Viscous heating correction for thermally developing flows in slit die viscometry

Abstract

In the thermally developing region, dπ yy /dx| y=h varies along the flow direction x, where π yy denotes the component of stress normal to the y-plane; y = ±h at the die walls. A finite element method for two-dimensional Newtonian flow in a parallel slit was used to obtain an equation relating dπ yy /dx/ y=h and the wall shear stress σω0 at the inlet; isothermal slit walls were used for the calculation and the inlet liquid temperature T0 was assumed to be equal to the wall temperature. For a temperature-viscosity relation η/η0 = [1+β(T−T0]−1, a simple expression [(hdπ yy /dx/ y=h )/σ w0] = 1−[1-F c(Na)] [M(χ)+P(Pr) ·Q(Gz −1)] was found to hold over the practical range of parameters involved, where Na, Gz, and Pr denote the Nahme-Griffith number, Graetz number, and Prandtl number; χ is a dimensionless variable which depends on Na and Gz. An order-of-magnitude analysis for momentum and energy equations supports the validity of this expression. The function F c(Na) was obtained from an analytical solution for thermally developed flow; F c(Na) = 1 for isothermal flow. M(χ), P(Pr), and Q(Gz) were obtained by fitting numerical results with simple equations. The wall shear rate $$\dot \gamma _{w0} $$ at the inlet can be calculated from the flow rate Q using the isothermal equation.