Prediction of multiphase flow pattern inside a 3D bubble column reactor using a combination of CFD and ANFIS

Abstract

This work presents a combination of Computational Fluid Dynamics (CFD) and Adaptive Network-based Fuzzy Inference System (ANFIS) developed for flow characterization inside a cylindrical bubble column reactor. An attempt has been made to predict the liquid flow pattern and gas dynamics for various ring sparger diameters (i.e., 0.07–0.16 m) and bubble column heights. Gas hold-up, Turbulent Kinetic Energy (TKE) and axial liquid velocity are the output parameters predicted by using the ANFIS method with respect to sparger diameter, axial coordination and radial coordination. Various architectures of the ANFIS method were constructed in order to achieve an accurate prediction model of the liquid flow behavior and gas dynamics inside the bubble column. ANFIS approaches were trained and tested by using CFD simulation results. The performance of the ANFIS approaches was examined by comparing the root mean square error and correlation coefficient values of the prediction models. The CFD simulation results are validated with existing experimental and numerical data and mathematical correlations. Both CFD simulation and ANFIS prediction results show that ring sparger diameter significantly changes the liquid flow pattern and gas dynamics, resulting in different amounts of the gas inside the column. Different ANFIS structures were selected for precise estimation of gas hold-up, TKE and axial liquid velocity. Eventually, the mathematical correlations of the proposed ANFIS approaches are presented with correlation coefficients of 0.9717, 0.9917 and 0.9877 for gas hold-up, turbulent kinetic energy and axial liquid velocity prediction models. Hence, the ANFIS approach is able to provide a prediction of the 3D bubble column hydrodynamics in a continuous domain.

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1 Introduction

Bubble column reactors are extensively used in several multiphase reaction processes within industrial applications such as chemical, biochemical and wastewater treatment.^1–7 They are preferred over other reactors due to high energy efficiency, low design costs, and simple structure and operation.^1,3 Furthermore, they have a good performance in phase mixing characteristics and heat and mass transfer (large interfacial areas).^1,3,8,9 They are often cylindrical or rectangular, including gas sparger (gas distributor) at the bottom. The sparger produces bubbles in a continuous liquid phase (stationary or flowing) or a liquid–solid suspension.^1,3,10–15

Design, optimization and manufacturing of these reactors highly depend on the complex dynamics of gas bubble interaction, liquid flow pattern and prevailing gas and liquid regime (i.e., homogeneous or heterogeneous).^{3,11–13,16–20} One of the main challenges in enhancing the design and manufacturing of the bubble column is to properly predict and measure the hydrodynamics properties, while the complex behavior of the gas and liquid movement, including the interaction between bubbles are unavoidable.^{1,3,11,12,15,21,22} Several experimental methods such as Particle Image Velocimetry (PIV), Laser Doppler Anemometry (LDA) and radioactive particle tracking have been developed in recent years to develop an insight into this arguably complex and nonlinear behavior of gas and liquid dynamics, particularly liquid flow pattern and amount of gas inside bubble column reactors.^23–26

Apart from experimental techniques, many CFD approaches and mathematical calculations are available to predict bubble column hydrodynamics.^{3,7,13–15,27–29} Nowadays, high performance computers have enabled the use of detailed mathematical and computational approaches to study the liquid flow pattern and gas dynamics on a feasible time span and space. There are two main CFD approaches i.e., the Eulerian–Eulerian and Eulerian–Lagrangian to model multiphase fluid flow.^{3,6,10,13–15,19,27–30} In the Eulerian–Lagrangian approach (discrete particle model), each bubble is separately tracked inside the bubble column by solving forces acting on the bubbles, while continuum description is considered for the liquid phase.⁶ In this approach, the interaction between bubbles i.e., coalescence, break-up and collisions can be observed. However, this framework is limited for large bubble columns with high number of bubbles due to solving more equations in large domains. On the other hand, Eulerian–Eulerian approach (the two fluid model), considers gas bubbles and liquid in the Eulerian framework as two interpenetrating fluids. Unlike the discrete particle model, the Eulerian framework is an appropriate method to solve the large bubble column with high superficial gas velocity, particularly in industrial bubble column reactors.^{13–15,22,29}

Although several experimental, numerical and mathematical methods have been used to measure and estimate the flow pattern and bubbles dynamics,^{3,5–7,9,31–39} there are some difficulties to completely predict the liquid flow pattern and gas dynamics (bubble coalescence, break-up, velocity, shape, size and gas hold-up) at each point of 3D bubble column reactors when the operation conditions (i.e., superficial gas velocity, column dimensions, gas and liquid properties and sparger parameters), flow regime and operation time change. For instance, measuring the fluid flow parameters inside the 3D bubble column reactor during experiment is extremely expensive and required much measurement equipment. Computation time and computer capability are the major limitations of the computational approaches in numerically simulation of the large bubble column and various operation conditions. Because of these limitations, soft computing methods have been developed to estimate the bubble column hydrodynamics in various conditions that have not been simulated or experimented at every point of the bubble column.³⁷

There are several soft computing techniques (e.g., neural networks, support vector machines, evolutionary algorithms, and adaptive neuro fuzzy inference system) proposed in many studies to estimate phenomena behaviour in the real life applications.^37,40–47 Among these techniques, Adaptive Network-based Fuzzy Inference System (ANFIS) has attracted researchers because of its ability to learn complex relationships and its vast application has been illustrated in numerous studies.^41,48–50 The accuracy of the ANFIS approach can be altered by changing prediction model structure and adapted on the basis of the relationship complexity.^{40,41,51–58} ANFIS method can use either simulation or experimental results as training data to learn the phenomena behavior. An appropriate set of training data is required to successfully train ANFIS model.

Azwadi et al.⁴¹ used CFD results for training ANFIS method to estimate the temperature and flow fields in a 2D lid-driven cavity. They found that the result of ANFIS method is in good agreement with temperature and flow field obtained by CFD simulation. Recently, Pourtousi et al.³⁷ employed this methodology to predict multiphase flow inside a bubble column reactor. They utilized bubble column hydrodynamics data (i.e., liquid velocity components, turbulent kinetic energy and gas hold-up), obtained by CFD (Eulerian method) simulation, at the bubble column bulk region for ANFIS learning process. It was found that the combination of CFD and ANFIS is a robust methodology to predict the bubble column hydrodynamics properties in a continuous domain. They showed that ANFIS method can be a favourable replacement with CFD simulation to predict the complex behaviour of multiphase flow inside the bubble column reactor when the flow regime is homogeneous.

In this study we develop the recent methodology (Pourtousi et al.’s research³⁷) to propose an intelligent approach which is able to model multiphase flow inside the bubble column reactor for various sparger diameters. In addition, an attempt has also been made to improve the overall predictive capabilities of liquid flow pattern and gas hold-up using the combination of CFD and ANFIS methods. A new mathematical correlation is proposed to predict the bubble column hydrodynamics as the ring sparger diameters varied from 0.07 to 0.16 m. The effect of ring sparger diameter on liquid flow velocity, turbulent kinetic energy and gas hold-up is investigated using ANFIS and CFD results. Various ANFIS structures were constructed to realize the most accurate structure for each output. The accuracy of all prediction models was compared by two common error evaluation formulas; root means square error and correlation coefficient. The results of selected ANFIS models were compared to the CFD simulation results to illustrate the capability of the ANFIS approach.

2 Methodologies

2.1 Geometrical structure

In this CFD simulation study, a 3D cylindrical bubble column reactor with 2.6 m height and 0.288 m diameter is used to produce the multiphase flow (bubbling process). The bubble column is filled with stationary water at ambient condition. At the bottom of the column, 20 similar holes (sparger) whose diameter varied from 0.07 to 0.16 m are symmetrically defined a circle with diameter of 0.7 m. The superficial gas velocity for all simulation cases are 0.005 m s⁻¹, resulting in homogeneous flow regime with almost uniform bubble sizes, shapes and velocities.

2.2 Combination of CFD and ANFIS methods

The prediction of the 3D bubble column hydrodynamics is started by CFD simulation of 10 case studies with different sparger diameters. The CFD simulation results are employed to train the ANFIS approaches. Fig. 1 portrays the schematic of prediction procedure of bubble column using the combination of CFD and ANFIS.

Fig. 1 Schematic of prediction procedure of bubble column using the combination of CFD and ANFIS.

2.3 CFD modeling

In the current CFD simulation study, the Euler–Euler approach as incorporated in the commercial ANSYS CFX-13 software is employed to model the multiphase flow inside the bubble column. The Eulerian framework of the gas and liquid flow represents a point volume fraction for the gas and liquid individually. This method is based on the notion of pseudo-continuum. Both the liquid and gas phases (multi-bubbles) are formulated in the Eulerian framework as interpenetrating continua. The formulations of the Eulerian model, selected in the present CFD study, are based on ensemble-averaged mass and momentum transport equations for the gas and liquid separately. The continuity and momentum transfer equations for the Euler–Euler multi-phase framework are represented respectively (solved for the gas and liquid phases individually) as follows:

The mass conservation equation for both liquid and gas is shown as follows:

where ∈_k and u_k indicate the volume fraction and average velocity of phase k respectively.

In the present numerical investigation, the control volume method is used to discretize the conservation equations. There are several solution methods (such as finite difference,⁵⁹ Lattice Boltzmann,^60–63 finite volume method,^{13–15,27,28} etc.) in the CFD to solve the fluid flow problems. The most robust, reliable and the one, on which CFX is based, is called finite volume discretization method. Based on the finite volume discretization method, the momentum transfer formulation for multi-bubbles and liquid phases can be described as:

The right side of the momentum transfer formulation consists of the stress, pressure gradient, gravity and the momentum interfacial exchange between gas bubbles and liquid. In this equation, the stress term of phase k is represented as follows:

where μ indicates the liquid phase effective viscosity, comprising of three terms: molecular viscosity, turbulence viscosity and viscosity based on bubble induced turbulence.

μ_eff,I = μ_L + μ_T,L + μ_BI,L (4)

The effective gas viscosity is formulated based on the effective viscosity of liquid and it can be described as follows:

In the current CFD simulation study, the model of Sato and Sekoguchi is employed for the extra term due to bubble induced turbulence, containing a constant value of C_μ,BI = 0.6 . The viscosity due to the turbulence induced by the gas bubble flow has been described by ref. 64 and 65. They demonstrated a model taking account the turbulence induced by bubble agitation inside the liquid phase. In general, to predict momentum of bubble flow it is crucial to describe the turbulent structure of the continuous liquid phase, which may result in how to describe the contribution of bubble existence to the flow characteristics. Sato and Sekoguchi,⁶⁴ reported that the turbulent shear stress in bubble flow is affected by two terms. Firstly, the inherent liquid turbulence which is independent of relative motion of bubbles in the liquid phase. Secondly, the additional liquid turbulence term, producing by bubble agitation (bubble motion).

μ_BI,L = ρ_LC_μ,BI∈_G d_B|u_G − u_L| (6)

The last term in the momentum transfer equation is the total interfacial force. This term can be described as follows:

M_I,L = −M_I,G = M_D,L + M_TD,L (7)

The total interfacial forces, illustrated above, indicate the drag and turbulent dispersion force when the lift and virtual mass are neglected. The interphase momentum transfer between gas bubble and liquid phase due to drag force is shown as follows:

where the C_D and d_B are the drag coefficient and bubble diameter respectively. In general, the drag coefficient and bubble diameter can be assumed as a constant value due to uniform behaviour of bubble size and shape in homogeneous regime. The drag coefficient and bubble diameter are selected as 0.44 and 4 mm diameter, respectively, based on the literature suggestions,^3,13,14,37 experimental observation and numerical setting of Pfleger and becker's study.¹³

Turbulent dispersion force model is used for current CFD investigations based on literature studies^{3,15,22,37,66} to improve the flow field prediction towards the walls. This model, formulated by Lopez de Bertodano,⁶⁷ is on the basis of the analogy with molecular movement and interaction. It approximates a turbulent diffusion of the bubbles by the liquid eddies and can be described as:

M_TD,L = −M_TD,G = −C_TDρ_Lk∇∈_L (9)

where k and C_TD indicate the liquid TKE and turbulent dispersion coefficient. Various values of turbulent dispersion coefficient have been recommended in the previous studies.^{3,15,22,37,66} In our simulation setting, turbulent dispersion coefficient of 0.3 is used based on the sensitivity study.^3,15,37,66 The sensitivity study has been carried out for turbulent dispersion coefficients from 0.2 to 0.5 and the result shows a small difference in results of flow pattern and gas hold-up, while the value of 0.3 results in marginal improvement of flow pattern results.

In addition to interfacial forces, a proper selection of turbulence model is necessary to appropriately predict the bubble column hydrodynamics.^{3,12–16,22,29,37,38,66} For the disperse bubbly phase a zero equation turbulence model is used. However, the standard k–ε model is applied for the continuous phase which have been used and recommended in prior CFD studies due to obtaining average results, simplicity and low computation time.^{3,12–16,22,27–29,37,66,68} As k–ε is employed for turbulence modelling, the turbulent eddy viscosity is calculated using the standard k–ε turbulence model, where k represents the turbulent kinetic energy and ε its dissipation rate in the liquid phase. k and ε determine the energy in turbulence and the scale of the turbulence, respectively. The turbulent eddy viscosity can be defined as follows:

The turbulent kinetic energy (k) and its energy dissipation rate (ε) are calculated based on the following governing equations:

Being k and ε calculated from their conservation equations. The k–ε model is applied, in this work, with its standard constants values (model parameters): C_μ = 0.09, σ_k = 1, σ_ε = 1, C_ε1 = 1.44, C_ε2 = 1.92. These constants, although not universal, are commonly used in the case of single-phase flow.^19,69 The selection of these values based on recommendation of prior numerical studies.^{13,15,22,37,70} The term G indicates the production of turbulent kinetic energy and can be represented as:

G = τ_L:∇u_L (13)

2.3.1 Grid. For meshing the cylindrical bubble column, the hexahedral grid is used throughout the column which is almost similar with study of Boutet et al.⁷¹ The circular cross section of bubble column is non-uniformly meshed, while the axial length of column is equally divided into 60 grids section. Fig. 2 shows a typical radial and axial grid layout for the 3D bubble column structure, containing 40 500 grids. On average, this grid structure has the orthogonal quality of 0.67, skewness of 0.62 and aspect ratio of 3.1. In addition, two denser meshes (i.e., 59 000 and 82 320 elements) with similar structure are used for grid sensitivity study.

Fig. 2 Grid intensity of the present CFD study containing 40 500 structural elements.

2.3.2 Boundary conditions. Instead of modelling the exact ring sparger for inlet condition, the mass source point is used for each sparger hole, calculating based on superficial gas velocity. At the top surface of the bubble column (outlet), a degassing boundary condition is treated, resulting in no penetration and slip condition for the liquid phase and an outlet for bubbles. In this case, the pressure remains variable on the top of the column, describing the various surface heights at different bubble column locations. On the walls, a no slip and free slip boundary conditions are used for the liquid and gas phases respectively. Considering the free slip boundary condition for the gas phase in Eulerian method describes the interaction between multi-bubbles and solid walls. As bubbles move towards the walls without fraction, they experience no interaction (freely movement) and the direct contacts between multi-bubbles and walls can be considered negligible. In this way, the velocity component parallel to the wall has a finite value, while both the velocity normal to the wall and the wall shear stress are set to zero.

2.3.3 Numerical methods. In order to mathematically discretize the conversation equations, the control volume technique is implemented throughout the 3D cylindrical bubble column. The velocity–pressure connection is achieved using SIMPLEC procedure. The high order differencing schemes of total variation diminishing (TVD) is used to decrease numerical diffusion in current CFD simulation study. The bubbling process is simulated for 1400 s and the results are time averaged over last 1300 s.

2.4 Adaptive-network-based fuzzy inference system (ANFIS)

ANFIS is an inference fuzzy system to accurately predict the behavior of complex and nonlinear systems.^{40,41,47,53,54,72} There are three different types of fuzzy reasoning in which Takagi and Sugeno proposed if-then rules are implemented in ANFIS structure.⁷³ Fig. 3 shows the structure of the employed ANFIS method on predicting the hydrodynamic characteristics in the 3D bubble column. In this paper (three inputs, sparger diameter, radial coordination and axial coordination) are taken to obtain the gas hold up, TKE and axial liquid velocity as output (see Fig. 3). The inputs are divided into various numbers of membership functions (MFs) in first layer. The incoming signals from first layer are multiplied according to AND rule as the node function for the second layer. For instance, the function of the i^th rule is as,

w_i = μ_Ai(D_s) × μ_Bi(x) × μ_Ci(H) (14)

where w_i is out coming signal of second layer's node and μ_Ai, μ_Bi and μ_Ci are incoming signals from implemented MFs on inputs, sparger diameter (D_s), axial coordination (H) and radial coordination (x), to second layer's node.

Fig. 3 Schematic of the ANFIS structure.

In layer three, the relative value of firing strength of each rule is calculated. This value equals to the weight of each layer over the total amount of all rules' firing strengths:

where w̄_i is called normalized firing strengths. The fourth layer applied the function of a consequence if-then rule proposed by Takagi and Sugeno.⁷³ Thus, the node function can be described as:

w̄_if_i = w̄_i(p_iD_s + q_ix + r_iH + s_i)

where p_i, q_i, r_i and s_i are the if-then rules' parameters and called consequent parameters. All incoming signals from layer four are aggregated to obtain the model output represents the estimation result.

A hybrid learning algorithm is utilized to update the parameters in which MFs parameters are updated by gradient descent method and consequent parameters are updated by Least Square Estimate (LSE) method.

2.4.1. Membership function selection. One of the main aspects of this research is the investigation of the best type and number of MFs for all inputs in terms of the Root Mean Square Error (RMSE) and Correlation Coefficient (CC). The equation of RMSE can be defined as:where N is the number of testing sets.

The equation of CC that provides the relationship strength between prediction and CFD simulation results is as:

where y_CFD(i) and y_CFD(m) are the CFD result of individual data and the mean value of CFD results. y_pre(i) and y_pre(m) are the ANFIS prediction result of each data and the mean value of the prediction results.

In this aim, bell-shaped, Gaussian and Sigmoidal MFs, and 27 combinations of number of MFs are employed to compare the estimation output errors. Table 1 portrays the equation of utilized MFs in the ANFIS model and Tables 2 and 3 depict the RMSE and CC values of various employed ANFIS structures.

Table 1 The equations of MFs used in the ANFIS model

Membership Function	Equation
Bell-shaped
Gaussian
Sigmoidal

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Table 2 RMSE of utilized ANFIS structures

Number of MF			Output	RMSE for			Number of MF			Output	RMSE for
Ds	H	x		Bell-shaped MF	Gaussian MF	Sigmoidal MF	Ds	H	x		Bell-shaped MF	Gaussian MF	Sigmoidal MF
2	2	2	εg	0.0059	0.0060	0.0060	3	3	4	εg	0.0041	0.0042	0.0040
			TKE	0.0025	0.0026	0.0030				TKE	0.0016	0.0017	0.0017
			Vy	0.0261	0.0263	0.0278				Vy	0.0131	0.0146	0.0162
2	2	3	εg	0.0049	0.0049	0.0051	3	4	2	εg	0.0056	0.0057	0.0057
			TKE	0.0021	0.0019	0.0022				TKE	0.0017	0.0021	0.0020
			Vy	0.0201	0.0207	0.0226				Vy	0.0189	0.0210	0.0225
2	2	4	εg	0.0046	0.0048	0.0048	3	4	3	εg	0.0041	0.0043	0.0043
			TKE	0.0022	0.0018	0.0022				TKE	0.0015	0.0016	0.0016
			Vy	0.0187	0.0182	0.0213				Vy	0.0131	0.0143	0.0174
2	3	2	εg	0.0059	0.0060	0.0059	3	4	4	εg	0.0039	0.0038	0.0039
			TKE	0.0022	0.0023	0.0024				TKE	0.0015	0.0015	0.0016
			Vy	0.0223	0.0237	0.0239				Vy	0.0111	0.0122	0.0137
2	3	3	εg	0.0047	0.0048	0.0047	4	2	2	εg	0.0057	0.0058	0.0058
			TKE	0.0019	0.0019	0.0018				TKE	0.0024	0.0026	0.0027
			Vy	0.0166	0.0178	0.0194				Vy	0.0255	0.0254	0.0265
2	3	4	εg	0.0046	0.0046	0.0044	4	2	3	εg	0.0045	0.0047	0.0047
			TKE	0.0019	0.0018	0.0018				TKE	0.0023	0.0023	0.0023
			Vy	0.0149	0.0152	0.0172				Vy	0.0192	0.0196	0.0216
2	4	2	εg	0.0058	0.0059	0.0059	4	2	4	εg	0.0044	0.0045	0.0044
			TKE	0.0020	0.0020	0.0026				TKE	0.0022	0.0018	0.0020
			Vy	0.0205	0.0222	0.0217				Vy	0.0168	0.0176	0.0198
2	4	3	εg	0.0044	0.0045	0.0047	4	3	2	εg	0.0057	0.0057	0.0057
			TKE	0.0017	0.0018	0.0018				TKE	0.0017	0.0020	0.0023
			Vy	0.0139	0.0163	0.0177				Vy	0.0218	0.0226	0.0225
2	4	4	εg	0.0043	0.0044	0.0040	4	3	3	εg	0.0042	0.0045	0.0043
			TKE	0.0017	0.0017	0.0018				TKE	0.0016	0.0017	0.0016
			Vy	0.0127	0.0109	0.0152				Vy	0.0153	0.0163	0.0180
3	2	2	εg	0.0057	0.0058	0.0058	4	3	4	εg	0.0037	0.0040	0.0039
			TKE	0.0024	0.0026	0.0029				TKE	0.0015	0.0016	0.0014
			Vy	0.0255	0.0255	0.0280				Vy	0.0124	0.0144	0.0148
3	2	3	εg	0.0045	0.0047	0.0047	4	4	2	εg	0.0055	0.0057	0.0056
			TKE	0.0024	0.0024	0.0020				TKE	0.0016	0.0018	0.0022
			Vy	0.0193	0.0196	0.0223				Vy	0.0204	0.0209	0.0203
3	2	4	εg	0.0044	0.0044	0.0045	4	4	3	εg	0.0040	0.0043	0.0042
			TKE	0.0024	0.0024	0.0019				TKE	0.0013	0.0015	0.0015
			Vy	0.0173	0.0179	0.0203				Vy	0.0130	0.0144	0.0163
3	3	2	εg	0.0057	0.0058	0.0058	4	4	4	εg	0.0037	0.0034	0.0032
			TKE	0.0018	0.0022	0.0021				TKE	0.0013	0.0013	0.0013
			Vy	0.0223	0.0224	0.0232				Vy	0.0094	0.0104	0.0127
3	3	3	εg	0.0043	0.0044	0.0043
			TKE	0.0017	0.0019	0.0017
			Vy	0.0157	0.0164	0.0189

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Table 3 CC of utilized ANFIS structures

Number of MF			Output	CC for			Number of MF			Output	CC for
Ds	H	x		Bell-shaped MF	Gaussian MF	Sigmoidal MF	Ds	H	x		Bell-shaped MF	Gaussian MF	Sigmoidal MF
2	2	2	εg	0.8997	0.8973	0.8984	3	3	4	εg	0.9524	0.9499	0.9552
			TKE	0.9598	0.9414	0.9124				TKE	0.9845	0.9776	0.9838
			Vy	0.9272	0.9265	0.9171				Vy	0.9821	0.9778	0.9728
2	2	3	εg	0.9331	0.9311	0.9282	3	4	2	εg	0.9110	0.9063	0.9088
			TKE	0.9771	0.9499	0.9750				TKE	0.9767	0.9653	0.9639
			Vy	0.9576	0.9548	0.9459				Vy	0.9627	0.9536	0.9465
2	2	4	εg	0.9399	0.9357	0.9342	3	4	3	εg	0.9531	0.9487	0.9486
			TKE	0.9775	0.9505	0.9752				TKE	0.9840	0.9831	0.9830
			Vy	0.9635	0.9655	0.9522				Vy	0.9823	0.9786	0.9682
2	3	2	εg	0.9008	0.8982	0.8993	3	4	4	εg	0.9582	0.9601	0.9572
			TKE	0.9704	0.9673	0.9533				TKE	0.9850	0.9826	0.9852
			Vy	0.9473	0.9406	0.9398				Vy	0.9874	0.9847	0.9804
2	3	3	εg	0.9379	0.9347	0.9373	4	2	2	εg	0.9080	0.9047	0.9043
			TKE	0.9798	0.9800	0.9770				TKE	0.9596	0.9237	0.9309
			Vy	0.9711	0.9669	0.9605				Vy	0.9310	0.9315	0.9252
2	3	4	εg	0.9413	0.9404	0.9460	4	2	3	εg	0.9422	0.9378	0.9386
			TKE	0.9813	0.9798	0.9796				TKE	0.9422	0.9401	0.9339
			Vy	0.9770	0.9759	0.9691				Vy	0.9616	0.9599	0.9507
2	4	2	εg	0.9025	0.8997	0.9016	4	2	4	εg	0.9459	0.9437	0.9455
			TKE	0.9749	0.9680	0.9586				TKE	0.9780	0.9490	0.9441
			Vy	0.9558	0.9480	0.9504				Vy	0.9707	0.9677	0.9589
2	4	3	εg	0.9460	0.9432	0.9386	4	3	2	εg	0.9081	0.9092	0.9064
			TKE	0.9811	0.9803	0.9783				TKE	0.9772	0.9740	0.9424
			Vy	0.9798	0.9724	0.9672				Vy	0.9498	0.9462	0.9465
2	4	4	εg	0.9493	0.9465	0.9556	4	3	3	εg	0.9511	0.9430	0.9491
			TKE	0.9822	0.9814	0.9801				TKE	0.9843	0.9812	0.9818
			Vy	0.9832	0.9877	0.9760				Vy	0.9757	0.9723	0.9660
3	2	2	εg	0.9067	0.9026	0.9037	4	3	4	εg	0.9611	0.9547	0.9573
			TKE	0.9537	0.9217	0.9281				TKE	0.9915	0.9832	0.9900
			Vy	0.9306	0.9308	0.9157				Vy	0.9840	0.9785	0.9774
3	2	3	εg	0.9422	0.9374	0.9372	4	4	2	εg	0.9126	0.9080	0.9096
			TKE	0.9651	0.9499	0.9762				TKE	0.9802	0.9708	0.9475
			Vy	0.9611	0.9596	0.9477				Vy	0.9564	0.9542	0.9567
3	2	4	εg	0.9457	0.9447	0.9440	4	4	3	εg	0.9553	0.9489	0.9508
			TKE	0.9440	0.9463	0.9824				TKE	0.9917	0.9893	0.9870
			Vy	0.9687	0.9664	0.9568				Vy	0.9824	0.9786	0.9722
3	3	2	εg	0.9082	0.9035	0.9056	4	4	4	εg	0.9612	0.9687	0.9717
			TKE	0.9695	0.9574	0.9601				TKE	0.9936	0.9930	0.9915
			Vy	0.9476	0.9470	0.9433				Vy	0.9909	0.9888	0.9834
3	3	3	εg	0.9494	0.9470	0.9473
			TKE	0.9824	0.9791	0.9816
			Vy	0.9742	0.9720	0.9625

---

Gaussian MF has two parameters (c and σ), bell-shaped MF has three parameters (a, b, and c), and Sigmoidal MF has four parameters (a₁, a₂, c₁, and c₂). These parameters are called premise parameters. Numbers of MF are varied from 2 to 4 for each input and ANFIS structures are configured from 2-2-2 configuration, which means 2 MFs for sparger diameter (D_s), axial coordination (H) and radial coordination (x), to 4-4-4 configuration.

As seen in Tables 2 and 3, the most accurate ANFIS structures in terms of RMSE and CC values are 4-4-4 configuration with Sigmoidal MF for gas hold-up, 4-4-4 configuration with bell-shaped MF for TKE and axial liquid velocity. Complexity of the model is another considerable parameter in selecting best ANFIS structure. In this aim, the total number of parameters, comprising premise and consequent parameters, is obtained for each model. This value is varied from 44 to 304 parameters.

Fig. 4, 6 and 8 portray gas hold-up, TKE and axial liquid velocity RMSE values of ANFIS structures in terms of number of parameters (premise and consequent) while Fig. 5, 7 and 9 show gas hold-up, TKE and axial liquid velocity CC values of ANFIS structures in terms of number of parameters. As seen in Fig. 4, RMSE results of gas hold-up illustrate an error reduction when the number of parameters increases. The RMSE results are distributed in two regions; high and low RMSE values. Different ANFIS configurations may have similar number of parameters which represents the complexity of the model.

Fig. 4 Gas hold-up RMSE values of ANFIS methods versus number of parameters.

Fig. 5 Gas hold-up CC of ANFIS methods versus number of parameters.

As instance, ANFIS models with configuration of 2-3-4, 2-4-3, 3-2-4, 3-4-2, 4-2-3, and 4-3-2 have 114, 123, and 132 parameters for Gaussian, bell-shaped, and Sigmoidal MFs, respectively. Type of MF has a slight influence on the gas hold-up prediction accuracy of gas hold-up while the number of parameters has a key role on the model precision. ANFIS models with configurations of 2-3-4, 2-4-3, 4-2-3, and 3-2-4 are much more accurate than models with configurations of 3-4-2 and 4-3-2. This comparison shows that radial coordination (x) needs to be split into more spaces by means of more MFs for x. Hence, ANFIS model with 4-4-4 configuration and Sigmoidal MF is selected as the best model to predict gas hold-up by comparing the RMSE and CC values and the influence of the ANFIS model configuration on the prediction accuracy.

Fig. 6 depicts the relationship between TKE RMSE values of diverse ANFIS models and numbers of parameters. In contrast with error result of gas hold-up prediction, TKE RMSE results are close together and reduction rate is lesser. Thus, a simpler ANFIS structure with less number of parameters can be chosen due to its simplicity and low RMSE value. ANFIS model with configuration of 4-4-4 has 280, 292, and 304 parameters for Gaussian, bell-shaped, and Sigmoidal MFs while ANFIS model with configuration of 4-4-3 for bell-shaped MF has 225 parameters. The RMSE values of these ANFIS models are close together. Therefore, the ANFIS model with lowest number of parameters can be selected as the best model to predict TKE. Fig. 7 illustrates that the ANFIS model with configuration of 4-4-3 and bell-shaped MF has an appropriate performance in terms of CC value and is selected as the best ANFIS model for estimation of TKE.

Fig. 6 TKE RMSE values of ANFIS methods versus number of parameters.

Fig. 7 TKE CC of ANFIS methods versus number of parameters.

Axial liquid velocity is another parameter that has been predicted by ANFIS method. Fig. 8 shows the RMSE values of diverse ANFIS structures versus number of parameters. As can be seen, the error result of utilizing Sigmoidal MF is higher than that of utilizing bell-shaped and Gaussian MFs. The RMSE and CC results of ANFIS model with 2-4-4 configuration and Gaussian MF are close to those of models with higher complexity such as 3-4-4 configuration with bell-shaped MF, 4-4-4 configuration with Gaussian and bell-shaped MFs (see Fig. 8 and 9). Hence, the best model in terms of both simplicity and accuracy is ANFIS model with 2-4-4 configuration and Gaussian MF to predict axial liquid velocity.

Fig. 8 Axial velocity RMSE values of ANFIS methods versus number of parameters.

Fig. 9 Axial velocity CC of ANFIS methods versus number of parameters.

3 Results and discussion

3.1 Validation of CFD method

As the first step, it is important to establish the validity of the CFD model for prediction of the flow pattern and amount of gas inside the bubble column reactor. Therefore, a comparison has been made with the previous experimental and numerical data (i.e., Pfleger and Becker¹³ and Diaz et al.¹⁹) and existing mathematical correlations (i.e., Joshi and Sharma,⁷⁴ Kumar et al.,⁷⁵ and Hughmark⁷⁶). Excellent comparison is obtained between the current CFD estimation and previous studies for overall gas hold-up. For instance, Fig. 10 shows the overall gas hold-up against different superficial gas velocities for the present CFD simulation and previous studies. The overall gas hold-up inside the bubble column reactor linearly rises when the superficial gas velocity increases. This figure shows that the present finding of gas hold-up is in satisfactory agreement with experimental and numerical studies of Pfleger and Becker¹³ (especially experimental finding) and Diaz et al.¹⁹ and mathematical correlations of Joshi and Sharma⁷⁴ and Hughmark,⁷⁶ when the superficial gas velocity alters from 0.0015 m s⁻¹ to 0.01 m s⁻¹. However, the numerical study of Pfleger and Becker¹³ overpredicts the gas hold-up almost for all superficial gas velocities.

Fig. 10 Overall gas hold up for current simulation.

Fig. 11 shows the planer averaged gas hold-up versus bubble column heights for Grids 1 and 3 and Pfleger and Becker's investigation.¹³ In comparison to Pfleger and Becker's numerical study, current numerical results are in excellent agreement with experimental data, especially near the sparger. The Pfleger and Becker¹³ showed an overprediction for gas hold-up at various column heights (particularly near the spargers). The figure shows that, towards the sparger, the numerical study of Pfleger and Becker¹³ could not accurately estimate gas hold-up (over prediction), while the current CFD results are in good agreement with experimental data. In addition, the figure also illustrates that Grid 1 presents better agreement in comparison with Grid 3 in all column heights, particularly near the sparger.

Fig. 11 Overall averaged gas hold up for CFD results, Grids 1, 3 and those from experiment and simulation in Pfleger and Becker at various heights.

3.2 Grid dependency

Three types of grids are used; Grid 1, Grid 2 and Grid 3 (mention in grid section), in order to investigate the effect of the mesh resolution on the results of flow pattern and amount of gas inside the bubble column. The CFD results of axial liquid velocity, based on three types of grids, are benchmarked against that from Pfleger and Becker experimental and numerical study.¹³ Fig. 12 portrays the time averaged axial liquid velocity versus the normalized radial coordinate for Grids 1, 2, and 3 and experimental and numerical results of Pfleger and Becker at 1.6 m column height. According to the figure, Grid 1 shows that axial flow is upward in the central region of column with higher gas hold-up, while a downward counter flow is observed towards the wall region with low gas hold-up. This hold-up gradient creates the density difference for liquid circulation to take place. The point of flow reversal is clearly seen at a radial location of around r/R = 0.6–0.8 m. In contrast, the results of Grids 2 and 3 show the asymmetric liquid profile in which Grids 2 and 3 tend to move towards the right and left walls respectively. The results of Grid 1 also are in good agreement with numerical data of Pfleger and Becker.¹³

Fig. 12 Average axial liquid velocity for CFD, Grids 1–3 and those from experiment and simulation in Pfleger and Becker at height 1.6 m.

In general, the grid dependency test shows a good agreement when the grid size increases (coarse mesh), while the numerical results with finer mesh differ more from experimental data of Pfleger and Becker¹³ (see Fig. 11 and 12). Buwa et al.⁶⁹ and Pfleger and Becker¹³ also reported that, as the grid size decreases, the agreement between numerical results of time averaged flow pattern and experimental finding deteriorates. Furthermore, Bech⁷⁷ showed that transient turbulence models produce new modes of instability in the plume oscillation when the grid size decreases. Based on results of grid dependency illustrated in Fig. 11 and 12, the coarse mesh, containing 40 500 grids, is used for all simulation cases in this study.

3.3 Time step optimization

The influence of the time-step size on the flow pattern and gas hold-up results is also studied. In order to investigate the size of time step, the Courant–Friedrichs–Levy (CFL) number, is used. The CFL order number of one is necessary to resolve the transient bubbling process inside the column. The CFL number can be described as follows:where |v|, Δy and Δt are the magnitude of the velocity vector's vertical component, characteristic dimension of the cell and time step, respectively. In order to study the effect of using different time steps on the accuracy of CFD results, time steps, ranging 0.1–0.01 are tested which results in small differences between results. In this study the time step of 0.1 is used for all CFD study. The selection of the time step is also evaluated by the CFL that the maximum CFL number must be less than one. Several investigations showed that when the CFL is smaller than 1, the numerical method can accurately predict the liquid flow pattern and gas dynamics and further refining of this parameter does not lead to significant changes on the results. However, using CFL larger than 1 results in inaccurate prediction results.^{17,29,38,78–82}

3.4 Prediction evaluations and discussions

In this section, an accuracy evaluation and comparison between CFD and ANFIS results are discussed. Three different ANFIS models are proposed to predict the gas hold-up, TKE and axial liquid velocity individually. The results are divided into three portions: prediction evaluation, surface rules and mathematical models.

ANFIS model has been trained by using CFD simulation results. To verify the trained model, four axial coordinates, 0.867, 1.3, 1.733 and 2.167 m, are employed to examine the prediction result. Fig. 13–15 depict both CFD simulation and ANFIS prediction results for gas hold-up, TKE and axial liquid velocity in the bubble column. These figures illustrate the high capability of ANFIS method to model the characteristics of bubbly flow.

Fig. 13 (a) Gas hold-up predicted and CFD values in sparger diameter of 0.08 m. (b) Gas hold-up predicted and CFD values in sparger diameter of 0.10 m. (c) Gas hold-up predicted and CFD values in sparger diameter of 0.12 m. (d) Gas hold-up predicted and CFD values in sparger diameter of 0.14 m.

Inside bubble column reactors, the amount of gas depends upon the regime operation (i.e., homogeneous/heterogeneous) which turn depends upon column dimensions, pressure, temperature, superficial gas velocity and sparger design parameters. Fig. 13(a)–(d) shows gas hold-up results of CFD simulation and ANFIS prediction model at various bubble column heights (0.867, 1.3, 1.73 and 2.16 m) and sparger diameters (0.08 m (Fig. 13(a)), 0.10 m (Fig. 13(b)), 0.12 m (Fig. 13(c)) and 0.14 m (Fig. 13(d)). According to this figure, both CFD and ANFIS methods show that gas bubbles tend to move towards the bubble column center due to centralized gas movement in the homogeneous regime. An increase in sparger diameter from 0.08 to 0.14 m causes a significant decrease in the magnitude of gas hold-up at the column center. Among all sparger diameters, the sparger diameter 0.14 m produces almost flatter gas hold-up profile inside the bubble column reactor. Both CFD and ANFIS methods show that the gas hold-up profile is most likely uniform near the bubble column outlet. The figures show that ANFIS prediction method can estimate gas hold-up profile almost identical with CFD method towards the column center for various sparger diameters and bubble column heights.

Another important parameter in prediction of bubble column is TKE. When bubbles travel to the column surface, the pressure energy is converted to turbulent kinetic energy. This parameter shows the intensity of turbulence inside the bubble column reactor. Fig. 14(a)–(d) compares CFD simulation and ANFIS prediction of TKE versus radial position in the column. According to the figure, TKE towards the column center is higher than other region, while near the wall region it reaches to zero due to higher fluctuations of turbulent fluid velocities at the column center. As the sparger diameter increases, TKE reduces particularly towards the column center. According to the figures, for all ring sparger diameters, TKE near the sparger region is much higher than bubble column outlet.

Fig. 14 (a) Turbulent kinetic energy predicted and CFD values in sparger diameter of 0.08 m. (b) Turbulent kinetic energy predicted and CFD values in sparger diameter of 0.10 m. (c) Turbulent kinetic energy predicted and CFD values in sparger diameter of 0.12 m. (d) Turbulent kinetic energy predicted and CFD values in sparger diameter of 0.14 m.

Axial liquid velocity is the last parameter that has been estimated by ANFIS approach. The ANFIS prediction result is compared with CFD simulation results in Fig. 15(a)–(d). As the sparger diameter rises, the liquid centerline velocity reduces and results in flatter liquid velocity profile. Additionally, the centerline liquid velocity decreases, as the column height increases. This is attributed to the fact that, near the sparger region, swarm bubbles supply higher energy to the liquid than bulk region, and resulting in higher turbulent dissipation energy. This energy transferring shows the critical bubble column location where bubble plume breaks and bubbles split to smaller bubbles (break-up). In summary, the results show that ANFIS approach beside the CFD method is a capable prediction methodology to estimate the local hydrodynamics parameters at various column locations and operation conditions.

Fig. 15 (a) Axial velocity predicted and CFD values in sparger diameter of 0.08 m. (b) Axial velocity predicted and CFD values in sparger diameter of 0.10 m. (c) Axial velocity predicted and CFD values in sparger diameter of 0.12 m. (d) Axial velocity predicted and CFD values in sparger diameter of 0.14 m.

This section provides information about 3D bubble column hydrodynamics surface plots in order to understand the flow pattern and gas dynamics throughout the bubble column for all column heights and sparger diameters. The predicted gas hold-up contour has been presented for various column heights and sparger diameters in Fig. 16. This figure shows the predicted gas hold-up for sparger diameters of 0.085 m (Fig. 16(a)), 0.115 m (Fig. 16(b)) and 0.145 m (Fig. 16(c)). According to the figure, the gas hold-up profile has a non-uniform behavior towards the sparger, while at the middle and surface region (e.g., 1.3 and 2.5 m) gas phase disperses uniformly. In general, the gas hold-up has a maximum value at the central region while the amount of gas decreases as sparger diameter increases around this region (see Fig. 16). Fig. 17 portrays the predicted axial liquid velocity in different column heights and sparger diameters. As seen, maximum liquid velocity occurs near the sparger region at almost every column height, while the liquid direction changes and results in two recirculation area near walls. Fig. 18 depicts the predicted TKE for various column heights and sparger diameters (0.085 m (Fig. 18(a)), 0.115 m (Fig. 18(b)) and 0.145 m (Fig. 18(c))). According to this figure, TKE towards the sparger is significantly higher than other regions. The figure shows that, with an increase in sparger diameter, TKE decreases.

Fig. 16 Surface rules of selected ANFIS method in gas hold-up at sparger diameters of (a) 0.085 (b) 0.115 and (c) 0.145.

Fig. 17 Surface rules of selected ANFIS method in axial velocity at sparger diameters of (a) 0.085 (b) 0.115 and (c) 0.145.

Fig. 18 Surface rules of selected ANFIS method in turbulent kinetic energy at sparger diameters of (a) 0.085 (b) 0.115 and (c) 0.145.

The ANFIS method can predict the BCR hydrodynamics with different operational conditions in less computational time and provide continuous results. In order to examine the prediction ability, the BCR hydrodynamics are predicted for different column heights. All predicted results are compared with CFD results which are not used in training process.

The ANFIS method is used to predict the results of ϵ_g at column heights (Y mesh coordinate) of 0.43, 0.86, 1.3, 1.73, 2.16 and 2.56 m. The number of prediction data in X and Z mesh coordinate is increased from 705 (CFD data) to 4800 nodes. For ANFIS training and model development, 70% of the actual data, which is the CFD results from benchmark case, at column heights of 0.217, 0.47, 0.73, 0.997, 1.3, 1.56, 1.8, 2.08, 2.3 and 2.6 m is given as an input. In addition, the number of data in X and Z mesh coordinate for the input is reduced to three quarter or to only 490 nodes. Please take note, the prediction is for the column heights that is not given as an input data to ANFIS model and hence the ability of the model is tested. Fig. 19 shows the predicted contour plot of gas hold-up at various column heights (i.e., 0.432, 0.86, 1.3, 1.73, 2.17 and 2.56 m) for ANFIS and CFD method. According to the figure, the ANFIS results are in good agreement with CFD results almost for all column heights. The ANFIS method predicts the circular gas hold-up distribution almost for all column heights, which is similar with CFD results. Both CFD and ANFIS show the higher gas hold-up at the center region of column, ranging 0.02–0.03, while this parameter reaches to zero value near the walls. Towards the sparger region (h = 0.432 m), the sparger has influence on the gas distribution, resulting ring shape gas fraction (with 0.0012 m inner and 0.09 m outer diameter). As the column height increases, this influence diminishes and results in uniform distribution of gas. In comparison to CFD results, ANFIS method slightly over predicts the gas hold-up towards the walls at 2.56 m. This may attribute to the fact that, ANFIS method cannot accurately recognize gas behaviour near the BCR boundary (particularly outlet). In order to enhance this over prediction, different ANFIS setting parameters or data filtering are required.

Fig. 19 Contour of gas hold-up at various column heights for ANFIS and CFD method.

In summary, the combination of ANFIS and CFD prediction framework shows that, in case of a proper ANFIS learning process with CFD results, ANFIS approach can adequately predict bubble column hydrodynamics. In comparison to the CFD simulation, ANFIS approach provides the approximated bubble column hydrodynamics in a continuous domain. When the specific range of ring sparger diameters and column heights are trained in the ANFIS method, it can smartly approximate the flow pattern and gas dynamics within these particular ranges. On the other hand, in CFD simulation, the CFD simulation needs to be implemented for any changes in operation conditions due to production of discrete results. Therefore, providing a complete set of result for various conditions such as different sparger diameter requires computational efforts, resulting in large computational time.

In the last portion of this section, we propose the mathematical models of the liquid flow pattern and gas dynamics that have been estimated by selected ANFIS models. These mathematical correlations show the axial liquid velocity, gas hold-up and TKE at different column heights, radiuses and sparger diameters. The method of gaining mathematical models from ANFIS approach was described in section 2.4. The formula of the relationship between gas hold-up and effective variables, which are sparger diameter, axial coordination and radial coordination, can be written as,

in which

First subscription of μ shows the input number and the second subscription depicts the MF number. The values of MFs' parameters (premise parameters) are shown in Table 4. Subscription of m represents the rule number ranged from 1 to 64. Table 5 shows the values of p_m, q_m, r_m, and s_m (consequent parameters). Each MF has four parameters and linear portion of every rule has four parameters. As seen in Table 4, the values of a₁ and a₂ for all MFs of individual input are approximately equal.

Table 4 Premise parameters of gas hold-up's ANFIS prediction model

Input	MF	a1	c1	a2	c2
Sparger diameter	1	4.444 × 10^2	5.495 × 10^−2	4.444 × 10^2	7.955 × 10^−2
	2	4.444 × 10^2	9.655 × 10^−2	4.444 × 10^2	1.161 × 10^−1
	3	4.444 × 10^2	1.067 × 10^−1	4.444 × 10^2	1.333 × 10^−1
	4	4.444 × 10^2	1.571 × 10^−1	4.444 × 10^2	1.750 × 10^−1
Axial coordination	1	1.538 × 10	−4.332 × 10^−1	1.539 × 10	−6.853 × 10^−2
	2	1.539 × 10	1.943 × 10^−1	1.538 × 10	1.286
	3	1.538 × 10	1.197	1.538 × 10	2.172
	4	1.538 × 10	2.160	1.538 × 10	3.033
Radial coordination	1	1.390 × 10^2	−1.919 × 10^−1	1.390 × 10^2	−4.025 × 10^−2
	2	1.390 × 10^2	−5.078 × 10^−2	1.390 × 10^2	3.719 × 10^−2
	3	1.390 × 10^2	3.270 × 10^−2	1.390 × 10^2	6.373 × 10^−2
	4	1.390 × 10^2	5.365 × 10^−2	1.390 × 10^2	1.918 × 10^−1

---

Table 5 Consequent parameters of gas hold-up's ANFIS prediction model

Rule	p	q	r	s	Rule	p	q	r	s
1	2.06 × 10^−1	−9.83 × 10^−2	−3.13 × 10^−1	−4.81 × 10^−2	33	3.82 × 10^−1	2.60 × 10^−1	8.93 × 10^−1	4.96 × 10^−2
2	−5.34 × 10^−1	1.29	−1.77	4.37 × 10^−2	34	−6.72 × 10^−1	−1.27 × 10^−2	6.73 × 10^−1	8.78 × 10^−2
3	1.55	8.96 × 10^−1	−1.35 × 10	6.23 × 10^−1	35	−1.65	8.76 × 10^−1	6.47	−5.24 × 10^−2
4	−5.06 × 10^−1	−1.17 × 10^−1	−4.00 × 10^−1	9.51 × 10^−2	36	2.24 × 10^−1	2.53 × 10^−2	1.13 × 10^−1	−4.47 × 10^−2
5	−5.94 × 10^−1	3.96 × 10^−3	1.23 × 10^−1	5.75 × 10^−2	37	−2.40 × 10^−2	1.98 × 10^−3	3.64 × 10^−1	4.88 × 10^−2
6	−3.63 × 10^−1	7.86 × 10^−3	3.59 × 10^−1	7.16 × 10^−2	38	−4.16 × 10^−1	−1.05 × 10^−2	8.52 × 10^−2	9.34 × 10^−2
7	2.50	2.43 × 10^−2	6.92 × 10^−1	−1.90 × 10^−1	39	8.82 × 10^−1	6.61 × 10^−3	1.75 × 10^−1	−9.68 × 10^−2
8	−6.96 × 10^−2	8.56 × 10^−3	−1.13 × 10^−1	1.52 × 10^−2	40	−1.48 × 10^−1	5.68 × 10^−3	−2.07 × 10^−1	4.29 × 10^−2
9	−3.33 × 10^−1	3.06 × 10^−3	2.04 × 10^−1	4.73 × 10^−2	41	−9.09 × 10^−2	1.57 × 10^−3	2.63 × 10^−1	4.72 × 10^−2
10	5.32 × 10^−1	−5.28 × 10^−3	1.87 × 10^−1	1.17 × 10^−2	42	−2.72 × 10^−1	−4.60 × 10^−3	3.95 × 10^−2	7.16 × 10^−2
11	1.43	3.94 × 10^−3	3.02 × 10^−1	−9.52 × 10^−2	43	2.12 × 10^−1	−7.98 × 10^−4	7.83 × 10^−2	−5.00 × 10^−3
12	2.25 × 10^−1	4.51 × 10^−3	−2.32 × 10^−1	8.20 × 10^−3	44	2.08 × 10^−2	3.11 × 10^−3	−2.27 × 10^−1	2.46 × 10^−2
13	−2.26 × 10^−1	9.50 × 10^−3	2.00 × 10^−1	2.31 × 10^−2	45	−3.84 × 10^−2	5.75 × 10^−3	2.29 × 10^−1	2.65 × 10^−2
14	2.48 × 10^−1	3.12 × 10^−3	1.37 × 10^−1	1.05 × 10^−2	46	−2.20 × 10^−1	−5.82 × 10^−3	5.25 × 10^−2	6.97 × 10^−2
15	9.01 × 10^−1	−3.44 × 10^−3	2.06 × 10^−1	−3.86 × 10^−2	47	9.88 × 10^−2	3.85 × 10^−3	5.65 × 10^−2	−1.62 × 10^−3
16	1.43 × 10^−1	5.88 × 10^−3	−2.23 × 10^−1	8.75 × 10^−3	48	4.17 × 10^−2	6.62 × 10^−3	−2.09 × 10^−1	1.09 × 10^−2
17	5.02 × 10^−1	1.48 × 10^−2	4.36 × 10^−2	−4.27 × 10^−2	49	−6.29 × 10^−2	2.26 × 10^−1	9.12 × 10^−1	1.15 × 10^−1
18	−1.14	6.72 × 10^−1	−1.65	9.57 × 10^−2	50	−3.45 × 10^−1	8.55 × 10^−2	6.66 × 10^−1	3.97 × 10^−2
19	−2.40	4.16 × 10^−1	−1.50	4.44 × 10^−1	51	−4.37 × 10^−1	−5.38 × 10^−2	−5.82 × 10^−1	5.43 × 10^−2
20	2.88 × 10^−1	1.11 × 10^−2	3.96 × 10^−1	−7.65 × 10^−2	52	−3.72 × 10^−2	1.73 × 10^−1	−9.34 × 10^−1	1.23 × 10^−1
21	−1.99 × 10^−1	5.30 × 10^−3	2.05 × 10^−1	4.28 × 10^−2	53	1.17 × 10^−2	5.71 × 10^−3	3.95 × 10^−1	4.61 × 10^−2
22	−6.42 × 10^−1	−4.30 × 10^−3	1.13 × 10^−1	1.10 × 10^−1	54	−5.77 × 10^−1	−1.10 × 10^−2	−1.41 × 10^−1	1.14 × 10^−1
23	9.91 × 10^−1	6.76 × 10^−3	7.56 × 10^−1	−1.05 × 10^−1	55	−1.13	−3.36 × 10^−3	−2.71 × 10^−2	2.05 × 10^−1
24	6.10 × 10^−3	7.05 × 10^−4	−1.77 × 10^−1	2.19 × 10^−2	56	3.13 × 10^−1	6.17 × 10^−4	−3.96 × 10^−1	3.84 × 10^−3
25	−2.00 × 10^−1	2.86 × 10^−3	2.33 × 10^−1	4.68 × 10^−2	57	6.16 × 10^−2	2.01 × 10^−3	2.95 × 10^−1	3.02 × 10^−2
26	−1.51 × 10^−1	−6.19 × 10^−3	6.05 × 10^−2	6.00 × 10^−2	58	−3.37 × 10^−1	−2.57 × 10^−3	−4.69 × 10^−2	7.39 × 10^−2
27	4.69 × 10^−1	−2.02 × 10^−4	1.72 × 10^−1	−2.64 × 10^−2	59	−2.38 × 10^−1	−3.14 × 10^−3	−1.39 × 10^−1	7.15 × 10^−2
28	8.65 × 10^−2	1.51 × 10^−3	−2.22 × 10^−1	2.04 × 10^−2	60	1.64 × 10^−1	1.06 × 10^−3	−2.82 × 10^−1	1.50 × 10^−2
29	−1.21 × 10^−1	6.74 × 10^−3	2.17 × 10^−1	2.77 × 10^−2	61	6.03 × 10^−2	5.18 × 10^−3	2.35 × 10^−1	1.60 × 10^−2
30	−1.56 × 10^−1	1.43 × 10^−3	5.58 × 10^−2	4.38 × 10^−2	62	−2.50 × 10^−1	−6.52 × 10^−3	−2.41 × 10^−2	7.24 × 10^−2
31	2.01 × 10^−1	5.99 × 10^−3	1.40 × 10^−1	−1.50 × 10^−2	63	−1.22 × 10^−1	−5.83 × 10^−3	−1.27 × 10^−1	5.98 × 10^−2
32	4.97 × 10^−2	4.99 × 10^−3	−2.05 × 10^−1	1.43 × 10^−2	64	1.08 × 10^−1	4.57 × 10^−3	−2.37 × 10^−1	1.05 × 10^−2

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ANFIS model with 4-4-3 configuration and bell-shaped MF is select among 81 ANFIS structures for prediction of turbulent kinetic energy. This model can be represented mathematically as,

in which

Table 6 depicts premise parameters and illustrates that the values of b for all MFs are pretty close to each other while a and c values are different. Forty-eight rules were constructed on the basis of the selected ANFIS structure and the values of p_m, q_m, r_m, and s_m parameters in these rules are portrayed in Table 7.

Table 6 Premise parameters of turbulent kinetic energy's ANFIS prediction model

Input	MF	a	b	c
Sparger diameter	1	3.393 × 10^−3	2.000	8.292 × 10^−2
	2	6.082 × 10^−3	2.001	1.227 × 10^−1
	3	1.200 × 10^−2	2.000	1.804 × 10^−1
	4	3.643 × 10^−3	2.000	1.551 × 10^−1
Axial coordination	1	4.189 × 10^−1	1.996	−2.915 × 10^−2
	2	4.684 × 10^−1	2.001	8.275 × 10^−1
	3	4.452 × 10^−1	2.000	1.731
	4	4.316 × 10^−1	2.000	2.604
Radial coordination	1	1.028 × 10^−1	2.000	−1.210 × 10^−1
	2	1.036 × 10^−1	1.999	9.861 × 10^−3
	3	1.078 × 10^−1	1.999	1.299 × 10^−1

---

Table 7 Consequent parameters of turbulent kinetic energy's ANFIS prediction model

Rule	p	q	r	s	Rule	p	q	r	s
1	2.80 × 10^−1	−1.33 × 10^−2	6.51 × 10^−2	−1.84 × 10^−2	25	−1.19	−1.45 × 10^−1	−7.61 × 10^−1	5.80 × 10^−2
2	−4.17 × 10^−1	1.94 × 10^−1	−1.85 × 10^−1	2.40 × 10^−2	26	2.03	7.35 × 10^−2	−6.50 × 10^−1	−2.77 × 10^−1
3	1.07 × 10^−1	−4.24 × 10^−2	−2.82 × 10^−1	4.06 × 10^−2	27	−8.98 × 10^−1	6.99 × 10^−2	−7.17 × 10^−1	2.47 × 10^−1
4	1.60 × 10^−1	2.09 × 10^−4	6.26 × 10^−1	8.82 × 10^−2	28	−1.18	−2.83 × 10^−3	−1.10 × 10^−1	1.56 × 10^−1
5	−8.39 × 10^−2	2.38 × 10^−3	3.88 × 10^−1	4.28 × 10^−4	29	−2.52	1.10 × 10^−2	−1.51 × 10^−1	3.12 × 10^−1
6	−1.53 × 10^−1	−5.04 × 10^−3	−2.20 × 10^−1	3.61 × 10^−2	30	1.79	−7.02 × 10^−3	−1.02	−8.78 × 10^−2
7	7.07 × 10^−2	−1.14 × 10^−3	5.45 × 10^−1	8.36 × 10^−2	31	−1.05 × 10^−1	2.31 × 10^−3	1.94 × 10^−1	5.19 × 10^−2
8	2.65 × 10^−1	2.77 × 10^−3	2.38 × 10^−1	−3.78 × 10^−2	32	−3.10	1.05 × 10^−2	4.82 × 10^−1	4.13 × 10^−1
9	−2.50 × 10^−1	−4.47 × 10^−3	−2.82 × 10^−1	6.38 × 10^−2	33	1.11	−1.03 × 10^−2	−2.47 × 10^−1	−1.15 × 10^−1
10	−1.36 × 10^−2	−6.68 × 10^−4	3.57 × 10^−1	6.19 × 10^−2	34	1.80 × 10^−1	1.41 × 10^−2	1.36 × 10^−1	−3.22 × 10^−2
11	−2.73 × 10^−2	−1.92 × 10^−3	1.88 × 10^−1	3.26 × 10^−3	35	−6.42 × 10^−1	1.17 × 10^−2	1.98 × 10^−1	6.21 × 10^−2
12	−5.67 × 10^−2	7.82 × 10^−3	−1.84 × 10^−1	7.49 × 10^−3	36	8.92 × 10^−2	−2.20 × 10^−2	−1.53 × 10^−1	5.87 × 10^−2
13	1.78 × 10^−1	6.15 × 10^−3	3.50 × 10^−1	3.15 × 10^−2	37	8.98 × 10^−1	5.27 × 10^−2	6.83 × 10^−1	−3.04 × 10^−2
14	−7.65 × 10^−2	1.20 × 10^−1	1.18 × 10^−1	−1.25 × 10^−2	38	−1.75	5.31 × 10^−2	3.79 × 10^−1	2.31 × 10^−1
15	3.50 × 10^−2	−1.32 × 10^−2	−2.33 × 10^−1	2.75 × 10^−2	39	1.27	−1.36 × 10^−2	−1.27 × 10^−1	−1.84 × 10^−1
16	1.65 × 10^−1	−1.60 × 10^−3	7.61 × 10^−1	1.04 × 10^−1	40	9.84 × 10^−1	−4.15 × 10^−3	9.44 × 10^−1	1.32 × 10^−2
17	−2.55 × 10^−1	3.30 × 10^−3	3.92 × 10^−1	−3.91 × 10^−4	41	1.59	3.29 × 10^−3	5.41 × 10^−1	−2.92 × 10^−1
18	9.51 × 10^−3	−3.17 × 10^−3	−3.84 × 10^−1	4.86 × 10^−2	42	−1.44	−9.89 × 10^−3	−3.76 × 10^−1	2.67 × 10^−1
19	7.87 × 10^−2	−1.26 × 10^−3	5.14 × 10^−1	7.51 × 10^−2	43	9.84 × 10^−2	−3.64 × 10^−3	4.83 × 10^−1	7.11 × 10^−2
20	−1.78 × 10^−1	−2.31 × 10^−3	1.73 × 10^−1	3.81 × 10^−3	44	2.00	9.41 × 10^−4	1.84 × 10^−1	−3.23 × 10^−1
21	−6.91 × 10^−2	−6.62 × 10^−4	−3.70 × 10^−1	6.18 × 10^−2	45	−9.52 × 10^−1	−4.91 × 10^−3	−2.55 × 10^−1	1.87 × 10^−1
22	2.01 × 10^−2	−2.73 × 10^−3	3.56 × 10^−1	6.24 × 10^−2	46	−3.14 × 10^−1	−1.11 × 10^−2	3.23 × 10^−1	1.28 × 10^−1
23	−2.22 × 10^−1	−1.68 × 10^−3	1.36 × 10^−1	1.70 × 10^−2	47	1.22 × 10^−1	9.00 × 10^−3	1.33 × 10^−1	−5.63 × 10^−2
24	3.18 × 10^−2	−1.55 × 10^−3	−2.62 × 10^−1	3.70 × 10^−2	48	6.67 × 10^−2	−5.35 × 10^−3	−1.83 × 10^−1	2.88 × 10^−2

---

Axial liquid velocity is another element predicted by ANFIS approach. The selected ANFIS model to estimate the axial liquid velocity has the highest simplicity among chosen ANFIS models for prediction of gas hold-up and TKE. The equation of this model as follows:

in which

The values of premise and consequent parameters for TKE are shown in Tables 6 and 7. Overall, there are 20 premise parameters and 128 consequent parameters that have been refined by using CFD simulation axial liquid velocity results (Tables 8 and 9). Providing these types of mathematical models assists in improving the knowledge of flow pattern (flow field) and gas dynamics for various operation conditions. In addition, this model can predict much smarter when the number of trained data increases as input parameters (i.e., bubble column dimension in X, Y and Z direction, superficial gas velocity, gas and liquid properties).

Table 8 Premise parameters of axial liquid velocity's ANFIS prediction model

Input	MF	σ	c
Sparger diameter	1	3.107 × 10^−2	8.611 × 10^−2
	2	1.996 × 10^−2	1.526 × 10^−1
Axial coordination	1	2.215 × 10^−1	−1.589 × 10^−1
	2	4.978 × 10^−1	8.053 × 10^−1
	3	4.685 × 10^−1	1.798
	4	9.646 × 10^−2	2.793
Radial coordination	1	5.538 × 10^−3	−1.434 × 10^−1
	2	8.356 × 10^−2	−7.177 × 10^−2
	3	6.069 × 10^−2	4.728 × 10^−2
	4	9.647 × 10^−2	3.347 × 10^−1

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Table 9 Consequent parameters of axial liquid velocity's ANFIS prediction model

Rule	p	q	r	s	Rule	p	q	r	s
1	7.86 × 10^−1	−2.80 × 10^−2	5.72	7.48 × 10^−1	17	4.05	−2.13 × 10^−1	−7.61	−1.70
2	−1.31	−2.58 × 10^−2	−9.81 × 10^−1	−1.61 × 10^−2	18	−6.29	2.47 × 10^−1	1.38 × 10^−1	9.77 × 10^−1
3	5.63 × 10^−1	3.97 × 10^−1	−1.06	2.45 × 10^−2	19	6.33	1.36 × 10^−3	8.17 × 10^−1	−1.04
4	1.44	−1.61	−1.14 × 10	1.80	20	−3.41	−2.88 × 10^−1	−2.54	8.45 × 10^−1
5	5.01 × 10^−2	−1.27 × 10^−2	−5.08	−6.45 × 10^−1	21	1.01	−8.13 × 10^−3	−4.67	−6.91 × 10^−1
6	−3.00 × 10^−2	1.28 × 10^−2	1.71	1.16 × 10^−1	22	−1.41	1.74 × 10^−2	4.10	6.10 × 10^−1
7	1.17 × 10^−1	7.43 × 10^−3	−2.49	2.31 × 10^−1	23	1.46	−2.23 × 10^−3	2.36	−3.01 × 10^−1
8	1.71	−7.45 × 10^−2	2.01 × 10	−2.85	24	2.63	1.01 × 10^−1	7.99 × 10	−1.27 × 10
9	−6.69 × 10^−2	−7.13 × 10^−3	−1.14 × 10	−1.54	25	5.03 × 10^−1	−1.76 × 10^−3	−1.08 × 10	−1.53
10	1.34 × 10^−1	6.32 × 10^−3	1.89	1.33 × 10^−1	26	−6.83 × 10^−1	5.31 × 10^−3	2.87	3.74 × 10^−1
11	−1.31 × 10^−1	−3.75 × 10^−3	−1.18	1.71 × 10^−1	27	5.11 × 10^−1	3.22 × 10^−3	1.15	−9.72 × 10^−2
12	1.13	−2.02 × 10^−2	2.77 × 10	−4.11	28	2.05	4.98 × 10^−2	5.42 × 10	−8.72
13	3.08 × 10^−1	1.54	6.78	−3.15	29	−1.40	5.06 × 10^−1	7.71	−1.30 × 10^−1
14	−1.05 × 10^−1	−1.78	−2.62	4.44	30	1.50	−4.56 × 10^−1	−3.53	6.58 × 10^−1
15	−1.95 × 10^−1	−2.42	2.93	6.00	31	−2.47	1.35 × 10^−1	2.20	−1.69 × 10^−1
16	−4.68 × 10^−1	1.01 × 10	−2.75 × 10	−2.23 × 10	32	−1.12	−6.44 × 10^−1	−3.55 × 10	6.94

---

4. Conclusions

This paper presents the combination of CFD and ANFIS to predict the 3D bubble column hydrodynamics for various ring sparger diameters at different bubble column heights. The Eulerian approach is used to simulate the liquid flow pattern and gas dynamics inside the 3D cylindrical bubble column reactor. The CFD simulation results are validated with existing numerical, experimental and mathematical correlations. After validation process, the CFD simulation results are used to train ANFIS approximation approach. The proper architectures of the ANFIS prediction models, in terms of number and type of MFs, are investigated to find the most accurate prediction model. Based on this model, the mathematical correlations for bubble column hydrodynamics are developed, as the sparger diameter and bubble column height change. The conclusions of this study are as follows:

• Both CFD and ANFIS prediction method show that the axial liquid velocity, turbulent kinetic energy and gas hold-up rise towards the column centre, while these parameters reach to zero value near the column walls for various gas sparger diameters and bubble column heights. The larger ring sparger diameter produces flatter gas hold-up profile in the bubble column cross-section. In addition, for all sparger diameters, the centreline velocity, gas hold-up and turbulent kinetic energy are higher near the sparger region.

• ANFIS approach can predict the bubble column hydrodynamics in a very short time and provide a non-discrete result, while the CFD simulation needs to be employed for any changes in operation condition.

• Evaluation of different ANFIS structures illustrates that the type and number of membership function significantly affect the precision of the prediction model.

• The ANFIS method contains a good ability to predict hydrodynamics parameters of bubble column reactor which are not used in the training process. This will show that, this method can be used as assistance tools together with CFD methodology to predict parameters and minimize computational efforts, and numerical repetition.

Nomenclature

CD	Drag force coefficient (−)
CTD	Turbulent dispersion coefficient (−)
Cε1	Model parameter in turbulent dissipation energy equation (−)
Cε2	Model parameter in turbulent dissipation energy equation (−)
Cμ	Constant in k–ε model (−)
Cμ,BI	Constant in bubble induced turbulence model (−)
dB	Bubble diameter (m)
d0	Sparger hole diameter (m)
D	Diameter of the column (m)
Ds	Sparger diameter (m)
g	Gravitational constant (m s^−2)
G	Generation term (kg m^−1 s^−2)
H	Height (m)
k	Turbulent kinetic energy per unit mass (m^2 s^−2)
MI	Total interfacial force acting between two phases (N m^−3)
MD	Drag force (N m^−3)
P	Pressure (N m^−2)
r	Radial distance (m)
R	Column radius (m)
ReB	Reynolds number (= dBVS/v) (−)
VG	Superficial gas velocity (m s^−1)
Vy	Axial liquid velocity (m s^−1)
TKE	Turbulent kinetic energy
MF	Membership function
RMSE	Root mean square error

Greek symbols

ε	Turbulent energy dissipation rate per unit mass (m^2 s^−3)
∈	Fractional phase hold-up (−)
∈	Average fractional phase hold-up (−)
μ	Molecular viscosity (Pa s)
μBI	Bubble induced viscosity (Pa s)
μeff	Effective viscosity (Pa s)
ρ	Density (kg m^−3)
μT	Turbulent viscosity (Pa s)
σ	Surface tension (N m^−1)
σε	Prandtl number for turbulent energy dissipation rate (−)
σk	Prandtl number for turbulent kinetic energy (−)
τk	Shear stress of phase k (Pa)
εg	Air fraction/Gas hold-up

Subscripts

G	Gas phase
L	Liquid phase

Acknowledgements

This research is financially supported by University of Malaya, Ministry of Higher Education High Impact Research (UM.C/HIR/MOHE/ENG/13) and the University of Malaya research grant (UMRG:RP031C-15AFR).

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