Droplet Size Reduction Prediction

Reference ID: MET-9C25 | Process Engineering Reference Sheets Calculation Guide

Introduction & Context

Droplet size reduction prediction is a critical calculation in process engineering, particularly within the food, pharmaceutical, and chemical industries. It is primarily used to design and optimize high‑pressure valve homogenizers, which are essential for creating stable emulsions and facilitating emulsion formation in liquid‑liquid extraction. By predicting the final Sauter mean diameter of dispersed droplets, engineers can ensure product consistency, shelf‑life stability, and desired rheological properties. This calculation is typically applied during the scale‑up of homogenization processes or when evaluating the performance of existing equipment under varying operating pressures.

Methodology & Formulas

The prediction of droplet size reduction relies on an empirical power-law correlation that relates the final droplet diameter to the pressure drop across the homogenizer valve. The physics of the system is governed by the competition between disruptive turbulent forces and the stabilizing interfacial tension of the droplets. The dispersed phase volume fraction \(\phi\) must be below the dilute limit (typically \(\phi < 0.05\)) to suppress coalescence and ensure the correlation remains valid.

The fundamental relationship for the final Sauter mean diameter \(d\) is defined as:

\[ d = d_{0} \cdot \left( \frac{\Delta P_{0}}{\Delta P} \right)^{n} \]

To validate the regime of the breakup, a simplified Weber number for homogenization is calculated to ensure that turbulent inertial forces are sufficient to overcome interfacial tension. In high‑pressure homogenizers, the disruptive stress scales with the pressure drop \(\Delta P\), yielding the practical form used in this correlation:

\[ We_{\Delta P} = \frac{\Delta P \cdot d}{\sigma} \]

Note on the Weber number: The standard Weber number is defined as \(We = \rho_{c} v^{2} d / \sigma\). In a homogenizer valve, assuming complete conversion of pressure energy to kinetic energy (Bernoulli: \(\Delta P = \tfrac{1}{2}\rho_{c} v^{2}\)), the standard Weber number becomes \(We = 2\Delta P d / \sigma = 2\,We_{\Delta P}\). The simplified form \(We_{\Delta P}\) is widely adopted in homogenization practice because the continuous phase density \(\rho_{c}\) cancels algebraically under this Bernoulli simplification, and the critical threshold is calibrated empirically. For aqueous systems (\(\rho_{c} \approx 1000\ \mathrm{kg/m^{3}}\)), the standard Weber number is approximately twice the value computed here.

Where the pressure drop is converted to SI units (Pascals) using the conversion factor \(1\ \mathrm{bar} = 10^{5}\ \mathrm{Pa}\), the diameter is maintained in meters, and the interfacial tension \(\sigma\) is in N/m.

To determine whether the system operates in the turbulent inertial or viscous regime, the Ohnesorge number provides a complementary criterion:

\[ Oh = \frac{\mu_{c}}{\sqrt{\rho_{c} \cdot \sigma \cdot d_{0}}} \]

When \(Oh \ll 1\) (typically \(Oh < 0.1\)), turbulent inertial breakup dominates and \(n = 0.6\) is appropriate. As \(Oh\) increases, viscous stresses become significant, shifting the exponent toward \(n = 0.4\) to \(0.5\).

Regime / Parameter	Condition / Value	Engineering Significance
Turbulent Inertial	\(n = 0.6\)	Dominant mechanism in high-pressure homogenization; low-viscosity continuous phase.
Viscous Regime	\(n = 0.4\) to \(0.5\)	Occurs with high continuous phase viscosity or low turbulence; verified via Ohnesorge number.
Dilute Limit	\(\phi < 0.05\)	Required to prevent coalescence and hindered breakup; \(\phi\) is dispersed phase volume fraction.
Breakup Validity	\(We_{\Delta P} > 100\)	Confirms turbulent inertial breakup is the primary mechanism (empirical threshold for the simplified Weber number).
Pressure Range	\(10\ \mathrm{bar} \leq \Delta P \leq 300\ \mathrm{bar}\)	Empirical bounds for standard valve homogenizer models.
Ohnesorge Number	\(Oh < 0.1\)	Supplementary check confirming the turbulent inertial regime; uses \(\rho_{c}\) and \(\mu_{c}\).

The exponent \(n\) in the power-law correlation \(d = d_{0} (\Delta P_{0} / \Delta P)^{n}\) is determined by the dominant droplet breakup mechanism:

Turbulent inertial regime (\(n = 0.6\)): Applies when disruptive turbulent pressure fluctuations govern breakup. This is the default for low-viscosity continuous phases at high pressure drops. Verify via the Ohnesorge number \(Oh = \mu_{c} / \sqrt{\rho_{c} \cdot \sigma \cdot d_{0}} < 0.1\) and the simplified Weber number \(We_{\Delta P} = \Delta P \cdot d / \sigma > 100\).
Viscous regime (\(n = 0.4\) to \(0.5\)): Applies when viscous shear stresses in the continuous phase contribute significantly to breakup. Select \(n = 0.4\) for highly viscous continuous phases (\(Oh > 0.1\)) and \(n = 0.5\) for transitional conditions.

Always confirm the regime using both the Ohnesorge number and the simplified Weber number before selecting the exponent.

To perform an accurate droplet size reduction prediction, the following input parameters must be provided:

\(d_{0}\) – Initial Sauter mean diameter of the dispersed phase droplets (typically in µm).
\(\Delta P_{0}\) – Reference pressure drop corresponding to the known initial diameter (in bar).
\(\Delta P\) – Operating pressure drop across the homogenizer valve (in bar, within 10–300 bar).
\(n\) – Power-law exponent (0.6 for turbulent inertial; 0.4–0.5 for viscous regime).
\(\sigma\) – Interfacial tension between the dispersed and continuous phases (in N/m).
\(\phi\) – Dispersed phase volume fraction (must be < 0.05 for the dilute limit).
\(\rho_{c}\) – Continuous phase density (in kg/m³, required for Ohnesorge number verification).
\(\mu_{c}\) – Continuous phase dynamic viscosity (in Pa·s, required for Ohnesorge number verification).

The predicted droplet size is only reliable when all validity checks are satisfied. If any condition fails, the empirical correlation may produce inaccurate results:

Dilute limit (\(\phi < 0.05\)): If the dispersed phase fraction exceeds 0.05, coalescence effects become significant and the correlation underpredicts the actual droplet size. Consider reducing \(\phi\) or applying a coalescence correction factor.
Pressure bounds (10–300 bar): Operating outside this range extrapolates beyond the empirical data set. For \(\Delta P < 10\ \mathrm{bar}\), breakup may be insufficient; for \(\Delta P > 300\ \mathrm{bar}\), cavitation and valve damage may occur, invalidating the correlation.
Weber number (\(We_{\Delta P} > 100\)): Values below 100 indicate that turbulent inertial forces may not dominate breakup. Re-check the Ohnesorge number and consider whether the system is in the viscous regime (\(n = 0.4\)–\(0.5\)).
Ohnesorge number (\(Oh < 0.1\)): If \(Oh > 0.1\), the continuous phase viscosity is significant, and the exponent \(n = 0.6\) may be inappropriate. Switch to the viscous regime exponent and re-evaluate.

If multiple validity checks fail, treat the prediction as an order-of-magnitude estimate only and consider pilot-scale testing to calibrate the correlation for your specific system.

Worked Example: Droplet Size Reduction in a Dairy Homogenizer

Knowns

Initial Sauter mean diameter, \(d_{0} = 10.0\ \mu\mathrm{m}\)
Reference pressure drop, \(\Delta P_{0} = 10.0\ \mathrm{bar}\)
Operating pressure drop, \(\Delta P = 200.0\ \mathrm{bar}\)
Exponent (turbulent inertial regime), \(n = 0.6\)
Dispersed phase volume fraction, \(\phi = 0.02\)
Interfacial tension, \(\sigma = 0.02\ \mathrm{N/m}\)
Continuous phase density, \(\rho_{c} = 1000\ \mathrm{kg/m^{3}}\) (aqueous dairy system)
Continuous phase viscosity, \(\mu_{c} = 0.001\ \mathrm{Pa{\cdot}s}\) (for Ohnesorge number verification)

Step-by-Step Calculation

Compute pressure ratio
\[ \frac{\Delta P_{0}}{\Delta P} = \frac{10.0}{200.0} = 0.05 \]
Apply the power law correlation
\[ d = d_{0} \cdot \left( \frac{\Delta P_{0}}{\Delta P} \right)^{n} = 10.0\ \mu\mathrm{m} \times (0.05)^{0.6} \]

Using the numerical result from the calculation, the final diameter is \(d = 1.657\ \mu\mathrm{m}\) (rounded to three decimal places).
Calculate the simplified Weber number
\[ We_{\Delta P} = \frac{\Delta P \cdot d}{\sigma} \]

With \(\Delta P = 200\ \mathrm{bar} = 2.0 \times 10^{7}\ \mathrm{Pa}\) (using \(1\ \mathrm{bar} = 10^{5}\ \mathrm{Pa}\)), \(d = 1.657 \times 10^{-6}\ \mathrm{m}\), and \(\sigma = 0.02\ \mathrm{N/m}\), the computed simplified Weber number is \(We_{\Delta P} = 1657\). This value is well above 100, confirming dominant turbulent inertial breakup. For reference, the standard Weber number is \(We = 2 \cdot We_{\Delta P} = 3314\) for this aqueous system.
Supplementary regime check: Ohnesorge number
\[ Oh = \frac{\mu_{c}}{\sqrt{\rho_{c} \cdot \sigma \cdot d_{0}}} = \frac{0.001}{\sqrt{1000 \times 0.02 \times 10.0 \times 10^{-6}}} = \frac{0.001}{\sqrt{2.0 \times 10^{-4}}} = 0.071 \]

Since \(Oh = 0.071 < 0.1\), the turbulent inertial regime is confirmed, supporting the choice of \(n = 0.6\).
Validity checks
- Dispersed phase volume fraction: \(\phi = 0.02 < 0.05\), within the dilute regime.
- Pressure drop: \(\Delta P = 200\ \mathrm{bar}\) lies within the empirical range 10–300 bar.
- Exponent: \(n = 0.6\) is consistent with the turbulent inertial regime, verified by \(Oh = 0.071 < 0.1\).
- Simplified Weber number: \(We_{\Delta P} = 1657 > 100\), satisfying the minimal turbulent breakup criterion.
All conditions are met, supporting the validity of the calculation.

Final Answer

The final Sauter mean diameter is 1.657 µm, with a simplified Weber number of 1657 and an Ohnesorge number of 0.071, confirming the turbulent inertial regime.

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