Introduction & Context

In the production of flatbreads and crispbreads, the extrusion process is critical for defining the final product geometry and texture. The configuration of the slit die determines the uniformity of the dough sheet before it enters the baking tunnel, and careful die shape selection for forming is essential to optimize flow characteristics. This calculation is essential for Process Engineers to ensure that the extruder pressure remains within mechanical limits while achieving the target sheet thickness, accounting for non-Newtonian rheological behavior and die swell phenomena.

Methodology & Formulas

The following methodology utilizes the Power Law model to characterize the dough's non-Newtonian flow through a slit die geometry. The process assumes laminar flow conditions and accounts for the Rabinowitsch correction to determine the true wall shear rate.

1. Volumetric Flow Rate
The volumetric flow rate is derived from the mass throughput and the density of the dough:

\[ Q = \frac{\dot{m}}{\rho} \]

2. Die Gap Determination
To achieve a target sheet thickness, the die gap must be adjusted for the die swell ratio, which represents the elastic recovery of the dough upon exiting the die:

\[ H_{\text{die}} = \frac{H_{\text{sheet}}}{B} \]

3. Shear Rate Calculations
The apparent shear rate for a slit die is calculated first, followed by the Rabinowitsch correction to account for the shear-thinning nature of the dough:

\[ \dot{\gamma}_{\text{app}} = \frac{6 \cdot Q}{W \cdot H_{\text{die}}^2} \] \[ \dot{\gamma}_w = \dot{\gamma}_{\text{app}} \cdot \frac{2n + 1}{3n} \]

4. Viscosity and Wall Shear Stress
Using the Power Law consistency index and flow behavior index, the wall viscosity and resulting wall shear stress are determined:

\[ \eta = K \cdot \dot{\gamma}_w^{\,n-1} \] \[ \tau_w = \eta \cdot \dot{\gamma}_w \]

5. Land Length Calculation
The required land length of the die is calculated based on the maximum allowable pressure drop across the die to ensure the extruder operates within safe mechanical limits. The force balance for a slit die yields:

\[ L = \frac{\Delta P_{\text{max}} \cdot H_{\text{die}}}{2 \cdot \tau_w} \]

Parameter	Condition / Threshold
Shear Rate	10 s^-1 ≤ γ̇_app ≤ 1000 s^-1
Power Law Index	0.2 ≤ n ≤ 0.6
Die Swell Ratio	1.1 ≤ B ≤ 1.5
Die Temperature	30 °C ≤ T_die ≤ 40 °C
Aspect Ratio	W / H_die > 20 (preferably > 100)
Land-to-Gap Ratio	20 ≤ L / H_die ≤ 100

To ensure optimal machineability and prevent tearing during the sheeting process, process engineers should maintain the following moisture parameters:

Target hydration levels between 34% and 38% depending on flour protein content.
Maintain a dough temperature of 22°C to 25°C to prevent premature yeast activity.
For shelf-stable crispbread and flatbread products, monitor the water activity index (a_w) to ensure it remains below 0.65 after baking to prevent mold growth during storage.

Adjusting the oven profile requires a balance between belt speed and thermal input. Follow these guidelines:

Increase belt speed for thinner flatbreads to prevent over-baking and brittleness.
Lower the top zone temperature by 5°C to 10°C if the product exhibits excessive blistering.
Ensure the dwell time is calibrated to achieve a final moisture content of less than 12%.

Rapid cooling is essential to prevent mold growth and packaging deformation. Process engineers should implement the following:

Utilize a multi-stage cooling conveyor to reduce product temperature from 90°C to 30°C within 8 minutes.
Maintain an ambient humidity level below 50% in the cooling tunnel area.
Verify that the final core temperature is within 2°C of the ambient room temperature before sealing.

Worked Example: Slit Die Configuration for Crispbread Production

In this example, we design a slit die for producing crispbread sheets at a target thickness of 3.0 mm and width 500 mm, with a mass throughput of 200 kg/hr. The die is isothermal at 30 °C, and the dough follows a power-law rheology with consistency index K = 5000 Pa·sⁿ and flow index n = 0.4. The maximum allowable die pressure drop is 50 bar (5×10⁶ Pa). A die-swell ratio of 1.3 is assumed for the crispbread dough.

Known Parameters

Target sheet thickness: \(H_{\text{sheet}} = 0.003 \ \text{m}\)
Sheet width: \(W = 0.5 \ \text{m}\)
Mass throughput: \(\dot{m} = 200 \ \text{kg/hr} = 0.0556 \ \text{kg/s}\)
Dough density: \(\rho = 1100.0 \ \text{kg/m}^3\)
Consistency index: \(K = 5000.0 \ \text{Pa·s}^n\)
Power-law index: \(n = 0.4\)
Die-swell ratio: \(B = 1.3\)
Maximum die pressure drop: \(\Delta P_{\text{max}} = 5\,000\,000 \ \text{Pa}\)
Die temperature: \(T_{\text{die}} = 30.0 \ \text{°C}\)

Step-by-Step Calculation

Step A – Volumetric flow rate:
\[ Q = \frac{\dot{m}}{\rho} = \frac{0.0556}{1100.0} = 5.055 \times 10^{-5} \ \text{m}^3/\text{s} \]
Step B – Die gap accounting for die swell:
\[ H_{\text{die}} = \frac{H_{\text{sheet}}}{B} = \frac{0.003}{1.3} = 0.002308 \ \text{m} \]
Step C – Apparent and true shear rates:
Apparent shear rate: \[ \dot{\gamma}_{\text{app}} = \frac{6 \cdot Q}{W \cdot H_{\text{die}}^2} = \frac{6 \times (5.055 \times 10^{-5})}{0.5 \times (0.002308)^2} = 113.9 \ \text{s}^{-1} \] Rabinowitsch correction for a slit die: \[ \dot{\gamma}_w = \dot{\gamma}_{\text{app}} \cdot \frac{2n + 1}{3n} = 113.9 \times \frac{2(0.4) + 1}{3(0.4)} = 113.9 \times \frac{1.8}{1.2} = 170.8 \ \text{s}^{-1} \]
Step D – Viscosity and wall shear stress:
Viscosity at the wall: \[ \eta = K \cdot \dot{\gamma}_w^{\,n-1} = 5000.0 \times (170.8)^{0.4 - 1} = 5000.0 \times (170.8)^{-0.6} = 228.8 \ \text{Pa·s} \] Wall shear stress: \[ \tau_w = \eta \cdot \dot{\gamma}_w = 228.8 \times 170.8 = 39\,080 \ \text{Pa} \]
Step E – Required land length under pressure limit:
\[ L = \frac{\Delta P_{\text{max}} \cdot H_{\text{die}}}{2 \cdot \tau_w} = \frac{5\,000\,000 \times 0.002308}{2 \times 39\,080} = 0.1476 \ \text{m} \] Rounded to the nearest millimetre: \(L = 0.148 \ \text{m}\) (148 mm).
Step F – Geometry validation:
Aspect ratio: \[ \frac{W}{H_{\text{die}}} = \frac{0.5}{0.002308} = 216.7 \] Land-to-gap ratio: \[ \frac{L}{H_{\text{die}}} = \frac{0.1476}{0.002308} \approx 64.0 \] Both ratios are well within the recommended ranges (aspect ratio > 100 for sheet uniformity, land-to-gap ratio between 20 and 100).

Final Answer

The slit die should have a gap of 2.31 mm (2.308 mm nominal), a width of 500 mm, and a land length of 148 mm. With a maximum die pressure drop of 50 bar, this geometry produces a crispbread sheet of 3.0 mm thickness at a throughput of 200 kg/hr.

"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