Flow-rate based method for velocity of fully developed laminar flow in tubes

Abstract

This work proposes an explicit method to determine velocity profiles of non-Newtonian fluids flowing in the laminar fully developed regime through a straight tube with a circular cross section. An integral expression for local velocity is derived by introducing the concept of a core-flow rate at a point in the tube as the rate of the partial flow passing through a coaxially centered circular cross section with a radius equal to the radial position of that point. In this approach, the velocity is expressed as the difference between the mean core velocity from the core-flow rate and the mean velocity of the flow through a virtual tube with its wall at the corresponding radial position. First, this method has been verified using velocities for the Newtonian and power law models. Then, it has been applied to velocity calculations for fluids obeying the Carreau, Cross, and Phan–Thien–Tanner (PTT) models. Moreover, the velocity result for the Carreau model has been applied to the viscosity reconstruction by the inverse parameter estimation from the measured velocity. Specifically, for the PTT model case, the slip boundary condition has also been considered. The mean velocity of the virtual tube flow can be determined by the difference between the measured and the core velocities. Therefore, a measured velocity profile allows the acquisition of the apparent shear rates of many virtual tube flows. These virtual tube flows have different wall shear stresses because of the differences in radii despite having the same pressure gradient. With the apparent shear rate and the wall shear stress, the Rabinowitsch correction has been conducted to retrieve the true wall shear rate, which facilitates accurate estimation of the viscosity and the Reynolds number. It has been found that the obtained Reynolds number closely follows the generalized Reynolds number.

Effective viscosity and Reynolds number of non-Newtonian fluids using Meter model

“…Here, η 0 and η ∞ are zero and infinite shear viscosity respectively (Meter and Bird, 1964). Although Sochi (Sochi, 2015) and Kim (Kim, 2018) proposed analytical solutions for Carreau and Cross fluid flow through a circular tube and Peralta et al, (Peralta et al, 2014(Peralta et al, , 2017 proposed analytical solution for flow over free-draining vertical plate, the exact analytical solution is absent for estimation of the radial velocity profile, average velocity and volumetric flow rate of fluid flow in a circular tube/micro-capillary obeying Cross, Carreau, Meter, or Steller-Ivako model. The Reynolds number of non-Newtonian fluids in a circular tube/capillary is commonly defined using the viscosity of the fluid at the wall (Escudier et al, 2005;Kim, 2018), the zero-shear viscosity (Ferrás et al, 2020) or Metzner and Reeds equation (Metzner and Reed, 1955). The shear viscosity of non-Newtonian fluids vary along the radial direction in a fully developed circular capillary.…”

A pore network modelling approach to investigate the interplay between local and Darcy viscosities during the flow of shear-thinning fluids in porous media

“…where l T;0 and l T;1 are the low shear rates and high shear rates viscosity plateaus, c T is the consistency and n T is the shearthinning index. This model allows for the development of simple analytical solutions to relate effective viscosity to pressure drop during the flow of a non-Newtonian fluid in a capillary, which is not the case of Carreau fluids despite recent progress [11,15,27,28]. Nonetheless, whereas the truncated power law approximates reasonably well Carreau model under low shear rates, the differences are important at the highest shear rates [3].…”

Optimization of Polymer Extrusion Die Based on Response Surface Method

“…The inlet is a fully developed profile with a volumetric flow rate of 5 × 10 −3 m 3 /sec. Velocity is calculated based on the method given by Kim [17].Velocity and temperature profiles are adapted from Wei and Michaeli [5,18,19]. In this method, velocity is given as follows: (9) where:…”

“…In these equations, shear rate and shear stress are substituted by the Carreau-Yasuda viscosity in Equation (8) as mentioned in the study by Kim [17]. Integrations of Equations ( 10) and (11) are solved numerically by the trapezoidal method.…”