Engineering guide
This module deals with the formation and movement of droplets and bubbles in technical apparatus according to Section L4.1 of the VDI-Wärmeatlas (VDI Heat Atlas, 12th edition, 2019). Among other things, it calculates the characteristic bubble or droplet diameters at the outlet of nozzles and capillaries – separated by the dominant force mechanism in each case: surface tension force, inertia force or viscous force. The ratios of these diameters allow you to judge which regime governs at the actual operating point.
Such data are needed wherever one phase is dispersed in another: at sparger nozzles in bubble columns and fermenters, in liquid atomization, in extraction and scrubbing columns, or when sizing droplet separators whose effectiveness depends directly on the droplet size produced. The primary bubble or primary droplet diameter determines the mass transfer area and the rise or settling velocity of the dispersed phase.
The module distinguishes between Newtonian and non-Newtonian liquids and covers both periodic single-bubble formation at low flow rates and jet break-up at high flow rates.
Scope
- Formation and movement of droplets and bubbles in technical apparatus
Typical workflow
Define the material system and rheology
Densities of the continuous and the dispersed phase, interfacial or surface tension and viscosity are entered. The Newtonian/non-Newtonian selection (V30) defines the constitutive law of the liquid, since non-Newtonian behavior changes the detachment conditions.
Specify nozzle or capillary geometry and flow rate
The orifice diameter of the nozzle or capillary and the gas or liquid flow rate determine whether periodic single-bubble formation, bubble chains or jet break-up occurs.
Calculate the limiting diameters of the individual force regimes
For each dominant mechanism a characteristic diameter is determined: the diameter for the dominant surface force from the equilibrium of buoyancy and the retaining force of surface tension, the diameter for the dominant inertia force from the momentum of the inflowing fluid, and the diameter for the dominant viscous force from the viscous resistance of the surrounding liquid.
Identify the governing regime
The diameter ratios d-inertia to d-surface (V26) and d-viscosity to d-surface (V27) show which mechanism controls the detachment process; the largest of the three diameters is usually the governing primary diameter.
Assess the result
With the governing primary diameter, the rise velocity, residence time and mass transfer area of the dispersed phase can then be estimated; at high flow rates it must be checked whether jet break-up with significantly smaller secondary bubbles already occurs.
Frequently asked questions
When does the simple force balance of buoyancy and surface tension apply?
Only for quasi-static, periodic bubble formation, i.e. at very low gas flow rates where the bubble grows slowly at the nozzle mouth and detaches as soon as the buoyancy exceeds the retaining force of the surface tension. At higher flow rates inertia effects enlarge the primary bubble; in very viscous liquids the viscous force dominates – the module provides a dedicated diameter for each case.
Why are three different bubble diameters reported?
Because depending on the operating point a different force controls the detachment process. The three diameters (surface tension, inertia and viscous regime) are limiting-case solutions; the reported ratio values show which limiting case dominates in the specific situation. If the ratios are close to one, you are in the transition region and should conservatively use the largest diameter.
Does the method also apply to droplets in gases or in a second liquid?
Yes, the force balances can be formulated analogously; instead of the surface tension, the interfacial tension of the material pair and the density difference of the two phases must then be used. In liquid-liquid systems the interfacial tensions are often considerably lower than against air, which leads to smaller primary droplets – measured values of the actual material system are preferable to estimates here.
What special considerations apply to non-Newtonian liquids?
In shear-thinning or viscoelastic media (e.g. fermentation broths, polymer solutions) the effective viscosity depends on the shear rate at the bubble edge; bubbles detach later and coalesce more easily. The module accounts for this via the regime selection V30, but the prediction accuracy is nevertheless lower than for Newtonian media.
Worked example
Air bubbles are generated in water at 20 °C at a sparger nozzle with an orifice diameter of dD = 1 mm. The gas flow rate is so small that quasi-static, periodic bubble formation occurs (surface force dominates). This worked example finds the primary bubble diameter at detachment.
Given
| Nozzle diameter dD | 1.0 mm |
|---|---|
| Surface tension water/air σ (20 °C) | 0.0728 N/m |
| Density of water ρl | 997 kg/m³ |
| Density of air ρg | 1.2 kg/m³ |
| Gravitational acceleration g | 9.81 m/s² |
Solution
Force equilibrium at the detachment point
For quasi-static bubble formation the bubble detaches when the buoyancy force reaches the retaining force of the surface tension along the nozzle perimeter:
(π/6) · dp³ · Δρ · g = π · dD · σ
Solved for the bubble diameter:
dp = (6 · σ · dD / (Δρ · g))1/3
Numerical calculation
Density difference: Δρ = 997 − 1.2 = 995.8 kg/m³
dp = (6 · 0.0728 · 0.001 / (995.8 · 9.81))1/3 = (4.471 · 10−8 m³)1/3 ≈ 3.55 · 10−3 m
The primary bubble diameter is about 3.5 mm – the bubble is therefore considerably larger than the nozzle mouth, which is typical of the surface force regime.
Result
| Primary bubble diameter dp,σ | ≈ 3.55 mm |
|---|
Technical content source: LV-Solved-Problems-Content 628d313247e7