Amorphous Polymer Melt Rheology and Extrusate Swell Dynamics in Sheet Extrusion
Amorphous extrusate swell dynamics depend on first normal stress differences; controlling land L/H ratio and calender drawdown balances web gauge and shrinkage.
Tanner Equation
Meaning
Mathematical models describe the expansion of polymer melts once they exit a die into a surrounding low pressure zone. The tanner equation relates the ratio of the extrudate diameter to the die diameter based on the stored elastic stresses within the fluid. It governs the engineering of die profiles to ensure that the finished item hits the nominal size target on the first attempt.
The calculation applies specifically to viscoelastic fluids where the first normal stress difference is a known factor during the extrusion process. It excludes simple fluids like oils where no elastic recovery exists and no expansion beyond the exit occurs.
Expansion Mechanics
Melted plastic holds memory of the previous deformations it experienced while flowing inside the metal tool. Use of the tanner equation allows scientists to quantify how the diameter of a rod or profile will grow once it enters the cooling phase. The input values rely on the relationship between the shear stress and the normal stress components of the specific resin grade.
If the resin exhibits high elasticity, the calculated expansion is greater than for shorter or simpler molecular chains. This prediction helps avoids multiple expensive rework cycles of the steel die plate. It provides a formal target for die designers who otherwise would rely entirely on trial and error with actual resin on the machine.
This mathematical approach shortens the transition between digital concept and physical reality for new extruded profiles.
Practical Limits
Formula accuracy depends on steady state conditions where the material temperature and flow speed remain consistent during characterization. If these variables drift, the tanner equation might overestimate the real world expansion due to temperature variations. It serves as a strong guide for initial sizing but is often used alongside empirical data from bench tests.
High shear rates can push resins beyond the limits where simple models stay reliable for long durations. Designers adjust the die geometry to move the final part closer to the center of the tolerance window. Factors such as molecular weight and additive loading shift the elastic response and thus shift the result of the calculation.
Successful use of this tool leads to faster tool sign offs and more consistent wall thicknesses across the manufacturing cycle.
Engineering Impact
Tooling budgets benefit from precise predictions that minimize the need for multiple manual tweaks to the metal orifice. Applying the tanner equation correctly reduces the consumption of setup material during the start of a production shift. Moulders use it to find the ideal processing parameters that minimize waste and maximize the visual quality of the extrudate.
The equation also helps identify if a resin will be too difficult to control before thousands of meters of profile are run incorrectly. It highlights the importance of elastic properties in the sourcing process for high specification industrial materials. Companies that rely on this analytical depth tend to have lower rejection rates in the profiles they ship to downstream customers.
The process ends once the cooling geometry matches the calculated dimensions of the equation output.