Analysis of Steady-State Temperature Distribution in Moving Webs in Roll-to-Roll Manufacturing

Abstract

Abstract. Heat-transfer processes are integral to roll-to-roll (R2R) manufacturing, where flexible substrates (webs) are continuously transported through operations such as printing, coating, embossing, and lamination. This article presents mathematical models for the steady-state temperature distribution in moving webs by considering various heat-transfer mechanisms and associated boundary conditions. Analytical solutions to the steady-state advection–diffusion equation in two dimensions (2D) are derived for the temperature distribution across the thickness and length of the web. In addition, conditions are established for the validity of one-dimensional (1D) approximations (along the length or transport direction), providing a framework for simplified analysis under specific operating scenarios of R2R applications. The models incorporate diverse boundary conditions representative of R2R systems, including free web spans exposed to convective heating/cooling, sections of the web span passing through a heating/cooling zone, and regions wrapped around heated/chilled rollers. These developments are useful for optimizing the thermal management of moving webs and predicting thermal strains that influence transport dynamics (including web tension and velocity), thereby improving process and transport control strategies to minimize web defects. Comprehensive discussions based on parametric analysis are provided to offer guidance on model selection and operational conditions for various R2R scenarios.