Helical flow of molten polymers in a cylindrical annulus

Summary The paper is concerned with an analytical investigation of helical flow of a non-Newtonian fluid through an annulus with a rotating inner cylinder. The shear dependence of viscosity is described by a power law and the temperature dependence by an exponential function.Velocity and temperature profiles, energy input and shear along the stream lines, pressure drop, and torque are presented for the range of input parameters encountered in polymer extrusion.The results of the study can be applied to a mixing element in a screw extruder and for a device to control extrudate temperature and output.Nomenclature a thermal diffusivity m²/s] - b temperature coefficient K^–1], see eq. 4] - c heat capacity J/kg K] - h slot width m] - I ₁,I ₂,I ₃ invariants of the rate of deformation tensor, see eq. 5] - k thermal conductivity J/m s K] - l, L = 1/h length of the slot - l _T ,l _K thermal and kinematic entrance length - m power law exponent, see eq. 3] - M torque m N] - p pressure N/m²] - P dimensionless pressure gradient, see eq. 24] - P _R,P _RZ dimensionless components of the shear stress tensor, see eq. 25] and eq. 26] - r, R = r/r _wa radial coordinate - r _wa, r_wi outer and inner radius of annulus m] - t time s]; dwell time in the annulus - T temperature K] - v , v_r, V_z velocity components m/s] - v ₀ angular velocity at inner wall m/s] - average velocity inz-direction m/s] - V , V_R, V_Z dimensionless velocity components,v /v₀, v_r/v₀, v_z/v₀ - V _z velocity ratio, helical parameter - Y coordinate inr-direction, see eq. 20] - z, Z = z/h Pe axial coordinate - deformation - rate of deformation tensor s^–1] - apparent viscosity N s/m²], see eq. 3] - dimensionless temperature,b (T – T ₀) - azimuth coordinate - ratio of radii,r _wi/r_wa - density kg/m³] - , kl shear stress tensor N/m²] - fluidity m²w/Nw s], see eq. 4] - Gf Griffith number, see eq. 12] - Pe Péclet number, see eq. 13] - Re Reynolds number, - 0 initial state, reference state - equilibrium state - e entrance - wi, wa at surface of inner or outer wall - r, R, z, Z, coordinates - i, j radial and axial position of nodal point in the grid - k, l tensor components Presented at Euromech 37, Napoli 6. 20–23. 1972.With 15 figuresDedicated to Prof. Dr.-Ing. G. Schenkel on his 60th birthday