state variables representing the mass balance of ultimate BOD (BOD_{U}) and the total sludge mass in an AFL. His model assumes first-order reactions for substrate utilization rates (aerobic and anaerobic). In this model, it is considered that a fraction of the influent substrate instantly disperses in the lagoon, while the rest settles on the bottom layer, where the accumulation and fermentation of sludge occurs; a fraction of the fermented sludge is released as substrate towards the liquid phase of the lagoon. The model considers one CSTR with continuous flow and was applied to the “Portage la Prairie” AFL-based WWTP sited in Manitoba, Canada. The model equations are:

where _{U} concentration (mg/L); _{U} dispersed within the body of the lagoon (dimensionless); S_{i} is the influent BOD_{U} concentration (mg/L); R is the lagoon retention time (d); K_{s} is the aerobic degradation rate (d^{−1}); _{U} entering lagoon liquid phase after sludge fermentation (dimensionless); k_{L} is the sludge fermentation reaction rate (d^{−1}); L_{t} is the total sludge mass (lbs BOD_{U}); V is the effective lagoon volume (ft^{3}); V_{0} is the initial lagoon volume (ft^{3}); G is the mass density of water (lbs/ft^{3}) and Q is the wastewater influent flowrate (ft^{3}/d).

It can be observed that the model has dimensional inconsistencies. Some refer to the International System of Units and others to the English System. A detailed analysis was performed and there was the need to make some modifications to the model for the mass balance presented by Bemister (1978) to have dimensional consistency and to present the expressions in BOD terms; such modifications were:

where S_{i} and S are the influent and effluent BOD concentration (mg/L); _{t} is the total sludge

mass (kg BOD_{U}); f_{BODu} is the BOD_{U} to BOD ratio; V is the effective lagoon volume (m^{3}); V_{0} is the initial lagoon volume (m^{3}); Q is the wastewater influent flowrate (m^{3}/d). f_{l} is the sludge accumulation index (dimensionless) and γ_{l} is the sludge specific weight (considered as 1050 kg/m^{3}).

This model corresponds to a modified version of the stabilization pond model of [

According to model author _{2S} is the carbon dioxide saturation concentration in the lagoon; CO_{2} is the actual carbon dioxide concentration in the lagoon (mg/L); FC is the fecal coliforms concentrations per volume unit (not defined); Q is the flowrate (m^{3}/d); V is the lagoon volume (m^{3}); R_{12} is the transformation rate from organic carbon to inorganic carbon (d^{−1}); R_{21} is the transformation rate from inorganic carbon to organic carbon (d^{−1}); R_{20} is the atmospheric CO_{2} re-aeration rate (d^{−1}); R_{1S} is the organic carbon net loss rate (d^{−1}); K_{SC} is the half-saturation constant for carbon; R_{8S} is the overall fecal coliform decay rate (d^{−1}). Since there were no data available for CO_{2} concentration in the lagoon, (CO_{2S}-CO_{2}) was considered constant and calculated in parameter estimation. Similarly, K_{SC} was managed as an estimated parameter, since its value was not specified in [_{i} as a constant fraction of influent BOD, f_{l}, calculated in parameter estimation; for this, Equation (3) and (4) were modified into Equation (3a) and (4a) expressed in BOD terms. Since the primary AFL of the “Planta Norte” is 1.7 times deeper than the “As-Samra” lagoon, then the R_{20} value was changed from 23.86 to 41.02 d^{−1}, according to the [

where S_{i} and S are the influent and effluent BOD concentration (mg/L); f_{i} is the IC to BOD ratio.

A fed-batch model of an aerated facultative lagoon at pilot scale is proposed, consisting of three state variables: substrate (expressed as soluble COD), biomass (expressed as VSS) and the lagoon volume; the pilot study was applied to treat wastewater samples from a winery industry of Molina city, Chile. The model equations are:

where, according to the model authors: V is the lagoon volume (m^{3}); F is the volumetric flowrate (m^{3}/d); μ_{m} is the maximum specific microbial growth rate (d^{−1}); S_{0} and S are the influent and effluent substrate concentrations (mg sCOD L^{−1}); S_{nb} is the non-biodegradable substrate concentration (mg sCOD L^{−1}); X is the cellular or biomass concentration (mg VSS L^{−1}); Y is the cell yield coefficient (g VSS g^{−1} COD); K_{S} is the saturation constant (mg sCOD L^{−1}).

The model was adapted to a completely-mixed lagoon with equal influent and effluent volumetric flowrate. In this case, dV/dt = 0, so Equation (6) can be discarded. The substrate and biomass balances resulted in the same differential equations, i.e., Equation (7) and Equation (8).

Since there were no data available for sCOD, it was considered proportional to the measured BOD, S_{nb} concentration was assumed as a constant fraction of influent BOD, f_{nb}, and calculated in parameter estimation. Based on these considerations S is expressed in BOD terms, of course, expressing the kinetic parameters in terms of BOD also.

All the mentioned models were built and implemented in MATLAB^{®}/Simulink^{®}, their calibrations were performed with the Parameter Estimation tool, by the conjugated gradient optimization method, and using the sum of squared errors of the effluent oxygen demand (measured vs. estimated) as cost function. For this purpose, a period of two consecutive data years (2007-2008) was taken as calibration period and the rest of the time series was used as validation period, this calibration period was chosen due to during this, abrupt oscillations in the influent flowrate occurs with less frequency, in comparing to the rest of the time series.

The daily input data for the proposed models, corresponding to Q, and S, were obtained from the Proagua records of the Planta Norte WWTP corresponding to seven years of measurements, covering the period: January 2004-December 2010. The lagoon temperature input data (T_{w}) were estimated with the models described below.

Influent temperature was estimated using the time-series model developed in a former study [

The lagoon temperature evolution through the 2004-2010 period was estimated using a dynamic version of the steady state thermal model of [

All models were applied to the operational data of the Planta Norte primary lagoon, starting with the parameter values proposed by the author of each model. In order for the results to be comparable, the period 2007-2008 was established as calibration period for all models. The model fit evaluations for calibrations and validations were performed by obtaining the coefficient of determination (proportion of variation explained by the model, R^{2}) and the root mean squared error (RMSE). In the figures showing the variations of the measured and estimated concentrations at the lagoon effluent, the dots correspond to the measured data while the solid line show the model estimates, the thick frame within the graph corresponds to the calibration period (2007-2008), while the model validation period comprehends the two sub-periods outside the frame.

Once resolved the dimensional inconsistencies, the model was executed and its parameters were calibrated. The measured and estimated BOD concentrations at the effluent of the lagoon are presented in ^{2} with RMSE of 11.9 mg/L for calibration, and 0.345 with 17.6 mg/L for validation, respectively.

In

Model | Parameter | ||||||
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b | f_{BODu} | f_{l} | i_{p} | m | k_{L,}_{20} | s_{p} | |

(d^{−1}) | (gBOD_{U}g^{−}^{1} BOD) | (-) | (-) | (d^{−1}/˚C^{−1}) | (d^{−1}) | (-) | |

Bemister’s original | 0.003 | 1.47 | 0.70 | 0.85 | 0.012 | 0.002 | 0.400 |

Calibrated, this work | 0.028 | 1.30 | 0.90 | 0.871 | 0.003 | 0.002 | 0.400 |

Adapted, recalibrated, this work | −0.010 | 1.20 | 0.930 | 0.726 | 0.004 | 0.010 | 0.191 |

part of the simulation period (2004-2008), the fluctuations in the measured data are not represented by the model, especially in the specific cases of sudden increases or decreases (pulses) in the effluent BOD concentration of the lagoon, which are associated to rain-induced influent flow peaks. Furthermore, in the last part of the time series (2009-2010), where the concentration variation range is the widest of all the time series, the model clearly overestimates the effluent concentrations. This occurs through both higher-than-observed peaks and shallower-than-observed valleys, as well as through periods of general model overestimation. Regarding the parameters of the model,

While the Bemister model [^{2} and RMSE. This is especially true for the validation period and―visibly―around the sharp perturbations appearing in several years during summer/ fall, the latter being associated to intense rain episodes or periods. Another fact to consider is that the Bemister model [_{O}_{,H} = 0.2 mg/L [

The model considers k_{L}_{,20} = 0.002 and s_{ρ}= 0.4, according to a study by [^{2} with RMSE of 11.4 mg/L for calibration, and 0.362 with 17.2 mg/L for validation, respectively; _{L}_{,20} and s_{ρ} with different values than originally proposed) is partially correct.

The measured and estimated effluent BOD concentrations are presented in

It can be seen, in

The results of measured and estimated BOD and VSS concentrations for the lagoon effluent are presented in ^{2} of 0.242 with RMSE of 16.8 mg/L for the calibration period, and 0.230 with 19.6 mg/L for the validation, respectively; while for VSS, it obtains R^{2} of 0.078 with 35.8 mg/L of RMSE for the calibration period, and 0.236 with 29.7 mg/L for validation, respectively. The model parameters

Model | Parameter | |||||
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(CO_{2S}-CO_{2}) | K_{SC} | R_{12} | R_{21} | R_{1S} | f_{i} | |

(mg/L) | (mg/L) | (d^{−1}) | (d^{−1}) | (d^{−1}) | (mg IC mg^{−1} BOD) | |

Original | 46.0^{*} | - | 0.010 | 0.070 | 0.16 | - |

This work | 36.0 | 200 | 0.100 | 0.010 | 0.016 | 0.020 |

^{*}Data estimated from [

values (initial and adjusted) are shown in

This model presents the poorest fit value of the study. In _{d} (0.60 d^{−1}, very high) and μ_{m} (1.17 d^{−1}). This can give some relief to the BOD lack of fit, but results in a very poor representation of VSS.

Another cause in the lack of fit of this model may be the fact that it does not consider the variation of the parameters with respect to the lagoon temperature, being that according to [

The temperature effect on the model parameters was proved using the Arrhenius correction as follows, in accordance with [^{2} of 0.409 with RMSE of 14.0 mg/L for the calibration period, and 0.281 with 19.1 mg/L for the validation, respectively; while for VSS, it obtained an R^{2} of 0.062 with 21.7 mg/L of RMSE for the calibration period, and 0.273 with 24.9 mg/L for validation, respectively. Its calibrated parameters values are shown in ^{2}), mitigating the underestimation of the BOD (120.1 mg/L for estimates and 119.9 mg/L for measured data, on average) and, particularly, for VSS (107.8 mg/L for estimates and 104.4 mg/L for measured data,

Model | Parameter | ||||
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K_{d} | K_{S} | Y | μ_{m} | f_{nb} | |

(d^{−1}) | (mg/L) | (g VSS g^{−1} BOD) | (d^{−1}) | (-) | |

Montalvo’s original | 0.12 | 175.0 | 0.26 | 0.28 | - |

Calibrated, this work | 0.60 | 40.0 | 0.60 | 1.17 | 0.200 |

K_{d,}_{20} | K_{S,}_{20} | Y | μ_{m,}_{20} | f_{nb} | |

Modified, recalibrated, this work | 0.572 | 19.0 | 0.857 | 1.00 | 0.313 |

on average). However, the fit still remains from poor to moderate. The estimated concentrations at the lagoon effluent are also shown in

Finally, the three simple AFL models founded in the literature showed a poor to moderate fit when applied to the operational data of the “Planta Norte” (WWTP) in San Luis Potosí. This can be observed in ^{2}) obtained with each model as well as its associated error (as RMSE) for BOD determination and VSS for the Montalvo et al. model [

It is worth mentioning that two additional calibration periods (2004-2005 and 2005-2006) for the three models were tested, finding that it is in the selected period (2007-2008) in which a best fit was obtained in the description of the BOD behavior, this was verified at all three models.

The evaluation and comparisons made in this research allow confirming that oxygen demand behavior description in AFL requires further investigation for the development of better simple models that can be applied at full-scale plants for different objectives such as, effluent quality control or aeration control.

・ Most of the existing AFL models focus on hydrodynamic aspects and to a lesser extent to the composition of the wastewater and its transformation processes, and, due to their complexity, advanced wastewater characterization and specialized software are required for their application.

・ The Bemister model [

・ The Chagnon model [

・ The Montalvo et al. model [

・ In all three models, several not included aspects or processes can be responsible for their lack of fit. Some of these processes are: influence of dissolved oxygen and oxygen supply in the process, non-completely-mixed flow, nitrogen dynamics, and the effect of heavy rains in the lagoon dynamics. Integrating them without adding excessive complexity is a challenge in aerated facultative modeling.

Model | R^{2} | RMSE | ||
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(-) | (mg/L) | |||

BOD | ||||

Bemister [ | 0.528 | {0.345} | 11.9 | {17.6} |

Bemister adapted | 0.575 | {0.362} | 11.4 | {17.2} |

Chagnon [ | 0.489 | {0.290} | 13.0 | {18.9} |

Montalvo et al. [ | 0.242 | {0.230} | 16.8 | {19.6} |

Montalvo et al. [ | 0.409 | {0.281} | 14.0 | {19.1} |

VSS | ||||

Montalvo et al. [ | 0.078 | {0.236} | 35.8 | {29.7} |

Montalvo et al. [ | 0.062 | {0.273} | 21.7 | {24.9} |

Calibration period 2007-2008, the rest of the time series was utilized for validation.

・ The application, evaluation and comparisons made in this research confirm that a proper description of the oxygen demand behavior in aerated facultative lagoons requires further research for the development of better and simple models that can be applied at full-scale for multiple objectives.

The authors thank ProgramaMultidisciplinario de PosgradoenCienciasAmbientales, UASLP for the academic formation; GrupoProagua SA de CV and its staff for the data supply and Consejo Nacional de Ciencia y Tecnología (CONACYT) for Carlos Joel Ábrego-Góngora’s scholarship (No. 166201).

Carlos JoelÁbrego-Góngora,RobertoBriones-Gallardo,Luis ArmandoBernal-Jácome,AntoniEscalas-Cañellas,11, (2015) Evaluation of Three Dynamic Models for Aerated Facultative Lagoons. Journal of Water Resource and Protection,07,1131-1142. doi: 10.4236/jwarp.2015.714093