Moving Sheet With Variable Thermal Conductivity Emerging From a Slot

In this work, we treat the heat transfer process of a continuously moving flat sheet with variable thermal conductiviy, emerging from a slot in contact with a quiescent fluid. Due to the thermal conductivity of the sheet, strong longitudinal and transverse temperature gradients arise within it, requiring to solve simultaneously the energy equation of the sheet and fluid equations. The momentum and energy balance equations are reduced to a non-linear system of partial differential equations with four parameters: the Prandtl number, Pr, a non-dimensional sheet thermal conductance β, a non-dimensional parameter identifying the effect of the variable thermal conductivity γ and a suitable Peclet number, Pe. For finite values of the parameter γ we recognize the limits β≪1 and βPe²≪1 as the most relevant from a practical point of view. In this case, the problem is governed by an universal integro-differential equation in order to obtain the spatial evolution of the sheet temperature as a function of the nondimensional longitudinal coordinate.