Reference ID: MET-7A82 | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
In the field of food process engineering, the extrusion of viscoelastic materials such as pasta dough requires precise control over die geometry to achieve the desired final product dimensions. Due to the elastic memory of the dough, the extrudate undergoes die swell (also known as extrudate swell) upon exiting the die aperture. This phenomenon results in a cross‑sectional expansion that must be compensated for during the die design phase. Failure to account for this expansion leads to off‑specification product dimensions. This calculation is essential for determining the required die aperture side length to ensure the final product meets quality standards while verifying that the extrusion process remains within laminar flow regimes and equipment pressure constraints. For a comprehensive guide to cold extrusion for pasta, see cold extrusion of pasta dough.
Methodology & Formulas
The design process relies on the empirical swell ratio, which relates the final product dimension to the die aperture dimension. The following mathematical framework is used to determine the die geometry and validate the process feasibility, taking into account the flatbread production parameters.
The primary design equation for the die aperture side length is derived from the swell ratio B:
To ensure the validity of the extrusion process, the Reynolds number (Re) is calculated using the hydraulic diameter (Dh), which for a square duct is equivalent to the die side length; this calculation also supports product surface quality control by indicating the flow regime.
\[ D_{h} = a_{\mathrm{die}} \]
\[ Re = \frac{\rho \cdot V \cdot D_{h}}{\mu} \]
The pressure drop (ΔP) across the die is estimated using the friction factor (f) for laminar flow in a square duct:
\[ f = \frac{56.92}{Re} \]
\[ \Delta P = f \cdot \left( \frac{L}{D_{h}} \right) \cdot \left( \frac{\rho \cdot V^{2}}{2} \right) \]
Parameter
Constraint / Regime
Threshold
Swell Ratio (B)
Empirical Range
\( 1.1 \le B \le 1.3 \)
Reynolds Number (Re)
Laminar Flow
\( Re < 2000 \)
Pressure Drop (ΔP)
Equipment Limit
\( \Delta P < 100 \ \text{bar} \)
The swell ratio is an empirical parameter that must be measured experimentally for each dough recipe and processing condition. To determine B:
Extrude the dough through a die of known aperture dimensions.
Measure the cross‑sectional dimensions of the extrudate after it has fully relaxed (cooled and set).
Calculate B = (extrudate dimension) / (die dimension).
Repeat at the target extrusion temperature, moisture content, and throughput, as these factors significantly influence viscoelastic behaviour.
Maintain a database of swell ratios for your formulations; typical values for pasta dough fall between 1.1 and 1.3.
Ensuring laminar flow (Re < 2000) is important for several reasons:
The friction factor correlation \(f = 56.92/Re\) used to estimate pressure drop is valid only for fully developed laminar flow in a square duct.
Laminar flow promotes uniform velocity profiles across the die exit, which helps maintain consistent product dimensions and minimises internal stresses.
If the Reynolds number approaches or exceeds the transition value, secondary flows and turbulence can cause irregular die swell, surface defects, and unpredictable pressure losses.
Should the flow become transitional or turbulent, a different friction factor model (e.g. Colebrook or Blasius) would be required, and the simple design methodology presented here would no longer be conservative.
The die length plays a dual role in extrusion die design:
Pressure drop: \(\Delta P\) is directly proportional to the \(L/D_h\) ratio. A longer die increases the pressure required to push the dough through, which must remain below the extruder’s maximum operating pressure (e.g. 100 bar).
Flow development: A longer die allows the velocity profile to become fully developed and relax entrance effects. This leads to a more uniform velocity distribution at the die exit, reducing shape distortion.
Die swell control: Increased residence time in a longer die can partially relax elastic stresses, potentially reducing the swell ratio. However, excessive length may cause undesirable heating or degradation of shear‑sensitive materials.
Engineers must balance these factors: a die that is too short may produce inconsistent product geometry, while one that is too long may exceed the available pressure budget.
Worked Example: Die Shape Selection for Square Pasta
A pasta manufacturing plant requires a square die to produce pasta with a final side length of 10.0 mm. The extrusion line operates at a velocity of 0.02 m/s and uses a die length of 50.0 mm. The dough is modeled as a viscoelastic fluid with viscosity \(\mu = 5000.0 \, \text{Pa·s}\) and density \(\rho = 1000.0 \, \text{kg/m}^3\). Empirical tests indicate a swell ratio \(B = 1.2\) for this dough under the given conditions.
Knowns
Desired product side length: \(a_{\mathrm{product}} = 10.0 \, \text{mm}\)
State the product dimension. The final pasta must have side length \(a_{\mathrm{product}} = 10.0 \, \text{mm}\).
Choose the swell ratio. From empirical data, the swell ratio for the dough is \(B = 1.2\).
Compute the required die aperture side length. Using the relation \(a_{\mathrm{die}} = a_{\mathrm{product}} / B\):
\[
a_{\mathrm{die}} = \frac{10.0}{1.2} = 8.333 \, \text{mm}.
\]
Validate laminar flow. The hydraulic diameter for a square duct is the side length: \(D_h = a_{\mathrm{die}} = 8.333 \, \text{mm} = 0.008333 \, \text{m}\). The Reynolds number is
\[
Re = \frac{\rho V D_h}{\mu} = \frac{1000.0 \cdot 0.02 \cdot 0.008333}{5000.0} = 3.333 \times 10^{-5}.
\]
Since \(Re = 3.333 \times 10^{-5} < 2000\), the flow is laminar, validating the use of the laminar friction factor correlation.
Check pressure drop. For fully developed laminar flow in a square duct, the friction factor is \(f = 56.92 / Re\):
\[
f = \frac{56.92}{3.333 \times 10^{-5}} = 1.7076 \times 10^6.
\]
The pressure drop is then
\[
\Delta P = f \cdot \frac{L}{D_h} \cdot \frac{\rho V^2}{2}.
\]
With \(L = 50.0 \, \text{mm} = 0.05 \, \text{m}\) and \(D_h = 0.008333 \, \text{m}\):
\[
\Delta P = 1.7076 \times 10^6 \cdot \frac{0.05}{0.008333} \cdot \frac{1000.0 \cdot (0.02)^2}{2} = 20.491 \, \text{bar}.
\]
This is well below the equipment limit of 100 bar, confirming feasibility.
Specify the die. The die should have a square aperture with side length 8.333 mm (acceptable tolerance \(\pm 0.05 \, \text{mm}\)).
Final Answer
The required die opening side length is 8.333 mm. This value accounts for the die swell of 20% (\(B = 1.2\)) and ensures the final pasta dimension of 10.0 mm. All validation checks (laminar flow, pressure drop) are satisfied.
"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
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