Fluid dynamics of air in a packed bed: velocity profiles and the continuum model assumption

Fluid dynamics of air in a packed bed: velocity profiles and the continuum model assumption

A. L. NEGRINI¹, A. FUELBER², J. T. FREIRE³ and J. C. THOMÉO⁴

¹Engenharia Química/UFSCar, ²PPG-EQ/UFSCar

³DEQ/UFSCar. C. P. 676, CEP 13565-905, São Carlos - SP, Brasil

E-mail: [e-mail]

⁴DETA/IBILCE/UNESP, C. P. 136 CEP 15054-000, São José do Rio Preto - SP , Brasil

E-mail: [e-mail]

(Received: July 29, 1999; Accepted:September 19, 1999)

Abstract - Air flow through packed beds was analyzed experimentally under conditions ranging from those that reinforce the effect of the wall on the void fraction to those that minimize it. The packing was spherical particles, with a tube-to-particle diameter ratio (D/d_p) between 3 and 60. Air flow rates were maintained between 1.3 and 4.44 m³/min, and gas velocity was measured with a Pitot tube positioned above the bed exit. Measurements were made at various radial and angular coordinate values, allowing the distribution of air flow across the bed to be described in detail. Comparison of the experimentally observed radial profiles with those derived from published equations revealed that at high D/d_p ratios the measured and calculated velocity profiles behaved similarly. At low ratios, oscillations in the velocity profiles agreed with those in the voidage profiles, signifying that treating the porous medium as a continuum medium is questionable in these cases.

Keywords: Fluid Mechanics, Porous Media, Continuum Model, Fixed Bed, Velocity Measurements, Bed Structure.

INTRODUCTION

Hydrodynamic interactions between fluids and particles have fascinated researchers workers in several branches of engineering ¾ chemical, materials, food, mechanical and hydraulic, among others ¾ owing to their countless potential uses in industry and the variety of transport mechanisms involved. In the specific case of fixed porous matrices, applications range from grain dryers to filters and from catalytic fixed-bed reactors (CFBR) to adsorbers, among others, and a considerable number of research articles has been published on this subject. Meanwhile, there is such a wide diversity of experimental work and theoretical points of view that consensus is lacking, even on basic questions such as the validity of treating the porous medium as a continuum.

According to Bear (1972), a representative volume element (RVE) of a fixed porous matrix should include one particle and its neighborhood. In this approach, a homogeneous medium is roughly classified as one from which a pore could be withdrawn without significantly altering the overall void fraction. In beds where the ratio of the tube diameter (D) to the particle diameter (d_p) is high, e.g., above ten (as in ion-exchange resin columns), the removal of a single particle would indeed cause an almost imperceptible change in the radial void distribution (e). In this situation, the hydrodynamic, thermal and mass-based properties are sufficiently uniform throughout most of the column for the particles and interstices to be treated as a continuous medium. This facilitates analysis of a variety of cases enormously and helps in the collection of experimental data.

However, in beds with very low D/d_p ratios, such as CFBRs whose ratios can be as low as 1.2 (Freiwald, 1992), removing one particle dramatically alters the average voidage. In this case, discrete models, which take into account the intrinsic transport peculiarities of each phase, are more appropriate than homogeneous ones, although the difficulty of obtaining experimental data on the solid phase makes it hard to confirm these models.

It could be said that this doubt about the usefulness of the continuum model dates from the first descriptions of the structure of a porous medium bounded by a tube. Several authors (Benenati and Brosilow, 1962; Haughey and Beveridge, 1966; Zotin, 1985; Govindarao and Froment, 1986; Dixon, 1988; Mueller, 1991; Zou and Yu, 1995) have demonstrated that the radial void distribution in a packed bed of monodisperse particles undergoes damped oscillations, going from one, at the wall of the tube, towards a constant value at some distance from it, as can be seen in figure 1. For some D/d_p, this limiting value may never be reached, indicating that the medium is anisotropic and that the RVE selected may not be valid throughout the bed. Benenati and Brosilow (1962) concluded that in a packed bed of spheres a depth of 5 d_p is needed for the oscillations to subsidetotally, while Zotin (1985) claimed that when Raschig rings are used, the radial voidage profile attains a practically stable value at a mere 0.5 d_p.

Figure 1:
Experimental radial void fraction profiles (Zotin, 1985: n - spheres; l - Raschig rings)

To model this radial distribution in beds of spheres, Mueller (1991) proposed the following equation:

(1)

where J₀ is the Bessel function of the first kind of order zero;

and e_¥ is the damped voidage value, calculated by:

In order to simplify the work on models that include the radial variation of voidage, some authors have made use of the following simple exponential decay equation for:

(2)

where f is a factor which was adjusted to make e equal one at the wall, and g is equal to 6 for spherical and to 8 for nonspherical particles.

For low D/d_p beds, one of the hypotheses most often found in the literature on uniformity of the axial component of the velocity vector (v_z) makes no sense, as several authors (Morales et al., 1951; Schwartz and Smith, 1953; Marivoet et al., 1974; Lerou and Froment, 1977; Vortmeyer and Schuster, 1983; McGreavy et al., 1986; Hunt and Tien, 1988; Fuelber, 1997; Subagyo and Brooks, 1998) have been able to observe significant radial and angular dependence of v_z, as we will confirm below.

As far as experimental techniques to determine v_z are concerned, some authors make measurements above the bed, others within the pores. The former group uses a variety of devices: ring-shaped hot-wire anemometers (Morales et al., 1951; Schwartz and Smith, 1953), probe-shaped hot-wire anemometers (Marivoet et al., 1974; Lerou and Froment, 1977; Drahos et al., 1982) and Pitot tubes (Fuelber, 1997).

Vortmeyer and Schuster (1983) make mesurements inside the bed were made by using laser-Doppler anemometry. These authors point out that readings taken above the bed are considerably distorted and do not accurately reflect the flow pattern inside the porous medium. Possible perturbations of the flow provoked by the presence of the laser-Dopler probe were not mentioned. Thus, both choices of measurement technique should be considered.

As a result of these studies, some equations to predict the radial variation of v_z have been published. One of them was obtained by Fahien and Stankovic (1979) by fitting data from Schwartz and Smith (1953), which were collected in beds with D/d_p ratios ranging from 5.4 to 25.8. This empirical equation, shown bellow, predicts a maximum in the radial velocity profile at a distance of about one d_p from the wall:

(3)

where V is the ratio of superficial velocity as a function of radial position (u) to average superficial velocity ( v );

and a was defined in equation (1).

Later, Vortmeyer and Schuster (1983) solved the differential equation of motion proposed by Brinckman, including the wall-effect term, and used equation (2) to predict the radial distribution of e . This equation, which foresees the velocity peak at about 0.25 d_p from the wall, is written as follows:

(4)

where

and Re is the modified Reynolds number.

Despite differing in terms of the position of the velocity maximum, both equations agree in predicting that near the wall, where the void fraction is greatest, the fluid can reach higher velocities than it can in the central region, a phenomena called channeling.

In this article we examine the radial and angular velocity distributions in packed beds of spheres. To this end, velocities were measured above the bed with a Pitot tube. The profiles obtained are analyzed with the equations from Fahien and Stankovic (1979) and Vortmeyer and Schuster (1983) and also compared with the radial void fraction profiles predicted by the Mueller (1991) and exponential decay equations. In the light of these comparisons, the concept of a continuum in the porous medium is discussed.

MATERIALS AND METHODS

The experimental setup may be visualized in figure 2. Air, the working fluid, was forced to flow throughout the bed by a blower, and its flow rate was measured by a Venturi meter and controlled by a Globe valve. The stream entered the measuring cell through a 60⁰ cone. A detailed description of the equipment can be read in Fuelber (1997). The mean diameters of both the particles and the commercial steel tubes used in the packed beds are shown in table 1.

Figure 2:
Experimental setup (1- radial blower; 2- Venturi flow meter; 3- bypass gate valve; 4- manometer; 5- globe valve; 6- packed bed; 7- Pitot tube; 8- digital micromanometer)

Table 1: Diameters of particles and tubes used and main operating conditions (v – average velocity)

Bed	D/dp	D (cm)	dp (cm)	v(m/s)	Re
1	3.0	8.2	2.85	11.76	34,552
2	4.3	8.2	1.92	7.42	12,916
3	8.2	15.7	1.92	3.77	6,600
4	31.0	8.2	0.26	4.13	871
5	60.0	15.7	0.26	2.91	635

The particles were held in the tube by a steel screen and the height of the porous bed varied between 14 and 15 cm. To prevent fluidization, a screen was placed on top of the bed, and Fuelber (1997) demonstrated that the netting has practically no effect on the velocity measurements.

The particles were packed using a technique for dense packing, recommended by Zotin (1985) to guarantee reproducibility. Gas temperature was not controlled, and owing to the work done by the blower and the pressure drop in the porous medium, in some experiments it rose as high as 99⁰C, measured by a mercury thermometer at the exit. These temperature readings were necessary for correcting the physical properties of the air used to calculate the Reynolds number.

Velocity was measured with an Alnor Pitot tube with an outer diameter of 0.8 cm and a total pressure diameter tap of 0.3 cm. Eight orifices were used to obtain the static pressure, and they were set 6.77 cm back from the tip. The tube was set up with its tip 0.5 d_p above the bed top and connected to an Alnor digital micromanometer. Readings were taken at nine different radial positions and, for each of these, at 72 angular positions equally spaced at 5⁰ intervals for a total of 360⁰. Due to limitations inherent in the probe-moving set-up, it was not possible to maintain identical radial positions for all the beds.

For each of the radial positions, an arithmetic mean was taken of the velocities obtained at the 72 angular positions, and the relative standard deviation (Dv/v) was calculated, as shown in table 2. The resulting radial profiles were compared with those given by the velocity equations of Fahien and Stankovic (1979) and Vortmeyer and Schuster (1983). The experimental data were also matched with the radial void fraction profiles, obtained from equations (1) and (2), so as to check the coherence between available flowing space and fluid velocity.

Table 2: Mean velocity v (m/s) and relative standard deviation (Dv/v) at selected radii

			r/R (-)
Bed		0.9		0.43	0.22	0.15	0.075
1	v Dv/v	22.7 0.86		2.82 1.32	-	4.31 1.26	-
2	v Dv/v	11.97 0.84		6.06 1.35	-	-	9.57 1.07
3	v Dv/v	6.59 0.86		3.26 1.10	2.50 1.36	-	-
4	v Dv/v	8.51 0.31		6.15 0.74	4.62 0.57	-	-
5	v Dv/v	5.13 0.43		4.21 0.64	4.51 0.50	-	-

The data in the experimental curves were numerically integrated across the bed, using Simpson’s rule to derive flowrates from the average velocities. These flows could then be compared with rates measured with the Venturi meter.

RESULTS AND DISCUSSION

The disagreement between the measured and calculated flow rates obtained for beds 3 to 5 was in all cases less than 7.2%, and thus these velocity measurements can be taken as satisfactory. For beds 1 and 2, however, a divergence of around 16% was found, indicating problems in the technique. Freiwald (1992) makes the point that when hot-wire anemometers are used above the bed, the radial and angular components of the velocity vector significantly interfere with the axial velocity readings. The same could be true of readings taken with the Pitot tube. Even so, the magnitude of this interference is not great enough to disqualify either the measured radial profiles of v_z, some of whose values are 65% higher than the mean superficial velocity, or the angular profiles, in which velocities range from nil to 350% higher than the average, as can be confirmed in figures 3 and 4.

Figure 3:
Radial and angular velocity distribution (D/d_p=4.3)

Figure 4:
Radial and angular velocity distribution (D/d_p=60.0)

The main features of figure 3 are typical of the kind we intend to highlight in this work: clear zones of fluid stagnation next to velocity peaks. In figure 5a we note large regions of stagnant fluid or fluid in which the velocities are below the sensitivity of the Pitot tube, which are spread out over the entire transverse section of the bed, but are more intense close to the center of the tube. This region is probably occupied by a sphere, and the measurement was made above this particle. This may explain the existence of hot spots in CFBRs, as these stagnation zones would not take part in the removal of heat arising from highly exothermic chemical reactions. On the other hand, figure 5b shows that the very high velocities regions are more concentrated in the range of nondimensional radial position r/R between 0.7 and 0.9. For this bed (D/d_p = 3.0) the distance d_p/4 from the wall is located at r/R equals 0.83, and we would like to emphasize that this position corresponds to that of high velocity foreseen by the Vortmeyer and Schuster equation. This demonstrates that the velocity measurement is qualitatively valid. Figures 5a was constructed under assumption that between positions with null velocities remains unaffected the same is true for high velocities in figure 5b.

Figure 5:
Stagnant regions (a) and locations in which the velocities are higher than 100% of the average cross section velocity (b).

These phenomena are not compatible with the conception of the medium as a continuum, since in close regions, whose void fractions in principle have close values, the hydrodynamic results are very different. Hence, it would be unwise to put forward a single macroscopic equation based on an RVE representing the whole domain of the bed. The pseudo-homogeneous standpoint should be abandoned in favor of discrete modeling.

By contrast, when the D/d_p ratio is high, as in the case of bed 5, the angular oscillations are much lower in amplitude, as can be seen in table 2 and figure 3. Note that the relative velocity deviations for beds 1, 2 and 3 are actually greater than the velocities themselves in most cases. With beds 4 and 5, even though local velocity fluctuations are observable, the average hydrodynamic behavior is quite regular, so the use of an RVE to represent the whole porous medium is acceptable.

The experimental radial profiles of average angular velocity are plotted together with profiles calculated from equations 1 to 4 in figures 6 to 9. It can be observed that at low D/d_p ratios Mueller’s (1991) void fraction equation fits the experimental velocity data more closely, while at high D/d_p ratios the velocity equations are more appropriate. To construct figures 6 and ⁸, the porosity value obtained from equations 1 and 2 was normalized by the mean porosity of the bed, a value which was calculated by integrating the radial porosity profile.

Figure 6:
Radial velocity distribution (experimental data and profiles calculated from velocity equations for D/d_p=3.0)

Figure 7:
Radial velocity distribution (experimental data and profiles calculated from void fraction equations for D/d_p=3.0)

Figure 9:
Radial velocity distribution (experimental data and profiles calculated from void fraction equations for D/d_p=31.0)

In figure 6, values provided by the equation of Vortmeyer and Schuster (1983) are seen to be close to the experimental data near the wall of the bed, and indeed both this equation and that of Fahien and Stankovic (1979) were expressly proposed to reflect the preferential channeling of fluid through this region, which is more porous and less homogeneous than the center. The poor performance of equation 3 may be explained by the fact that it results from fitting data obtained over a wide range of D/d_p ratios above 4, which are higher than those in figure 6. Meanwhile, even Vortmeyer and Schuster’s equation wrongly predicts a flattened profile in the central region, since this flat behavior does not occur because the low value of D/d_p does not allow the void fraction to reach its stable value.

When the low D/d_p data are compared with curves from void fraction equations (figure 7) , equation 2 is clearly an over-simplification, while equation 1 reflects the relation between the amount of empty space and the peaks and troughs in the radial velocity distribution. This reinforces the relevance of discrete behavior in the system and the description of the medium as two well-defined phases.

At high D/d_p ratios, where the voidage profiles are flat along nearly the whole radius, the velocity profile still has a peak near the wall, and equation 4 represents this behavior best, as can be seen in ^{figures 8} and 9. The behavior of equation 1 is not suitable because of the perturbations close to the wall, which are not present in the experimental results. On the other hand, equation 2 does not represent the nonslip condition at the wall. In this case, to consider the medium homogeneous is not a strong imposition, and it is representative of a large portion of the bed.

CONCLUSIONS

The results of this study have clearly demonstrated that the applicability of the concept of a continuum for fluid flowing in a porous medium depends on its structure, in particular on the D/d_p ratio. At high ratios, the radial distribution of the axial component of velocity is almost flat across most of the cross section of the bed, even though its angular variation cannot be neglected. Under these conditions, a volume element chosen near the center can represent practically the whole radial domain, though it would be advisable to deal with the wall zone in isolation. When the D/d_p ratio is high, the radial velocity profile equations predict the actual behavior observed in this study adequately.

However, at low ratios, the structure of the porous matrix determines the flow behavior. Well-defined velocity plateaus are not seen, but rather oscillations that reflect the radial variation of the void fraction. In this case, there is a pronounced and highly discontinuous angular variation of axial velocity, causing the appearance of several stagnation regions in the bed. For this situation, it is preferable to analyze the bed in terms of discrete solid and fluid phases.

NOMENCLATURE

D diameter of bed [L]

d_p diameter of particles [L]

r radial position variable [L]

R radius of bed [L]

Re modified Reynolds number

u radial distribution of surface velocity [LT^-1]

v mean superficial velocity [LT^-1]

V dimensionless superficial velocity (V = u /v); [-]

x radial distance from wall [L]

Greek symbols

m viscosity [ML^-1T^-1]

e average radial void fraction [-]

r fluid density [ML^-3]

ACKNOWLEDGMENTS

J.C. Thoméo and J.T. Freire are grateful to FINEP (PRONEX program 41.96.0897.00) for financial support.

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Figure 8: Radial velocity distribution (experimental data and profiles calculated from velocity equations for D/d_p=31.0)

Publication Dates

• Publication in this collection
  01 Mar 2000
• Date of issue
  Dec 1999

History

• Accepted
  19 Sept 1999
• Received
  29 July 1999