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.