Reference ID: MET-6986 | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
In the field of Process Engineering, specifically within polymer processing and co‑extrusion design, the accurate prediction of pressure drop and flow characteristics is critical for ensuring uniform layer distribution and structural integrity of the final product. This calculation module focuses on the laminar flow regime of highly viscous polymer melts through circular conduits. By determining the pressure drop and head loss, engineers can size extrusion pumps, specify motor requirements, and optimize die geometry to prevent flow instabilities or excessive shear heating. For detailed guidance on estimating the pressure loss across the breaker plate, see our dedicated analysis of breaker plate pressure drop.
Methodology & Formulas
The analysis assumes a steady, fully developed, incompressible, and laminar flow of a Newtonian fluid. The following mathematical framework is utilized to characterize the hydraulic performance of the extrusion line:
The Reynolds number, which dictates the flow regime, is calculated as:
\[ Re = \frac{\rho \cdot v \cdot D}{\mu} \]
For laminar flow conditions, the pressure drop (\(\Delta P\)) is derived from the Hagen-Poiseuille equation, simplified for average velocity:
\[ \Delta P = \frac{32 \cdot \mu \cdot L \cdot v}{D^{2}} \]
The resulting head loss (\(h_{L}\)), representing the energy dissipation per unit weight of the fluid, is defined as:
\[ h_{L} = \frac{\Delta P}{\rho \cdot g} \]
Finally, the Darcy friction factor (\(f\)) for laminar flow is determined by the relationship:
\[ f = \frac{64}{Re} \]
Parameter
Condition/Regime
Criteria
Flow Regime
Laminar
\(Re \leq 2300\)
Flow Regime
Turbulent/Invalid
\(Re > 2300\)
Physical Validity
Non-zero Flow
\(Re > 0\)
To mitigate interfacial instability, process engineers should focus on balancing the rheological properties of the polymer melts. Key strategies include:
Matching the viscosities of adjacent layers at the specific shear rates encountered in the die.
Adjusting the temperature profile of individual extruders to align melt flow indices.
Optimizing the die geometry to ensure a streamlined flow path and uniform residence time distribution.
Reducing the velocity gradient at the interface by modifying the feedblock channel design.
Layer thickness non-uniformity is typically driven by thermal gradients or mechanical inconsistencies within the co-extrusion system. Common factors include:
Fluctuations in the melt pump speed or pressure stability.
Temperature differentials between the melt streams entering the feedblock.
Improper centering of the die mandrel or internal deckle adjustments.
Variations in the viscosity of raw materials due to batch-to-batch inconsistencies.
Preventing degradation requires careful management of the flow path and thermal environment. You should implement the following:
Eliminate stagnant zones or dead spots in the feedblock where material can accumulate and overheat.
Ensure that the residence time of the most heat-sensitive polymer is kept within its thermal stability limit.
Verify that the internal surfaces of the die and feedblock are polished to prevent material hang-up.
Perform regular purge cycles using a high-viscosity purging compound during material changeovers.
The viscosity ratio is a critical parameter that dictates the flow behavior and layer encapsulation tendencies. If the ratio is too high, the lower-viscosity material will tend to encapsulate the higher-viscosity material, leading to:
Poor control over the final layer distribution.
Increased risk of edge bead formation.
Unpredictable layer thickness ratios across the width of the die.
Worked Example: Laminar Flow in a Co-Extrusion Polymer Pipe
Scenario: A highly viscous polymer melt is transported through a straight pipe in a co-extrusion line. The flow is verified to be laminar (Re < 2300). We calculate the Reynolds number, pressure drop, head loss, and Darcy friction factor using the Hagen-Poiseuille correlation.
Knowns:
Density \(\rho = 950.0\; \text{kg/m}^3\)
Dynamic viscosity \(\mu = 150.0\; \text{Pa·s}\)
Gravity \(g = 9.81\; \text{m/s}^2\)
Pipe diameter \(D = 0.05\; \text{m}\)
Pipe length \(L = 10.0\; \text{m}\)
Average velocity \(v = 0.2\; \text{m/s}\)
Pipe roughness \(\epsilon = 4.5 \times 10^{-5}\; \text{m}\) (not used in laminar correlation)
Step-by-step calculation:
Reynolds number.
\[
Re = \frac{\rho \cdot v \cdot D}{\mu}
\]
With the values above, the result is \(Re \approx 0.06333\). (Because \(Re \ll 2300\), laminar flow is confirmed.)
Pressure drop (Hagen-Poiseuille).
\[
\Delta P = \frac{32 \cdot \mu \cdot L \cdot v}{D^{2}}
\]
Substituting the knowns gives \(\Delta P = 3\,840\,000.0\; \text{Pa}\).
Head loss.
\[
h_{L} = \frac{\Delta P}{\rho \cdot g}
\]
From the pressure drop, the head loss is \(h_{L} = 412.039\; \text{m}\).
Friction factor (laminar).
\[
f = \frac{64}{Re}
\]
Using the calculated Reynolds number, \(f = \frac{64}{0.06333\ldots} = 1010.526\).
Final Answer:
Reynolds number, \(Re = 0.06333\)
Pressure drop, \(\Delta P = 3\,840\,000.0\; \text{Pa}\)
Head loss, \(h_{L} = 412.039\; \text{m}\)
Darcy friction factor, \(f = 1010.526\)
Note: The very large friction factor and high head loss are characteristic of an extremely viscous melt moving at low velocity through a small-diameter pipe.
"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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