A New Mathematical Model for the Features of Bubble Collapse in Steam Cavitation Processes

Abstract

This study presents a novel mathematical model for bubble cavitation, demonstrating its application in the numerical simulation of steam bubble dynamics within hydrodynamic cavitation phenomena. While previous research has largely focused on the negative consequences of cavitation or its industrial applications, a key unresolved issue remains the physical mechanism of bubble destruction during collapse. This paper investigates the conditions leading to the instability of a spherical bubble’s surface, which in turn causes its irreversible collapse. The model is based on the hypothesis that a bubble is destroyed when its surface temperature exceeds a critical value (T_cr). The modified model, which accounts for heat and mass transfer processes at the bubble boundary, was used to analyse the behaviour of bubbles under different flow conditions. Our computational experiments show that the bubble collapses when the surface temperature surpasses the critical point, irrespective of its size. A comparison of theoretical and experimental data on bubble behaviour during hydrodynamic cavitation validates the proposed criterion. Specifically, the collapse of bubbles in the Venturi tube upon exceeding a critical temperature is shown, supported by experimental data with a maximum error of 6%.The results suggest that the hydraulic parameters of the flow are key factors determining the intensity of cavitation, and that the fulfillment of the condition T_s ≥ T_cr (T_cr = 647 K, p_cr = 22.5 MPa) can serve as a reliable criterion for bubble destruction.

1. Introduction

The phenomenon of cavitation, since its theoretical underpinnings were first established, has been a subject of intensive study by researchers aiming to understand its formation mechanisms and the dynamics of cavitation bubbles. Initially, a significant body of work focused on combating the detrimental effects of cavitation, such as erosion and noise, which are common in hydraulic machinery and other engineering systems [1,2,3]. However, a deeper understanding of the physics governing cavitation has sparked a growing interest in harnessing its energy and unique effects for beneficial purposes.

Currently, cavitation-based technologies are recognised as a highly effective approach for process intensification across various industrial applications, enabling superior energy and mass transfer. The significant potential of cavitation for industrial use lies in its distinctive effects: the collapse of bubbles generates high-velocity microjets, reaching speeds of several hundred metres per second [1,2], alongside localised zones of extremely high pressure, approximately 13.5 GPa [4,5,6], and temperatures reaching 5000 K [7,8]. Some authors, for instance in [9], indicate that these effects depend on the initial temperature of the liquid, although the qualitative nature of cavitation remains unchanged. This process releases a considerable amount of mechanical and thermal energy [10]. The authors of the review paper [11] analyse various cavitation effects in different types of cavitators and suggest that gases inside the cavitation bubble create a localised hotspot upon collapse, leading to energy release. Consequently, each cavity acts as a micro-reactor where both temperature and pressure reach extreme values. A structured review of cavitation effects is presented in [12], providing a foundation for enhancing cavitator designs. These powerful effects are also responsible for inducing intense local turbulence, microjets, and dynamic pressure loads, which significantly enhance mass and heat transfer rates [13]. The emergence of localised “hot spots” is also pointed out in several studies [13,14]. This hypothesis has been applied in practical investigations [15] focused on developing effective cavitator designs, as well as in theoretical works [16] aimed at justifying these effects. Studies have demonstrated that collapsing bubbles create “hot spots”, prompting extensive research into the role of imploding bubbles in augmenting heat transfer intensity [16]. While research into hydrodynamic cavitation and its impact on heat transfer is less abundant [17,18], existing studies consistently conclude that the turbulent cavitating flow is a primary driver in the enhancement of heat transfer [19]. Consequently, the effects of cavitation are now being widely and beneficially used in a multitude of manufacturing sectors for process intensification and as integral components of industrial technologies. A review by [20] analyses the applicability of cavitation phenomena in various technologies, highlighting that the industrial potential of cavitation hinges on its specific characteristics and developmental dynamics. Numerous publications have since emerged, dedicated to the practical applications of cavitation effects, for example in water and air purification [21] and chemical engineering [22], alongside theoretical justifications for these effects [23]. Despite this progress, one of the most persistent and unresolved challenges is the physical mechanism behind the destruction of bubbles during their collapse phase. The stability of the shape of cavitation bubbles has been a subject of investigation for many years. A multitude of analytical studies address this issue, predominantly assuming that bubble stability is altered by harmonic waves at various amplitudes [24]. In [25], it is shown that bubble shape distorts at high pressure amplitudes and larger bubble sizes, while [26] investigates the conditions for an equilibrium bubble size. The spherical stability of an acoustic cavitation bubble is modelled in [27], which largely generalises existing research. Linear stability theory suggests that during the intense compression of a spherical bubble, its surface begins to lose stability once the bubble reaches a certain size (R < 0.2R₀). This leads to a rapid increase in the amplitude of surface perturbations and, ultimately, to the bubble’s destruction. However, this theoretical mechanism frequently contradicts experimental observations of bubble collapse. Experiments have shown that the final closure of a bubble is often preceded by volume oscillations, and even when the condition R ≪ 0.2R₀ is satisfied, the bubble may not lose its spherical shape [28]. This discrepancy highlights a fundamental gap in our understanding of bubble dynamics and underscores the need for a more comprehensive model.

The present study addresses this research gap by introducing a new mathematical model that accurately describes the dynamics of steam bubbles in hydrodynamic cavitation. The central hypothesis of this work is that a bubble undergoes irreversible destruction when its surface temperature surpasses a critical value (T_s ≥ T_cr), a mechanism that more consistently aligns with experimental observations. We apply this model to numerically simulate the behaviour of steam bubbles, particularly focusing on the conditions that lead to their final collapse. This work aims to provide a reliable criterion for bubble destruction and to validate the model by comparing our theoretical findings with existing experimental results, thereby contributing to a more complete understanding of cavitation phenomena.

2. Mathematical Model and Methodology

This section details the mathematical model and numerical methods used to investigate the dynamics and collapse of cavitation bubbles. Our approach is based on a modification of an existing model [29,30], adapted to more accurately account for the extreme conditions and non-equilibrium processes at the vapour–liquid interface. The methodology is described with sufficient detail to allow for replication and further development of our findings.

The model is based on the behaviour of a local element, which includes a single bubble and the surrounding liquid layer. During the maximum compression stage of a bubble, the conditions within this element are extraordinary. Previous studies have shown [31] that the vapour temperature inside a bubble can reach 5000 K, the pressure can exceed 500 MPa, and the heating or cooling rate can surpass 109 K/s. The temperature and pressure within this superheated element can therefore significantly exceed the critical values for water (T_cr = 647 K and p_cr = 22.5 MPa). Our modified model aims to accurately simulate bubble behaviour under these extreme conditions.

The model is defined by a system of differential equations and incorporates approximations for the thermophysical parameters as a function of temperature. In all cases, the initial liquid temperature (T₀) and initial pressure (p₀) are assumed to be known, with subsequent changes in p₀ determining the bubble’s dynamic behaviour. Initial conditions for the temperature, density, and pressure of the vapour are derived from the thermodynamic equilibrium values of p₀ and T₀. A key aspect of our model is the consideration of gas content. While the initial presence of gas has a negligible effect on growing bubbles, for collapsing bubbles, it becomes a crucial factor, with even a small amount of foreign gas fundamentally altering the collapse dynamics.

2.1. Governing Equations

The core of our model is a system of equations that governs the dynamics of the bubble. These equations describe the change in the radial velocity of the liquid at the bubble surface [31]:

$${\displaystyle \frac{d{w}_R}{d\tau }}={\displaystyle \frac{p_r-p_f-1.5{\rho }_f\cdot {w_r}^2-\frac{2\sigma }{R}-\frac{4{\mu }_f\cdot w_r}{R}}{{\rho }_f\cdot R}}.$$

(1)

The bubble radius

$${\displaystyle \frac{dR}{d\tau }}=w_r+{\displaystyle \frac{J}{{\rho }_f}}.$$

(2)

The vapour temperature inside the bubble

$${\displaystyle \frac{d{T}_{st}}{d\tau }}={\displaystyle \frac{3}{\left({\rho }_g\cdot c_g+{\rho }_{st}\cdot c_{st}\right)}}\cdot \left(q-J\cdot c_{st}\cdot T_{st}-p_{st}\cdot {\displaystyle \frac{dr}{d\tau }}\right).$$

(3)

The change in vapour density

$${\displaystyle \frac{d{\rho }_{st}}{d\tau }}=-{\displaystyle \frac{3}{R}}\left(J-{\rho }_{st}\cdot {\displaystyle \frac{dr}{d\tau }}\right);$$

(4)

and the heat transfer to the bubble

$${\displaystyle \frac{d{H}_f}{d\tau }}=-4\pi R^2\cdot \left[J\cdot L\left(T_s\right)-q\right].$$

(5)

The initial conditions are defined as:

$$w_r\left(0\right)=w_{r0};R\left(0\right)=R_0;T_{st}\left(0\right)=T_{st0};{\rho }_{st}\left(0\right)={\rho }_{st0};{\rho }_g\left(0\right)={\displaystyle \frac{m_g}{4/3\cdot \pi R^3}}Hf\left(0\right)=H_{f0}.$$

(6)

The model is further supported by a set of supplementary equations.

Vapour pressure in the bubble

$$p_{st}={\displaystyle \frac{B\cdot {\rho }_{st}{T}_{st}}{M_{st}-{\rho }_{st}\cdot b_{st}}}-{\displaystyle \frac{a_{st}\cdot {{\rho }_{\mathit{st}}}^2}{{M_{\mathit{st}}}^2}}.$$

(7)

Gas pressure in the bubble

$$p_g={\displaystyle \frac{B\cdot {\rho }_g{T}_b}{m_g-{\rho }_g\cdot b_g}}-{\displaystyle \frac{a_g\cdot {{\rho }_g}^2}{{M_g}^2}}.$$

(8)

Interfacial mass transfer rate

$$J=0.25{\alpha }_m\cdot \left[{\rho }_s\cdot u_s\left(T_s\right)-{\rho }_v\cdot u_v\left(T_{st}\right)\right].$$

(9)

Hertz-Knudsen equation:

$$J={\displaystyle \frac{{\alpha }_m\cdot L\cdot {\rho }_{st}\cdot \Delta T}{\sqrt{2\pi \cdot B\cdot T_{st}}}}$$

(10)

Interfacial heat transfer rate

$$q=0.25\cdot \left({\rho }_{st}\cdot u_{st}\cdot c_{st}-{\rho }_g\cdot u_g\cdot c_g\right)\cdot \left(T_s-T_{st}\right)+j\cdot c_{st}\cdot T_s.$$

(11)

Molecular-kinetic velocity of vapour and gas molecules

$$u_{st}\left(T_{st}\right)={\left(8B{T}_{st}/\pi M_{st}\right)}^{0.5}: u_g\left(T_{st}\right)={\left(8B{T}_{st}/\pi M_{st}\right)}^{0.5}$$

(12)

Heat balance equation

$$\left(T_0-T_s\right)\cdot {\lambda }_f\cdot \left({\displaystyle \frac{2}{\delta }}+{\displaystyle \frac{1}{R}}\right)=-J\left(T_s\right)\cdot L\left(T_s\right)-q\left(T_s\right).$$

(13)

2.2. Modelling of Phase Transitions and Boundary Conditions

The non-equilibrium nature of phase transitions is addressed, where a temperature jump (ΔT) occurs at the interface. This jump, described by the well-known Hertz–Knudsen equation [31] (10), determines the intensity of the mass flow from the bubble wall into the gas phase during evaporation or condensation. The Rayleigh–Plesset equation [32] is the fundamental equation. It is commonly assumed that for water–vapour systems, the temperature jump can be neglected because the evaporation coefficient is very small (α_m << 1) [33,34]. Conversely, accounting for the temperature jump is considered important for substances with maximum values of α_m ≈ 1, such as during the boiling of viscous liquids [35,36]. This assumption is not valid when the bubble dynamics are considered with a strict accounting for non-equilibrium processes. In single-component liquid–vapour systems, both ΔT and α_m are primary parameters that characterise the intensity of mass transfer [37]. The authors of [38,39] emphasise that accounting for the non-isothermal effect is essential for problems involving varying phase transition regimes. An arbitrary choice of α_m in the range 0 ≤ α_m ≤ 1 automatically sets a specific value for ΔT, which may not correspond to the actual temperature jump but correctly describes the mass transfer rate in conjunction with the chosen α_m [40].

3. Results

3.1. Theoretical Analysis of Bubble Collapse Conditions

The transition to a supercritical state at maximum compression further complicates the situation [41,42]. When the temperature at the bubble interface, T_s, approaches the critical value, T_cr, the interfacial tension coefficient and the latent heat of vaporisation tend to zero. This implies that capillary effects are absent during maximum compression, and phase transitions occur without the absorption or release of heat, which significantly influences the nature of heat and mass transfer within the local element. Additionally, at T_s ≥ T_cr, the densities of the liquid and vapour become equal, a factor that must also be considered. Since we lack reliable information on the evolution of this local supercritical element, we can only hypothesise about its state.

It is possible that upon transitioning to a supercritical state, the phase boundary disappears. In this scenario, the bubble is replaced for a short time by a substance similar to a “thermic”—a small, localised region of superheated liquid in a metastable state [43,44]. Unlike a typical thermic, where superheating at normal pressure would initiate the formation and growth of vapour nuclei, the substance inside a bubble heated to supercritical temperatures is also under high pressure (p > 1000 MPa), which prevents the formation of a vapour phase. It is not impossible that with further development, the element could transform into a normal thermic if the pressure decreases much faster than the temperature. This could lead to the formation and growth of a vapour phase, which is visually recorded in experiments as a secondary expansion of the highly compressed bubble [45,46]. As shown in Figure 1, the expansion of the highly compressed superheated substance occurs when conditions such as p_st << p_cr and T > T_cr are met, which supports the possibility of such a mechanism.

A more probable characterisation of the superheated element’s evolution is the second scenario. The high local pressure and the absence of capillary forces at the phase boundary promote instability. Any weak external force or random fluctuation could lead to the irreversible destruction of the bubble. In this case, the former bubble gives way to a cloud of secondary microbubbles. These microbubbles, having exhausted the element’s thermal energy, then rapidly collapse in the cold surrounding liquid, releasing a packet of pressure pulses. The existence of such pressure pulsations after the collapse of a cavitation bubble has been recorded in experiments [47,48], although the reasons for their appearance were not analysed. It seems plausible that the report in [46] describes not the closure of a vapour cavity due to condensation, but its destruction, which results in the formation of numerous microbubbles.

3.2. Comparison with Experimental Data for Hydrodynamic Cavitation

Hydrodynamic cavitation has attracted the attention of researchers in a variety of technological applications [49,50]. Studies by [51,52] are dedicated to investigating the influence of temperature on the hydrodynamic parameters of flow in a Venturi tube to enhance the intensity of cavitation treatment. The results show that cavitation characteristics are affected by pressure, flow regime, and the thermal effects of cavitation. Authors in [53,54] examine the geometry of a Venturi nozzle to optimise the conditions for hydrodynamic cavitation within a Venturi tube. It is also worth noting several other works dedicated to the further development of approaches to investigate the influence of cavitator geometric parameters, which, according to the authors, provide a more accurate evaluation, for instance [55,56]. In these studies, the cavitation flow was modelled using the k-SST method for 3D calculations. These investigations open up possibilities for more accurate modelling of cavitation flows. Consequently, numerous modelling approaches exist, and we compare our computational results with established methods, such as the experimental findings of [57]. The work of M. Petkovšek and M. Dular [57] presents the results of an investigation into cavitation bubbles in a water flow within a glass Venturi tube. The flow velocity in the tube throat was 28 m/s, the inlet pressure was 0.56 MPa, and the liquid temperature was T_f = 28 °C. A cavitating flow was formed with bubble sizes of approximately 5–10 µm. The cavitation intervals were approximately 2 ms. For these conditions, we performed calculations using our developed model. The graphs showing the change in bubble diameter are presented in Figure 2.

During the tensile stresses in the narrow part of the Venturi tube, the nucleation and growth of individual vapour bubbles are observed. In the expanding part of the nozzle, the bubbles sharply collapse, releasing pressure pulses. In the experiments [57], the authors provided the profiles of the Venturi nozzles, along with the static pressure and velocity values at the inlet and in the throat for each experimental series. The magnitude and location of the pressure pulse upon bubble collapse were recorded using sensitive pressure sensors. Based on this information, a theoretical analysis of the experimental results was conducted using our model.

The growth of the bubble is caused by the tensile stress of the liquid in the vicinity of the cavitator. The bubble collapse occurs within 2 ms, which corresponds to the data in the photographs, where the time intervals and the distance (10 mm) over which the bubbles disappear after the flow pressure is fully restored can be seen.

The calculations from our model yield similar results. Furthermore, our model shows that by the beginning of the last period, the bubble size is still large enough (R_max ≈ 26 µm) for dampened oscillations to continue. This raises the question: why did the bubble collapse in this specific period and not in the previous compression periods?

As the calculation results in Figure 2 (line 2) show, only in this oscillation period did the surface temperature (T_s) at the maximum compression stage first exceed the critical value, which led to the bubble’s destruction due to thermal instability. Our computational experiments, using different flow parameters and cavitator shapes (and thus different static pressure distributions), showed that a bubble undergoes between 2 and 6 oscillations before complete disappearance, which is confirmed by experimental observations [57].

Another factor that we believe influences the dynamics and intensity of the collapse is the pressure (flow velocity) of the liquid in the Venturi tube. The maximum compression of the bubble occurs at the outlet of the expanding part of the Venturi nozzle. At this moment, the temperature of the bubble wall exceeds the value of T_cr by almost a factor of two, leading to its destruction. The experiment in [54] notes that at the location where such a bubble disappears, a region filled with a “grey fog” is observed for a short time. According to the author, this fog represents the finest vapour and gas microbubbles. This provides evidence that, in this case, the destruction of a sufficiently large bubble occurred due to the thermal instability of the interphase surface.

Our calculations with the new model indicate that at certain pressure and velocity values, the bubble will collapse in the first compression period. The dynamics of such a cavitation bubble are presented in Figure 3.

The ambient pressure in the calculations was 6 MPa, the jet velocity was 110 m/s, and the radius of the cavitation nuclei was 20 µm. In the initial stage, the bubble size decreased under the influence of high pressure, and then, as the liquid pressure decreased, the bubble expanded, reached its maximum radius, and was destroyed. When the bubble compressed to its minimum size, the pressure and temperature inside it increased.

During the collapse, the temperature reaches 2 × 10³ °C, which points to the enormous destructive potential of cavitation bubbles. Such a temperature is maintained for several nanoseconds.

4. Discussion

The numerical model presented in this study offers an advanced approach to understanding the extreme and non-equilibrium conditions that govern the dynamics and collapse of cavitation bubbles. The analysis highlights limitations of some traditional models, particularly those that neglect the temperature jump at the vapour–liquid interface. Our model suggests that this jump is a crucial parameter for accurately describing mass transfer kinetics under extreme conditions. By strictly accounting for non-equilibrium processes, we demonstrate that the commonly accepted assumption of a negligible evaporation coefficient for water–vapour systems may be inappropriate for the high-intensity phenomena observed during bubble collapse.

The analysis indicates a transition of the local bubble-liquid element into a supercritical state during maximum compression. Our calculations confirm that both the vapour temperature and pressure can significantly exceed the critical values for water, reaching approximately 5000 K and 500 MPa, respectively. This leads to a state where the densities of the vapour and liquid phases become equal, and the concepts of interfacial tension and latent heat of vaporisation may lose their physical meaning. This supercritical state fundamentally alters the dynamics, leading to two distinct, yet plausible, hypotheses regarding the bubble’s final state.

The first hypothesis suggests the temporary formation of a superheated substance in a metastable state, which could resemble a “thermic”. This mechanism could potentially explain the secondary expansion of the bubble observed in some experiments [45]. However, our model provides a basis for a second, and possibly more probable, hypothesis: the irreversible destruction of the bubble. We propose that high local pressure and the absence of capillary forces at the phase boundary may promote thermal instability. This instability could lead to the fragmentation of the original bubble into a cloud of secondary microbubbles, which then collapse, releasing pressure pulses. This mechanism appears to be consistent with experimental observations of a “grey fog” [54] and the presence of pressure pulsations after bubble collapse [46].

The presented model’s predictions align with several key experimental observations. Our calculations correctly show that bubbles undergo between 2 and 6 oscillations before their final destruction, which is consistent with experimental data from a Venturi nozzle [57]. Furthermore, the model provides a physical explanation for this phenomenon: the destruction of the bubble is initiated in the oscillation period where the surface temperature first surpasses the critical value, T_cr, due to thermal instability. This suggests the model’s utility in identifying the conditions for catastrophic bubble collapse.

In summary, this study provides a theoretical framework for understanding the behaviour of cavitation bubbles in extreme conditions. The model not only simulates bubble dynamics but also offers a plausible explanation for the destruction mechanism rooted in thermal instability. The immense temperatures (up to 2 × 10³ °C) generated during these events underscore the significant destructive potential of cavitation, with direct implications for various technological applications. Future work could focus on experimental techniques capable of directly observing these nanosecond-scale phenomena within the supercritical element to further validate these hypotheses.

5. Conclusions

In this study, a modified numerical model was successfully developed and applied to investigate the extreme dynamics and collapse of cavitation bubbles, with a particular focus on non-equilibrium phase transitions. The model provides a comprehensive framework for analysing phenomena under conditions where traditional assumptions, such as a negligible temperature jump at the interface, are no longer valid.

During the maximum compression stage, a cavitation bubble and the surrounding liquid layer can transition into a supercritical state, where both temperature and pressure significantly exceed critical values. In this state, physical concepts like interfacial tension and latent heat of vaporisation cease to be applicable.

To validate the model, an analysis of experimental data obtained by various authors in a Venturi cavitator was performed. For example, at a flow velocity of 12.2 m/s, a static pressure of 28.1 kPa, and a liquid temperature of 21 °C within a Venturi tube, the authors noted that the bubble underwent 4–6 oscillation cycles before its complete disappearance. Our data show satisfactory agreement with these experimental results throughout the entire lifespan of the bubble. The growth of the cavitation bubble is caused by a strong tensile stress in the liquid near the cavitator’s surface $p_f=-0.1 \mathrm{b}\mathrm{a}\mathrm{r}$ When the continuously increasing downstream pressure reaches a value of $p_f\left(\tau \right)=p_g\left(\tau \right)$, the bubble begins to compress. However, this compression does not lead to its final collapse because the external pressure has not yet been fully restored. Subsequent increases in flow pressure each time cause a more intense compression of the oscillating bubble, leading to higher amplitudes of the radiated pressure pulses. The final collapse of the bubble only occurs after the flow pressure has been fully restored, during the fourth oscillation period, where the pressure pulse reaches 52 MPa. Our calculations show that by the beginning of the fourth oscillation period, the maximum bubble size (R_max ≈ 1 mm) is still large enough for oscillations to continue. Yet, it is only in the fourth period, at the stage of maximum compression, that the bubble’s surface temperature (T_s) first exceeded the critical value. This seemingly led to the onset of interfacial instability and the destruction of the bubble.

The model’s predictions are validated by their consistency with experimental data, particularly in replicating the number of oscillations a bubble undergoes before its final collapse in a Venturi nozzle.