Introduction & Context
In single-screw extrusion, the metering section serves as the final stage where the material is homogenized and pressurized before exiting the die. The calculation of the shear rate in this zone is critical for process engineering, as it dictates the rheological behavior of non-Newtonian fluids, such as biopolymer melts. Because these materials are typically shear-thinning, the apparent viscosity is a function of the shear rate. Understanding this relationship is essential for predicting protein texturization, ensuring uniform product quality, and managing viscous dissipation, which can lead to thermal degradation if not properly accounted for.
Methodology & Formulas
The metering section is modeled as a drag-induced Couette flow within a rectangular channel. The following formulas define the physical state of the melt:
1. Unit Conversions and Barrel Velocity
The rotational speed of the screw is converted to revolutions per second, and the tangential velocity of the barrel relative to the screw is calculated as:
\[ V_{b} = \pi \cdot D \cdot N \]
2. Average Channel Shear Rate
Assuming a wide channel geometry, the shear rate is defined by the velocity gradient across the channel depth:
\[ \dot{\gamma} = \frac{V_{b}}{H} \]
3. Apparent Viscosity and Shear Stress
For shear-thinning fluids, the apparent viscosity is determined using the Power Law model, followed by the calculation of shear stress:
\[ \mu_{a} = K \cdot \dot{\gamma}^{n-1} \]
\[ \tau = \mu_{a} \cdot \dot{\gamma} \]
4. Thermal and Flow Regimes
To ensure the validity of the Couette flow assumption and to estimate thermal impacts, the Reynolds number and the estimated temperature rise due to viscous heating are calculated:
\[ Re = \frac{\rho \cdot V_{b} \cdot H}{\mu_{a}} \]
\[ \Delta T = \frac{\mu_{a} \cdot \dot{\gamma}^{2} \cdot t_{res}}{\rho \cdot C_{p}} \]
| Parameter |
Condition/Regime |
Engineering Significance |
| Geometry Ratio |
\( W/H < 5 \) |
Sidewall effects dominate; Couette assumption is invalid. |
| Flow Regime |
\( Re > 100 \) |
Flow is not strictly laminar; Couette assumption is invalid. |
| Texturization |
\( \dot{\gamma} < 10 \, \mathrm{s}^{-1} \) |
Insufficient shear for protein unfolding. |
| Texturization |
\( 50 \leq \dot{\gamma} \leq 500 \, \mathrm{s}^{-1} \) |
Ideal range for fibrous structure formation. |
| Texturization |
\( \dot{\gamma} > 1000 \, \mathrm{s}^{-1} \) |
Risk of molecular scission and excessive viscous burning. |
Worked Example: Shear Rate Estimation in a Single-Screw Extruder Metering Section for Protein Texturization
Scenario: A high-moisture extrusion cooking process is used to texturize plant protein. The metering section of a single-screw extruder is analyzed to determine if the shear rate is sufficient for protein unfolding and orientation. The channel is assumed to be fully filled with a non-Newtonian melt obeying the power-law model.
Knowns:
- Screw diameter, \( D = 50.0 \ \text{mm} \)
- Channel depth (metering section), \( H = 4.0 \ \text{mm} \)
- Channel width, \( W = 45.0 \ \text{mm} \)
- Screw speed, \( N = 300.0 \ \text{RPM} \)
- Consistency index, \( K = 5000.0 \ \text{Pa}\cdot\text{s}^n \)
- Power-law index, \( n = 0.4 \)
- Melt density, \( \rho = 1100.0 \ \text{kg/m}^3 \)
- Specific heat capacity, \( C_p = 2000.0 \ \text{J/kg}\cdot\text{K} \)
- Residence time, \( t_{res} = 10.0 \ \text{s} \)
Step-by-step calculation:
-
Convert units to SI.
\( D = 50.0 \times 0.001 = 0.05 \ \text{m} \)
\( H = 4.0 \times 0.001 = 0.004 \ \text{m} \)
\( W = 45.0 \times 0.001 = 0.045 \ \text{m} \)
\( N_{rps} = \frac{300.0}{60.0} = 5.0 \ \text{rps} \)
-
Check width-to-depth ratio for Couette flow validity.
\( \displaystyle \frac{W}{H} = \frac{0.045}{0.004} = 11.25 \)
Ratio \( > 5 \), so the wide-channel approximation is valid and sidewall effects are negligible.
-
Calculate barrel velocity.
\( V_b = \pi \cdot D \cdot N_{rps} \)
\( V_b = \pi \cdot 0.05 \cdot 5.0 = 0.7854 \ \text{m/s} \)
-
Estimate average shear rate in the channel.
\( \dot{\gamma}_{avg} = \frac{V_b}{H} \)
\( \dot{\gamma}_{avg} = \frac{0.7854}{0.004} = 196.35 \ \text{s}^{-1} \)
-
Calculate apparent viscosity using the power-law model.
\( \mu_a = K \cdot \dot{\gamma}_{avg}^{\,n-1} \)
\( \mu_a = 5000.0 \cdot (196.35)^{(0.4-1)} \)
\( \mu_a = 5000.0 \cdot (196.35)^{-0.6} \)
\( \mu_a = 210.452 \ \text{Pa}\cdot\text{s} \)
-
Compute shear stress.
\( \tau = \mu_a \cdot \dot{\gamma}_{avg} \)
\( \tau = 210.452 \cdot 196.35 = 41322.06 \ \text{Pa} \)
-
Determine viscous heating rate.
\( \dot{q} = \mu_a \cdot \dot{\gamma}_{avg}^{\,2} \)
\( \dot{q} = 210.452 \cdot (196.35)^2 = 8113567.44 \ \text{W/m}^3 \)
-
Verify laminar flow with Reynolds number.
\( Re = \frac{\rho \cdot V_b \cdot H}{\mu_a} \)
\( Re = \frac{1100.0 \cdot 0.7854 \cdot 0.004}{210.452} = 0.016 \)
\( Re \ll 1 \), confirming laminar Couette flow.
-
Estimate temperature rise due to viscous dissipation.
\( \Delta T = \frac{\mu_a \cdot \dot{\gamma}_{avg}^{\,2} \cdot t_{res}}{\rho \cdot C_p} \)
\( \Delta T = \frac{210.452 \cdot (196.35)^2 \cdot 10.0}{1100.0 \cdot 2000.0} = 36.88 \ ^\circ\text{C} \)
The significant temperature rise indicates a strongly non-isothermal process; iterative coupling between viscosity and temperature is required for accurate design.
Final answer:
The estimated shear rate in the metering section is \( 196.35 \ \text{s}^{-1} \). This value lies within the empirically validated range of \( 50 \)–\( 500 \ \text{s}^{-1} \) required for effective protein texturization. The corresponding shear stress of \( 41322.06 \ \text{Pa} \) is sufficient for protein unfolding and orientation, while the calculated Reynolds number (\( 0.016 \)) confirms a fully laminar, drag-dominated flow regime.
