Microbubble Physics in DAF Systems

1. Introduction: The Physics of Microbubble-Mediated Flotation

Dissolved Air Flotation (DAF) operates through the controlled nucleation of gaseous microbubbles from supersaturated aqueous solutions, creating a three-phase system where gas-liquid interfaces serve as transport vectors for solid particulates. Unlike dispersed-air systems producing millimetre-scale bubbles (700–1500 µm), DAF generates microbubbles (10–100 µm) through pressure-induced supersaturation followed by abrupt depressurization.

The efficacy of this separation technology hinges on fundamental interfacial physics governing:

• Thermodynamic nucleation: Gas-phase emergence from supersaturated solution
• Hydrodynamic transport: Low-Reynolds-number bubble rise and particle interception
• Colloidal attachment: Surface force-mediated bubble-particle adhesion

This article presents a physics-based framework connecting these domains through rigorous mathematical treatment of the governing equations.

2. Thermodynamic Foundations of Microbubble Nucleation

2.1 Henry's Law and Supersaturation Dynamics

The dissolution of air under pressurised conditions follows Henry's Law:

where p represents partial pressure of the gas, c the dissolved concentration, and k_H the Henry's constant (temperature-dependent). The thermodynamic driving force for microbubble formation is the supersaturation ratio S:

where C_press is the equilibrium dissolved-air concentration at operating pressure (typically 4–6 bar), and C_ambient represents baseline dissolved gas concentration. Heterogeneous nucleation requires S ≈ 1.3–1.5 to overcome the activation energy barrier for phase transition.

2.2 The Young-Laplace Pressure Effect

Microbubble stability and internal thermodynamics are governed by the Young-Laplace equation, relating pressure difference across the gas-liquid interface to bubble radius:

where γ is the surface tension of the liquid and R the bubble radius. This self-pressurizing effect creates elevated internal gas pressure that:

• Enhances gas dissolution kinetics at the bubble-liquid interface
• Stabilises bubbles against coalescence through increased surface energy barriers
• Creates size-dependent dissolution rates favouring smaller bubble longevity

For a 20 µm microbubble in water (γ = 72.8 mN/m), the internal pressure exceeds ambient by 7.28 kPa, fundamentally altering mass transfer characteristics compared to macroscopic bubbles.

3. Hydrodynamic Transport in the Stokes Regime

3.1 Terminal Rise Velocity and the Stokes Approximation

Microbubble transport in DAF systems occurs at low Reynolds numbers (Re ≪ 1), where viscous forces dominate inertial effects. Under these conditions, the terminal rise velocity V_t follows Stokes' Law:

where d is bubble diameter, g gravitational acceleration, ρ_l and ρ_g liquid and gas densities respectively, and μ_l the dynamic viscosity of the surrounding liquid.

3.2 Surface Force Dominance over Body Forces

The hydrodynamic behaviour of microbubbles exhibits a fundamental shift in force dominance as scale decreases. The body force (buoyancy) scales with volume (F_b ∝ R³), while surface forces scale with interfacial area (F_s ∝ R²):

As R → 0, the term (γ_water - γ_air)R/3K loses significance, reducing to:

This surface force dominance explains why microbubbles exhibit:

• Reduced coalescence tendencies (surface tension stabilisation)
• Enhanced response to local hydrodynamic shear
• Prolonged stagnation in quiescent zones critical for particle interception

4. Interfacial Physics: DLVO Theory and Bubble-Particle Attachment

4.1 Surface Force Balance

Bubble-particle attachment represents a colloidal interaction governed by the DLVO framework (Derjaguin-Landau-Verwey-Overbeek), which quantifies the net interaction energy F_total as the sum of attractive van der Waals forces and repulsive electrostatic double-layer forces:

For successful attachment, the total interaction energy must achieve negative values (net attraction), requiring the rupture of the intervening liquid film between bubble and particle surfaces—a process mediated by hydrophobic force interactions that become dominant at separations < 100 nm.

4.2 Contact Angle and Attachment Efficiency

The thermodynamic work of adhesion W_a depends on the three-phase contact angle θ:

where γ_lg is the liquid-gas interfacial tension. Higher contact angles (increased particle hydrophobicity) exponentially increase attachment probability while reducing detachment forces under turbulent conditions. Research indicates that reducing both floc and bubble sizes increases the effective contact angle, thereby improving attachment efficiency.

5. Collision and Attachment Kinetics

5.1 Single Collector Collision (SCC) Model

The probability of bubble-particle encounter in DAF systems is mathematically described by the Single Collector Collision model, which considers the hydrodynamic interception of particles by rising bubbles. The collision efficiency E_c depends on the size ratio λ = d_p/d_b (particle diameter to bubble diameter) and the flow regime.

For low-Reynolds-number conditions typical of DAF:

• Collision probability is directly proportional to floc size but inversely proportional to bubble size
• Smaller bubbles create higher surface area-to-volume ratios, increasing the probability of particle interception per unit gas volume
• The bubble radius represents the most significant factor influencing collision frequency in CFD simulations

5.2 The Dual-Size Dilemma: Nanobubbles vs. Microbubbles

Recent investigations reveal an optimal interaction between nanobubbles (NBs, < 1 µm) and microbubbles (MBs, 10–100 µm):

The flotation efficiency ε follows a kinetic model incorporating both bubble populations:

where E_c is collision efficiency, E_a attachment efficiency, and N the number concentration of respective bubble sizes. Maximum efficiency occurs at tailored bubble size distributions where NBs enhance attachment probability while MBs provide sufficient buoyancy for transport.

6. Multiphase Fluid Dynamics and System Modelling

6.1 Navier-Stokes Framework for Three-Phase Systems

DAF systems require solving the Navier-Stokes equations for three interacting phases (water, air, solid flocs). For the water phase:

Similar formulations govern the air and solid phases, with momentum exchange terms accounting for:

• Drag forces during relative motion between bubbles and flocs
• Pop and resistance forces during bubble coalescence and detachment
• Turbulent dispersion effects in the contact zone

6.2 Separation Zone Dynamics

In the separation zone, where flocs have grown larger than in the contact zone, the interaction physics shifts from"bubble-captures-floc" to"floc-captures-bubble" mechanisms. Large flocs exhibit:

• Periodic interaction cycles: collision → adhesion → coalescence → desorption
• Reduced unit buoyancy compared to small flocs due to decreased surface-area-to-mass ratios
• Higher desorption probabilities during mutual collisions

The unit buoyancy η (buoyant force per unit floc mass) follows:

where n_b is the number of attached bubbles, V_b bubble volume, and m_floc floc mass. Optimisation requires maximising η through microbubble size control.

7. Engineering Implications and Process Optimisation

7.1 Pressure-Temperature Relationship

Gas solubility in the saturator follows Henry's Law with temperature correction. At 6 bar operating pressure, dissolved air concentration increases substantially with decreasing temperature. This thermodynamic relationship dictates that:

• Lower temperatures increase supersaturation ratios
• Higher pressures (4–6 bar range) generate smaller microbubble distributions
• Bubble size distribution narrows with increased pressure release rates

7.2 The Microbubble Paradox

The physics presents an apparent paradox: smaller bubbles increase collision efficiency but provide insufficient individual buoyancy for coarse particle lifting. The solution lies in controlled size distribution engineering:

This staged approach leverages the inverse relationship between bubble size and attachment efficiency while ensuring adequate transport kinetics.

8. Conclusion

The importance of microbubbles in DAF systems emerges from fundamental physical principles governing interfacial thermodynamics, low-Reynolds-number hydrodynamics, and colloidal surface forces. The Young-Laplace pressure effect stabilises microbubbles against dissolution while enhancing surface reactivity. Stokes-regime transport kinetics (1–12 mm/s rise velocities) optimise residence time for particle interception, and DLVO-governed attachment mechanics favour smaller bubbles for enhanced contact angles and reduced detachment probabilities.

The collision efficiency paradox—where smaller bubbles increase attachment probability but decrease individual lifting capacity—is resolved through hybrid bubble population engineering, combining nanobubble surface modification with microbubble buoyant transport. Future DAF optimisation requires precise control of supersaturation thermodynamics (S = 1.3–1.5), pressure-release hydrodynamics, and bubble size distribution tailoring to maximise the dimensionless attachment efficiency parameter E_a while maintaining adequate separation zone rise velocities.

This physics-based framework demonstrates that microbubble dynamics constitute the critical control volume in DAF engineering, where thermodynamic nucleation, hydrodynamic transport, and interfacial chemistry converge to determine separation efficiency.