Data-Driven Gas Holdup Correlation in Bubble Column Reactors Considering Alcohol Concentration and Carbon Number

Abstract

Due to the complex relationship between various parameters affecting gas holdup in bubble column reactors, a unique correlation for gas holdup does not exist. The available correlations proposed in the literature for gas holdup prediction in aqueous alcohol solutions in bubble columns fail to predict gas holdup over a wide range of conditions. Therefore, based on around 1000 data points from the previous studies, an empirical correlation and a trained model were derived using the dimensionless numbers Reynolds, Froude, Eötvös to Morton ratio, and alcohol carbon number. The predictions were compared to experiments with different water–alcohol mediums at various concentrations to validate the correlation and trained model, and a good agreement was observed. However, the ML model was predicting more accurately, and it was indicated that the Reynolds number had the most significant impact on gas holdup, followed by the Eötvös to Morton ratio.

1. Introduction

Every multiphase process in process engineering requires a system that enables mass, momentum, and efficient energy transfer between continuous and dispersed phases [1,2]. Additionally, having benefits such as easy construction, a low number of mechanical components, adequate heat and mass transfer, high thermal stability, good mixing, low energy requirements, and low operating costs have become a must. Bubble columns include all these pros, which is why this type of reactor has become an attractive choice [3,4], and they are widely used in petrochemistry, biochemistry, wastewater treatment, flotation, etc. [5].

The dimensionless parameter, gas holdup $\left({\epsilon }_G\right)$ plays a critical role in designing and characterizing transport phenomena in a bubble column [6,7,8]. The ${\epsilon }_G$ is the volume fraction of the gas phase occupied by gas bubbles. Similarly, liquid and solid phase holdups are the volume fraction of liquid and solid phases [9]. ${\epsilon }_G$ is an essential parameter to describe bubble columns performance, which is why it is commonly studied [10,11]. The behavior of the ${\epsilon }_G$ depends on many different parameters, such as the physical properties of the fluid [12], the geometry of the column, the design of the gas sparger, and the operating variables, i.e., pressure, gas velocity, and temperature [6,8]. The physical properties of liquids, specifically viscosity and surface tension, impact gas holdup. As liquid viscosity increases, gas holdup decreases due to the promotion of bubble coalescence [13]. The magnitude of the cohesive forces present among the liquid molecules is responsible for the effect of surface tension on the total ${\epsilon }_G$. Increasing surface tension enhances the formation of more gas bubbles within the liquid, which prevents the deformation of the gas bubbles under the operating variables. Previous studies suggest that decreasing the surface tension will increase the ${\epsilon }_G$ because of smaller gas bubbles [14].

The addition of surface-active agents like alcohols reduces the surface tension of the liquid and leads to the production of lower-diameter bubbles, which in turn leads to an increase in the bubble residence time in the column and, consequently, an increase in ${\epsilon }_G$ [15,16]. The ${\epsilon }_G$ also increases due to the increase in the number of carbon atoms of the straight chain length for fatty alcohols [17,18]. Longer straight-chain fatty alcohols have larger molecular sizes and more complex structures, which can affect their interactions with the gas phase in a system. This may enhance gas retention in the system, leading to higher ${\epsilon }_G$ [19]. Therefore, gas holdup decreases in the following order [17]:

n-butanol > n-propanol > i-propanol > ethanol > methanol > water

In general, an increase in intermolecular stretching force enhances the tendency for bubble coalescence, forming larger bubbles. This hinders the transformation of bubbles under the prevailing operating conditions [7]. The addition of active surface agents, such as alcohols, which decrease the surface tension of the liquid, leads to the dominance of non-coalescence properties, resulting in the production of smaller bubbles. This increases the surface area available for mass transfer and the retention time of the bubbles within the column [20,21]. Consequently, the gas holdup is increased. The reduction in surface tension in the presence of alcohols as surfactants is not only a cause of variation in gas holdup but also of other parameters, such as the dynamics of bubbles and the structure of each bubble. These factors can provide a more detailed description of this behavior. In the presence of alcohol, the bubbles become denser and rise slightly due to remaining in a homogeneous flow regime, even at high aeration rates (0.1–0.08 m.s⁻¹) [21,22,23]. Due to the complexity of multiphase flows in bubble columns, it is critical to determine the most appropriate correlation to describe the behavior and performance of the bubble column reactor under various conditions. Although many correlations for gas holdup have been reported in previous studies, finding a unique correlation appears inaccessible [24,25].

Mouza et al. (2005) studied the effects of liquid properties on gas holdup in the air/n-butanol, glycerin, and water systems. They reported that an increase in the viscosity of the liquid leads to a reduction in gas holdup since an increase in viscosity increases the rate of bubble coalescence and prevents the bubbles from breaking. Also, alcohols with lower surface tension retained more gas than higher-surface-tension alcohols and water [3]. Kazakis et al. (2007) investigated the effect of liquid properties on gas holdup in the air/n-butanol, glycerin, water, and kerosene systems and reported that in a homogeneous flow regime, the liquid phase viscosity had little effect on gas holdup and gas holdup for the air/glycerin system was almost the same for air/water system in homogenous flow regime [26]. Up to now, many researchers have proposed many correlations for predicting gas holdup in aqueous alcohol solutions and its relationship with other mentioned parameters, and because of the variety of data, phases, and processes, gas holdup does not have any unique correlation for a wide range of conditions, various equations have been offered for predicting gas holdup in aqueous alcohol solutions in bubble column reactors. In

Table 1. The empirical correlations for gas holdup.

Ref.	Gas/Liquid/Solid	Correlation
[17]	Air/aqueous–alcoholic solutions	${\epsilon }_G={\displaystyle \frac{0.96U_{sg}^{0.58}{C}_N^{0.26}}{1+2.6U_L}}$	(1)
[3]	Air/water, n-butanol, glycerin	${\epsilon }_G=0.001{\left[{Fr}^{0.5}{Ar}^{0.1}{Eo}^{2.2}{\displaystyle \frac{d_s}{D_C}}\right]}^{2/3}$	(2)
[26]	Air/water, n-butanol, glycerin, kerosene	${\epsilon }_G=0.2{\left[{Fr}^{0.8}{Ar}^{0.2}{Eo}^{1.6}{\left({\displaystyle \frac{d_s}{D_C}}\right)}^{0.9}{\left({\displaystyle \frac{d_p}{d_s}}\right)}^{0.03}\right]}^{2/5}$	(3)
[27]	Air/glycerin/xanthan	${\epsilon }_G={2.2\left[{Fr}^{1.07}{Ar}^{0.84}{Eo}^{0.19}{\left({\displaystyle \frac{d_s}{D_C}}\right)}^{1.16}{\left({\displaystyle \frac{d_p}{d_s}}\right)}^{2.86}\right]}^{0.264}$	(4)

, some of these correlations are presented.

These correlations are functional in predicting the gas holdup in their respective systems. However, they cannot be applied to other systems to predict gas holdup due to the impact of medium rheology and properties. This means that accurate predictions of gas holdup in alternative systems cannot be easily transferred to these correlations [25].

With the increasing use of big data, predictive models are being developed through data-fitting algorithms and machine learning. Statistical learning is a broad range of techniques used to comprehend data. These techniques can be categorized as either supervised or unsupervised. Supervised statistical learning involves constructing a statistical model to predict or estimate an output based on one or more inputs. On the other hand, unsupervised statistical learning involves inputs without a supervising output. However, relationships and structure can still be learned from such data [28]. The strength of statistical learning is that it can deliver predictions on a wide range of parameters [29]. Since more complex phenomena exist in the bubble columns. The hydrodynamic complexities of bubble columns result from the interactions between the gas and liquid/slurry phases. These include the bubbles’ dynamics (size, shape, and velocity), the varying flow regimes (homogeneous and heterogeneous), and the deformable gas–liquid interface, which affects mass and heat transfer. Furthermore, the liquid circulation patterns driven by rising bubbles and gas distribution within the liquid (gas holdup) contribute to the system’s complexity. The sparger’s characteristics could control gas bubble coalescence and breakup. Several researchers have documented that the gas sparger has negligible influence on the bubble sizes and gas holdup when the orifice diameters exceed 1–2 mm. Furthermore, from these observations, it can be concluded that the ${\epsilon }_G$ is inversely proportional to the orifice diameter. Furthermore, the formation of small gas bubbles delays the transition from homogeneous to heterogeneous flow as the rate of bubble coalescence decreases. In addition to orifice diameters, the spacing between orifices also affects the gas holdup. Wider spacing makes bubbles smaller and more evenly spaced, which increases gas holdup. Smaller orifices produce smaller bubbles that rise slowly and increase gas holdup. Larger orifices generate larger bubbles that rise quickly and reduce gas holdup. Increasing the number of orifices enhances gas holdup by producing more small bubbles. The arrangement and resulting flow regimes are crucial, with optimized designs balancing these factors to achieve desired mass transfer rates, mixing efficiency, and operational stability [7,30]. Moreover, changing the physicochemical properties of the liquid within the bubble column and its aeration velocity significantly adds to the complexity of the whole system. These factors collectively influence the performance and efficiency of bubble columns in industrial applications [31]. Therefore, careful analysis must be applied. Machine learning (ML) has become a widely adopted method for identifying multiphase flow regimes within reactors [32,33]. ML is a field of study that enables machines to learn independently without explicit programming [34]. ML refers to a computer program’s ability to learn about specific tasks and performance metrics through practice. This occurs when the program’s task-based performance improves over time due to experience gained. By utilizing high-quality data samples and selecting a reasonable learning algorithm, an accurate black-box model can be built by machine learning to solve classification and prediction problems [35]. Although machine learning has many advantages, such as automation, the ability to complete complex tasks quickly, and a reduced number of errors, it also has some drawbacks. For example, machine learning requires extensive, unbiased, high-quality datasets, which can be challenging [36]. Utilization of machine learning models depends on factors such as the size of the dataset and the model’s accuracy [37]. SVM is based on the principle of constructing an optimal hyperplane, also known as a decision boundary or optimal boundary, which maximizes the distance between the nearest samples (support vectors) and the plane, effectively separating the classes. The model identifies the optimal separating hyperplane between classes by focusing on the training cases at the edge of the class distributions, the support vectors, while discarding the other training cases. Moreover, the support vector network (SVN) is a novel learning machine for two-group classification problems. The underlying concept of the SVN can be described as follows: input vectors are non-linearly mapped to a high-dimensional feature space, where a linear decision surface is constructed. The special properties of this decision surface ensure the learning machine’s high generalization ability. The idea behind the SVN was previously implemented for the restricted case where the training data can be separated without errors [38]. Consequently, this approach can achieve high accuracy with limited training data. Furthermore, Gandhi et al. (2008) demonstrated that the objective function in SVR is of a convex quadratic form, which possesses a single minimum [39]. This avoids getting trapped into a poor local minimum, thereby aiding in identifying the global minimum. Therefore, the SVM model can be employed for online predicting gas holdup in bubble column reactors under different operational conditions [39]. The objective function in SVM is of a convex quadratic form, which possesses a single minimum. This contrasts with the usage of ANN and the conventional non-linear least squares regression methodology for establishing the empirical correlations, where there is a high probability of getting trapped into a poor local minimum [37,39].

As gas holdup depends on many parameters, such as the system’s geometry, physio-chemical property of phases, sparger design, and operating conditions, a unique correlation cannot be achieved, and each system has its hydrodynamic characteristics. Even with all the correlations found in the literature, choosing the best correlation for each case is another challenge, as a large amount of scatter data does not allow for the utilization of a single correlation for gas holdup prediction [24]. Therefore, this work aims to develop a more generally applicable correlation for predicting overall gas holdup in bubble columns in aqueous alcoholic solutions and water. For this purpose, both data fitting and ML approaches were applied to a set of data extracted from literature (approximately 1000 measurements) [3,17,21,26,27,40]. The derived correlation and trained model were validated by comparing them to the experimentally collected data in the laboratory.

2. Material and Methods

2.1. Data Collection

Over the years, many scientists have attempted to characterize the hydrodynamics of bubble column reactors based on total gas holdup. Relying on the existing studies, the following databank was collected from the literature, as detailed in

Table 2. Sources and ranges of the gas holdup data extracted from the literature to train the model.

Author	System	${\mathit{U}}_{\mathit{s}\mathit{g}}$ $(\mathbf{m}.{\mathbf{s}}^{-1})$	${\mathit{\rho }}_{\mathit{L}}$ $(\mathbf{K}\mathbf{g}.{\mathbf{m}}^{-3})$	${\mathit{\mu }}_{\mathit{L}}$ $(\mathbf{m}\mathbf{P}\mathbf{a}.\mathbf{s})$	${\mathit{\sigma }}_{\mathit{L}}$ $(\mathbf{m}\mathbf{N}.{\mathbf{m}}^{-1})$	Geometry
[3]	Air/glycerin CN = 3, 33.3% wt	0.00076–0.026	1081	3.5	70	Rectangular
	Air/glycerin 50% wt	0.0007–0.008	1126	8.2	68
	Air/glycerin 66.7% wt	0.0007–0.008	1173	22.5	67
	Air/n-butanol CN = 4, 0.6% wt	0.00068–0.008	994	0.9	60
	Air/n-butanol 1.5% wt	0.0007–0.008	991	0.9	48
[27]	Air/glycerin, CN = 3, 70% wt	0.00004–0.038	1180	20	68	Tubular, (ID = 0.09 m)
[26]	Air/glycerin, 33.3% v.v−1	0.0024–0.0209	1080	3.6	70	Tubular, (ID = 0.09 m)
	Air/glycerin, 68% v.v−1	0.0023–0.0203	1182	23	67
	Air/n-butanol CN = 4, 0.75% v/v	0.00096–0.0255	992	0.9	60
	Air/n-butanol 1.5% v/v	0.000106–0.025	990	0.9	50
	Air/n-propanol CN = 3, 0.5% wt	0.027–0.269	994.8	0.85	66
[17]	Air/ethanol CN = 2, 0.5% wt	0.0274–0.257	997.3	0.83	68	(Tubular), ID = 0.154 m
	Air/methanol CN = 1, 0.5% wt	0.0234–0.265	997	0.83	70
[21]	Air/water	0.0150757–0.196993	997.086	0.8903	71.5	Tubular, (ID = 0.24)
	Air/ethanol 0.05% and 0.1% wt	0.00336134–0.191176	996.7 and 997.158	0.9–0.8917	71.66–71.5
	Air/MEG 0.05−80% wt	0.008–0.194	997.158−1094.801	0.8917−7.9655	71.5−50.2
[40]	Air/water	0.0302438–0.200337	998.2	1.0016	72.75	Tubular, (ID = 0.1)
	Air/pure ethanol	0.0205662–0.199878	789.2	1.28	22.4

. These data were used to fit the correlation and train the data-driven model.

Dimensionless numbers were then generated using physical forces determined from input variables. However, to choose the correct groups of dimensionless groups, the following criteria, which were introduced by Al-Dahan et al. (2002), were used [41]:

• The number of dimensionless groups should be minimized;
• Each group should be closely related to the output parameter (${\epsilon }_G$ in this case);
• The input parameters should have weak cross-correlations with each other;
• The input selected should provide the most accurate output prediction, which can be verified through statistical analysis (for example, squared regression).

Therefore, the dimensionless groups which were selected for this study were: (a) The Reynolds number of gas ($R{e}_g$), (b) The gas Froude number (${Fr}_g$), (c) Eötvös to Morton ratio (${\displaystyle \frac{Eo}{Mo}}$). These dimensionless groups are further explained. The number of carbon atoms of alcohol (C_N) was also considered an effective parameter.

The effect of superficial gas velocity and column diameter has been considered by defining a Froude number:

$$Fr={\displaystyle \frac{U_{sg}^2}{D_{C}g}}$$

(5)

where $D_C$ is the column diameter or, in the case of non-circular cross-sections, the hydraulic diameter and g acceleration of gravity (m.s⁻²).

Moreover, the fluid flow pattern was computed using the $R{e}_g$. Also, the effect of liquid phase properties has been considered using the ${\displaystyle \frac{Eo}{Mo}}$ as defined below [41]:

$$R{e}_g={\displaystyle \frac{D_C{u}_g({\rho }_L-{\rho }_g)}{{\mu }_L}}$$

(6)

$${\displaystyle \frac{Eo}{Mo}}={\displaystyle \frac{{\rho }_L^2{D}_C^2{\sigma }_L^2}{{\mu }_L^4}}$$

(7)

where $R{e}_g$, $Fr$, ${\displaystyle \frac{E_O}{M_O}}$, and $C_N$, are the Reynolds number of gas, Froud, Eötvös to Morton ratio, and number of carbons of alcohol, respectively. Also, the range of the dimensionless numbers for the data extracted from the literature that are used in this work is described in

Table 3. The range of effective parameters on gas holdup.

Author	${\mathit{F}\mathit{r}}_{\mathit{g}}$	${\mathit{R}\mathit{e}}_{\mathit{g}}$	$\frac{\mathit{E}\mathit{o}}{\mathit{M}\mathit{o}}$	${\mathit{C}}_{\mathit{N}}$
[17]	0.000136–0.003108	6.057895–34.40026	4.65 × 108–9.24 × 108	0−4
[26]	1.07 × 10−6–0.00074	0.004621–2.53209	181.5341–43,736,178	0−4
[3]	5 × 10−7–6.53E−05	0.076984–0.882485	240.9992–54,213,224	0−4
[27]	1.74 × 10−5–0.157236	0.002122–0.201577	325.9469	3
[21]	4.8 × 10−6–0.016482	0.116338–52.88619	43,216.35–4.66 × 108	0, 2
[40]	0.000431–0.040912	1.266133–19.94201	1,164,207–52,399,104	0, 2
Present study	5.27 × 10−5–0.000656	1.209073–5.103548	1.09 × 108–2.19 × 108	0−3

.

2.2. Mathematical Modelling

2.2.1. ML Model

The well-known statistical/ML model theory, namely the support vector machine (SVM), was utilized to construct the machine learning model. The goal is to accurately predict the outputs (y_i) corresponding to a new set of inputs (x_i) by fitting a regression function. To obtain the correct fit, the linear function of Equation (8) has been utilized [42]. The small dataset needs an acceptable overall generalization accuracy; therefore, the main dataset was divided into many subsets, trained on some and validated on others. This process in repeated until an acceptable average accuracy of the model is achieved.

$$f\left(x,\omega \right)=(w\varphi \left(x\right)+b)$$

(8)

where $\varphi \left(x\right)$ is the function termed feature and ($w\varphi \left(x\right)$) the dot product in the feature space, $F$, such that $\varphi \left(x\right)\to$ $F$, and $\omega \in F$. Therefore, after algebraic transformation, the model for SVR Equation (8) is converted to a convex optimization problem in its primal form:

$$Max(L\left({\alpha }_{i,j}^{*}\right)=\sum _{i=1}^N{y}_i\left({\alpha }_i-{\alpha }_i^{*}\right)-\epsilon \sum _{i=1}^N\left({\alpha }_i+{\alpha }_i^{*}\right)-{\displaystyle \frac{1}{2}}\sum _{i=1}^N\sum _{j=1}^N({\alpha }_i-{\alpha }_i^{*})({\alpha }_i-{\alpha }_j^{*})\left(\varphi \left(x_i\right)\varphi \left(x_j\right)\right)$$

(9)

As for constraints $C \ge {\alpha }_i$, ${\alpha }_i^{*}\ge 0$ and $\sum _{i=1}^{N_{SV}}\left({\alpha }_i^{*}-{\alpha }_i\right)=0, {\alpha }_i^{*}\in [0,C]$ where the hyperparameter ($C$) Equation (10) is used to balance the flatness of the regression function with the amount of tolerance for deviations larger than the loss function ($\epsilon$). The SVs $x_i$, and the corresponding non-zero Language multipliers ${\alpha }_i$ and ${\alpha }_i^{*}$ give the value of weight vector ($w$), which, by the expanded form of the SVR, is as follows:

$$\begin{gathered} Minimize{\displaystyle \frac{1}{2}}{‖w‖}^2+C\sum _{i=1}^N\left({\xi }_i+{\xi }_i^{*}\right) \\ \mathrm{Which} \mathrm{is} \mathrm{the} \mathrm{subject} \mathrm{to}: \\ {y}_i-\left(w.\varphi \left(x_i\right)+b\right) \le ϵ+{\xi }_i \\ \left(w.\varphi \left(x_i\right)+b\right)-y_i \le ϵ+{\xi }_i^{*} \\ {\xi }_i, {\xi }_i^{*} \ge 0 \end{gathered}$$

(10)

$$w=\sum _{i=1}^{N_{sv}}\left({\alpha }_i-{\alpha }_i^{*}\right)\varphi \left(x_i\right)$$

(11)

$$f\left(x,{\alpha }_i, {\alpha }_i^{*}\right)=\sum _{i=1 }^{N_{sv}}\left({\alpha }_i-{\alpha }_i^{*}\right)\left(\varphi \left(x_i\right)\varphi \left(x_j\right)\right)+b$$

(12)

Mercer’s condition must be satisfied if one tends to utilize kernel functions [43]. Mercer’s condition states that any positive, semi-definite, symmetric kernel function (K) could be expressed as a dot product in high-dimensional space. Several kernel functions are available, including linear, polynomial, and Gaussian radial basis functions (RBF). The RBF is the most commonly used kernel function and is defined as follows:

$$K\left(x_i,x_j\right)=\mathrm{e}\mathrm{x}\mathrm{p}\left({\displaystyle \frac{-{‖x_i-x_j‖}^2}{2{\sigma }^2}}\right)$$

(13)

where $\sigma$ shows the width of the RBF. If further information is needed about this approach, more details can be found from [37].

2.2.2. Data Fitting Model

DataFit (version 9.0) software is used for multivariable regression, and all the models are sorted by their $R^2$ and then displayed. Using these variables, one can reach a polynomial quadratic equation. However, for this study, the conventional multiple regression equation, $Y=a+b_1{X}_1+b_2{X}_2+\dots +b_n{X}_n$ was used, where Y is the dependent variable of the regression, b_n is the slope of the regression, X_n is the nth independent variable of the regression, and a is constant. Based on the generated results, the ones with the highest R² accuracies were chosen for investigation if they can be used as a proper correlation [44].

2.3. Experimental Section

Experiments are carried out in a cylindrical bubble column at room temperature (23 ± 2 °C). The bubble column was made of plexiglass and 0.14 m in internal diameter, 0.005 m in thickness, and 0.4 m in height. The height of the un-gassed liquid was 0.32 m. The gas distributor was a single nozzle with an internal diameter of 6 mm and was located at a distance of +2 cm from the bottom of the reactor. Figure 1 represents a schematic of the mentioned system.

The liquid phases used in this study were water and aqueous methanol, ethanol, and n-propanol solutions with concentrations of 0.25%, 0.5, and 1% v.v⁻¹. The physical properties of these solutions are presented in

Table 4. Liquid phase properties of the aqueous solution used in the present study.

Liquid Phase	$\mathbf{Density} {\mathit{\rho }}_{\mathit{L}}\left(\mathbf{K}\mathbf{g}/{\mathbf{m}}^3\right)$	$\mathbf{Viscosity} {\mathit{\mu }}_{\mathit{L}}(\mathbf{m}\mathbf{P}\mathbf{a}.\mathbf{s})$	$\mathbf{Surface} \mathbf{Tension} {\mathit{\sigma }}_{\mathit{L}}\left(\mathbf{m}\mathbf{N}/\mathbf{m}\right)$
Methanol 0.25% v/v	997.7	0.82	71.2
Methanol 0.5% v/v	997	0.83	70
Methanol 1% v/v	996.4	0.83	67
Ethanol 0.25% v/v	997.7	0.83	71
Ethanol 0.5% v/v	997.3	0.83	70
Ethanol 1% v/v	995	0.84	69.3
n-Propanol 0.25% v/v	995	0.85	68
n-Propanol 0.5% v/v	994.8	0.85	66
n-Propanol 1% v/v	994	0.85	60
Water	997	0.98	72

.

Air was used as the gas phase. The gas holdup was measured using the bed expansion method and determined using Equation (14) [45]:

$${\epsilon }_G={\displaystyle \frac{H_a-H_L}{H_a}}$$

(14)

The superficial gas velocity is defined as follows:

$$U_{sg}={\displaystyle \frac{Q}{A}}$$

(15)

where Q (m³.s⁻¹) is the gas flow rate, and A (m²) is the column cross-sectional area.

The fluctuation range in Table 3 and Table 4 shows articles and research regarding this issue. All the experiments were performed at ambient temperature (25 ± 3 °C) and pressure (~1 atm). The liquid’s height was marked at the beginning of the experiments with high accuracy, and the bubble column was filled every time to that specific point before aerating the system and a few minutes waited to pass before measuring the liquid’s height after the initiation of aeration to obtain the best results. After repeating each experiment three times, an average was taken and considered the final result. The experiment procedure was very similar to the one initiated by Pajoum Shariati et al. (2007) [12].

3. Results and Discussion

3.1. The Effects of Superficial Gas Velocity, Alcohol Concentration, and Number of Carbon Atoms on Gas Holdup

Figure 2 illustrates the results of the experiments and the effects of alcohol concentration and type. Moreover, ${\epsilon }_G$ ascends with the increase in $U_{SG}$ and alcohol concentration, which has also been confirmed by other published studies [3,26,27].

In corresponding $U_{SG}$ and alcohol concentrations, ${\epsilon }_G$. In n-propanol, the solution is more than the methanol solution, the methanol solution is more than the ethanol solution, and this solution is more than water. These results are supported by the literature [17,46]. As confirmed by many researchers, this is because increasing alcohol concentration in an aqueous solution leads to a reduction of surface tension, and hence, gas bubbles become more rigid, and bubble rise velocity decreases. The presence of alcohol in the system reduces the tendency of bubbles to interconnect and leads to the maintenance of a homogeneous flow regime and the production of small bubbles. As a result, an almost linear increase in gas holdup is associated with an increase in air velocity [26].

3.2. Gas Holdup Empirical Correlation

The results and data obtained from previous papers that have studied gas holdup in bubble columns containing alcoholic systems have been summarized in Table 4. Based on these data, the following correlation was developed to predict gas holdup in bubble column reactors with aqueous alcohol solutions and water:

$${\epsilon }_G=0.39\times {\left(Re\right)}^{0.539}·{\left(Fr\right)}^{0.0566}·{\left({\displaystyle \frac{Eo}{Mo}}\right)}^{-0.1077}·{\left(1+C_n\right)}^{0.19}$$

(16)

The R² of Equation (16) was 0.96, meaning that this equation can predict the gas holdup with high accuracy, and its mean standard error was estimated to be 0.0377. It should be noted that the accuracy of this equation is valid when the input data is within the range of data in which the correlation was generated; the relationship observed may not be accurate if being extrapolated. It is expected that the further one is from the known data, the less reliable the prediction becomes. To verify the correlation obtained in Equation (16) and compare it with correlations from previous research (as presented in Table 1), the data gathered from the present work’s experiment, as shown in Figure 2, was used. This dataset was called ‘blind points’, meaning it was new for both Equation (16) and the correlations from previous studies. Figure 3 shows the accuracy of all the correlations used in this research, with an error margin of ±15%.

Figure 3 shows that the correlation [17] developed is the closest match for predicting gas holdup in the present work. However, their correlation cannot predict gas holdup in pure water because it uses $C_N$ instead of ${1+C}_N$. Additionally, their correlation does not account for physical properties, which increases the error. Refs. [3,26] used n-butanol with a $C_N$ of 4 and high concentrations of glycerin in their correlations. In contrast, this work used lower alcohol concentrations and a $C_N$ that did not exceed 3. These differences may account for the inability of their correlations to predict this work’s data accurately.

Ref. [27] correlation could not predict the gas holdup in this work. The reason is that they developed the correlation to predict gas holdup in high concentrations of glycerin and for non-Newtonian fluids. Therefore, the predictions achieved by their correlation were totally out of range, as shown in Figure 3, because the current blind points were extracted from a Newtonian dilute fluid. In more recent studies, Fadili and Essadki (2021) have also studied the flow pattern, gas holdup, and gas–liquid mass transfer correlations in a bubble column [47]. However, reaching excellent results (especially regarding the effect of K_La), they have generated their correlation based on more limited systems (water, ethanol 0.05%, 2-propanol 0.05%, 1-butanol 0.05%, and SDS (10^–3)). Another similar work was performed by Nedeltchev et al. (2024), which reported a novel approach for the flow regime identification and a modified correlation for gas holdup prediction in a bubble column being operated with aqueous solutions of 2-pentanol at ambient conditions. They have developed an accurate correlation for predicting the gas holdup for DW-2 pentanol mixtures; however, had they not limited themselves to only DW-2 pentanol, they could have developed a more general correlation for gas holdup prediction [48].

To investigate the accuracy of Equation (16), almost 1000 data points were gathered from the literature, plus the data that was achieved in the lab were fed into Equation (16), which showed promising accuracy, as shown in Figure 4.

3.3. Machine Learning Results

Figure 5 indicates the model developed by ML with 70% of training points and 30% of testing points collected from previous studies.

Figure 6 shows the ML prediction of the blind points collected experimentally. The blind points were fed into the previously trained model, and the predicted values strongly correlated with the observed values. The ML model was able to predict the experimental blind points ${\epsilon }_G$ with R² = 0.99, which is 0.11 better than its data-fitting counterpart.

4. Conclusions

SVM models for bubble columns have the potential to be a valuable tool in the commercial sector for online monitoring and controlling point gas holdup in many industries, including beverages, pharmaceuticals, fermentation, and many more. Applying machine learning models to predict the behavior of alcoholic solutions in bubble columns can lead to significant enhancements in industrial processes. This is achieved by optimizing parameters such as gas and liquid flow rates and column dimensions, resulting in improved efficiency and yield. Furthermore, these models ensure optimal performance and consistent product quality. Additionally, they facilitate predictive maintenance, reducing downtime and unexpected failures. They also contribute to cost savings by optimizing resource consumption and minimizing waste. In conclusion, ${\epsilon }_G$ is a critical parameter in bubble columns and is heavily affected by the physical properties of the liquid phase. Therefore, adding alcohol to the liquid phase will alter these properties, leading to different ${\epsilon }_G$. This study compared the multivariable regression by Datafit with an ML model. To achieve this, a dataset was gathered from previous studies, and then a unique correlation was created for them using Datafit. Also, utilizing ML, a model was trained for the same data, with 70% of the data used for training the model and 30% used for testing it. The results showed that although ML predicted better than Datafit (99% and 96%, respectively), Datafit was able to yield a correlation based on the dimensionless numbers $R{e}_G$, $F{r}_G$, $Eo/Mo$, and $C_N$. ML also showed that in the mentioned systems, $R{e}_G$ has the most effect on the ${\epsilon }_G$, followed by $Eo/Mo$ and $F{r}_G$, and $C_N$ also had some effects on ${\epsilon }_G$, but not as much as the other parameters. To verify the correlation and the ML model, a set of blind points was gathered for the present work through experiments and were fed into the correlation and the ML model. Both the correlation and the ML model could predict these blind points. However, further investigations are needed for different gas/fluid systems and three-phase bubble columns.