Abstract

A numerical solution for the entrance region of non-Newtonian flow in annuli

M.C.A.MaiaI, ^* * To whom correspondence should be addressed ; C.A.Gasparetto^II

^IUniversidade Federal do Rio de Janeiro, Escola de Química Centro de Tecnologia, Phone: (55 21) 2562-7576, Fax: (55 21) 2562-7567, Bloco E, sala 203, CEP 21949-900, Rio de Janeiro - RJ, Brasil. E-mail: antun@eq.ufrj.br

^IIUniversidade Estadual de Campinas, Faculdade de Engenharia de Alimentos, Departamento de Engenharia de Alimentos, Campinas - SP, Brasil

ABSTRACT

Continuity and momentum equations applied to the entrance region of an axial, incompressible, isothermal, laminar and steady flow of a power-law fluid in a concentric annulus, were solved by a finite difference implicit method. The Newtonian case was solved used for validation of the method and then compared to reported results. For the non-Newtonian case a pseudoplastic power-law model was assumed and the equations were transformed to obtain a pseudo-Newtonian system which enabled its solution using the same technique as that used for the Newtonian case. Comparison of the results for entrance length and pressure drop with those available in the literature showed a qualitative similarity, but significant quantitative differences. This can be attributed to the differences in entrance geometries and the definition of asymptotic entrance length.

Keywords: annular geometries, velocity, pressure, finite differences, pseudoplastics.

INTRODUCTION

When a viscous fluid in laminar isothermal flow enters an annulus, the velocity changes from a flat profile at the inlet to a fully developed condition at a given distance downstream known as hydrodynamic entrance length. Generally the entrance region length is assumed to be the distance at which velocity reaches 98 or 99% of its value for developed flow.

The study of flow in annuli is of great significance because of its important engineering applications, such as in extrusion systems, heat exchange devices, axial-flow turbomachinery, and the perforation and production of oil and natural gas. A wide variety of fluids of practical interest are non-Newtonian, so the study of non-Newtonian systems in the entrance region of an annulus is of great practical interest.

In spite of the importance of non-Newtonian flow in the entrance region of annular ducts, relatively little experimental and theoretical work has been published. This is due to the difficulty of solving the equation of motion with its nonlinear inertia terms. The nonlinear equations are not amenable to exact solution. Many researchers have extended the approximation methods for the circular entrance flow to the annular cross section. Several techniques have been applied by different researchers to solve entrance flow problems. Fan and Hwang (1966) provided an extensive bibliography of the entrance problem and Fargie and Martin (1971) provided a review of the methods adopted for solving the entrance flow problems for Newtonian fluids; the same was done by Gillard and Bellet (1980) for non-Newtonian fluids. The studies concerned with entrance region flow of non-Newtonian fluids in annuli are summarized in Table 1.

Recently Matras and Nowak (1983) presented a new approximative method for predicting changes in pressure drop for the laminar flow in

the entrance region with power-law fluids in circular tubes. Their solution utilizes a "pseudo-Newtonian" model of the flow under consideration. This technique was used successfully for power-law fluids by Gupta and Agarwal (1993) for developing flow through a circular tube and by Gupta (1987, 1990) for the entrance flow in a straight channel.

FORMULATION OF THE PROBLEM

Consider a viscous, incompressible power-law fluid characterized by

where n is the flow behavior index and K is the consistency index.

The fluid comes from a very large reservoir and enters a horizontal concentric annular duct with internal radius r_i and external radius r_e and having edges at right angles with the reservoir, according to Figure 1. The flow is incompressible, steady, laminar and isothermal. It is assumed that the radial and axial velocity components are both zero at the walls of the annulus and that at the inlet section the axial velocity profile is flat and the radial velocity component is zero.

The boundary layer equations describing the present system are

continuity:

momentum:

The boundary conditions are

For the present analysis u₀ is assumed to be uniform. Substitution of equation (1) into equation (3) yields

or

THE PSEUDO-NEWTONIAN MODEL

Using a suitable transformation, Matras and Nowak (1983) arranged the momentum equation for the power-law case in the same form as that for a Newtonian fluid. They used this approach to solve numerically the problem of the entrance region in a circular tube.

Subsequently, the technique of transformation is developed for the present case of power-law fluid in the entrance region of a concentric annulus.

The Reynolds number is defined in equation (6) according to Kozicki, Chou and Tiu (1966) for the annulus of arbitrary cross sectional area

where e₀ and e₁ are characteristic of the flow geometry and z is the r_i / r_e ratio. The above parameters assume the values e ₁=1/4 and e ₀ =3/4 for tubes and e₁ =1/2 and e₀=1 for slits. For the annulus the values of e ₀ and e ₁ vary with the z ratio, as shown in Table 2.

In the concentric annular flow, the ratio between mean and maximum velocities for developed flows is

where

is the dimensionless radial position of maximum velocity. Equation (7) is valid only for developed flows.

Let us define the transformation factor F such as that

and

similar to the expression of bulk velocity for the power-law model presented by Kozicki, Chou and Tiu (1966). From now on, u*, r*, K* etc. are the transformed values of u, r, K etc, indicating a new flow system which is pseudo-Newtonian.

The mass flow rate is assumed to be the same in both flows; thus

so

and

where z ^* is the new radii relation for r_i* e r_e *:

Governing equations (2) and (3) become respectively

Using the dimensionless variables

equations (23) and (24) may be expressed in the following dimensionless form:

with boundary conditions

where

and

.

THE FINITE DIFFERENCE METHOD

The above equations were solved using the finite difference method as presented by Hornbeck, Rouleau and Osterle (1965). A grid is now imposed on the flow field as shown in Figure 1, and the representations used for the various terms in equation (31) at a point (j,k) are

and equation (31) may be written in the following finite difference form:

where:

For equation (30) the representations used are

and equation (30) becomes

The finite difference form of equation (44) may be written as

Using trapezoidal rule and the boundary conditions

equation (44) may be written as

where =1

Equation (46) may be rewritten as

or

Equation (48) for k=1(1)m with equation (31) give a set of (i+1) simultaneous equations for (i+1) unknowns, viz., U_j+1,k at n mesh points and P_j+1 at section j+1 in terms of known values of U_j,k and P_j at section j. Thus this step-by-step technique determines the velocity and pressure field in the entire flow. Having obtained U values at mesh points, the radial component of velocity, V_j+1,k, at these mesh points is determined from equation (38) and the process is continued downstream. Equations (36) and (48) form a (i+1)x(i+1) matrix, which is of a tridiagonal type except for the row corresponding to equation (48).

RESULTS

Values for F and z *

Table 3 shows the values of F (equation 8) and z * (equation 15) for different values of n and z . The values of F for n=1 were obtained from equation (8) and are very close to one, as was expected. F and z * decrease with the decrease in n, except for z =0.9 because the solution by KOZICKI et al. [14] is not suitable for use in equation (8).

Entrance Length

The entrance length was calculated for the fluids referred to in Tiu and Battacharyya (1974). The results are shown in Table 4. The longest entrance length is associated with the largest Rey, as expected, but the dimensionless results do not directly show this fact. This can only be seen with the dimensional results.

Pressure Drop

Figure 3 shows the pressure drop with X for z = r_i / r_e = 0.1 to 0.9. It can be seen that for the same X value dP/dX increases with the increase in z , as can be observed from the slope of the curve.

The entrance effects on pressure drop are bigger for small values of z , as can be seen by the intersection of the curve with the abciss; dP/dX is smaller for z =0.1 than for z =0.3, but the entrance length is longer in the first case so the total effect of pressure drop in the entrance region is bigger for z =0.1.

As shown in Figure 3, the pressure difference is the same for any value of n because of the Matras and Nowak transformation.

The following expressions were chosen to fit the pressure difference solution:

entrance flow:

developed flow:

The classical proposal for overall pressure loss is given in equation (51), and Table 5 shows the values of the coefficients from equations (49) and (50).

A comparison of equations (50) and (51) gives E(x)=2^A, and Table 6 shows the values of E(x) obtained in this work and others found in the literature. The values of E(x) obtained in this work are lower but all suggest that the pressure drop due to entrance effects is higher for small values of z .

Mishra and Mishra (1977) report that the values obtained by Tiu and Battacharyya (1973), using a linearization method with the geometric parameters of Kozicki, Chou and Tiu (1966), are inconsistent with their results and the results of Lundgren, Sparrow and Starr (1964). But data from Tiu and Battacharyya (1973) obtained by the momentum-energy integral technique are smaller than those obtained through linearization by Sparrow and Lin (1964), which are smaller than those from Lundgren, Sparrow and Starr (1964). Mishra and Mishra (1977) quote that E(x) values obtained by linearization are bigger than those obtained by the momentum-energy integral technique. From the above it is clear that results of the different techniques do not agree, thus resulting in the difficulty to choose any of them as a reference.

The loss coefficient, f , for developed flow is calculated from equation (52) and shown in Table 7, where it is clear that for the same average velocity, f increases as z increases, thus generating a higher dP/dx.

As an illustration, Table 8 shows the results for some real fluids.

Table 9 shows a comparison with experimental work by Tiu and Battacharyya (1974). The agreement between this work, E(x) (1), and their analytical solution, E(x) (2), is important. Since this indicates a good agreement between this work and the only research reported in the literature with a similar geometry.

CONCLUSIONS

The generalized definition of the Reynolds number and the technique of transformation of the equations to obtain a pseudo-Newtonian system enabled its solution using the same technique as that used for the Newtonian case while providing a more simple solution.

Comparison of the results for entrance length and pressure drop with those available in the literature showed a qualitative similarity, but significant quantitative differences. This is attributed to the differences in entrance geometries and the definition of asymptotic entrance length.

NOMENCLATURE

Received: September 4, 2001

Accepted: January 3, 2003

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