The flow of power-law fluids in concentric annuli: A full analytical approximate solution

Highlights

• A new analytical approximate expression for pseudo-plastic annular flow calculations.

• Mathematical demonstration of the zero shear rate position's expression.

• A good precision for a large range of geometric and rheological parameters.

• Simple and explicit expression for further sensitivity or parametric investigations.

Abstract

This work deals with the zero shear rate (maximum velocity) position parameter λ of a steady laminar axial annular flow of power-law fluids especially polymeric ones.  λ is involved in the shear rate, velocity profile and the flow rate calculations, which are essential for studies such as viscous dissipation, convective heat transfer and pressure drop prediction in annuli. However, the analytical explicit expression of λ remains unsolved despite some approximate solutions. In this paper, we will provide a simple and analytically demonstrable expression of λ  so that it can be used for parametric investigations, identifications, sensitivity analysis etc. in terms of industrial or laboratory applications.

Introduction

The concentric or eccentric annular geometry with moving or static walls is used in numerous industrial fields such as drilling of oil wells, circulating muds [1], food processing [2], plastics processing [3] and heat transfer equipment [4]. The channel can be helical or curved in some applications for example bio-fluid mechanics or chemical reactors. Combined with different flow behaviors such as Bingham, power-law or viscoelastic models, the problem is sometimes difficult to solve analytically. This paper deals with the use of pseudo-plastic flow behavior in a concentric annular geometry. The mastery of the annulus flow in the industry and laboratories requires the simplest expression of the flow equation in order to help designing geometries and allow parametric investigations between independent parameters [5].

The most simple and representative behavior law used in polymer processing models is the so-called power law [6], [7], [8], which has the advantage to require only 2 parameters as shown in the Eq. (1),

$$\eta =K{\dot{\gamma }}^{n-1}$$

with η being the viscosity, K the consistency coefficient, $\dot{\gamma }$ the absolute value of shear rate and n the power-law index (0 < n  <  1 for pseudo-plastic materials such as polymer [9], blood [10] and whipped cream [11]).

The shortcoming of the power-law model to describe the viscosity at the zero shear rate was pointed out by Frederickson [12]. Nevertheless Bird [13] and McEachern [14] showed that this model describes well the rheological behavior of a laminar axial flow in annuli by comparing it to some experimental data, especially in the shear rate ranges used in polymer processing. Escudier & al. [1,15] further confirmed the use of the power-law model with some experimental data and sensitivity analysis.

The first analysis of flow through annulus was done by Volarovich & Gutkin [16], Laird [17] and in the case of a power-law fluid by Frederickson & Bird [18]. The first analytical relation between the flow rate and the pressure drop was achieved by Frederickson & Bird [18] in an integral form. The results of Frederickson & Bird were substantiated by the measurement of Tiu & Bhattacharyya [19]. However, Frederickson & Bird's solution gave an analytical explicit expression either for integer values of 1/n ratio with a cumbersome power series or for a thin annular slit case. The expression of the thin annular slit case has proved to be surprisingly accurate [20], [21], [22] for a large range of n and σ: the ratio between the inner and the outer radius R of an annular flow (Fig. 1). Later, Bird & al. [9] improved this expression by refining the approximation.

In 1979, Hanks & Larsen [23] solved analytically Frederickson & Bird's integral. They obtained a general analytical explicit flow rate expression depending on the lambda value: the ratio between the zero shear rate (maximum velocity) position λR and the outer radius R (Fig. 1). Nevertheless, Hanks & Larsen's solution requires the use of a numerical procedure in order to calculate the value of lambda.

Several analytical expressions (2), (3) and (4) of lambda exist for respectively n = 1, n = 0 and n tending to infinity [9,18].

$$\lambda \left(n=1,\sigma \right)=\sqrt{\frac{1}{2}\frac{{\sigma }^2-1}{ln\left(\sigma \right)}}$$

$$\lambda \left(n=0,\sigma \right)=\sqrt{\sigma }$$

$$\underset{n\to +\infty }{lim}\lambda \left(n,\sigma \right)=\frac{1}{2}\left(1+\sigma \right)$$

For a more general solution, Frederickson & Bird [18] proposed tabulated values of lambda which were obtained from the numerical integration and from the interpolation between the non-integer values of the 1/n ratio. Many authors like McEachern, Pinho and Daprà [14,24,25] used numerical calculations to determine lambda. Hanks & Larsen [23] also used a table of computed values for lambda depending on the n and σ parameters. Wein & al. [26] found an analytic differential equation for lambda, but the result is a recursive formula based on approximate values determined by the tangent method. Ilicali & Engez [2] proposed to use the Newtonian case ($n=1$) lambda value in the Hanks & Larsen's flow rate expression, when the radius ratio σ is greater than 0.3 Fig. 1). However, their experiments were performed on materials with a power-law index n ranging from 0.62 to 0.97 only. Based on the Eqs. (2) and ((3), David & al. [27] found a pseudo-plastic lambda's approximate expression (5) as one of the most recent and accurate lambda models.

$${\lambda }_{David}\left(n,\sigma \right)=\sqrt{\sigma }+n^{0.37+0.28{\left(1-n\right)}^2+\frac{1}{100\sigma }}\left(\sqrt{\frac{1}{2}\frac{{\sigma }^2-1}{ln\left(\sigma \right)}}-\sqrt{\sigma }\right)$$

With the same numerical approach, David & Filip [27], [28], [29] also obtained other approximate expressions for a dilatant fluid's lambda and for the flow rate. All expressions of lambda are either limited to a small range of (n, σ) values or cannot be fully demonstrated with analytical methods. Besides, semi-analytical solutions constructed with numerical values are cumbersome for analytical use.

In this paper we present a mathematical procedure to obtain a new analytical lambda for pseudo-plastic fluids. The proposed expression of lambda is simple enough in order to allow the parametric analysis and the identification process. A comparison is carried out between the new lambda's expression and numerical values. The precision of the flow rate calculated with our lambda expression is also evaluated against some other solutions for the validation of our model in a large range of power-law index n and radius ratio σ.