Non-isothermal flow of polymer melts in a curved tube

The results of a numerical study (using finite differences) of heat transfer in polymer melt flow is presented. The rheological behaviour of the melt is described by a temperature-dependent power-law model. The curved tube wall is assumed to be at constant temperature. Convective and viscous dissipation terms are included in the energy equation. Velocity, temperature and viscosity profiles, Nusselt numbers, bulk temperatures, etc. are presented for a variety of flow conditions.Br — Brinkman number - c specific heat, J/kg K - De — Dean number - E dimensionless apparent viscosity, eq. (14d) - G dimensionless shear rate, eq. (19) - k parameter of the power-law model, °C^–1, eq. (7) - mass flow rate, kg/s - m₀ parameter of the power-law model, Pa · sⁿ, eq. (7) - n parameter of the power-law model, eq. (7) - Nu 2r_p/ — Nusselt number, eqs. (28,31) - p pressure, Pa - Pe — Péclet number - P —(p/)/r_c — pressure gradient, Pa/m - dissipated energy, W, eq. (29) - total energy, W, eq. (30) - r radial coordinate, m - r_c radius of tube-curvature, m, fig. 1 - r_p radius of tube, m, fig. 1 - r_t variable, m, eq. (6) - R dimensionless radial coordinate, eq. (14a) - R_c dimensionlessr_c, eq. (14a) - R_t dimensionlessr_t, eq. (14a) - t temperature, °C - bulk temperature, °C, eq. (27) - t₀ inlet temperature of the melt, °C - t_w tube wall temperature, °C - T dimensionless temperature, eq. (14c) - T_w dimensionless tube wall temperature - T dimensionless bulk temperature - u₁ variable, s^–1, eq. (4) - u₂ variable, s^–1, eq. (5) - U₁ dimensionlessu₁, eq. (18) - U₂ dimensionlessu₂, eq. (18) - v velocity in-direction, m/s - average velocity of the melt, m/s - V dimensionlessv, eq. (14b) - dimensionless, eq. (15) - z r_c — centre length of the tube, m - Z dimensionlessz, eq. (14e) - heat transfer coefficient, W/m² K - shear rate, s^–1, eq. (8) - — shear rate, s^–1 - apparent viscosity, Pa · s, eq. (7) - ₀ — apparent viscosity, Pa · s - angular coordinate, rad, fig. 1 - thermal conductivity, W/m K - melt density, kg/m³ - axial coordinate, rad, fig. 1 - rate of strain tensor, s^–1, eq. (8) - (—p) pressure drop, Pa