Unified Modelling of Turbulent Droplet and Bubble Fragmentation through a Novel Breakup Constraint

In multiphase chemical and environmental processes, the breakup of fluid particles—bubbles and droplets—lies at the heart of transport, mixing, and reaction efficiency. Systems such as bubble columns, stirred tanks, and gas–liquid reactors rely on the continuous renewal of interfacial area, which is directly determined by how large particles disintegrate into smaller ones under turbulence. Predicting this fragmentation accurately remains one of the most persistent challenges in computational multiphase flow modelling. Although CFD–PBM (Computational Fluid Dynamics coupled with Population Balance Models) has become a powerful framework for simulating multiphase systems, its quantitative reliability hinges on a physically consistent description of breakage frequency and daughter-size distribution (DSD). Existing models have struggled to achieve universality—what succeeds for droplets often fails for bubbles—because of the different interfacial dynamics, density ratios, and deformation modes that govern the two. Historically, four categories of breakup constraints have been explored: energy-based, energy-density-based, force-balance, and time-scale criteria. Each offers valuable insight but also clear limitations. The classic energy-increase model of Luo and Svendsen could reproduce certain daughter-size trends, yet it drastically overpredicted the formation of fine fragments. Later, surface-energy-density models refined this concept, but they were generally tuned to droplet systems and failed to capture the asymmetric fragmentation patterns observed in bubble flows. Force-balance approaches, while conceptually intuitive, introduced numerous semi-empirical coefficients that reduced generality, and time-scale constraints worked only within narrow flow regimes. Previous work has shown that the turbulent energy spectrum exerts decisive influence on fragmentation and studies assumed that only eddies within the inertial subrange were effective, but high-speed imaging and direct numerical simulations revealed significant contributions from both energy-containing and dissipation scales. Moreover, the so-called bottleneck effect—a kinetic-energy pile-up near the inertial-dissipation transition—can notably shift the predicted breakup rate. Neglecting this feature compromises quantitative accuracy, particularly for particles whose sizes fall within that spectral region. To this account, new research paper published in Chemical Engineering Science and led by Professor Shenggao Gong and Professor Ningning Gao from the Hunan Institute of Technology, School of Chemical and Environmental Engineering, the researchers developed two interlinked models: a novel breakup-constraint model based on the available energy density transferred from turbulent eddies, and a comprehensive multiple-fragmentation model integrating that constraint with bottleneck-corrected energy spectra and eddy–particle interaction dynamics. Together, these frameworks allow simultaneous, physically consistent prediction of droplet and bubble breakup frequencies and daughter-size distributions. Their approach eliminates the need for distinct empirical formulations for each phase type and markedly improves agreement with experimental data across various turbulent regimes.

The researchers constructed the new model from first principles of turbulent energy exchange. They idealized the flow as locally isotropic—a common assumption supported by prior DNS evidence—and simplified the particles as spheres to isolate hydrodynamic rather than geometric effects. Breakup was assumed to generate no more than four fragments, a statistically justified limit since experimental observations show that such modes account for over 90 % of events. Central to the formulation was the fragmentation criterion and instead of relying on absolute energy or stress balances, the authors introduced the concept of available average energy density, defined as the kinetic energy per unit volume that a particle of size d₀ can acquire from an eddy of size λ. Breakup occurs when this available energy exceeds the increase in surface-energy density required to create new interfaces. This principle was extended systematically from binary to ternary and quaternary fragmentation through calibrated parameters (Cₜ = 0.039 and Cq = 0.11) representing the ease of secondary breakup of deformed intermediates.

To link this microscopic criterion to turbulence statistics, Gong and Gao derived an expression for the eddy energy actually transferable during particle–eddy interaction. They developed interaction-time corrections that distinguish small from large eddies and accounted for eddy–particle discrepancy through a probabilistic collision factor based on previous DNS data. Combined with an improved kinetic-energy distribution function—formulated from a three-dimensional Gaussian velocity field rather than a two-dimensional approximation—the model produced a new breakage probability law devoid of arbitrary constants. They obtained complete breakup frequency by integrating the collision frequency and probability over the entire energy spectrum modified by the bottleneck correction of Pope’s model. Predictions were benchmarked against extensive experimental data covering single-droplet, droplet-swarm, and bubble-swarm breakage in various reactors. The model successfully reproduced both qualitative and quantitative behaviors: the non-monotonic evolution of breakage frequency with particle size, the observed shift of peak frequency between bubbles and droplets, and the correct dependence on energy-dissipation rate and surface tension. It also captured the pronounced difference in DSD: droplets tending toward near-equal binary splits, while bubbles exhibited strongly asymmetric fragmentation with more numerous small daughters. Across pump-mixer, stirred-tank, and jet-reactor conditions, the agreement between prediction and experiment was superior to classical Luo–Svendsen and Coulaloglou–Tavlarides formulations, confirming the general validity of the new model.

In conclusion, the study of Professor Shenggao Gong and Professor Ningning Gao establishes the first comprehensive breakup model capable of accurately describing both droplet and bubble fragmentation within a turbulent field. By reframing the breakup constraint in terms of available energy density rather than idealized energy or time-scale thresholds, the authors achieved a physically meaningful bridge between distinct fragmentation regimes. The model reconciles prior contradictions in breakage behavior by acknowledging that only a portion of eddy kinetic energy is effectively utilized, and that this portion depends on interaction duration and spatial alignment—features long overlooked in simpler treatments. The implications for multiphase reactor engineering are substantial. Reliable breakup models underpin the prediction of interfacial area, gas–liquid mass-transfer coefficients, and even reaction selectivity in processes ranging from Fischer–Tropsch synthesis to carbon-capture operations. The unified approach developed here enables CFD–PBM simulations to maintain predictive continuity across gas–liquid and liquid–liquid systems without retuning empirical constants. Incorporation of the bottleneck-modified energy spectrum further ensures that energy transfer at transitional scales is represented realistically, improving design optimization for both fine emulsification and large-bubble dispersion.

It is important to mention the mechanistic significance the model provides and its ability to reproduce the asymmetric daughter-size distributions of bubbles and the near-symmetric splits of droplets offers a coherent explanation grounded in physical parameters—density contrast, surface tension, and available eddy energy—rather than in adjustable functions. This advance both elevates the predictive accuracy of population-balance solvers and deepens our understanding of how turbulence interacts with interface physics at multiple scales. Future extensions could merge this formulation with coalescence kinetics or with data-driven corrections derived from high-fidelity DNS which will allow dynamic adjustment of model parameters to local flow conditions. Furthermore, the new framework could aid environmental and biomedical systems where bubble or droplet fragmentation controls transfer rates, such as in aeration tanks, oceanic gas exchange, or aerosol generation. In essence, Gong and Gao’s contribution transforms the empirical practice of breakup modelling into a more mechanistic, spectrum-resolved science. Their novel formulation restores physical consistency across diverse flow domains while retaining computational tractability—an achievement likely to influence how multiphase turbulence is modelled for years to come.