MATHEMATICAL MODEL OF A ROLLER PAIR FOR APPLICATION OF POLYMER COATING

Abstract

The paper presents a mathematical model of a roll pair used for applying polymer coatings onto textile substrates. The study focuses on the hydrodynamic and viscoelastic mechanisms governing pressure distribution in the inter-roll gap and the formation of the final coating thickness. Roll-type machines, including rolls and calenders, are widely employed in polymer processing; however, predictive models that quantitatively link roll geometry, rheological properties of polymers, and coating thickness remain insufficiently developed.

The proposed model is based on the lubrication approximation for thin-gap flow, reducing the governing equations to a Reynolds-type formulation describing pressure evolution within the roll gap. The geometry of the inter-roll clearance is defined as a function of roll radius, allowing the investigation of diameter variation in the range of 5–20 cm. The polymer material is treated as an incompressible viscoelastic medium. To account for time-dependent deformation and elastic recovery after exiting the contact zone, a Kelvin–Voigt viscoelastic formulation is incorporated into the model.

A engineering approach is introduced by limiting the maximum contact pressure to realistic industrial values (1–5 MPa) and by employing effective viscoelastic parameters corresponding to polyurethane-based coating systems. The final coating thickness is determined as a combination of the geometric gap and pressure-induced viscoelastic recovery, which depends on the contact time between the polymer layer and the rolls. The contact time is shown to increase with roll diameter, leading to a nonlinear dependence of coating thickness on roll size.

Numerical simulations demonstrate that smaller roll diameters generate steeper pressure gradients and higher peak pressures, which intensify elastic recovery and may cause instability in coating thickness. Conversely, larger diameters provide smoother pressure distributions and improved thickness control. The model predicts realistic coating thickness values in the range of 90–145 μm for typical processing parameters, which correspond to industrial textile coating operations.

The developed mathematical framework establishes a quantitative relationship between roll geometry, rheological characteristics of the polymer, operating speed, and resulting coating thickness. The proposed approach can be used for the optimization of roll pair design and process parameters in continuous coating lines. Furthermore, the model provides a basis for the development of closed-loop thickness control systems and for future extension toward two-dimensional simulations including substrate deformability. The obtained results contribute to improving the accuracy of engineering calculations for roll coating systems and enhance the theoretical understanding of viscoelastic effects in thin-layer polymer deposition processes.