Development of Hybrid Models for a Vapor-Phase Fungi Bioreactor

This study is aimed at the development of a model for an experimental vapour-phase fungi bioreactor, which could be derived in a simple way using the available measurements of a pilot-plant reactor, without the development of ad hoc experiments for the evaluation of fungi kinetics and the estimation of parameters related to biofilm characteristics. The proposed approach is based on hybrid models, obtained by the connection of the mass balance equation (used in traditional phenomenological models) with a feedforward neural network (used in black-box modelling), and the proper use of statistical tools for the model assessment and system understanding. Two different hybrid models were developed and compared by proper performance indexes, and their capability to predict the biological complex phenomena was demonstrated and compared to that of a first-principle model.


Introduction
The harmful effects of the emissions of volatile organic compounds (VOCs) on the environment and human health have prompted the development of a wide range of offgas treatment technologies [1].The degradation of pollutant compounds by means of biological systems is attractive for several reasons: low cost of the process, absence of toxic by-products, ambient condition for operation, and high efficiency of the process.
Among the different biodegradation methods, biofiltration is emerging because of its efficiency to treat large volume of air contaminated by volatile organic compounds (VOCs) [1][2][3].In this context, fungi-based biofiltration can support an enhanced mass transfer of hydrophobic VOCs due to the high hydrophobicity of the fungal cell wall and the ability of fungi to colonize with their aerial hyphae the empty space in the biofilter [4].The possibility of using such processes for industrial applications depends on the availability of good knowledge of the complex phenomena occurring in the biological systems; therefore a considerable effort in bioprocess modeling has been done [5][6][7][8][9].The implementation of system models is a support for design, management, and process control purposes.
Traditionally, biofilter models are based on mass balance principles and require a good knowledge of the underlying physics of the process such as information on specific growth rate of microbes, biofilm thickness and density, values of diffusivity, partition coefficient, yield, and biofilm distribution [9].Significant efforts have been done on developing methods and equations to estimate key design parameters of biofiltration processes such as Henry constants, interfacial areas, and active biomass in biofilters [10].It is worth noticing that the achievement of a precise and reliable model requires a high experimental effort, sometimes with elaborate technical facilities and expertise in order to properly estimate model parameters.For instance, Dorado et al. [11] evidenced that the determination of kinetic parameters is a demanding task, due to the difficulty to reproduce the experimental system and the necessity of calibrating each model for each specific experimental condition.Even if the use of first-principle models proposed in literature is generally able to give a good description of the considered biofilter system [8][9][10][11], a limited understanding of the biological processes along with the high uncertainties in the model parameter estimation may restrain the applications of such models.
The aim of the present work is to propose a simple modeling approach minimizing the measurement and experimental effort, while retaining model efficiency.Within this purpose, data-driven (or black-box) models are a possible alternative to describe the biological complex systems, with the advantage that, in principle, they can be constructed without any knowledge on the process, if a proper amount of data is available.In particular, neural networks (NNs) have shown good reconstruction capabilities in different application fields [12][13][14][15] and, recently, in the modeling of biological wastegas treatment systems [16][17][18][19][20][21].As well underlined by Rene et al. [19], the main issues with black-box models concern (i) the availability of a large amount of data from which it is possible to extract the information on the underlying system, (ii) the choice of the network architecture, and (iii) the selection of input variables from which it is possible to infer the process outputs.Sometimes, it may be convenient to take advantage of the a priori knowledge of the nonlinear system using the macroscopic balances (i.e., mass, energy balances) and introducing NNs only for the description of some phenomena.The resulting model is called hybrid [19,[22][23][24], and it has proved to be successful for dynamic systems, with better generalization features and, in addition, identifiable with a reduced set of data with respect to a blackbox equivalent model.To our best knowledge, such hybrid models have been recently applied to waste-water treatment bioreactors [25] and food technology [26,27] but never to a gas-phase bioreactor.
In previous works [9,28] the authors applied a phenomenological model to a bioreactor, where hexane was removed from a contaminated stream, and the obtained results were compared with experimental data.The biofiltration process was conducted in a vapor-phase fixed-bed bioreactor, containing a biological phase, the fungus Aspergillum niger, immobilized on a support.Different experiments were conducted both on the lab-scale reactor and in batch conditions for assessing, respectively, fluid dynamic properties and kinetic parameters.Even if the first-principle model allowed a good reconstruction of the average removal efficiency of the biofilter at different pollutant loads, model uncertainty was evidenced through sensitivity analysis, showing that partition coefficient, maintenance coefficient, and available specific surface, which had been determined partly by theoretic and simulation approach, were the parameters with greatest influence on the final removal efficiency of the bioreactor.
Following these premises, the aim of this work is to propose an alternative modeling approach based on the available measurements easily acquired in a vapor-phase fungi pilot plant.For this purpose, the first-principle model previously developed [28] has been modified by introducing a neural network for the description of the removal rate of the pollutant, which represents the most difficult modeling task, because it requires the determination of the specific kinetic rate expression (which is usually a complex function of state variables) and of the distribution of the nonuniform biofilm coverage.This information on the reacting biosystem was extracted from the input-output experimental data of the pilot plant, using a neural network in two different hybrid models: (i) a heterogeneous model where the neural network describes the kinetic rate in the biological phase and (ii) a homogeneous model where the neural networks approximate the flux of hexane at the biolayer/air interface.The two different structures have been compared and the assessment of the models has been aided by using proper statistical tools.

Experimental Apparatus and Conditions
A lab-scale bioreactor inoculated with a strain of Aspergillus niger was used for the treatment of an artificially polluted hexane airstream.The system consisted of two identical columns connected in series; therefore reactor performance could refer either to each single column (single configuration) or to a unique reactor of double length (double configuration).Each column was a jacketed glass column of overall height 0.40 m, internal diameter 25 mm, with a stainless steel net at 40 mm from the bottom to sustain the packing material (expanded clay in granular form with average Ø 3-5 mm) and sampling ports for the substrate and air supply and for the outlet gas flow.The contaminated air-stream was artificially created by mixing two distinct flows supplied by a compressor: the first one was passed through a humidifying system; the second one was made air sparging in a vessel containing liquid hexane at 30 ∘ C. The gas flow rate was set to 4⋅10 −3 m 3 /h, corresponding to an empty bed residence time (EBRT) of 159 sec.Optimal EBRT should be in the range from 15 to 60 sec, but the minimum value is actually dictated by the given set of off-gas composition and filter conditions, the pollutant RE, or maximum outlet concentration allowed by regulations.In our case preliminary trials with a double flow rate and corresponding EBRT of 80 sec had brought to an almost zero RE and poor mycelium development [28].
Even if the system was located in a conditioned room to work as close as possible to a constant temperature of 30 ∘ C, optimal value for the fungal growth, daily monitoring temperature showed that it varied from 19 ∘ C to 30 ∘ C.
More details about the bioreactor inoculation procedure, the development of the experiments, and the results obtained from this reactor were already reported [9,28].
Reactor performance was represented in terms of removal efficiency (RE) calculated from the inlet ( 0 ) and outlet (  ) gas concentration, as reported in the following:

Modeling Approach
Starting from the first-principle model of the biofiltration system, a hybrid model approach has been proposed where empirical models (e.g., neural networks) are used to describe the most critical phenomena of the considered biosystem.
3.1.First-Principle Model.Before describing the hybrid models developed in the present paper, the first-principle model previously proposed [9] to describe the biofiltering system of Section 2 is recalled: with the following boundary conditions: In the above equations   and   are the dimensionless pollutant concentrations, respectively, in the gas phase and in the biofilm phase, obtained by dividing the actual concentrations,   and   , for the inlet concentration  0 ;   is the superficial gas velocity;  is the dimensionless reactor height calculated with respect to the reactor length (ℎ/, with ℎ being the position in the column); and  is the dimensionless biolayer thickness calculated with respect to the effective one (/ * , with  being the position in the biolayer and  * the effective biolayer thickness).The model has six parameters related to the system fluid dynamics, biofilm characteristics, and mass transfer:  is the dispersion coefficient in the reactor; V is the bed porosity;   is the effective diffusion coefficient of the pollutant in the biolayer;  is the biolayer surface area per unit volume of the reactor;  is the fraction of  covered by the biofilm; and  is the pollutant air/biofilm distribution coefficient.For the specific growth rate expression of the fungi growing on hexane, Monod kinetics with an Andrews type inhibition was assumed where   is the saturation constant;   is the inhibition constant;   is the maintenance coefficient;  max is the maximum specific growth;   is the biofilm density; and  is the biomass yield coefficient.The model has many parameters, which had been evaluated [9] partly through a trial-and-error method, partly using the knowledge on the process obtained through experimental data and from the literature (Table 1).The proposed model showed good performance capabilities, but sensitivity analysis evidenced model uncertainty, principally due to parameters related to the biological phase, which are difficult to be experimentally evaluated.The main critical experimental points concern the evaluation of biodegradation kinetics and the prediction of biomass film distribution, which strongly influence degradation and imply large data variability.This aspect unfavorably plays in the construction of a bioreactor  model, which should be robust with respect to data variation or presence of outliers.It is also important to notice that some parameters are quite difficult to be experimentally estimated, like the partition coefficient , while biofilm thickness and biofilm surface are impossible to measure and might be only adjusted by fitting.Furthermore, the inoculated fungus Aspergillus niger can develop as filamentous mycelium and spores, and in the previous study it was not possible to recognize which form was responsible for hexane degradation.

Neural Networks.
A feedforward neural network (FNN) has been used here coupled with the first-principle model, in order to describe phenomena occurring in the biofilter systems and related with pollutant degradation kinetics.A general FNN is represented in Figure 1, for  1 inputs, represented by the vector z 1 , and one output, z 3 .In more detail, the input signals z 1 are scaled by the adjustable parameters, called weights,  1 (, ); then all the contributions are summed and processed by the activation function   (3a).The resulting signal, z 2 , is scaled by the weight  2 (, ).Its components are summed and mapped into the vector z 3 by the activation function  0 (3b): where  represents the bias term.
Neural networks are data-driven models and, in principle, it is not necessary to have a deep knowledge on the physicochemical phenomena governing the process.They are universal approximators [29].Therefore they should be able to model any nonlinear system if the proper network structure is used, that is, the right network input variables and the number of hidden neurons.
In this work, a knowledge-based approach supported by statistical tools has been used to identify the inputs of the network, which are the variables affecting the consumption of reactant.With regard to hidden neurons, it is important to underline that they cannot be determined from the knowledge of the process, because they elaborate signals that have lost the physical meaning of the inputs.In this case, a trialand-error method has been used for the choice of the number of hidden units: starting from the general consideration that a parsimonious model is preferred, the number of hidden units has been evaluated from the simplest model, with only one hidden neuron and adding one more neuron until a significant change in the model performance was observed.Then, the input and hidden layers were augmented with an extra neuron, the bias, which provides a constant output signal equal to one.

Hybrid Models.
Considering the drawbacks underlined for the first-principle model in Section 3.1, the aim of the present work is finding a simple modeling approach, which could be easily applied also to industrial bioreactors, where the possibility of accomplishing experimental measurements for biomass characterization is scarce.The approach proposed here is based on hybrid models, obtained by the integration of a FNN in the first-principle model ((2a)-(2c)).Aiming at the best compromise between simplicity of description and prediction capabilities of the model, two different approaches have been considered to reconstruct the biological system behavior: a heterogeneous and a homogenous model.

Heterogeneous Hybrid Model (GM1).
The identification of the kinetic law occurring in the biological phase is one of the most critical points when describing biological systems, because the kinetic constants may vary significantly with process conditions [30].The neural network, indicated as  NN1 (z 1 ), has been therefore introduced in the reactor model ((2b)-(2c)) to estimate hexane reaction rate along with the term     ; therefore the following heterogeneous or twophase hybrid model is obtained ((4a) and (4b)): along with the boundary conditions reported in Section 3.1.
In the above formulation z 1 is the input vector of the network (cf. Figure 1), which consists of independent selected variables affecting the reaction rate.It will be defined later.
The use of a neural model has the immediate consequence that it can be applied directly on the experimental data obtained in the reactor configuration used for the biofiltration of the polluted stream, without the necessity of conducting ad hoc experiments for kinetics identification and parameter estimation, as those previously carried out [9].

Homogeneous Hybrid Model (GM2).
A further simplification of the model has been obtained by using the neural network to describe the flux of the hexane at the gasbiological phase interface, avoiding also the estimation of the parameters related to the biofilm characteristics.As a result, a homogeneous model is obtained, where the derivative of concentration in the biological phase is modeled by a neural network, indicated as  NN2 (z 1 ), as reported in In this case only the fluid dynamic characteristics of the system need to be estimated, and the reactor efficiency is obtained by integrating (5) along with the boundary condition with respect to , reported in Section 3. Again, the input network variables, indicated with z 1 , will be defined later.

Development of the Neural Network
The experimental data available for parameter estimation and neural model validation were 290 outlet concentration values, at constant gas flow rate of 4⋅10 −3 m 3 h −1 , according to the single and double configuration, at different inlet concentration and temperature values, spanning, respectively, from 1 to 20 g m −3 and from 19 to 30 ∘ C. All the used concentration values were collected at regime condition, that is, when, after an adaptation period of about two weeks, biomass development was not visually observed anymore and steadystate conditions could be assumed [9].
Data have been randomized and divided into training (90%) and test (10%) set.The former series has been further divided into two different sets, one set for parameter estimation (80%) that means the proper training and one set (10%) for cross-validation.The latter data set is used in order to assure generalization capability of the neural model [29].In more detail, the training phase is stopped when the objective function calculated on the cross-validation data reaches the minimum.
The number of data points used for the development of the hybrid models is on average a high number compared to the other artificial neural network models developed for different waste-gas treatment systems [19] and it matches the requisite that data-driven model needs a high number of experimental patterns in order to give a good estimate of the process.
The training of the network has been developed by means of the Levenberg-Marquardt optimization algorithm.Unlike more common applications of neural network modeling, the variables to be estimated, which are reaction rate (GM1) and pollutant flux at the air/biolayer interface (GM2), are not experimentally measured; therefore FNN parameter estimation has been based on the error between the experimental and calculated concentration at the reactor exit.The latter is obtained through the integration of the reactor model, assuming that the error between the experimental and calculated concentration is exclusively due to reaction rates for GM1 and due to the hexane flux for GM2.In particular, the following objective function Φ(w) has been used for both models: where w represents the weights (parameters) of the neural model,   is the number of data used for the training, and    () and    () are, respectively, the experimental and calculated concentration at reactor exit.
The minimum search was accomplished for every network structure considering one hundred initial w vector values, randomly generated, and considering those weights leading to the lowest error calculated on cross-validation set.This approach solves the problem of generalization, that is, the obtainment of good performance with data that do not belong to the training set [29].The cross-validation was also used to examine the prediction capabilities of neural network with respect to sample variation and to assess model robustness.
The development of the neural kinetic model should capture, as well as possible, the essential characteristics of the functional relationships between inputs (i.e., concentration and temperature) and outputs (reaction rate).In other words, the neural kinetic model must also provide consistent derivatives of the reaction rate with respect to the concentration and temperature.

Selection of the Model
The construction of the two neural models has been accomplished selecting the model inputs, the number of hidden neurons, and the activation functions.The selection of the best structure has been based on the following performance indexes evaluated on the training data: (i) Coefficient of determination  2 , which measures the variance explained by the model and defined as where   is the average concentration value calculated for the   experimental points of the training set; (ii) Standardized residuals, (), which indicates if there are deterministic features that have not been predicted by the model, defined as [31] where   is the number of neural model parameters.
Nonrandom behavior of residuals with respect to calculated RE reveals the inadequacy of the model to capture system behavior.The Kolmogorov-Smirnov test is also used to assess the goodness of fit [32][33][34].

Selection of NN Structure in GM1.
The use of neural network in the two-phase model GM1 allows using a priori information on the reactor fluid dynamics and biofilm phase.
In particular, the following parameters previously reported [9] have been used: ,   , , ,  * , and .The kinetic law has been, on the other hand, extracted from the experimental data using the proper input variables, in particular hexane concentration in the biofilm and reactor temperature.The results obtained for the training showed that temperature does not affect the reaction rate in the considered experimental conditions, confirming the correctness of isothermal assumption in the phenomenological model [9].The selection of the hidden neurons has been established by varying the number of neurons in the range 1-4 and the best structure, consisting of two hidden neurons, has been selected on the basis of residual analysis and  2 .Training results evidenced better model behavior when a sigmoidal and a linear activation function were used, respectively, for input and hidden neurons instead of using the nonlinear one for both the layers.The analytical forms of the activation functions are reported in A summary on the structures and performance indexes for different network is reported in Table 2, while Figure 2(a) shows the results obtained for the training data set in terms of comparison between experimental removal efficiency (RE) and calculated ones.
It is worth noting that the variability of the data is quite large due to the variations of biomass activity with respect  to inlet pollutant concentration, as evidenced in [28].The hybrid model GM1 follows a medium trend of the RE with respect to the inlet hexane concentration.Modeling errors may exceed the measurement error that is equal to 5%, but it is important to underline that the estimated RE correctly tends to unity, for both reactor configurations, as  0 tends to zero.
Furthermore, a measure of the quality of the fitting is obtained plotting the standardized residuals with respect to the calculated RE (Figure 2(b)).The trend appears without a deterministic structure, indicating that the obtained model captures the essential features of the data.The data contained in the region (−2, 2) are more than the 95% of the total amount of data used for training, indicating that residuals can be reasonably modeled as an outcome of a random normal value.This is also corroborated by the Kolmogorov-Smirnov test [33,34] which does not reject the null hypothesis of Gaussian assumption of the residuals with a significance level of 5%.

Selection of the NN Structure for GM2.
The neural network used in the homogeneous model has to describe two steps of the hexane degradation; in particular it should extract from the experimental data the phenomenon of both adsorption and reaction in the biological phase.On the other hand, GM2 needs less information on the biological phase and kinetics, therefore requiring a minor experimental effort in terms of ad hoc experiments and sophisticated system analysis.In particular, only the dispersion coefficient () has been used (5).
The selected network inputs are inlet hexane concentration and concentration of hexane along the reactor.Again, results do not evidence temperature effects on RE, leading to a network with two inputs.
The number of hidden neurons leading to the best results has been, in this case, equal to two and, again, a sigmoidal and a linear activation function have been used, respectively, in the input and hidden neurons.
The principal aspects of GM2 model are reported in Table 2, and the results for the training set are shown in Figure 3(a).As for GM1, modeling errors may be higher than measurement errors, but it is important to note that residuals (Figure 3(b)) again seem randomly disposed with respect to the estimated variable, and the amount of residuals enclosed in the (−2, 2) region is less than 95%.Applying the Kolmogorov-Smirnov test [33,34] to residuals with a significance level of 5% the null hypothesis of Gaussian assumption is again not rejected.

Test Results
The ability of the two hybrid models to predict the biofiltration system behavior has been evaluated by comparing calculated values with the experimental data selected for testing purposes.To analyze the model performance for a wide range of pollutant inlet concentration, the interval of variation of  0 1-20 g m −3 has been divided into nineteen subintervals of unitary length.Test data (10% of all the data points) have been randomly selected with more than one sampling from intervals with larger amount of data.Because those data have not been used during the network training, this comparison shows the performance of the developed models in unknown situations.
The test of GM1 is reported in Figure 4, where the removal efficiency calculated with the hybrid model for the single (Figure 4(a)) and double (Figure 4(b)) configuration reactor is compared with the experimental data, with a mean square error for the total points equal to 0.0059.Considering the significant variability of the experimental RE used for training and validation of the model, the capability reconstruction of the hybrid model is quite good.This is also evidenced by the closeness of GM1 estimation with the RE calculated using the first-principle model [28], which has a mean square error equal to 0.0064.This means that the information contained in the data available at reactor exit allows a good reconstruction of the kinetic law, which rules hexane degradation in the biofilm phase.As a further confirmation of the capability of the hybrid model approach, the kinetic rate predicted by the neural model and that obtained in [9] (cf.(2b)) are reported in Figure 5.Even if a mismatch exists in the estimation of the two models, the qualitative behavior of the kinetic law is well reproduced using only concentration data calculated in one point of the reactor.This result indicates the possibility of using this approach to obtain information on the functional form of a kinetic law, when this is unknown for the problem at hand.
The GM2 test results are reported in Figures 6(a) and 6(b), where the comparison with the experimental data shows again discrete performance of the model.In this case the distance between the hybrid model and the experimental data is smaller than in the previous case, the mean square error being equal to 0.004, but there is bigger mismatch with the firstprinciple model.This is because GM2 used only information on the system fluid dynamics, extracting all the information regarding biofilm characteristics, kinetics, and mass transfer phenomena from the experimental data (Table 2), with the advantage of reducing ad hoc experiments and avoiding the use of difficult analytical procedures to calculate the model parameters related to the biofilm characteristics.

Conclusions
The modeling problem of a gas-phase bioreactor was solved by resorting to hybrid models, a combination of material balances (used in traditional phenomenological models) and neural networks used to describe the most complex phenomena present in the process.Two different options were investigated: (i) a heterogeneous model where the neural network describes the rate of consumption of the pollutant in the biological phase and (ii) a homogenous model where the neural network reconstructs the flux of the hexane at the interphase, avoiding the integration of the mass balance in the biological phase.Model construction was based on a large amount of experimental data, characterized by a large variability, from which both the hybrid models were able to capture the deterministic features of the observations (as demonstrated by residual analysis).It is worth noting that the neural model has not been used in the traditional way, where it is trained to model directly observed variables.In fact, in the present paper the kinetic rate or the mass transfer term was not measured; therefore the neural networks had to learn their functionality by using an indirect measure, which was the outlet pollutant concentration.
The main contribution of the proposed modeling approaches is the reduction of time and resources necessary to conduct experimental activities aimed at the identification of kinetic law and mass transfer phenomenon for the bioreactor, which are the most demanding task when modeling biological systems.The satisfactory reconstruction capabilities of the neural networks in both the heterogeneous and homogeneous model have been demonstrated by comparing hybrid models predictions with the ones obtained using the first-principle model.The good match between the predictions indicates that neural networks are able to extract information on specific complex phenomena from the simple observations of the laboratory reactor exit concentration.Furthermore, through the analysis of residuals [31,32] and Kolmogorov-Smirnov [33,34] test applied to the hybrid models, it was possible to corroborate the isothermal assumption used in the phenomenological model.
Results evidenced that the behavior of the heterogeneous model was closer to that of first-principle model compared to the homogenous one, with good prediction of the reaction rate law.This model should, then, be recommended.On the other hand, the homogenous model used less a priori information on the biosystem, which might imply a possible reduction of performance when used at different operating conditions (e.g., changes of gas flow rate).To overcome this limitation, a proper experimental campaign (i.e., with ad hoc trials at different flow rates) followed by model recalibration could be carried out to improve description model capabilities and obtain a wider applicability.This approach is easier to be implemented in a full-scale industrial plant making the homogenous model preferable to the heterogeneous one.
The proposed methodology allows the construction of parsimonious models, which are useful for on-line application, and also requires less computational time during training.In particular, hybrid models based on neural networks can be useful for process monitoring purposes in case of delayed concentration measurements, where it is advantageous to have a simple tool to fast recognize if the system is going out of control and take corrective measures.Furthermore, the availability of the model can give an effective tool for on-line optimization of the process.Effective diffusion coefficient of the pollutant in the biolayer (m 2 /h) : Reactor height (m)   : Inhibition constant (g/m 3 )   : Saturation constant in the specific growth rate expression of a culture growing on the pollutant (g/m Dimensionless reactor height (ℎ/) z 1 : Input vector of neural network z 2 : Signal vector of neural network from the hidden to the output layer z 3 : Output vector of neural network.

Figure 1 :
Figure 1: Scheme of the feedforward neural network.

Nomenclature𝐴:
Biolayer surface area per unit volume of the reactor (m −1 )   : Concentration of the pollutant in the air at position ℎ along the biofilter (g/m 3 )   : Average concentration value calculated for the experimental points of the training set (g/m 3 )    : Experimental concentration    : Calculated concentration  0 : Concentration of the pollutant in the air at the inlet of the biofilter (g/m 3 )   : Concentration of the pollutant in the air at the outlet of the biofilter (g/m 3 )   : Concentration of the pollutant at a position  in the biolayer at a point ℎ along the column (g/m 3 ) : Dispersion coefficient in the reactor (m 2 /h)   :

Table 1 :
Estimation of parameters in the first-principle model.

Table 2 :
Network structure and performance indexes of the two hybrid models.
/ 0 )   : Dimensionless pollutant concentration in the biological phase (  / 0 )   : Superficial gas velocity (m/h) : Yield coefficient of the culture on the pollutant (g-biomass/g-compound)   : Biofilm density (g-dry cells/m 3 biofilm) : Bias term of the neural model   : Activation function of the hidden neurons   : Activation function of the output neurons  NN1 (): Neural network for the heterogeneous hybrid model  NN2 (): Neural network for the homogeneous hybrid model ℎ: Position in the column; ℎ = 0 at the entrance and ℎ =  at the exit   : Maintenance coefficient (g-hexane/g-biomass/ h) : Pollutant air/biofilm distribution coefficient   : Parameter vector related to kinetics  1 (, ): Weight connecting th input and th hidden neuron  2 (, ): Weight connecting th hidden neuron and th output neuron : Dimensionless biolayer thickness (/ * ) :