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 σ_ω0 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.