Morton Number (Mo): Property-Based Control of Bubble and Drop Dynamics
The Morton number (Mo) captures how fluid properties dictate the shape and motion of bubbles and droplets rising or falling in a surrounding medium. Defined by Mo = g μ⁴ (ρL − ρG) / (ρL² σ³), where g is gravitational acceleration, μ the continuous-phase viscosity, ρL and ρG the liquid and gas densities, and σ the surface tension, Mo distinguishes regimes dominated by surface tension, viscosity, or buoyancy. Unlike the capillary number, Mo depends solely on material properties, making it invaluable for selecting working fluids before conducting experiments.
This article reviews the Morton number’s origin, explains its role in bubble regime maps, outlines measurement and property-determination techniques, and surveys applications in chemical reactors, energy systems, and environmental processes. Pair these insights with the Ohnesorge number guide when designing sprays and with the Bond number article to evaluate gravitational deformation effects.
Definition, Units, and Regime Interpretation
The Morton number is dimensionless: viscosity μ appears to the fourth power, balancing density and surface-tension terms to yield a pure number. High Mo values (≫ 10) correspond to viscous, low-surface-tension fluids such as glycerol–water mixtures, producing spherical-cap bubbles and sluggish rises. Low Mo values (≪ 10⁻³) typify clean water or liquid metals, where surface tension dominates and bubbles remain nearly spherical even at high Reynolds numbers. Intermediate Mo ≈ 10⁻³–10 controls transitions between wobbling ellipsoidal bubbles and spherical-cap shapes.
Engineers often couple Mo with the Eötvös number (Eo = g (ρL − ρG) L² / σ) and the Reynolds number based on bubble slip velocity. Plotting Mo against Eo yields regime maps originally proposed by Grace, showing boundaries between spherical, ellipsoidal, wobbling, and skirted bubbles. Because Mo embeds fluid properties, altering temperature or adding surfactants can shift a system into different regions without changing apparatus geometry. These maps aid scale-up from bench experiments to industrial bubble columns or flotation cells.
Historical Background and Standardisation
The Morton number honours Frederick C. Morton, a British engineer who, with G. Taylor and W. R. Grace, investigated bubble rise in the 1950s at Imperial College London. Their studies sought to predict the drag and shape of gas bubbles in viscous liquids—critical for chemical reactors and nuclear coolant systems. Morton proposed the parameter now bearing his name to collapse property variations across experiments. Grace later popularised Mo through regime maps that became standard in reactor design textbooks.
Subsequent decades saw Mo integrated into chemical engineering curricula and standards. The American Institute of Chemical Engineers (AIChE) incorporated Mo into design correlations for bubble columns and airlift reactors. ISO 80000-11 lists Mo among characteristic numbers, reinforcing notation and symbol usage. Modern computational fluid dynamics (CFD) validation studies still reference Grace’s data, demonstrating the enduring influence of Morton’s work on multiphase-flow metrology.
Conceptual Foundations and Scaling Relationships
Balancing forces on rising bubbles
A bubble rising through a liquid experiences buoyancy (∝ (ρL − ρG) g), viscous drag (∝ μ), and surface tension resisting deformation (∝ σ). Morton’s grouping arises from dimensional analysis seeking dimensionless combinations that remain invariant under scaling. Setting characteristic length L via bubble diameter and velocity via terminal slip, the governing Navier–Stokes equations reveal that Mo determines the relative importance of viscous to capillary forces independent of flow speed, while Eo introduces buoyancy-to-capillary effects.
Link to Ohnesorge and capillary numbers
Although Mo excludes velocity, it influences the Ohnesorge number through property selection: for a given nozzle diameter, increasing μ or decreasing σ raises both Mo and Oh, shifting droplet breakup from inertial to viscous regimes. Similarly, Mo informs expected capillary numbers because higher μ tends to increase Ca under the same velocity, enabling engineers to plan flow rates before experimentation.
Temperature and composition dependence
Viscosity and surface tension strongly depend on temperature, pressure, and composition. For example, heating water from 20 °C to 80 °C lowers μ from 1 mPa·s to 0.36 mPa·s and σ from 72 to 62 mN·m⁻¹, reducing Mo by nearly an order of magnitude. Adding surfactant may reduce σ to 30 mN·m⁻¹, increasing Mo dramatically. Consequently, process control strategies often target property adjustments (temperature ramps, additive dosing) to position Mo within desired ranges. Reporting Mo alongside temperature and composition ensures traceability.
Measurement and Property Determination
Because Mo relies on accurate property data, laboratories deploy a suite of instruments. Viscosity is measured via capillary viscometers, rotational rheometers, or oscillatory techniques, with attention to shear rate and potential non-Newtonian behaviour. Density measurements use pycnometers, oscillating U-tubes, or hydrometers calibrated to traceable standards. Surface tension determinations employ pendant-drop, du Noüy ring, or Wilhelmy plate methods. Each measurement should include temperature control (±0.1 K) because small deviations propagate strongly through μ⁴ and σ³ terms.
When evaluating complex mixtures—fermentation broths, polymer solutions, ionic liquids—property data may be scarce. Researchers combine experimental measurement with predictive correlations such as the Arrhenius viscosity model or Guggenheim–Katayama surface-tension correlation. Uncertainty analysis is crucial: because μ enters as μ⁴, a 5% error in viscosity yields ~20% error in Mo. Reporting expanded uncertainty and calibration traceability aligns with ISO/IEC 17025 expectations and supports comparability across laboratories.
Applications Across Industries
Chemical and biochemical reactors
Bubble columns, airlift reactors, and fermentation vessels rely on Mo-guided design to ensure appropriate bubble sizes and gas holdup. High-Mo broths, such as viscous biopolymer cultures, require sparger designs that generate sufficient buoyancy-driven circulation despite damped bubble rise. Engineers adjust gas velocity, sparger pore size, and surfactant dosing to shift effective Mo and maintain oxygen transfer. Coupling Mo with Nusselt-number correlations helps link bubble behaviour to heat-removal strategies.
Mineral processing and wastewater treatment
Froth flotation cells use surfactants to tune σ, thereby adjusting Mo to optimise bubble size for particle capture. Wastewater aeration basins manage viscosity changes caused by biomass concentration; monitoring Mo assists in maintaining oxygen transfer efficiency and avoiding coalescence that reduces surface area. Chemical dosing to modify surface tension offers a lever for controlling bubble regimes without altering mechanical equipment.
Energy systems and climate technologies
In boiling water reactors and next-generation molten-salt systems, Mo helps assess bubble departure sizes and two-phase stability. Carbon capture systems employing gas–liquid contactors (spray columns, packed towers) use Mo-derived insights to select solvents and operating temperatures that balance mass transfer with hydraulic performance. Geothermal power plants track brine properties affecting Mo to predict phase-separation behaviour in separators and turbines.
Food, cosmetics, and consumer products
Aerated foods (whipped creams, foams) and cosmetic emulsions depend on controlling bubble or droplet stability. Formulators adjust viscosity with hydrocolloids and surface tension with emulsifiers to target Mo ranges that stabilise textures while preventing coalescence. Quality-control protocols record Mo alongside rheological curves to ensure batch-to-batch consistency and regulatory compliance.
Strategic Importance and Future Directions
Digital twins and advanced CFD models increasingly incorporate Morton-number scaling to predict multiphase reactor performance under dynamic operating conditions. Machine-learning algorithms trained on Mo, Eo, and Re features can forecast bubble size distributions, enabling real-time control. Emerging materials—ionic liquids, deep eutectic solvents, bio-based surfactants—offer new levers to manipulate Mo for sustainable processes.
Future research explores coupling Mo with microscale imaging (x-ray tomography, high-speed laser scanning) to derive constitutive models bridging pore-scale physics and plant-scale design. As industries pursue decarbonisation and resource efficiency, mastering the Morton number ensures that multiphase contactors deliver high throughput, selectivity, and energy efficiency. Keep this reference alongside the ISO 80000-11 characteristic-number overview to maintain rigorous notation and reporting across collaborations.