This paper develops governing equations for material strain and tension based on a temperature distribution model when the flexible materials (often called webs) are transported on rollers through heat transfer processes within roll-to-roll (R2R) processing machines. Heat transfer processes are employed widely in R2R systems that contain process operations such as printing, coating, lamination, etc., which require heating/cooling of the moving web material. The heat transfer processes introduce the thermal expansion/contraction of the material and changes in the elastic modulus. Thus, the temperature distribution in the moving material affects the strain distribution in the material. Because of change in strain as well as modulus as a function of temperature, tension in the material resulting from elastic strain is also affected by heating/cooling of the web. To obtain the temperature distribution, two basic heat transfer modes are considered: web wrapped on a heat transfer roller and the web span between two consecutive rollers. The governing equations for strain is then obtained using the law of conservation of mass considering the temperature effects. Subsequently, a governing equation for web tension is obtained by assuming the web is elastic with the modulus varying with temperature; an average modulus is considered for defining the constitutive relation between web strain and tension. Since it is difficult to obtain measurement of tension using load cell rollers within heat transfer processes, a tension observer is designed. To evaluate the developed governing equations, numerical simulations for a single tension zone consisting of a heat transfer roller, a web span, and a driven roller are conducted. Results from these numerical model simulations are presented and discussed.

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