1. Introduction
Anaerobic digestion (AD) is an eco-friendly biological process which is universally used for the treatment of agricultural, industrial and municipal wastewater around the world [
1,
2,
3,
4]. Its utilization is increasingly widely, due to its capacity for producing methane, which can be used afterwards as a heat source or for electricity generation, taking part within the low-carbon energy technologies and circular bio-economy [
5]. In this context, anaerobic lagoons (AL) are natural wastewater treatment systems with a long hydraulic retention time, suitable for small communities due to the low energy demand and the operating costs [
6,
7,
8,
9]. By applying this kind of technology, the mechanical equipment, used for mixing processes in conventional plants, are avoidable. In addition, AL offer a number of advantages, such as the establishment of concentration profiles along the reactor, a buffering capacity in cases of overloads and greater protection against acidification [
10]. However, due to the fact that AD is strongly influenced by temperature, there is a close dependence between AL and weather conditions, so its implementation may be limited in cold or low solar radiation areas [
8,
11].
The application of mathematical models builds understanding for both microbial-related dynamic and kinetic processes, reveals optimisation possibilities, which lastly improves the digester’s performance [
12,
13]. The IWA Anaerobic Digestion Model No.1 (ADM1) [
14], created in 2002 to stablish a common platform for the modelling of AD processes [
15], has been widely applied in waste treatment processes, due to its high feasibility, considering the fact that most of the processes of AD are included within ADM1 [
16]. However this model has merely been applied to completely mixed reactors. The approach of models based on the ADM1 for unstirred waste water treatment systems has been little studied. In these models, complexity is increased and the effect of boundary conditions is essential. Moreover, the mathematical complexity required by these models does not entail a significant issue, due to the increasing technological and computational development [
12].
In the past twenty years many researches based on mathematical models for treatment processes in lagoons have been carried out. Fleming [
17] created the first models applying computational fluid dynamic (CFD) for the prediction of the performance of full-scale incompletely mixed anaerobic digesters. Wu and Chen [
8] developed a CFD model for AL which combines physical and biological processes, and includes both heat conduction and solar radiation by a thermal model. In this model, a single-phase incompressible Newtonian fluid is considered. Goodarzi et al. [
18] determined the effect of ambient and inlet temperature variations on the hydraulic performance of a typical rectangular pond. In all these described models, the biological processes are depicted by a single equation depending on the concentrations of the influent and effluent. Brito-Espino et al. [
19] defined advection, diffusion and reaction phenomena for wastewater treatment in anaerobic plug flow reactors by non-linear, second order, partial differential equations. ADM1 is implemented within this model, and both biochemical and physical–chemical reactions of ADM1 are calculated by a flowchart for sequential processes. In this method, temperature is not considered. Nevertheless, very few researches have been conducted to develop a comprehensive model which integrates fluid flow, heat transfer, and cells behaviour in AL.
The aim of this work is to set-up a theoretical framework for wastewater treatment in unstirred flow anaerobic lagoons, by a model which allows the integration of fluid flow, heat transfer and cells behaviour, for the purpose of describing processes occurring in AL. The implementation of the ADM1 into the model and the consideration of the influence of the local thermal weather, identified with the boundary conditions, allows the model to portray the processes taking place in reality more precisely than [
19]. In order to do this, an improved two dimensional mathematical model, based on the coupling of a set of parabolic partial differential equations (PDEs) and related to the phenomena associated to AL, has been developed. In addition, Dirichlet, Neumann and Robin boundary conditions have been established on the differential equations. This model combines the parametrization of different processes within the lagoon and its environment with the finite element analysis. Finally, the parallelization of the resulting algorithm has been performed in the simulation, therefore allowing an improved computational efficiency than the resulting form sequential processes in [
19]. Thanks to the help of FreeFemm++ and the parallel solver package, available for this software, the processing of each one of the variables related to AD processes and the simultaneous exchange of the data has been feasible. Having said this, we conclude that the novelty of this study resides in the following aspects. Firstly, in the implementation of the ADM1 and the heat transfer phenomenon in a mathematical model which describes a unstirred fluid flow, in order to predict the spatial distribution of the different variables that take part in the processes within the AL. Furthermore, secondly, in the optimisation and designing of the algorithm, by parallel method, providing an accurate forecast of the real behaviour of the process, as is shown in the ADM1. In the simulation, two different scenarios have been chosen as examples; the first corresponds to a conventional AL which is subjected to the ambient temperature, and the second includes heat sources, induced by solar assisted [
20] or through the biofuel recovery in the anaerobic process [
21,
22].
4. Conclusions
In this paper, we have proposed and assessed a methodology for anaerobic cells performance for wastewater treatment, in AL, under the influence of the temperature. It has been studied in terms of biomass and substrate concentrations. The model couples a series of PDEs, related to the phenomena associated to AL (ADRE, ADM1, Stokes and heat transfer), to each other.
Diffusion for horizontal and vertical directions, the movement of the bulk of the concentrations in accordance with a gradient, external temperature interactions, biochemical and physical–chemical reactions, and a set of boundary values were considered in this study.
This model builds understanding for microbial community’s behaviour along the lagoon as a function of the temperature. Applying heat load in different points of the system, it has been possible to establish correlations through the graphics, as well as the comparison between diverse scenarios according to their corresponding boundary values. The results give us the possibility to obtain effective designs adapted to each circumstance, avoiding energy loss.
This methodology allows the optimization of unstirred flow systems, taking into account that the advantages of these systems make them more suitable for specific applications. The model can be used in the prediction of the effluent quality and in the design of AL to achieve better performances.
In view of the results, it can be concluded that this methodology has significant potential as a tool for both the design of AL, and the interactive learning of the microbial ecology in plug flow systems.