Growth of a Single Bubble Due to Super-Saturation: Comparison of Correlation-Based Modelling with CFD Simulation

Abstract

This paper investigates and assesses the potential applicability of global mass transfer coefficients derived from large-scale experiments to the bubble growth of a single bubble in a super-saturated flow $(\sigma =9)$. Therefore, it presents, for a specific flow velocity $(u=1{\displaystyle \frac{\mathrm{m}}{\mathrm{s}}}, Re=\mathrm{10,678})$, a comparison between correlation-based modelling and 3D Large Eddy Simulation–Volume of Fluid (LES-VOF) Computational Fluid Dynamics (CFD) simulations (minimum cell size of 10 µm, Δt = 10 µs). After the verification of the CFD with pool nucleation bubbles, two cases are regarded: (1) the bubble flowing in the bulk and (2) a bubble on a wall with a crossflow. The correlation-based modelling results in a nearly linear relationship between bubble radius and time; meanwhile, theoretically, the self-similarity rule offers $r~{Bt}^{0.5}$. The Avdeev correlation gives the best agreement with the CFD simulation for a bubble in the flow bulk (case 1), while the laminar approach for calculation of the exposure time of the penetration theory shows good agreement with the CFD simulation for the bubble growth at the wall (case 2). This preliminary study provides the first quantitative validation of global mass transfer coefficient correlations at the single-bubble scale, suggesting that computationally intensive CFD simulations may be omitted for rapid estimations. Future work will extend the analysis to a wider range of flow velocities and bubble diameters to further validate these findings.

1. Introduction

When the concentration of dissolved gas molecules ($C_k$) exceeds the solubility ($C_{ks}$), a state termed super-saturation, the system will strive for equilibrium by forming bubbles. The corresponding level of super-saturation is calculated as $\sigma =C_k/C_{ks}-1$. Examples of bubble formation due to super-saturation are prevalent in both everyday and extreme environments. Common instances include the sparkle observed in carbonated beverages, like beer and sparkling wine [1,2], and the formation of air bubbles in heating systems, where a rise in temperature reduces the solubility of gases [3]. On a more dramatic scale, the explosive eruptions of liquid magma, driven by the sudden outgassing of dissolved gases during rapid pressure drops or temperature increases, exemplify the powerful effects of super-saturation in geological processes [4].

The process of bubble formation has been extensively studied and categorized into five distinct stages: nucleation, growth, detachment, rise, and coalescence, as outlined by Lavenson [5]. This intricate process, central to various natural and industrial phenomena, has been further elucidated in the comprehensive reviews by Jones [6], Liger-Belair [7], and Lubetkin [8], who provide deeper insights into the underlying physics of bubble evolution in super-saturated systems.

In process engineering, the phenomenon of outgassing is of significant consequence, with the velocity of bubble growth representing a crucial factor. The growth of bubbles in a vessel or tank has been a subject of considerable research [7,9,10,11,12]. On the one hand, Scriven [9] derived for a free and motionless spherical bubble in a quiescent liquid a parabolic relationship between the bubble radius ($r$) and time ($t$), based on Fick’s law, indicating that the radial growth rate follows the relation $r~t^{0.5}$. On the other hand, Liger-Belair [7,13] showed that CO₂ bubbles rising in a quiescent liquid exhibit a constant growth rate, resulting in a linear relationship: $r~t$. This linear growth pattern has also been supported by the findings of Shafer and Zare [10], who observed a similar increase in radius over time for rising bubbles.

These studies have significantly advanced our understanding of bubble dynamics under no-flow conditions. While the dynamics of bubble growth in quiescent liquids have been intensely investigated, the influence of a flow on bubble growth presents a new frontier for research. In flowing systems, the growth process diverges from that in a quiescent liquid due to the convective transport of dissolved gas molecules to the bubble surface [14]. A few investigations have been already carried out in this field. Kwak et al. [15] saturated water with methane gas at around 68 bar at room temperature. They subsequently reduced the pressure to ambient pressure and analyzed the process of bubble nucleation in the presence of shear flow. Unfortunately, no correlations for bubble growth were derived from the investigations. Furthermore, Groß et al. [16] derived a Sherwood relation from the simplified advection–diffusion mass transfer equation. However, it is worth noting that this relation only represents a lower limit for the calculation of the diffusion-driven mass flow. In Groß’s model, the size of the surface of the mass transfer remains constant, despite the actual growth of the bubble. Groß [16] therefore suggests that bubble growth in a flow should rather be analyzed with a CFD simulation, which also solves the mass transfer equation. Additionally, a number of correlations were derived based on the results of experiments, with an overview provided in [17]. In all experimental work, the global mass transfer coefficient was determined, representing the mean mass transfer coefficient to all bubbles present in the respective pipe flow. However, it remains unclear to what extent this correlation can be applied to the growth of a single bubble. Furthermore, the correlations often yield significantly different mass transfer coefficients [17].

In summary, the existing studies either treat quiescent liquids or report global mass-transfer coefficients. To date, no work has resolved the time-dependent growth of a single bubble in a super-saturated flow—whether wall-attached or freely convected—and directly compared a full CFD solution of the advection–diffusion field with a correlation-based model.

In this context, the present paper aims to provide a preliminary investigation into this gap in the literature. On the one hand, innovative Computational Fluid Dynamics (CFD) simulations of bubble growth, as demanded by Groß et al. [16], are carried out for a specific flow velocity in a rectangular flow channel. On the other hand, the results of the simulation are compared to those of a simplified correlation-based model. Since this study considers only one flow velocity, it serves as an exploratory assessment of the accuracy and applicability of existing correlations of the global mass transfer coefficient for modelling single-bubble growth. Two cases are considered here: (1) the growth of a bubble that is in motion and flowing with the bulk liquid and (2) the growth of a bubble on the wall with a crossflow. The findings presented here establish a foundation for future studies, in which a broader range of parameters will be explored to further assess the general validity of the approach.

A direct motivation for the present study is the cooling circuit of proton-exchange membrane (PEM) electrolysis stacks. At the anode, molecular oxygen is generated at rates that exceed its equilibrium solubility in water; the circulating coolant therefore becomes locally super-saturated, and bubbles nucleate and grow in situ. The oxygen-rich stream then exits the cell and passes through an external cooling loop. Along this path—up to and including the gas–liquid separator and plate heat exchanger—super-saturation can persist, so bubble evolution continues to affect pressure drop and heat-transfer coefficients. Accurately predicting whether the coolant remains super-saturated when it reaches these downstream components is essential for reliable thermal–hydraulic design. By analyzing single-bubble growth under controlled super-saturation and crossflow conditions, our work supplies the mechanistic understanding and correlation framework needed to forecast bubble behavior in PEM-electrolyzer cooling circuits and, ultimately, to optimize separator sizing and heat-exchanger performance [18,19].

2. Materials and Methods: Modelling of Bubble Growth

2.1. Problem Description and Boundary Conditions

The comparison between the CFD simulation, outlined in Section 2.2, and the correlation-based modelling of bubble growth, detailed in Section 2.3, is conducted for a water flow super-saturated with oxygen through a rectangular flow channel illustrated in Figure 1. The height and length of the flow channel are $a=6 \mathrm{m}\mathrm{m}$ and $b=50 \mathrm{m}\mathrm{m}$, respectively.

The hydraulic diameter, $d_h$, which is necessary for modelling, is defined according to the following equation.

$${\mathit{d}}_{\mathit{h}}={\displaystyle \frac{\mathbf{2}\mathbf{a}\mathit{b}}{\mathbf{a}+\mathit{b}}}=\mathbf{10.7} \mathbf{m}\mathbf{m}$$

(1)

Overall, two distinct cases of bubble growth due to super-saturation of a single bubble are identified: the growth of a bubble flowing with the bulk liquid (Section 2.3.1), where the bubble’s initial radius is set to $r_{flow,0}=0.1 \mathrm{m}\mathrm{m}$, and a bubble on the wall (Section 2.3.2). In this instance, the initial diameter is $r_{site,0}=0.1 \mathrm{m}\mathrm{m}$, and the static contact angle is set to be $\mathsf{\theta }=60°$. For both cases, the configurations of temperature, $T=20 °\mathrm{C}$; the pressure, $p=1 \mathrm{b}\mathrm{a}\mathrm{r}$; and the flow velocity, $u=1{\displaystyle \frac{\mathrm{m}}{\mathrm{s}}}$, are set. The resulting physical properties are summarized in

Table 1. Physical properties for temperature of $T=20 °\mathrm{C}$, pressure of $p=1 \mathrm{b}\mathrm{a}\mathrm{r}$, and velocity of $u=1{\displaystyle \frac{\mathrm{m}}{\mathrm{s}}}$.

Surface Tension	Concentration of Dissolved Oxygen	Solubility	Kinematic Viscosity Water	Density Oxygen	Diffusion Coefficient	Reynolds Number
$\gamma \mathrm{in} {\displaystyle \frac{\mathrm{N}}{\mathrm{m}}}$	$C_k \mathrm{in} {\displaystyle \frac{\mathrm{m}\mathrm{g}}{\mathrm{L}}}$	$C_{k,s} \mathrm{in} {\displaystyle \frac{\mathrm{m}\mathrm{g}}{\mathrm{L}}}$	${\nu }_{H2O}$$\mathrm{in}$${\displaystyle \frac{{\mathrm{m}}^2}{\mathrm{s}}}$	${\rho }_{O2} \mathrm{in}{\displaystyle \frac{\mathrm{k}\mathrm{g}}{{\mathrm{m}}^3}}$	$D \mathrm{in} {\mathrm{m}}^2/\mathrm{s}$	$Re$
$0.0727$	$440$	$44$	$1.003\cdot {10}^{-6}$	$1.5$	$2.192\cdot {10}^{-9}$	$\mathrm{10,678}$

. In the following sections, the modelling of the bubble growth with the CFD simulation and based on the correlations are explained.

2.2. CFD-Based Modelling of Bubble Growth

In this work, CFD simulation is employed to address two primary challenges: accurately resolving the concentration gradient-driven bubble growth and capturing bubble dynamics within a centimeter-scale channel with a measurable crossflow. To tackle these challenges, we use wall-modelled Large Eddy Simulation (WMLES) combined with the Volume of Fluid (VOF) method—an approach that balances physical fidelity and computational cost [20,21]. The parameters used in the Large Eddy Simulation are specified in

Table 2. Parameters used in the Large Eddy Simulation.

Parameter	Default Value	Description
Wall model type	S-omega	The wall model used for WMLES
LES turbulence model	Dynamic Smagorinsky	Subgrid-scale model for LES turbulence
Matching location (y+)	200	Dimensionless wall distance where wall model connects with LES region
Wall model turbulent viscosity	S-omega model	Based on the S-omega equation
Damping function	Enabled	Helps transition from RANS to LES
Blending function	Enabled	Smooth transition between wall model and LES
Turbulent Prandtl number	0.9	Ratio of momentum diffusivity to thermal diffusivity
Wall heat transfer modelling	Enabled	Uses wall function approach for heat transfer if energy equation is solved
Wall roughness	Smooth wall	No roughness (can be adjusted)
Wall model time stepping	Implicit	Temporal integration method for wall model equations
Time averaging for wall model	Enabled	Averages flow quantities used in wall model
Averaging time window	User-defined or 0.1–0.5 s	Duration for time-averaging (if enabled)
SST damping switch	Off	Only used for specific variants

. Furthermore, Figure 2 presents exemplary results to demonstrate the simulation’s capability. It illustrates the sliding motion of a growing bubble on the wall and a flowing bubble in the liquid domain. Although the imposed bulk inlet velocity is 1 m s⁻¹, the flow accelerates locally to about 1.4 m s⁻¹.

LES resolves the large, energy-containing eddies that dominate mixing, momentum transfer, and pressure fluctuations, all of which directly influence bubble growth, while the subgrid-scale (SGS) model represents only the smallest eddies. Compared with fully resolved Direct Numerical Simulation (DNS), LES reduces the mesh size and computational cost by orders of magnitude, making centimeter-scale channel simulations practical. Near the wall, however, LES would normally demand an extremely fine grid; WMLES circumvents this by modelling the unresolvable near-wall eddies, yet it still resolves the outer-layer structures that control mass transfer, allowing for accurate predictions at an affordable cost [22]. VOF, meanwhile, tracks the liquid–gas interface. This WMLES–VOF combination therefore delivers (i) the turbulence scales that set the concentration boundary layer and (ii) the interface motion that governs bubble growth, making it sufficient for the present problem. That said, the approach still requires validation against corresponding experimental data—a task already planned as part of our ongoing work.

In this paper, we present preliminary results that use the WMLES–VOF framework to evaluate some of the most popular global mass-transfer correlations. Demonstrating the capability of bubble-resolved WMLES paves the way for future large-scale reactor models in which detailed local mass-transfer phenomena must be captured efficiently.

The Volume of Fluid (VOF) method [23,24] is a widely used simulation technique to capture the bubble interface in liquid flows [25]. A scalar field, which represents the fraction of the phases in the local cell, characterizes the liquid and gas phases, ranging from 0 to 1. The interface is located at the intermediate scalar value (typically 0.5). The scalar field satisfies the advection equation. To accurately determine the position of the interface, two approaches were developed: VOF interface-reconstruction and VOF interface-sharpening methods [26]. However, due to the basic concept of the VOF method, the mesh cell size impacts the interface position until the mesh is sufficiently fine. To balance computational resources, computing time costs, and simulation accuracy, in this work, the adaptive mesh refinement method is employed in ANSYS FLUENT (version 23 R1) [27]. The adaptive mesh refinement method is primarily focused on the liquid/gas interface region where mass transfer occurs. Together with the interface-sharpening method, the adaptive mesh refinement on the interface allows the simulation to capture the interfaces of up to hundreds of bubbles with acceptable resource requirements (~200 CPU cores in a cluster) and time consumption (~days). However, due to bubble size and position changes, the interface position moves continually. This means the mesh on the interface may need to be refined every time step, and the time step should be as small as possible, which contradicts the objective of a fast computation. After evaluation of the balance, the refinement was set to be performed every second time step, with a time step of $\Delta t=$ 10 µs and targeted refined cell size of $h_{min}=$ 10 µm. This balance allows for an acceptable calculation time but introduces interface position fluctuations. Nonetheless, due to the interface-sharpening and refinement method always strictly following mass conservation, this fluctuation does not significantly impact the accuracy of the simulation.

• Governing equations

To describe the dynamics of Newtonian fluid (incompressible), it is necessary to solve numerically the continuity equation and momentum equation as follows:

$$\mathsf{\nabla }·\mathit{u}=\mathbf{0}$$

(2)

$$\mathit{\rho }{\displaystyle \frac{\mathsf{\partial }\mathit{u}}{\mathsf{\partial }\mathit{t}}}+\left(\mathit{\rho }\mathit{u}·\mathsf{\nabla }\right)\mathit{u}=-\mathsf{\nabla }\mathit{p}+\mathit{\mu }{\mathsf{\nabla }}^{\mathbf{2}}\mathit{u}+\mathit{\rho }\mathit{g}$$

(3)

where $u$ represents the velocity, $\rho$ is the density, $p$ stands for the pressure, $\mu$ is the dynamic viscosity, and $g$ the body (gravity) force. For the instantaneous Navier–Stoke equation with a built-in filtered approach via LES, the equation can be written as follows:

$$\mathsf{\nabla }·\overline{\mathit{u}}=\mathbf{0}$$

(4)

$$\mathit{\rho }{\displaystyle \frac{\mathsf{\partial }\overline{\mathit{u}}}{\mathsf{\partial }\mathit{t}}}+\left(\mathit{\rho }\overline{\mathit{u}}·\mathsf{\nabla }\right)\overline{\mathit{u}}=-\mathsf{\nabla }\overline{\mathit{p}}+\mathit{\mu }{\mathsf{\nabla }}^{\mathbf{2}}\overline{\mathit{u}}+\mathit{\rho }\mathit{g}+{\displaystyle \frac{\mathsf{\partial }\mathit{\tau }}{\mathsf{\partial }\mathit{x}}}$$

(5)

where $\tau$ or ${\tau }_{ij}$ represents the subgrid-scale stress (SGS), defined by ${\tau }_{ij}\equiv \rho \cdot \left(\overline{u_i{u}_j}-\overline{u_i}\overline{u_j}\right)$. Since the Boussinesq hypothesis is employed, it can be further computed from the following:

$${\mathit{\tau }}_{\mathit{i}\mathit{j}}=-{\displaystyle \frac{\mathbf{1}}{\mathbf{3}}}{\mathit{\tau }}_{\mathit{k}\mathit{k}}{\mathit{\delta }}_{\mathit{i}\mathit{j}}-{\mathit{\mu }}_{\mathit{t}}\left({\displaystyle \frac{\mathsf{\partial }\overline{{\mathit{u}}_{\mathit{i}}}}{\mathsf{\partial }{\mathit{x}}_{\mathit{j}}}}+{\displaystyle \frac{\mathsf{\partial }\overline{{\mathit{u}}_{\mathit{j}}}}{\mathsf{\partial }{\mathit{x}}_{\mathit{i}}}}\right)=-{\displaystyle \frac{\mathbf{1}}{\mathbf{3}}}{\mathit{\tau }}_{\mathit{k}\mathit{k}}{\mathit{\delta }}_{\mathit{i}\mathit{j}}-{\mathbf{2}\mathit{\mu }}_{\mathit{t}}\overline{{\mathit{S}}_{\mathit{i}\mathit{j}}}$$

(6)

where ${\delta }_{ij}$ stands for the Kronecker delta, ${\mu }_t$ represents the turbulent viscosity, and $\overline{S_{ij}}$ is the resolved strain-rate tensor. Due to the different definitions of ${\mu }_t$, various SGS models are proposed. The subgrid-scale turbulence model wall-modelled Large Eddy Simulation (WMLES $S–\Omega$) in ANSYS FLUENT was selected, where $S$ represents the strain-rate tensor, and Ω is the rotation-rate tensor. The WMLES $S–\Omega$ model [28] is a hybrid RANS/LES approach that uses the RANS method for the finer near-wall region and transitions to the LES method when the grid spacing is adequate to resolve local scales. Unlike traditional models, WMLES $S–\Omega$ computes the LES portion using $abs(S–\Omega )$ instead of $S$, effectively capturing the transition between laminar and turbulent flows by accounting for zero eddy viscosity in constant shear flow.

Overall, the implementation of the adaptive mesh refinement provides a finer mesh, which is required by the WMLES $S–\Omega$ method. The WMLES and k-ω models have been extensively tested and validated under various conditions. A sensitivity analysis of the employed model will be considered in our next works with corresponding planned experiments. Furthermore, a modified dynamic contact angle model based on Cox–Voinov Model [29] and our findings [30] was implemented here to consider the wall-wetting dynamics.

In the present CFD simulation, three phases are considered: the liquid phase (water), the dissolved-gas phase, and the continuous-gas phase. The dissolved-gas phase is modelled with a local concentration in the liquid phase using the mass transfer equation.

$${\displaystyle \frac{\mathsf{\partial }{\mathit{C}}_{\mathit{k}}}{\mathsf{\partial }\mathit{t}}}+\mathit{u}·\mathsf{\nabla }{\mathit{C}}_{\mathit{k}}-\mathit{D}{\mathsf{\nabla }}^{\mathbf{2}}{\mathit{C}}_{\mathit{k}}={\mathit{S}}_{{\mathit{C}}_{\mathit{k}}}$$

(7)

where $C_k$ represents the (weight) concentration; $u$ is the velocity; and $D$ is the diffusion coefficient of dissolved gas, $k$. Typically, in the concentration driving bubble growth, the phase change rate of species from dissolved-gas phase to the continuous-gas phase is described by the local mass flux through the bubble surface by Fick’s first law:

$$\mathit{J}=-\mathit{D}\mathsf{\nabla }{\mathit{C}}_{\mathit{k}}$$

(8)

However, in the computational fluid dynamics, $S_{C_k}$ from Equation (7) is a volumetric value for local discretized cells. It needs an additional method to transfer the local mass flux, $J$, to a volumetric source term. By applying a mesh refinement method in our simulation, a very fine mesh is achieved at the liquid/gas interface region, where mass transfer occurs. Furthermore, the bubble surface is assigned a local pressure-dependent saturation concentration, $C_{k,s}$. However, in the VOF method, the interface is represented not as a sharp line but as several layers of cells with a gas fraction, ${\alpha }_g$, ranging from 0 to 1. The transition layers of the bubble interface are divided into two regions: gas region (0.5–1) and liquid region (0–0.5). Two steps are executed to maintain mass conservation. First, it needs a mass projection to the gas region to balance the mass flux due to the forced saturation concentration on the interface. This is achieved by calculating the source term, $S_{C_k}$, by applying a central scheme, which balances the efficiency and accuracy:

$${\mathit{S}}_{{\mathit{C}}_{\mathit{k}}}^{\mathit{t}}=\mathbf{\Delta }\mathit{V}\left({\displaystyle \frac{\left({\mathit{C}}_{\mathit{k}}^{\mathit{i}-\mathbf{1}}-{\mathit{C}}_{\mathit{k},\mathit{s}}^{\mathit{i}-\mathbf{1}}\right)}{\mathbf{2}\mathbf{\Delta }\mathit{t}}}+{\displaystyle \frac{\left({\mathit{C}}_{\mathit{k}}^{\mathit{i}+\mathbf{1}}-{\mathit{C}}_{\mathit{k},\mathit{s}}^{\mathit{i}+\mathbf{1}}\right)}{\mathbf{2}\mathbf{\Delta }\mathit{t}}}\right)$$

(9)

where $\Delta V$ represents the cell volume, the superscript $i$ is the time step, $i-1$ means the previous time step, and $\Delta \mathrm{t}$ is the time step length. Moreover, $J$ in Equations (2)–(8) is resolved by the concentration gradient in the liquid region at time step $i-1$.

• Model verification

To verify the CFD model, we considered the limiting case of zero crossflow, for which the analytical Epstein–Plesset equation [31] offers an exact benchmark. Simulations were run on three grids with minimum cell sizes of $h_{min}=0.1 \mathrm{\mu }\mathrm{m}, 1 \mathrm{\mu }\mathrm{m}, \mathrm{a}\mathrm{n}\mathrm{d} 10 \mathrm{\mu }\mathrm{m}$ at two super-saturation levels, σ = 9 and σ = 99. As Figure 3 demonstrates, the numerically predicted bubble-growth curves match with the Epstein–Plesset solution, confirming the model’s accuracy in the absence of crossflow. Because no visible differences arise between the three meshes—and finer grids demand prohibitively smaller time steps to satisfy the Courant stability criterion—we adopt the coarsest grid ($h_{min}=10 \mathrm{\mu }\mathrm{m}$) and the time step $\Delta t=10 \mathrm{\mu }\mathrm{s}$ for the remainder of the study to keep computational costs low.

2.3. Correlation-Based Modelling of Bubble Growth

2.3.1. Bubble Growth in a Flow

The growth of a bubble flowing with the bulk liquid can be described with a simplified correlation-based model. The governing dimensionless numbers for forced-convection mass transfer are the Reynolds number, the Schmidt number, and the Sherwood number. The following steps are applied iteratively for each time step, $i$, with the corresponding bubble radius, $r_i$, whereby the difference between each time step is set to be $\Delta t={10}^{-4} \mathrm{s}$.

(1)

The pressure inside the bubble is calculated by applying the Laplace equation [7]. Afterwards, the density of the gas inside the bubble, ${\rho }_{bubbl{e}_i}$, is derived.

$${\mathit{p}}_{\mathit{b}\mathit{u}\mathit{b}\mathit{b}\mathit{l}{\mathit{e}}_{\mathit{i}}}=\mathit{p}+\mathbf{2}\cdot {\displaystyle \frac{\mathit{\gamma }}{{\mathit{r}}_{\mathit{i}}}}$$

(10)

(2)

Calculation of the bubble surface area, $A_{bubbl{e}_i}$, and the bubble volume, $V_{bubbl{e}_i}$.

$$\begin{gathered} {\mathit{A}}_{\mathit{b}\mathit{u}\mathit{b}\mathit{b}\mathit{l}{\mathit{e}}_{\mathit{i}}}=\mathbf{4}\cdot \mathit{\pi }\cdot {\mathit{r}}_{\mathit{i}}^{\mathbf{2}} \\ {\mathit{V}}_{\mathit{b}\mathit{u}\mathit{b}\mathit{b}\mathit{l}{\mathit{e}}_{\mathit{i}}}={\displaystyle \frac{\mathbf{4}}{\mathbf{3}}}\cdot \mathit{\pi }\cdot {\mathit{r}}_{\mathit{i}}^{\mathbf{3}} \end{gathered}$$

(11)

(3)

Determination of the mass, $m_{bubbl{e}_i}$, of the bubble.

$${\mathit{m}}_{\mathit{b}\mathit{u}\mathit{b}\mathit{b}\mathit{l}{\mathit{e}}_{\mathit{i}}}={\mathit{V}}_{\mathit{b}\mathit{u}\mathit{b}\mathit{b}\mathit{l}{\mathit{e}}_{\mathit{i}}}\cdot {\mathit{\rho }}_{\mathit{b}\mathit{u}\mathit{b}\mathit{b}\mathit{l}{\mathit{e}}_{\mathit{i}}}$$

(12)

(4)

The mass flow, ${\dot{m}}_{bubbl{e}_i}$, of dissolved oxygen molecules from the liquid bulk to the bubble can be described using a classical transport approach.

$${\dot{\mathit{m}}}_{\mathit{b}\mathit{u}\mathit{b}\mathit{b}\mathit{l}{\mathit{e}}_{\mathit{i}}}={\mathit{A}}_{\mathit{b}\mathit{u}\mathit{b}\mathit{b}\mathit{l}\mathit{e}}\cdot \mathit{k}\cdot \left({\mathit{c}}_{\mathit{\infty }}-{\mathit{c}}_{\mathit{n}\mathit{s}}\right) \mathit{w}\mathit{i}\mathit{t}\mathit{h} \mathit{k}=\mathit{f}\left({\mathit{d}}_{\mathit{h}},\mathit{R}\mathit{e}\right) \mathit{a}\mathit{n}\mathit{d} \mathit{R}\mathit{e}={\displaystyle \frac{\mathit{u}\cdot {\mathit{d}}_{\mathit{h}}}{{\mathit{\nu }}_{\mathit{H}\mathbf{2}\mathit{O}}}}$$

(13)

(5)

The mass transfer coefficient, $k$, in Equation (13) is calculated using the penetration theory. In a previous publication [17,32], it was summarized that the small eddy approach is best suited for calculating the required exposure time for bubble growth in the flow. For this reason, various correlations based on the small eddy approach for the exposure time of the penetration theory are summarized in Figure 4. In each case, the authors adjusted the coefficients of the fundamental equation in such a way that the equation most closely aligned with their experimental data.

(6)

The mass flow, ${\dot{m}}_{bubbl{e}_i}$, of dissolved gas molecules through the bubble surface leads to an increase of the mass of the gas inside the bubble. This increase results in a new mass, $m_{bubbl{e}_{i+1}}$, after the time interval, $\Delta t$.

$${\mathit{m}}_{\mathit{b}\mathit{u}\mathit{b}\mathit{b}\mathit{l}{\mathit{e}}_{\mathit{i}+\mathbf{1}}}={\mathit{m}}_{\mathit{b}\mathit{u}\mathit{b}\mathit{b}\mathit{l}{\mathit{e}}_{\mathit{i}}}+{\dot{\mathit{m}}}_{\mathit{b}\mathit{u}\mathit{b}\mathit{b}\mathit{l}{\mathit{e}}_{\mathit{i}}}\cdot \mathbf{\Delta }\mathit{t}$$

(14)

(7)

Derivation of the radius, $r_{i+1}$, of the next time step, which corresponds to a spherical bubble with the mass $m_{bubbl{e}_{i+1}}$.

$${\mathit{r}}_{\mathit{i}+\mathbf{1}}={\left({\displaystyle \frac{{\mathit{m}}_{\mathit{b}\mathit{u}\mathit{b}\mathit{b}\mathit{l}{\mathit{e}}_{\mathit{i}+\mathbf{1}}}}{{\mathit{\rho }}_{\mathit{b}\mathit{u}\mathit{b}\mathit{b}\mathit{l}{\mathit{e}}_{\mathit{i}+\mathbf{1}}}}}\cdot {\displaystyle \frac{\mathbf{3}}{\mathbf{4}\cdot \mathit{\pi }}}\right)}^{\mathbf{1}/\mathbf{3}}$$

(15)

Steps (1) to (7) are repeated for each radius, $r_i$, in order to derive the time-dependent course of the radius. The results are compared with the CFD simulation in Section 3.1. Before discussing the outcomes, the next section first highlights the adjustments made to model bubble growth at the nucleation site.

2.3.2. Bubble Growth at the Wall

In contrast to Section 2.3.1, a bubble on the channel wall is now considered. The super-saturated crossflow leads to the bubble growth. To model this, the velocity profile, $u\left(y\right)$, is required. For this purpose, the single-phase flow expression proposed by Reichardt for the boundary layer of a turbulent pipe flow, initially put forth by Zeng et al. [22] in the context of bubble detachment, is employed. The values of the constants are $\kappa =0.4, \chi =11$, und $c=7.4$ [39].

$$\mathit{u}\left(\mathit{y}\right)={\mathit{u}}^{\ast }\cdot \left({\displaystyle \frac{\mathbf{1}}{\mathit{\kappa }}}\mathit{ln}\left(\mathbf{1}+\mathit{\kappa }{\displaystyle \frac{\mathit{y}\cdot {\mathit{u}}^{\ast }}{\mathit{\nu }}}\right)+\mathit{c}\left(\mathbf{1}-\mathit{exp}\left(-{\displaystyle \frac{\mathit{y}\cdot {\mathit{u}}^{\ast }}{\mathit{\chi }\cdot \mathit{\nu }}}\right)-{\displaystyle \frac{\mathit{y}\cdot {\mathit{u}}^{\ast }}{\mathit{\chi }\cdot \mathit{\nu }}}\mathit{exp}\left(-\mathbf{0.33}{\displaystyle \frac{\mathit{y}\cdot {\mathit{u}}^{\ast }}{\mathit{\nu }}}\right)\right)\right)$$

(16)

The applied friction velocity, $u^{\ast }$, is derived from the following equations, where the Fanning friction coefficient, $f$, is calculated according to Filonenko’s relationship for smooth pipes [40].

$$\begin{gathered} {\mathit{u}}^{\ast }=\sqrt{{\displaystyle \frac{{\mathit{\tau }}_{\mathit{W}}}{\mathit{\rho }}}}=\mathit{u}\cdot \sqrt{{\displaystyle \frac{\mathit{f}}{\mathbf{2}}}} \\ \mathit{f}={\displaystyle \frac{\mathbf{1}}{{\left(\mathbf{1.819}\cdot \mathit{log}\left(\mathit{R}\mathit{e}\right)-\mathbf{1.64}\right)}^{\mathbf{2}}\cdot \mathbf{4}}} \end{gathered}$$

(17)

The same procedure as outlined for the bubble growth in the flow in Section 2.3.1 is applied. Besides this, three adaptations are made. Firstly, for the sake of simplicity, the velocity profile shown in Equations (2)–(16) is reduced to an average velocity. This is achieved by applying the approach of Al-Hayes and Winterton [41], who determined the fluid velocity at distance $y_i$ from the wall.

$${\mathit{y}}_{\mathit{i}}={\mathit{r}}_{\mathit{i}}\cdot {\displaystyle \frac{\mathbf{1}+\mathit{c}\mathit{o}\mathit{s}\mathbf{\theta }}{\mathbf{2}}}$$

(18)

As illustrated in Figure 5, the position $y_i$ corresponds to the widest part of the bubble at the nucleation site. It is therefore used as a characteristic point to determine the average velocity of the water flowing over the bubble. A growing bubble results in a larger distance, $y_i$, from the wall and, accordingly, a larger average flow velocity. Secondly, a part of the bubble is covered by the wall. Consequently, the calculation of the bubble surface area in step 2 (Equation (11)) is corrected by a factor to represent solely the area that is actually in contact with the liquid.

$${\mathit{A}}_{\mathit{b}\mathit{u}\mathit{b}\mathit{b}\mathit{l}{\mathit{e}}_{\mathit{i}}}=\mathbf{4}\cdot \mathit{\pi }\cdot {\mathit{r}}_{\mathit{i}}^{\mathbf{2}}\cdot {\displaystyle \frac{\mathbf{1}+\mathit{c}\mathit{o}\mathit{s}\mathbf{\theta }}{\mathbf{2}}}$$

(19)

Thirdly, the mass transfer coefficient is calculated in accordance with the penetration theory. In contrast to the bubble that flows with the bulk liquid, the exposure time is calculated for the bubble at the nucleation site according to the laminar approach [42,43]. Hereby, the average velocity, $\overline{u\left(y_i\right)}$, at the position $y_i$ is used as the relative velocity between the bubble surface and the water flow.

$${\mathit{k}}_{\mathit{i}}={\displaystyle \frac{\mathbf{2}}{\sqrt{\mathit{\pi }}}}\sqrt{{\displaystyle \frac{\mathit{Ɗ}\cdot \overline{\mathit{u}\left({\mathit{y}}_{\mathit{i}}\right)}}{\mathbf{2}\cdot {\mathit{r}}_{\mathit{i}}}}}$$

(20)

In light of the aforementioned adjustments, steps (1) to (7) outlined in Section 2.3.1 are also applicable to each radius, $r_i$, of the bubble at the nucleation site, allowing for the derivation of the time-dependent course of the radius. The findings will be compared with those of the CFD simulation in the subsequent section.

3. Results and Discussion

3.1. Bubble Growth in a Flow

The outcomes of the correlation-based modelling and the CFD simulation for the bubble growth in the flow are illustrated in Figure 6. The correlation-based modelling indicates a linear dependency between the bubble radius and time, regardless of the correlation used for the mass flow coefficient. This linearity can be explained by the relationship between the change in the mass of the bubble, $\Delta m_{bubble}$, and the radius of the bubble. Based on Equation (14), the difference of the mass of the bubble can be derived as follows:

$$\mathbf{\Delta }{\mathit{m}}_{\mathit{b}\mathit{u}\mathit{b}\mathit{b}\mathit{l}\mathit{e}}={\dot{\mathit{m}}}_{\mathit{b}\mathit{u}\mathit{b}\mathit{b}\mathit{l}\mathit{e}}\cdot \mathbf{\Delta }\mathit{t}$$

(21)

As the bubble’s volume increases cubically with the radius, the mass also increases to the third power of the radius, assuming constant density. This also applies consequently to the difference in mass: $\Delta m_{bubble}~r^3$. However, according to Equation (13), the mass flow, ${\dot{m}}_{bubble}$, to the bubble is governed by the concentration difference, $\left(c_{\mathrm{\infty }}-c_{ns}\right)$; the mass transfer coefficient, $k$; and the bubble surface area, $A_{bubble}$. Since the concentration difference remains constant and the mass transfer coefficient is independent of the radius—being influenced instead only by the turbulence of the flow (see Figure 4)—the mass flow depends only on the bubble surface area, which increases quadratically with the radius. This squared relationship between mass flow and radius (${\dot{m}}_{bubbl\mathrm{e}}~r^2$), combined with the cubic relationship between bubble mass and radius, ultimately leads to a linear relationship between the radius and time. The larger the mass flow coefficient from the correlation, the greater the slope of the curve.

The various correlations yield differing mass flow coefficients, which in turn result in different courses of radius for the correlation-based modelling. The underlying cause of this discrepancy can be attributed to the diverse methodologies employed in deriving the correlations, which exhibit significant divergence from one another. Ge and Fsadni [3,35] employed two high-speed cameras, positioned at a distance of 2.2 m apart, to quantify the shrinking of bubbles in under-saturated water following its departure from the boiler. However, the authors only considered a very small degree of under-saturation, namely 0.89, which resulted in a very small driving concentration difference. Kress and Keyes [37] investigated the oxygen mass transfer to artificially created helium bubbles. They observed that the Sherwood number varied linearly with bubble diameter over the range of 0.25 mm to 1.27 mm. Consequently, the factor $d_{bubble}/d_h$ was introduced. However, there are strong indications that the correlation is only applicable within the aforementioned range of the bubble diameter [37]. Given that the bubble under consideration is smaller, the calculated mass transfer coefficient is found to be relatively low, resulting in a slow bubble growth. Furthermore, although Kawase’s correlation [36] was applied by Lezhnin [34] to a pipe flow, it is based on data obtained from a 40 L bubble column and a 1000 L pilot plant fermenter. As a result, its direct application in a flow remains questionable. The correlation by Lamont and Scott was derived through an analytical consideration of the small scales of turbulent motion, with the objective of calculating the required exposure time for the penetration theory (small eddy approach) [38]. The model demonstrates a quantitative agreement within a factor of two with respect to their own experiments [44], in which they considered the dissolution of CO₂ bubbles in de-aerated flow. The approach was further developed by Avdeev [33] through the introduction of a correction factor accounting for the gaseous phase in the formulation of the turbulent energy dissipation [45]. The confidence interval of ±20% of the resulting correlation encompasses 95% of the experimental data points reported by Lamont and Scott [44].

With respect to Figure 6, Avdeev’s correlation demonstrates indeed the most precise alignment with the CFD simulation. Both courses result in bubble growth of a similar order of magnitude. This agreement suggests that the globally formed mass transfer coefficient could also be employed to estimate the individual bubble growth.

3.2. Bubble Growth at the Wall

The simulation and correlation-based modelling of the bubble growth at the wall are compared in this section, with the results for the course of the radius shown in Figure 7. The modelling based on the penetration theory with the exposure time calculated according to the laminar approach again indicates a nearly linear dependency between the bubble radius and time. This can be attributed to the applied average velocity, $\overline{u\left(y_i\right)}$, calculated at the position $y_i$, which increases with the growing radius. Despite the turbulent flow profile, the velocity gradient near the wall is almost constant, following $u~r$. In this case, the radius appears in both the numerator and denominator of Equation (20), cancelling out. As a result, the mass transfer coefficient is independent of the bubble radius for bubbles near the wall. The justification for the linear relationship between radius and time, given a constant mass transfer coefficient, was previously discussed in Section 3.1 and applies here as well.

The penetration-theory model (red curve) predicts a noticeably faster bubble-growth rate than the CFD simulation. The gap stems from a physical assumption: the penetration-theory correlation is derived from a bubble that stays pinned at its nucleation site and therefore does not account for sliding. In the simulation, however, the drag exerted by the liquid exceeds the surface-tension anchoring force, so the bubble slides along the wall. Sliding reduces the relative velocity between the bubble and the surrounding flow, which in turn lowers the mass-transfer rate and slows growth. To incorporate this sliding effect while still using the penetration-theory framework, we replaced the idealized relative velocity with the value observed in the CFD data. The mean bubble velocity is 0.346 m/s; the mean liquid velocity in its vicinity is 0.44 m/s. Their difference gives an average relative velocity of $\overline{u}=0.44 \mathrm{m}/\mathrm{s}-0.346 \mathrm{m}/\mathrm{s}=0.094 \mathrm{m}/\mathrm{s}$. Using $\overline{u}$ in Equation (20) to evaluate the mass-transfer coefficient yields the blue curve in Figure 7, which aligns closely with the simulation. This agreement confirms that penetration theory—combined with the laminar exposure-time formulation—accurately describes wall-attached bubble growth, provided the relative velocity reflects any sliding that occurs in practice.

4. Limitations and Future Work

An area of potential weakness in the present study is the determination of the relative velocity between the bubble and the surrounding flow. In the current model, this parameter was derived directly from the CFD simulation. However, to enable a simulation-independent application of the model, the relative velocity should be determined independently—potentially through a force balance approach [32]; however, implementing this would exceed the scope of the present work. Moreover, while the study presents promising results, it is currently limited to a single flow velocity and should therefore be regarded as an initial assessment. These results serve as a first step toward establishing the feasibility of applying global mass transfer coefficients to describe local bubble growth. Future work should extend the analysis to a broader range of flow velocities to verify the general applicability of the globally derived mass transfer correlations. One potential approach is to conduct additional CFD simulations at varying velocities. These simulations should also be supported by experimental investigations of single-bubble growth in super-saturated flow to validate the model. The findings from such studies could then be used to develop a novel, broadly applicable correlation for bubble growth under flow conditions. This would enable future predictions using a correlation-based model, significantly reducing the reliance on computationally intensive CFD simulations. Furthermore, more accurate models for estimating the relative velocity between a wall-attached bubble and the crossflow should be developed to expand the applicability of the correlation-based approach both to bubble growth at nucleation sites and to bubbles sliding along surfaces.

5. Conclusions and Outlook

In this preliminary study, the growth of a single bubble in a super-saturated flow using both correlation-based modelling and CFD simulations was compared for a single flow velocity. Two cases were regarded: (1) the bubble flowing with the bulk liquid and (2) a bubble on a wall with a crossflow. For both cases, the correlation-based modelling revealed a nearly linear relationship between the bubble radius and time. For the first case, the CFD simulation aligned closely with Avdeev’s correlation. This agreement suggests that the globally formed mass transfer coefficient of Avdeev can also be employed to estimate the individual bubble growth in a flow. For the bubble on the wall (second case), sliding occurred in the CFD simulations. By applying the mean relative velocity between the bubble and the flow from the simulation in the modelling of bubble growth with the laminar approach of the penetration theory, it was possible to demonstrate a high degree of agreement with the simulation results. This further demonstrates that an initial estimation of the individual bubble growth is feasible with the global mass transfer coefficient calculated from the laminar approach of the penetration theory.