Introduction & Context
The feed section capacity calculation is a fundamental procedure in polymer and food processing engineering. It determines the throughput potential of a single‑screw extruder by modeling the feed zone as a drag‑induced conveyor. Unlike pressure‑driven flow in the metering section, the feed section relies on the friction between the material and the barrel wall to transport granular or powdered solids forward. This calculation is critical for sizing equipment, selecting motor power, and ensuring that the extruder is not starved or overfed, which could lead to process instability or mechanical failure; a comparable methodology is applied in the belt extractor capacity calculation for downstream material handling.
Methodology & Formulas
The calculation follows a deterministic approach based on the geometric displacement of the screw flights, adjusted by empirical efficiency factors to account for real-world material behavior and slip conditions.
First, the helix angle θ is determined by the screw pitch t and the barrel diameter D:
\[ \theta = \arctan\left(\frac{t}{\pi \cdot D}\right) \]
The theoretical volume displaced per single revolution of the screw Vrev is obtained by multiplying the cross-sectional area of the channel normal to the flight by the helical length of one revolution. This simplifies to:
\[ V_{\mathrm{rev}} = \pi \cdot D \cdot H \cdot (t - e) \]
The theoretical volumetric feed rate Qtheo is derived from the screw speed N:
\[ Q_{\mathrm{theo}} = V_{\mathrm{rev}} \cdot \left(\frac{N}{60}\right) \]
To obtain the actual volumetric feed rate Qactual, the theoretical rate is adjusted by the fill efficiency ηfill and the slip efficiency ηslip:
\[ Q_{\mathrm{actual}} = Q_{\mathrm{theo}} \cdot \eta_{\mathrm{fill}} \cdot \eta_{\mathrm{slip}} \]
Finally, the mass feed rate ṁ is calculated using the bulk density of the material ρ_bulk, which can be significantly influenced by the effect of particle size on bulk density.
\[ \dot{m} = Q_{\mathrm{actual}} \cdot \rho_{\mathrm{bulk}} \]
| Parameter |
Description |
Typical Range / Condition |
| Pitch-to-Diameter Ratio (t/D) |
Geometric constraint for feed section |
0.8 – 1.2 |
| Flight Depth Ratio (H/D) |
Geometric constraint for feed section |
0.15 – 0.25 |
| Screw Speed (N) |
Operational range |
50 – 500 RPM |
| Overall Efficiency (ηfill · ηslip) |
Combined empirical correction factor |
0.3 – 0.8 |
Worked Example: Feed Section Capacity of a Single-Screw Extruder
Scenario: Calculate the maximum mass feed rate of wheat flour through a 50 mm single-screw extruder with a grooved barrel and gravity feeding.
Knowns
- Barrel inner diameter, \(D = 0.05 \, \text{m}\)
- Flight depth, \(H = 0.01 \, \text{m}\)
- Screw pitch (square), \(t = 0.05 \, \text{m}\)
- Flight land width, \(e = 0.005 \, \text{m}\)
- Screw speed, \(N = 200.0 \, \text{RPM}\)
- Bulk density of wheat flour, \(\rho_{\mathrm{bulk}} = 600.0 \, \text{kg/m}^3\)
- Fill efficiency (gravity feeding), \(\eta_{\mathrm{fill}} = 0.5\)
- Slip efficiency (grooved barrel), \(\eta_{\mathrm{slip}} = 0.85\)
Step-by-Step Calculation
-
Compute the helix angle \(\theta\):
\[
\theta = \arctan\left( \frac{t}{\pi D} \right) = \arctan\left( \frac{0.05}{\pi \times 0.05} \right) = 17.657^\circ
\]
(The helix angle is used for geometry checks, though the volumetric displacement formula does not contain a \(\cos\theta\) factor.)
-
Theoretical channel volume per revolution \(V_{\mathrm{rev}}\):
\[
V_{\mathrm{rev}} = \pi D H (t - e) = \pi \times 0.05 \times 0.01 \times (0.05 - 0.005) = 7.07 \times 10^{-5} \, \text{m}^3/\text{rev}
\]
-
Theoretical volumetric feed rate \(Q_{\mathrm{theo}}\):
\[
Q_{\mathrm{theo}} = V_{\mathrm{rev}} \cdot \frac{N}{60} = (7.07 \times 10^{-5}) \cdot \frac{200.0}{60} = 2.36 \times 10^{-4} \, \text{m}^3/\text{s}
\]
-
Actual volumetric feed rate \(Q_{\mathrm{actual}}\):
\[
Q_{\mathrm{actual}} = Q_{\mathrm{theo}} \cdot \eta_{\mathrm{fill}} \cdot \eta_{\mathrm{slip}} = (2.36 \times 10^{-4}) \cdot 0.5 \cdot 0.85 = 1.00 \times 10^{-4} \, \text{m}^3/\text{s}
\]
-
Mass feed rate \(\dot{m}\):
\[
\dot{m} = Q_{\mathrm{actual}} \cdot \rho_{\mathrm{bulk}} = (1.00 \times 10^{-4}) \cdot 600.0 = 0.060 \, \text{kg/s}
\]
Converting to kg/h:
\[
\dot{m} = 0.060 \, \text{kg/s} \times 3600 = 216 \, \text{kg/h}
\]
-
Empirical range checks:
- Pitch-to-diameter ratio \(t/D = 1.0\) (within 0.8–1.2)
- Flight depth ratio \(H/D = 0.2\) (within 0.15–0.25)
- Overall efficiency \(\eta_{\mathrm{overall}} = \eta_{\mathrm{fill}} \cdot \eta_{\mathrm{slip}} = 0.425\) (within the typical 0.3–0.8 range)
- Screw speed \(N = 200.0 \, \text{RPM}\) (within 50–500 RPM)
Final Answer
The maximum mass feed rate of wheat flour through the feed section is approximately \(\dot{m} = 0.060 \, \text{kg/s}\), which equates to 216 kg/h. This result satisfies the empirical bounds and is consistent with the given geometry and operating conditions.
