Introduction & Context
This engineering reference sheet provides a standardized framework for evaluating the physical characteristics of nano-ingredients post-milling. In process engineering, achieving a target particle size is critical for maximizing bioavailability and ensuring product stability. By integrating geometric surface area analysis with fluid dynamics, this methodology allows engineers to predict how nano-scale particles will behave within a carrier fluid, specifically regarding their suspension stability and settling behavior.
Methodology & Formulas
The evaluation process follows a two-stage analytical approach: determining the geometric surface area enhancement and assessing the stability of the resulting suspension via terminal velocity calculations.
1. Specific Surface Area (SSA)
The SSA represents the total surface area per unit mass, which is a primary indicator of the potential reactivity and bioavailability of the milled ingredient.
\[ SSA = \frac{6}{\rho_p \cdot D} \]
2. Stokes Law (Terminal Velocity)
To determine the settling velocity of particles within a carrier fluid, we apply Stokes Law. This calculation assumes the particle is moving through a viscous medium under laminar flow conditions.
\[ V_t = \frac{g \cdot D^2 \cdot (\rho_p - \rho_f)}{18 \cdot \mu} \]
3. Validity and Regime Constraints
The accuracy of these calculations is dependent on the flow regime. The following table outlines the criteria for validity and the physical constraints applied to the model.
| Parameter |
Formula / Condition |
Constraint |
| Reynolds Number |
\( Re = \frac{\rho_f \cdot V_t \cdot D}{\mu} \) |
\( Re < 1.0 \) |
| Particle Diameter |
\( D > 0 \) |
Must be positive |
| Viscosity Conversion |
\( \mu = \mu_{cP} \cdot 0.001 \) |
\( Pa \cdot s \) |
| Temperature Conversion |
\( T_K = T_C + 273.15 \) |
Kelvin |
Note: If the calculated Reynolds number exceeds the threshold of 1.0, the flow is no longer in the Stokes regime, and the terminal velocity calculation will require correction for inertial drag effects. Furthermore, this model assumes spherical particle geometry and Newtonian fluid behavior.
Worked Example: Post-Milling Analysis of Nano-Encapsulated Curcumin
A process engineer is developing a bioactive ingredient where curcumin has been ultrafine milled to nanoscale. The objective is to quantify the increase in specific surface area for bioavailability and assess the suspension stability in a water-based syrup carrier using Stokes' Law.
Known Input Parameters:
- Target particle diameter, \( D \) = \( 200.0 \, \text{nm} \)
- Particle density, \( \rho_p \) = \( 1200.0 \, \text{kg/m}^3 \)
- Carrier fluid density, \( \rho_f \) = \( 1050.0 \, \text{kg/m}^3 \)
- Carrier fluid viscosity, \( \mu \) = \( 5.0 \, \text{cP} \)
- Process temperature, \( T \) = \( 25.0 \, ^\circ\text{C} \)
Step-by-Step Calculation:
-
Unit Conversions:
- Diameter: \( D = 200.0 \, \text{nm} \times 10^{-9} = 2.000 \times 10^{-7} \, \text{m} \) (using \( D\_M\_R = 2 \times 10^{-7} \))
- Viscosity: \( \mu = 5.0 \, \text{cP} \times 0.001 = 0.005 \, \text{Pa} \cdot \text{s} \) (using \( MU\_PAS\_R = 0.005 \))
- Temperature: \( T = 25.0 \, ^\circ\text{C} + 273.15 = 298.15 \, \text{K} \) (using \( TEMP\_K\_R = 298.15 \))
-
Calculate Specific Surface Area (SSA):
Formula: \( SSA = \frac{6}{\rho_p \cdot D} \)
Using \( \rho_p = 1200.0 \, \text{kg/m}^3 \) and \( D = 2.000 \times 10^{-7} \, \text{m} \), the calculation yields \( SSA = 25000.0 \, \text{m}^2/\text{kg} \) (from \( SSA\_R = 25000.0 \)).
-
Calculate Terminal Settling Velocity (\( V_t \)) for Stability Assessment:
Formula (Stokes’ Law): \( V_t = \frac{g \cdot D^2 \cdot (\rho_p - \rho_f)}{18 \cdot \mu} \)
Using constants and converted values: \( g = 9.81 \, \text{m/s}^2 \), \( D^2 = 4.000 \times 10^{-14} \, \text{m}^2 \) (from \( D\_SQ = 4.000 \times 10^{-14} \)), \( \rho_p - \rho_f = 150.0 \, \text{kg/m}^3 \) (from \( RHO\_DIFF = 150.0 \)), and \( \mu = 0.005 \, \text{Pa} \cdot \text{s} \).
The computed velocity is \( V_t = 6.540 \times 10^{-10} \, \text{m/s} \). Rounded to three decimal places, \( V_t \approx 0.000 \, \text{m/s} \) (from \( V\_T\_R = 0.0 \)).
-
Validate Stokes’ Law Regime with Reynolds Number:
Formula: \( Re = \frac{\rho_f \cdot V_t \cdot D}{\mu} \)
Plugging in \( \rho_f = 1050.0 \, \text{kg/m}^3 \), \( V_t = 6.540 \times 10^{-10} \, \text{m/s} \), \( D = 2.000 \times 10^{-7} \, \text{m} \), and \( \mu = 0.005 \, \text{Pa} \cdot \text{s} \), the result is \( Re = 2.747 \times 10^{-11} \). Rounded, \( Re \approx 0.000 \) (from \( RE\_R = 0.0 \)).
Since \( Re < 1 \), the flow is in the Stokes regime, validating the use of Stokes’ Law.
Final Answer:
For the nano-encapsulated curcumin after ultrafine milling:
- The specific surface area is 25000.0 m²/kg, indicating a high surface area for enhanced bioavailability.
- The terminal settling velocity is approximately 0.000 m/s, confirming that the particles will remain effectively suspended in the carrier fluid, ensuring formulation stability.
