Introduction & Context

The Weber number We is a dimensionless group that quantifies the ratio of disruptive hydrodynamic stresses to the cohesive interfacial stress acting on a droplet. In high-pressure valve or rotor–stator homogenisers it is the key scaling parameter used to predict whether the supplied mechanical energy is sufficient to overcome the Laplace pressure and cause droplet break-up. Process engineers use We to size equipment, set operating pressure, select surfactant systems, and guarantee the final droplet-size distribution without resorting to costly trial-and-error campaigns.

Methodology & Formulas

Velocity estimation from valve pressure drop
Assuming one-dimensional, incompressible, inviscid flow across the valve gap, the specific kinetic energy is obtained from the pressure potential via Bernoulli’s equation: \[ v = \sqrt{\frac{2\,\Delta p}{\rho}} \] where Δp is the pressure drop across the valve and ρ is the density of the continuous phase.
Weber number definition
The instantaneous Weber number for a droplet of diameter d is \[ We = \frac{\rho\,v^{2}\,d}{\sigma} \] with σ the equilibrium interfacial tension. Break-up is initiated when the dynamic pressure ½ρv² exceeds the capillary pressure 4σ/d; the factor 4 is absorbed into the critical value.
Critical Weber number for low-viscosity systems
For dispersed-to-continuous viscosity ratios μ_d/μ_c ≪ 1 the classical literature value is \[ We_{\text{crit}} = 12 \]
Viscosity-corrected critical Weber number
When the dispersed phase is viscous, additional energy is dissipated internally and the critical value increases according to \[ We_{\text{crit}} = We_{\text{base}}\left[1+1.3\left(\frac{\mu_{d}}{\mu_{c}}\right)^{0.65}\right] \] with We_base = 12 for turbulent inertial break-up.

Regime	Condition	Interpretation
Break-up guaranteed	We ≥ We_crit	Droplet diameter will reduce until viscous stresses balance interfacial stress.
Break-up marginal	We < We_crit	No size reduction; increase Δp, reduce σ (surfactant), or decrease feed droplet size d.

The Weber number (We) is a dimensionless ratio of disruptive hydrodynamic stress to the stabilizing interfacial tension. In homogenization it predicts whether a droplet or cell will break up:

We ≪ 1 → surface tension dominates; little size reduction.
We ≥ critical value (typically 6–12 for emulsions, 1–3 for cells) → droplet rupture and size reduction occur.

Matching your operating We to the critical value lets you hit target size distributions while minimizing passes and energy.

Use the valve-gap form: We = ρ v² d / σ where

ρ = continuous-phase density (kg m^-3)
v ≈ 2.5–3.5 times the mean velocity through the gap (m s^-1) from CFD or vendor data
d = initial droplet diameter (m) (use D_3₂ if known)
σ = interfacial tension (N m^-1)

If you only know homogenizing pressure P, estimate v ≈ C_v √(2P/ρ) with C_v ≈ 0.7 for most valve trims.

Yes—both ρ and σ drop as temperature rises, but σ usually falls faster, so We increases.

Raising T from 20 °C to 60 °C can cut σ by 30 % and raise We by ~40 %, allowing lower pressure for the same droplet size.
Include σ(T) in your dimensionless group when scaling from pilot (small valve gap) to plant (larger gap) to keep We constant.

Yeast/bacteria disruption: We 1–3 is enough; higher We wastes energy and heats the product.
Mammalian cells: We 0.5–1 avoids complete lysis if you want only intracellular product release.
Nano-emulsions (<200 nm): We 8–12 plus multiple passes; ensure We stays above critical each pass by adjusting pressure or interfacial tension with surfactant.

Worked Example – Estimating Weber Number for a High-Pressure Homogenizer Valve

A dairy plant is designing a single-stage valve homogenizer to reduce the average fat-globule diameter in whole milk from 2 µm to 0.4 µm. A lab trial has shown that the target size is reached when the valve pressure drop is 18 MPa. The process engineer needs to confirm that the Weber number in the valve gap exceeds the critical value of 12, above which secondary break-up of oil droplets (milk fat) is expected.

Knowns

Continuous phase density (skim milk, 20 °C): ρ_c = 998 kg m^-3
Interfacial tension (milk fat / skim milk, with added emulsifier): σ = 0.010 N m^-1
Target droplet (fat-globule) diameter after break-up: d = 0.4 µm = 0.4 × 10^-6 m
Average velocity in the valve gap (from 18 MPa pressure drop): v = 135 m s^-1

Step-by-Step Calculation

Write the Weber number definition for a droplet in turbulent flow:
\[ We = \frac{\rho_c\,v^2\,d}{\sigma} \]
Insert the known quantities, converting units so that all terms are in SI base units:
ρ_c = 998 kg m^-3
v = 135 m s^-1
d = 0.4 × 10^-6 m
σ = 0.010 N m^-1
Compute the numerator:
ρ_c v² d = 998 × (135)² × 0.4 × 10^-6 = 7.276 kg m s^-2 = 7.276 N
Divide by the interfacial tension to obtain the Weber number:
We = 7.276 N / 0.010 N m^-1 = 727.6

Final Answer

We ≈ 728 (dimensionless). Because 728 ≫ 12, the flow conditions in the valve are well above the critical Weber number, confirming that the selected 18 MPa pressure drop is sufficient to achieve the desired fat-globule size reduction.

"Un projet n'est jamais trop grand s'il est bien conçu." — André Citroën

"La difficulté attire l'homme de caractère, car c'est en l'étreignant qu'il se réalise." — Charles de Gaulle