Viscous heating correction for thermally developing flows in slit die viscometry

In the thermally developing region, d_yy/dx|_y=h varies along the flow direction x, where _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 d_yy/dx/_y=h and the wall shear stress ₀ 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 /₀ = [1+(T–T₀]^–1, a simple expression [(hd_yy/dx/_y=h)/_w0] = 1–[1-F_c(Na)] [M()+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; 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(), P(Pr), and Q(Gz) were obtained by fitting numerical results with simple equations. The wall shear rate at the inlet can be calculated from the flow rate Q using the isothermal equation.Notation x,y Cartesian coordinates (Fig. 2) - , dimensionless spatial variables [Eq. (16)] - dimensionless variable, : = Gz(x)^–1 - dimensionless variable [Eq. (28)] - t,t^* time, dimensionless time [Eq. (16)] - , velocity vector, dimensionless velocity vector - _x, velocity in x-direction, dimensionless velocity - _y, velocity in y-direction, dimensionless velocity - V average velocity in x-direction - _yy, ^* normal stress on y-planes, dimensionless normal stress - shear stress on y-planes acting in x-direction - _w, _w^* value of shear stress stress at the wall, dimensionless wall shear stress - _w0, _w0^* wall shear stress at the inlet, dimensionless variable - , ^* rate-of-strain tensor, dimensionless tensor - wall shear rate, wall shear rate at the inlet - Q flow rate - T, T₀, temperature, temperature at the wall and at the inlet, dimensionless temperature - h, w half the die height, width of the die - l,L the distance between the inlet and the slot region, total die length - T₂, T₃, T₄ pressure transducers in the High Shear Rate Viscometer (HSRV) (Fig. 1) - P, P2, P3 pressure, liquid pressures applied to T₂ and T₃ - , ₀, ^* viscosity, viscosity at T = T₀, dimensionless viscosity - viscosity-temperature coefficient [Eq. (8)] - k thermal conductivity - C_p specific heat at constant pressure - Re Reynolds number - Na Nahme-Griffith number - Gz Graetz number - Pr Prandtl number