Abstract
The objective of this work is to study the effect of confining walls on the free settling of spherical particles along the axes of five vertical cylindrical tubes in Newtonian and non-Newtonian liquids. Experimental results were predominantly obtained in the particle flow region between the Stokes and the Newton regimes (intermediate region) and displayed Reynolds numbers in the ranges 0.7<Re<311 for aqueous glycerine solutions and 0.2<Re<340 for aqueous carboximethilcellulose solution considering the range of 0<<FONT FACE="Symbol">b</font><0.55. These results allowed to analyze the procedure for estimating vt<FONT FACE="Symbol">¥ </font>as a function of b based on vt data and present correlations for estimating the wall effects on the fall of spheres in Newtonian liquids as well as for the characteristic shear rate of spheres in non-Newtonian fluids.
wall effects; spherical particles; Newtonian and non-Newtonian liquids
Wall effects on the terminal velocity of spherical particles in Newtonian and non-Newtonian fluids
C. H. ATAÍDE, F. A. R. PEREIRA and M. A. S. BARROZO
Universidade Federal de Uberlândia, Programa de Pós Graduação em Engenharia Química, Bloco K,
Campus Santa Mônica, 38400 089, Uberlândia - M.G, Brazil
E-mail chataide@ufu.br
E-mail masbarrozo@ufu.br
(Received: August 3, 1999; Accepted September 13, 1999)
Abstract - The objective of this work is to study the effect of confining walls on the free settling of spherical particles along the axes of five vertical cylindrical tubes in Newtonian and non-Newtonian liquids. Experimental results were predominantly obtained in the particle flow region between the Stokes and the Newton regimes (intermediate region) and displayed Reynolds numbers in the ranges 0.7<Re<311 for aqueous glycerine solutions and 0.2<Re<340 for aqueous carboximethilcellulose solution considering the range of 0<b<0.55. These results allowed to analyze the procedure for estimating vt¥ as a function of b based on vt data and present correlations for estimating the wall effects on the fall of spheres in Newtonian liquids as well as for the characteristic shear rate of spheres in non-Newtonian fluids.
Keywords: wall effects, spherical particles, Newtonian and non-Newtonian liquids.
INTRODUCTION
It is well known that the presence of walls or finite boundaries exerts a retarding effect on the terminal velocity of particles in a viscous medium. Knowledge of this effect is needed to deduce the net hydrodynamic drag on the particle due solely to the relative motion between the particle and the fluid medium. It is customary to introduce a wall factor, fw, to quantify the extent of wall effects on the terminal velocity of a particle. One of the simplest definitions of the wall factor is the ratio of the terminal velocity of a particle in a bounded medium to that of a particle in an unbounded medium: fw = vt / vt¥ .
Works developed using Newtonian fluids can be cited from the literature, e.g., Newtons classic work and the study proposed by Munroe (Fidleris and Whitmore, 1961), which as early as 1888 investigated the wall effects on the free-settling velocity of spherical particles in Newton regime (Re > 1000). In 1921, Faxen quantified the retarding effect of the wall on the motion of spherical particles in Stokes regime (Re <0.10) under another condition; however, this study was developed for a narrow range of b (0 < b < 0.2), (Fidleris and Whitmore, 1961). Not long afterwards, Francis (1933) established a correlation for the same flow region (Re< 0.10), though over a broad range of diameter ratios 0< b<0.8. Gurel corroborated Francis study, stating that this could be extended over the range of 0 < b < 0.9
(Fidleris and Whitmore, 1961).
In a study based on extensive experimental work, Fidleris and Whitmore (1961) compared several equations for the prediction of wall effects. As a result of their study, the correlations proposed by Munroe and Francis were established. However, there is little information concerning the prediction of wall effects within the intermediate region (0.1<Re<1000). In this study the analyses for this region were based on graphic interpretations.
All the aforementioned studies present experimental results for fw as a function of b ; however, the correlations in these studies are restricted to either a specific particle flow region or a range of ratios between diameters (b ). Almeida (1995) presented an equation which estimates the effect of finite boundaries for the Reynolds number within the intermediate range for Newtonian fluids, considering not only the b effect, but also the particle shape and the influence of the Reynolds number.
For non-Newtonian fluids, the work related to the study of particle motion developed by Massarani and Silva Telles (1978) and Laruccia (1990) are worth mentioning. Using another methodology, Chhabra et al. (1981) quantified the effect of rigid boundaries on the fall of spherical particles moving in creeping regime, for the b range of 0 <b< 0.5.
However, there is little information in the literature about the fall of particles experiencing wall effects in regions other than the laminar flow in non-Newtonian fluids. Taking these facts into account, the goals of this work are as follows:
(a) to analyze the methodologies for estimating vt¥ by extrapolating vt expressions as functions of b and compare the results obtained with the values predicted by a classic correlation available in the literature.
(b) to acquire an experimental data set and compare it with Almeidas expression (1995) for the intermediate region in Newtonian fluids.
(c) to obtain another set of experimental data, and by using a non-Newtonian fluid and the effective viscosity concept, propose a correlation for the prediction of the characteristic shear rate experienced by the fluid due to particle motion.
METHODS AND MATERIALS
Experimental Unit
With the purpose of evaluating the wall factor (fw =vt/vt¥ ) as a function of the diameter ratio (b =dP/DT), an experimental apparatus composed of four glass tubes with a length of 1500 mm and different inside diameters was used, as can be seen in Figure 1. The terminal velocity (vt) in each fall tube can thus be determined and utilized to estimate the terminal velocity in an unbounded medium (vt¥ ).
Experimental Settings
Careful determination of terminal velocity is of crucial importance, since occasional errors at this stage would propagate and jeopardize the final findings. Bearing this in mind, some care must be taken with the following:
(a) Inlet length (L1). Defined as the distance that the particle should travel before reaching an equilibrium of forces (weight, buoyancy and drag), thereby beginning free-settling at a constant velocity. The inlet length used in this work was 850 mm; the choice of this value for L1 was empirical for both sorts of fluid behaviour.
(b) Fall times. The precision of the measurement of the velocity is directly related to the precision of the measurement of the time taken by the particle to travel a known distance (after traveling L1). Aiming to assure the precision of the time measurement and to eliminate the human error, the unit was coupled with eight digital stopwatches, commanded by photosensors (Muccillo-8800â) and reading up to 10-3 s.
(c) Absence of air bubbles. During the filling of the tubes with the test fluid, countless air bubbles are trapped. In order to remove them, the fall tubes were filled with the test fluids 24 hours prior to beginning the drop tests.
(d) Thermal equilibrium. To avoid occasional temperature oscillations in the fluid during the experiments, the experimental apparatus was set in a cooled compartment in which the temperature could be controlled. In order to allow thermal equilibrium to be reached, the cooling system was turned on 24 hours prior to the commencement of the tests.
Fluids
In the present work, Newtonian fluids - distilled water and aqueous glycerine solutions - and non-Newtonian fluids - aqueous carboximethil-cellulose - were used as the test fluids.
The viscosity of the Newtonian fluids and the rheological properties of the polymeric solutions were obtained using a cone-plate Brookfieldâ rheometer, model RVDV-III, coupled with a previously gauged constant temperature bath. The power-law model was chosen as a model of rheological representation, due to its suitability to the solution employed (r2>0.99), simplicity (two-parameter model) and widespread applicability in the literature. The densities of the solutions utilized were measured by means of pycnometers.
Particles
In this work 30 spherical particles of several sizes ranging from 6.92 to 35.00 mm and made of materials such as Teflon, glass, PVC, steel, brass, lead, ceramic and porcelain were used. The diameters of the particles were measured using a digital micrometer, whereas the density of each test particle was measured by its respective mass/volume ratio.
Fifth Tube
With the purpose of obtaining experimental data closer to the situation where b tends to zero, a fifth glass tube with a length of 1500 mm and an inside diameter of 300 mm was added to the experimental apparatus. The fall times of particles in this tube were obtained manually, because the light bundle that commands the stopwatches is not intense enough to cross the liquid medium for the distance of the tube diameter.
For the manually performed measures it was established that each particle would enter the reading area, set at 500 mm for this work, after traveling the inlet length (L1). Four operators equipped with digital stopwatches (brand: Mondaine), reading up to 10-2 s, registered the fall times. The data obtained in this tube, however, were useful for elucidative purposes only, and were not incorporated the data used for estimating vt¥ .
Effective Viscosity
To determine particle flow in non-Newtonian fluids, it is usual to extend the definition of the classic Reynolds number by substituting dynamic viscosity for effective viscosity:
(1)
Adopting the power-law model for function f(l*) enables incorporation of effective viscosity in the Reynolds number as follows:
(2)
Characteristic Shear Rate
The characteristic shear rate can be determined by the experimental measure of the terminal velocity of the particle with the aid of a correlation of the Reynolds number as a function of the drag coefficient. In this approach, Laruccia (1990) and Almeida (1995) proposed expressions to estimate the shear rate that the fluid experiences owing to the fall of the particle.
Studying the dynamics of isolated spherical and non-spherical particles, Larrucia (1990) presented the following correlation for the estimate of l* (Stokes regime):
Similarly, investigating the motion of spheres along the main axes of tubes, Almeida (1995) obtained the expression below for l * prediction (for b <0.5):
(4)
RESULTS AND DISCUSSION
Terminal Velocity
Based on a set of 220 experimental data measurements of velocity, the typical result of the variation in terminal velocity with particle diameter may be represented. Figure 2 illustrates the terminal velocities of eleven steel particles in free fall through a glycerine solution in a tube with an inside diameter equal to 39.50 mm.
In analyzing Figure 2, one can notice two regions of wall effects. The first is for b<0.5, where an increase in velocity occurs with the increase in particle diameter. The second region corresponds to b>0.5, where terminal velocity decreases as particle diameter increases.
Another way of elucidating the same effects exerted by the wall on velocity is to consider a particle moving through an aqueous glycerine medium inside five fall tubes with different inside diameters, as shown in Figure 3. The same tendency as that observed previously is verified in this figure.
Terminal Velocity in an Unbounded Medium
In order to analyze the methods adopted in the literature to estimate terminal velocity in an unbounded medium, we compared two techniques (linear and non-linear extrapolation) with the expression proposed by Haider and Levenspiel (1989), described by Equation (5). Table 1 presents the results obtained for a glycerine solution.
where
(5)
For the case of non-Newtonian fluids, the results of terminal velocities obtained for spheres in a CMC solution displayed a deviation of 13.84% between the two techniques of vt¥ determination.
Average deviations between the techniques for vt¥ estimation, compared with the Haider and Levenspiel correlation.
WALL EFFECTS
Newtonian Fluid
By analyzing the equations proposed by Munroe (Fidleris and Whitmore, 1961) and Francis (1933), Equations (6) and (7), respectively, the strong non-linearity of the wall factor in relation to the diameter ratio (b) can be verified.
(6)
(7)
Figure 4 presents the experimental results of fw, obtained using the non-linear technique for vt¥ determination, together with the plots of Equations (6) and (7).
Based on the behaviour of the curves plotted in Figure 4, it can be concluded that the Stokes regime presents a curvature with upward concavity, whereas the opposite is verified for Newton regime. The intermediate range between the two regions presents a two tendencies (see Figure 3), i.e., for b<0.5 the tendency of Newton regime curve is observed, while for b>0.5 the curvature is similar to that of Stokes regime. To improve the precision of the vt¥ estimation by a second-order polynomial (non-linear), only values displaying b<0.5 were used.
The results obtained were compared with the study proposed by Almeida (1995), according to the following equation:
where
; 
(8)
The values of the experimental wall factors presented deviations of 16% when compared to the above-mentioned correlation. After re-estimation of the parameters, the new expression (Equation 9) presented a quadratic correlation coefficient (r2) of 0.96 and an average deviation of 3.85% between the predicted values and the experimental data.
where
; 
(9)
The ranges of validity of the parameters for the new equation are 0.38<Re<310.7 and 0<b<0.61.
The fluid in this case was an aqueous glycerine solution with a dynamic viscosity of 0.370 Pa s, as determined by a Brookfield viscometer.
Correlation for Re as a Function of CD and b
According to the methodology proposed by Massarani (1997), it is possible to determine l* with the CD values. Based on this principle, an expression for the Reynolds number similar to that proposed by Massarani (1997), was employed in this study. The experimental data points obtained in this work were fitted to a new equation, which is a function of drag coefficient and diameter ratio (r2=0.90) for values of K1=1.0 and K2=0.43.
(10)
The use of this methodology eliminates dependence on vt¥ for analyses in studies of this nature.
Non-Newtonian Fluids
As a way of extending the results obtained for Newtonian fluids, the methodology proposed by Massarani (1997) was employed. An expression was obtained to estimate the characteristic shear rate of the particle on the fluid (r2 =0.88), as given by the following equation:
(11)
The validity ranges of the parameters are 0.07 <vt/dp <65.55 s-1, 0 <b< 0.55 and 2.18 <rs/r<11.60.
The fluid used for this case was a CMC solution. The rheological parameters for the power-law model were K=3.90 Pa sn and n=0.465.
In this equation it can be observed that l* as a function of vt/dp was influenced by b and another dimensionless number [(rs-r )/r]. The incorporation of the latter is due to the differentiated behaviour of l* for different flow ranges.
CONCLUSIONS
Regarding the findings of this work, the conclusions can be stated as follows:
(a) a correlation for estimating the Reynolds number as a function of the drag coefficient and the spherical particle diameter/tube diameter ratio was proposed. The use of this correlation can eliminate the troublesome dependence on vt¥ for analyses in similar studies.
(b) a correlation for estimating the wall factor in Newtonian liquids, similar to that proposed by Almeida (1995), was presented. The re-estimated parameters resulted in an equation with a good fit (r2=0.96) and a small average deviation (3.85%).
(c) an expression was presented for the prediction of the characteristic shear rate associated with the physical situation of falling spheres in non-Newtonian liquids, which accounted for not only for the particle and tube diameters, but also particle and fluid densities.
NOMENCLATURE
A parameter in Equation (8)
B parameter in Equation (8)
CD drag coefficient
dp diameter of particle (m)
d* parameter in Equation (5)
DT inside diameter of fall tube (m)
fw wall factor: fw = vt / vt¥
g local acceleration due to gravity (m/s2)
K power law consistency index of the fluid (Pa sn)
K1 parameter in Equation (10)
K2 parameter in Equation (10)
n power law flow behaviour index of fluid flow
r2 quadratic correlation coefficient
Re Reynolds number
vt terminal velocity (m/s)
vt¥ unbounded medium falling velocity (m/s)
Greek letters
b diameter ratio: b =dp/DT
f sphericity
l* characteristic shear rate (s-1)
rs particle density (kg/m3)
r liquid density (kg/m3)
m fluid dynamic viscosity (Pa s)
mef fluid effective viscosity (Pa s)
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Publication Dates
-
Publication in this collection
01 Mar 2000 -
Date of issue
Dec 1999
