Effect of temperature-dependent consistency index on the exiting sheet thickness in the calendering of power-law fluids

Abstract

In this work we treat theoretically the calendering process of inelastic (power-law fluid) sheets of finite initial thickness, taking into account that the consistency index of the fluid is a well-defined function of the temperature. We predict the influence of the temperature-dependent consistency index on the exiting sheet thickness in the calendering process. The mass, momentum and energy balance equations, based on the lubrication theory, were nondimensionalized and solved for the velocity, pressure and temperature fields by using perturbation and numerical techniques, where the exiting sheet thickness represents an eigenvalue of the mathematical problem. When the above variables were obtained, the exiting sheet thickness in the calendering process was determined, considering the influence of the temperature variations in the process. The mentioned governing equations contain four dimensionless parameters: the Graetz number, Gz, a geometrical aspect ratio, β, the power law index of the fluid, n, and a parameter that takes into account the effect of the variable consistency index as a function of the temperature and ϵ, defined as the ratio of the Nahme–Griffith number, Na, to the Graetz number. Using the limit of ϵ ≪ 1, the dimensionless exiting sheet thickness of the calendering process have been obtained as a function of the involved dimensionless parameters. The numerical results show that the inclusion of temperature-dependent consistency index effect modifies the dimensionless exiting sheet thickness in about 6.91%.