A New Adaptive Neuro-Fuzzy Inference System Approach for Modelling a Biofilter Treating Hexane

1. Introduction

Among the various contaminated-air treatment methods, biological treatment requires comparatively low investment and operating costs and is also environmentally friendly [1]. Biofiltration is a biological purification method in which contaminated air is passed through a packed bed and a microbial biofilm formed on the bed particles degrades pollutants transferred from the air.

Water availability is one of the key factors affecting microbial degradation rate, so moisture fluctuations can produce unsteady-state conditions. Microorganisms die and lose activity with gradual reduction of bed moisture, reducing biofilter performance and eventually stopping contaminant removal. Thus, bed moisture content is an important regulating factor for optimum performance [2]. One proposed solution for reducing bed drying is the use of water superabsorbents. These low-cross-linked hydrophilic polymers can absorb and store water for long periods and gradually release water and minerals [3].

From an industrial viewpoint, modelling of biofiltration is valuable for scale-up and performance prediction. Two broad modelling approaches are found in the literature: mechanistic modelling and artificial-intelligence methods. Mechanistic biofiltration models have progressively incorporated Monod-type substrate kinetics [4], gas-phase dispersion [5], adsorption and dynamic bed behaviour [6], bed drying [7], and drying effects associated with water-superabsorbent use [8]. Despite successful descriptions of particular experiments, increasingly detailed mathematical models require additional parameters that must be measured accurately.

Artificial intelligence has been applied to complex problems such as climate forecasting [9] and autonomous-vehicle systems [10]. More broadly, neural, fuzzy, and machine-learning approaches have been applied to short-term load forecasting [11], clinical and health prediction [12,13,14,15,16], and general multiclass machine-learning software [17]. These studies demonstrate the capacity of data-driven models to learn complex nonlinear input–output relationships, which motivates their use for nonlinear environmental and bioprocess systems such as biofiltration.

Related pattern-recognition research has addressed face localization, detection, and recognition [18,19,20,21], palmprint and signature recognition [22,23], three-dimensional deformable texture matching [24], and retinal vessel segmentation [25]. These developments form part of the broader evolution of computational vision and representation-learning methods [26].

Computational modelling and optimization methods have also been investigated in broadband wireless computing [27], antenna design [28], mobile-robot control [29], and scalable distributed-data processing [30]. Quantitative modelling has additionally been applied outside engineering to investigate relationships among complex variables, including thinking styles and self-efficacy [31]. AI-based biofiltration models can reduce computational burden and avoid difficult parameter measurements. Previous studies have used multilayer perceptron neural networks to predict hydrogen-sulphide removal [32,33], neural-network models for immobilized-cell biofilters [34,35], differential neural networks for fungal biofiltration [36], and back-propagation models for dichloromethane, styrene, hydrogen sulphide, and ammonia biofilters [37,38,39]. Neural modelling has also been applied to hexane biofiltration [40], while radial-basis-function networks have been investigated for toluene biofilter performance prediction [41].

In this work, a mechanistic model and a neuro-fuzzy method are developed to predict the dynamic performance of a hexane biofiltration system using water superabsorbents. After bed moistening stops, inlet air progressively dries parts of the bed, causing outlet pollutant concentration to increase. The two approaches are compared to assess their modelling capabilities.

2. Materials and Methods

2.1. Adaptive Neuro-Fuzzy Inference System

ANFIS combines artificial neural-network learning with a fuzzy inference system [42]. It was introduced by Jang [43] for Takagi–Sugeno–Kang fuzzy inference systems [44,45]. ANFIS modelling and control techniques are described in [46,47], with applications including time-series prediction [48,49,50] and earlier kick assessment in oil wells [51].

For a first-order TSK fuzzy model, the consequent of rule i is

$$f_i=p_{i}x+q_{i}y+r_i,i=1,2,\dots ,n.$$

(1)

The five-layer ANFIS structure is represented as follows.

Layer 1: membership degrees:

$$\begin{array}{cc}\hfill O_i^1& ={\mu }_{A_i}\left(x\right),i=1,2,\dots ,n,\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill O_i^1& ={\mu }_{B_i}\left(y\right),i=1,2,\dots ,n.\hfill \end{array}$$

(3)

Layer 2: rule firing strengths:

$$O_i^2=w_i={\mu }_{A_i}\left(x\right){\mu }_{B_i}\left(y\right).$$

(4)

Layer 3: normalized firing strengths:

$$O_i^3={\overline{w}}_i={\displaystyle \frac{w_i}{{\sum }_{j=1}^n{w}_j}}.$$

(5)

Layer 4: defuzzification:

$$O_i^4={\overline{w}}_i(p_{i}x+q_{i}y+r_i).$$

(6)

Layer 5: overall output:

$$O^5={\displaystyle \sum _{i=1}^n}{\overline{w}}_i{f}_i.$$

(7)

2.2. Gravitational Search Algorithm

The Gravitational Search Algorithm (GSA) is a heuristic optimization algorithm inspired by Newtonian laws of gravity and motion [52]; it has also been applied to image-processing problems [53]. For a system of s masses, the position of agent i is

$$X_i=(x_i^1,\dots ,x_i^d,\dots ,x_i^n),i=1,2,\dots ,s.$$

(8)

For a minimization problem,

$$q_i\left(t\right)={\displaystyle \frac{fi{t}_i\left(t\right)-worst\left(t\right)}{best\left(t\right)-worst\left(t\right)}},$$

(9)

and the normalized mass is

$$M_i\left(t\right)={\displaystyle \frac{q_i\left(t\right)}{{\sum }_{j=1}^s{q}_j\left(t\right)}}.$$

(10)

The best and worst fitness values are

$$\begin{array}{cc}\hfill worst\left(t\right)& =\underset{j\in \{1,\dots ,s\}}{max}fi{t}_j\left(t\right),\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill best\left(t\right)& =\underset{j\in \{1,\dots ,s\}}{min}fi{t}_j\left(t\right).\hfill \end{array}$$

(12)

The total force and acceleration in dimension d are

$$\begin{array}{cc}\hfill F_i^d\left(t\right)& =\sum _{\begin{array}{c}j\in kbest\\ j\ne i\end{array}}ran{d}_{j}G\left(t\right){\displaystyle \frac{M_j\left(t\right)M_i\left(t\right)}{R_{ij}\left(t\right)+\epsilon }}\left(x_j^d\left(t\right)-x_i^d\left(t\right)\right),\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill a_i^d\left(t\right)& ={\displaystyle \frac{F_i^d\left(t\right)}{M_i\left(t\right)}}.\hfill \end{array}$$

(14)

Velocity and position are updated by

$$\begin{array}{cc}\hfill v_i^d(t+1)& =ran{d}_i{v}_i^d\left(t\right)+a_i^d\left(t\right),\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill x_i^d(t+1)& =x_i^d\left(t\right)+v_i^d(t+1).\hfill \end{array}$$

(16)

The gravitational constant decreases with time as

$$G\left(t\right)=G_0{e}^{-\alpha t/T}.$$

(17)

2.3. Mechanistic Model Development

A mechanistic model is developed through mass balances to predict variations in outlet concentration of the hexane biofiltration system using water superabsorbent. When bed moistening is stopped, water evaporation increases and unsteady-state conditions govern the biofilter.

The model considers gas and biofilm phases, with z and x coordinates used for the gas and biofilm phases, respectively. Biofilm formed on bed particles contains microorganisms responsible for pollutant biodegradation. Diffusion into the biofilm and microbial reaction constitute important biofiltration limitations. Moisture content and axial dispersion are included, and a moisture-depletion pattern influences bed porosity and particle surface area. The equations are discretized by the finite-volume method and solved numerically.

2.3.1. Model Assumptions

• Cartesian coordinates are used because biofilm thickness is small compared with bed-particle dimensions.
• Temperature changes are negligible; thermal equilibrium exists between phases and physical properties are constant.
• Bed particles consist of perlite and water superabsorbent, considered spherical with an identical average diameter when fully water-saturated and treated as non-porous.
• Radial concentration changes are neglected and gas-phase axial dispersion is represented through a dispersion coefficient.
• The biofilm is homogeneous; biological reactions occur only within the biofilm. Its thickness and density are uniform and constant, and it covers bed particles but not superabsorbent particles.
• Biofilm reaction rate is represented by a Monod equation. Moisture effects are treated independently through a coefficient.
• Gas-phase and interfacial resistances are neglected, and contaminant concentrations at the gas/biofilm interface are related by Henry’s law.
• Bed particles exchange water only through their surfaces; capillary effects are neglected and particles do not adsorb pollutant.
• Water evaporation from superabsorbent is modelled as evaporation from the surface of a droplet of identical diameter.
• Bed porosity varies as water is withdrawn from perlite pores and superabsorbent particles shrink; shrinkage also reduces superabsorbent surface area.

2.3.2. Description of the Model

The pollutant mass balance in the gas phase is

$${\displaystyle \frac{\partial \left({\epsilon }_g{C}_g\right)}{\partial t}}=D^{\prime }{\displaystyle \frac{{\partial }^2{C}_g}{\partial z^2}}-u_g{\displaystyle \frac{\partial C_g}{\partial z}}+A_s{D}_{\mathrm{hex}}{\left.{\displaystyle \frac{\partial C_b}{\partial x}}\right|}_{x=0}.$$

(18)

The biofilm pollutant balance is

$${\displaystyle \frac{\partial C_b}{\partial t}}=D_{\mathrm{hex}}{\displaystyle \frac{{\partial }^2{C}_b}{\partial x^2}}-{\displaystyle \frac{X_v}{Y_{x/s}}}{\displaystyle \frac{{\nu }_m{C}_b}{K_s+C_b}}.$$

(19)

The source document specifies the interface and initial/boundary conditions as

$$\begin{array}{cc}\hfill C_b(0,z,t)& ={\displaystyle \frac{C_g(z,t)}{m}},\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill {\left.{\displaystyle \frac{\partial C_b}{\partial x}}\right|}_{x=x_n}& =0,\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill C_b(x,z,0)& =C_{b0}(x,z).\hfill \end{array}$$

(22)

The moisture balance in the gas phase is represented by

$${\displaystyle \frac{\partial \left({\epsilon }_g{H}_g\right)}{\partial t}}=D^{\prime }{\displaystyle \frac{{\partial }^2{H}_g}{\partial z^2}}-u_g{\displaystyle \frac{\partial H_g}{\partial z}}+K_{c}a(H^{*}-H_g).$$

(23)

Water contents in perlite and superabsorbent are described by

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial m_p}{\partial t}}& =-K_c{a}_p(H^{*}-H_g),m_p\left(0\right)=m_{p0},\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial m_s}{\partial t}}& =-K_c{a}_s(H^{*}-H_g),m_s\left(0\right)=m_{s0}.\hfill \end{array}$$

(25)

The resulting equations were discretized using the finite-volume method. Model parameters were obtained from the literature or calculations, using biofiltration data and hexane parameters reported in [3,54,55].

Table 1. Model parameters of bacterial biofiltration [54,55].

Parameter	Value	Unit
Hexane partition coefficient in water, $m_{\mathrm{Hex}}$	9.14	–
Specific surface area, $A_s$	2600a	m−1
Biomass yield coefficient, $Y_{x/s}$	1.314c	–
Effective pollutant diffusivity, $D_{\mathrm{Hex}}$	$2.58\times {10}^{-10}$	m2 s−1
Biofilm thickness, $x_n$	387	$\mu$m
Dispersion coefficient, $D^{\prime }$	$1.22\times {10}^{-4}$	m2 s−1
Bed porosity, ${\epsilon }_g$	0.45	–
Biofilm density, $X_v$	9744	g m−3
Maximum specific growth rate, ${\nu }_m$	$5.83\times {10}^{-5}$	h−1
Saturation constant, $K_s$	0.02	g m−3

^a Fitted value; ^c calculated value.

Table 1. Model parameters of bacterial biofiltration [54,55].

Parameter	Value	Unit
Hexane partition coefficient in water, $m_{\mathrm{Hex}}$	9.14	–
Specific surface area, $A_s$	2600a	m−1
Biomass yield coefficient, $Y_{x/s}$	1.314c	–
Effective pollutant diffusivity, $D_{\mathrm{Hex}}$	$2.58\times {10}^{-10}$	m2 s−1
Biofilm thickness, $x_n$	387	$\mu$m
Dispersion coefficient, $D^{\prime }$	$1.22\times {10}^{-4}$	m2 s−1
Bed porosity, ${\epsilon }_g$	0.45	–
Biofilm density, $X_v$	9744	g m−3
Maximum specific growth rate, ${\nu }_m$	$5.83\times {10}^{-5}$	h−1
Saturation constant, $K_s$	0.02	g m−3

^a Fitted value; ^c calculated value.

2.4. Development of ANFIS for Modelling Hexane Biofiltration

A total of 71 data points were used. The dataset was divided into a testing set of 15 points (26%) and a training set of 56 points (74%). Training minimizes error between the measured target and ANFIS output. Subtractive clustering (SC) was used to formulate the ANFIS. The model was implemented in MATLAB, and GSA was used to identify the cluster-radius value that minimized mean squared error on the training data. The selected GSA parameters were $\alpha =20$, $G_0=100$, and $N=50$ agents.

In conventional fuzzy inference systems, membership functions and fuzzy-rule structures may depend on trial-and-error selection and expert knowledge [42]. In the proposed ANFIS-based methodology, parameters are tuned automatically during learning, allowing membership functions to represent nonlinear system behaviour.

2.5. Estimation of Error

Mean squared error (MSE) and coefficient of determination ($R^2$) are used for comparison:

$$MSE=\left({\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n}{(Y_{p,i}-Y_{o,i})}^2\right)\times 100,$$

(26)

and

$$R^2=1-{\displaystyle \frac{{\sum }_{i=1}^n{(Y_{p,i}-Y_{o,i})}^2}{{\sum }_{i=1}^n{(Y_{o,i}-{\overline{Y}}_o)}^2}},$$

(27)

where $Y_{p,i}$ and $Y_{o,i}$ denote predicted and observed concentrations, respectively.

3. Results and Discussion

The two modelling approaches were assessed for prediction of dynamic biofilter performance. When bed watering stops, evaporation reduces microorganism activity and outlet contaminant concentration increases. Inlet air with moisture below saturation progressively dries the bed.

In the ANFIS method, day, inlet concentration, and biofiltration flow rate were used as inputs, while outlet concentration was the output. During learning, suitable membership functions were generated and tuned using error-correction training with back propagation; constant parameters of the linear output functions were adapted using recursive least squares.

For inlet concentrations of 0.5 and 1 g/m³ at a flow rate of 0.3 L/min, both the mechanistic and ANFIS-based approaches fitted the experimental data reasonably well. For 0.5 g/m³, the mechanistic and ANFIS-based models reached the inlet concentration at approximately days 64 and 77, respectively. For 1 g/m³, the corresponding values were approximately days 79 and 91.

For inlet concentrations of 1 and 1.5 g/m³ at a flow rate of 0.5 L/min, the mechanistic and ANFIS-based methods reached the inlet concentration at approximately days 63 and 69 for the 1 g/m³ case and days 63 and 67 for the 1.5 g/m³ case.

At a flow rate of 0.7 L/min and inlet concentration of 1 g/m³, the mechanistic and ANFIS-based methods reached the inlet concentration at approximately days 56 and 76, respectively. The mechanistic method did not model performance as effectively in this case, potentially because parameters such as pollutant diffusion coefficient were treated as constant despite changes associated with the higher airflow rate.

Overall, the ANFIS-based method showed more precise predictive capability. It does not provide the same mechanistic description of the system, but it can estimate outputs without requiring solution of the full physical model. The mechanistic approach remains useful for interpreting heat- and mass-transfer mechanisms and for scale-up and optimization.

Table 2. MSE and $R^2$ over test and training data.

Metric	Dataset	GSA-SC-ANFIS	Mechanistic
MSE	Test	0.028%	0.499%
MSE	Training	0.0003%	0.400%
$R^2$	Test	0.998	0.992
$R^2$	Training	0.999	0.989

Table 2. MSE and $R^2$ over test and training data.

Metric	Dataset	GSA-SC-ANFIS	Mechanistic
MSE	Test	0.028%	0.499%
MSE	Training	0.0003%	0.400%
$R^2$	Test	0.998	0.992
$R^2$	Training	0.999	0.989

Both methods predicted outlet concentrations accurately on the test and training data, but the ANFIS-based method achieved the higher predictive accuracy.

4. Conclusions

Modelling of biofiltration is a valuable tool for understanding system behaviour and predicting performance. This paper presented two approaches for predicting variations in outlet concentration of a hexane biofiltration system under unsteady-state conditions with water superabsorbent used to improve water supply. The ANFIS-based approach was compared with a mechanistic model. Both approaches showed strong predictive performance, with higher accuracy obtained by the ANFIS-based method. ANFIS has potential for generalization to other biofiltration systems under suitable conditions, whereas the mechanistic approach provides system description and insight into underlying mechanisms.