The drag coefficient for particles in aerosols flowing through a horizontal conduit

The drag coefficient for particles in aerosols flowing through a horizontal conduit

L. PEREIRA, W.D. MARRA JR. and J.R. COURY

Departamento de Engenharia Química, Universidade Federal de São Carlos,

C.P. 676, 13565-905, São Carlos - SP, Brazil

(Received: July 30, 1999; Accepted:September 10, 1999)

Abstract - The drag coefficient, C_D, was experimentally determined during sedimentation of aerosol particles in air flowing horizontally in a conduit of rectangular cross section, and its relation to the Reynolds numbers for the particle, Re_p, and the conduit, Re_c, was found. The method used to obtain the frictional force on the particles is based on observation of the trajectories of the particles being deposited on the bottom wall of the conduit. The diameter (d_p) and point of deposition of the particles were determined by examining small glass slides distributed along the floor of the apparatus at given positions. The diameter of particles adhering to these laminae can be observed by microscope, but a factor must then be applied to convert these values to the diameters in suspension (d_p), since the particles are liquid and undergo flattening as they collect on the glass. Results were compared with the aerodynamic diameter of the particle, obtained independently, and the discrepancies that appeared will need to be investigated further. The velocity profiles of the air inside the apparatus were also recorded. A correlation of the form C_D = f(Re_p, Re_c) is proposed.

Keywords: drag coefficient measurement, aerosol particles.

INTRODUCTION

In gas-particle suspensions flowing in channels, several forces act on the particles, namely the force of gravity, buoyancy and the drag or frictional force. The forces due to gravity and buoyancy can be worked out from the particle and gas densities and the volume of the particle. The drag and the related drag coefficient must be found experimentally, as they vary with the Reynolds number characterizing the flow configuration.

In previous research, one of the authors (Marra Jr, 1998) required detailed values for the drag coefficient of particles sedimenting in an aerosol flowing horizontally through a conduit of rectangular cross section. While the literature is rich in correlations for the drag coefficient on falling particles in stationary fluids, none was found which could be used in this particular case. It was thus decided to measure the drag coefficient under conditions close to those of interest.

MATERIALS AND METHODS

Figure 1 is a diagram of the apparatus constructed to determine the drag coefficient (ADDC). It basically consists of a conduit of rectangular cross section, a honeycombed entrance area to stabilize the velocity profile inside the conduit and a horizontal central slit where the aerosol is added. The inside of the rectangular conduit is 13 cm wide by 8.5 cm high, while the gap in the central slit is 0.5 cm.

Figure 1:
Sketch of apparatus used to determine the drag coefficient (ADDC).

The ADDC is built of transparent acrylic and glass, 10 mm and 5 mm thick, respectively. The construction ensures it can be dismounted, which is important for cleaning and transport.

An expression for the drag coefficient can be obtained from a balance of the forces acting on the particle in the y-direction during its flight, after it has left the central slit. Figure 2 is a sketch of this trajectory.

Figure 2:
Trajectory of a particle in the field of gravity.

Consider a particle of diameter d_p, mass m_p, and density r_p being carried by a gas of density r_f and average flow velocity U₀ in the z-direction. The forces on the particle in the y-direction are weight, P, drag, F_d and buoyancy, E_p. Since P = F_d + E_p, we have,

(1)

where C_D is the drag coefficient and g is the acceleration due to gravity.

The velocity of fall of the particle, v_y, can be expressed as a function of the gas velocity, U₀, taken as the total flow rate of the gas divided by the area of cross section of the ADDC. The fall time of the particle (Y/v_y) is equal to the time it take for to travel distance Z (see Figure 2); hence

(2)

It is important to realize that Equation (2) is equally valid for a uniform gas velocity profile (plug flow) or for a general parallel flow profile, in the following form:

(3)

where K and n are constants.

Substituting Equation (2) into (1), we get

(4)

Rearranging (4), we have

(5)

It can be seen from this equation that the drag coefficient can be calculated from the known physical properties of the particle and the gas, the mean velocity of gas flow and the distances, Z and Y, traveled by the particle.

Determination of C_D

The drag coefficient was determined as follows. An aerosol was introduced into the ADDC by the central slit and the downward motion of the particles was then followed by measuring their displacement in the z-direction at deposition, starting from the point where they left the slit, as well as the velocity of the air. The apparatus used is outlined in Figure 3 and essentially consists of the ADDC, an aerosol generator connected to its intake and a suction pump to the outlet tube.

Figure 3:
Apparatus used to obtain the velocity profile and drag coefficient.

The aerosol was generated by a vibrating-orifice device, TSI model 3450, capable of producing a monodisperse aerosol of liquid particles of uniform diameter from a solution, and injected into the ADDC via the central slit. To study the behavior of the droplets inside the ADDC, it was necessary to use a nonvolatile liquid solute in the solution, given that the particles had to be liquid during flight and, on reaching the lower wall of the ADDC, to stick to it and avoid being lifted off by the air stream. It was shown by Liu et al. (1982) and Olan-Figueroa et al. (1982) that, in this aerosol generator, dioctyl phthalate (DOP) could be used as solute and ethanol as solvent. The density of DOP is 980 kg/m³.

To obtain the mean diameter of particles adhering to the wall, it was decided that glass slides should be laid inside the ADDC along the z-axis and held in place by adhesive tape. Particles would then land on the slides, which could be taken out and inspected under an Olympus BX-60 optical microscope linked to a microcomputer running an image analysis program.

The slides were placed at five different positions along the lower wall at the following distances from the entry point of the aerosol: 46, 58, 83.5, 98.5, 111 and 126 cm. It was possible to measure particle diameter on the glass, but because it was liquid the droplet became flatter upon sticking to the slide, so the diameter observed was larger than its diameter in the air.

According to Olan-Figueroa et al. (1982), DOP droplets with a diameter smaller than 50 mm adhering to a glass surface coated with the fluorocarbon FC-721, an oleophobic surfactant made by 3M, have a diameter around 40% larger than that particles in flight. Thus, to facilitate estimation of particle size, the diameters of particles collected on clean glass slides, d_L, were compared with those collected on FC-3532 (3M), an oleophobic product equivalent to the one used by Olan-Figueroa et al. (1982). This test provided a conversion factor between the observed diameters on glass and those of the particles in the air.

In each of these comparative tests, five clean glass slides and five slides coated with FC-3532 were used to collect the DOP droplets as they left the aerosol generator. The proportion of DOP to ethanol was varied so as to generate droplets with different diameters. All the diameters observed by microscopic examination of each treated slide were divided by 1.4 to arrive at the diameter in suspension, d_pL. This value was then used to determine the conversion factor, F_L, for clean glass by dividing the observed diameter, d_L, by d_pL and repeating for all droplets on each slide:

(6)

The test conditions for measurement of C_D are listed in Table 1. The Reynolds numbers of the particle (Re_p) and conduit (Re_c) are defined as follows:

where D_h is the hydraulic diameter of the conduit and v_t is the terminal velocity of the particle.

Table 1: Experimental conditions for C

U0 (m/s)	dL (m m)	Z (m)	Rep	Rec
0.26	59.13	0.84	7.9x10-2	1,620.7
0.26	59.87	0.99	8.1x10-2	1,620.7
0.26	45.02	1.11	4.1x10-2	1,620.7
0.26	46.64	0.84	4.5x10-2	1,620.7
0.26	43.05	0.99	3.7x10-2	1,620.7
0.33	57.32	0.84	7.4x10-2	2,068.5
0.33	56.08	0.99	7.0x10-2	2,068.5
0.33	52.41	1.11	6.0x10-2	2,068.5
0.33	46.08	1.26	4.4x10-2	2,068.5
0.40	54.15	0.99	6.5x10-2	2,521.3
0.40	58.16	1.11	7.7x10-2	2,521.3
0.40	52.45	1.26	6.0x10-2	2,521.3
0.13	14.09	0.84	1.9x10-3	810.4
0.13	27.66	0.84	1.2x10-2	814.9
0.13	34.10	0.58	2.0x10-2	814.9
0.26	44.61	0.99	4.1x10-2	1,638.9
0.26	43.51	1.11	3.8x10-2	1,638.9
0.26	42.05	1.26	3.5x10-2	1,638.9
0.21	40.90	0.99	3.3x10-2	1,338.6
0.21	43.20	1.11	3.8x10-2	1,338.6
0.21	20.09	1.26	5.1x10-3	1,338.6

The Aerodynamic Diameter

For the sake of comparison, a piece of equipment designed to measure the aerodynamic diameter of aerosol particles was used to obtain an independent measure of d_p. This Aerodynamic Particle Sizer (APS), model TSI-3320, measures particle diameter (d_pA) by sampling in situ. By comparing these values with the diameters on clean glass, a correlation between d_pA and d_L was found.

Velocity Profile in the Conduit

Experiments were carried out to measure the velocity profiles along the horizontal conduit. To measure the air velocity at a given point, a hot-wire anemometer probe, Cole-Parmer Tri-Sense 37000-00, was introduced into the ADDC.

Initially the air flow rate was adjusted to maintain the velocities at points 1, 2 and 3 of axial position 1(z = 6 cm) as nearly constant as possible. Then, the velocity profiles at three additional z-positions, as indicated in Figure 4, were recorded, by reading velocities at seven y-positions (heights) for each of the z-positions, 2, 3 and 4. All readings were taken in the central plane (x=0).

Figure 4:
Sketch of velocity recording positions and dimensions of apparatus used for determination of drag coefficient (side view).

RESULTS AND DISCUSSION

Determination of C_D

A typical photomicrograph of DOP droplets collected on a glass slide is shown in Figure 5. The flattening of the drops is evident in the figure.

Figure 5:
Photomicrograph of DOP particles on clean slide (50x).

In Figure 6, the value of the correction factor, F_L, for each DOP droplet collected on clean glass is plotted against its observed diameter, d_L. By fitting a linear equation to these data, the following correlation between F_L and d_L was found:

F_L = 0.0063d_L + 1.302 (7)

Figure 6:
Conversion factor for particle diameter on clean slides.

Combining Equations (6) and (7) and eliminating F_L, we have the calculated particle diameter, d_pL, in terms of d_L alone:

(8)

To obtain the drag coefficient from Equation (5), the suspended particle diameter, d_p, must be determined. In this study, the value calculated from the glass slide observations was used, and d_pL was taken as d_p. The drag coefficient values so obtained were designated C_DL. In Figure 7, experimental values of C_DL are plotted against the particle's Reynolds number, Re_pL.

Figure 7:
Variation of C_DL with Reynolds number of particle.

It can be seen in the above figure that the values of the coefficient are higher than those predicted by the ratio C_D=24/Re_p, applied normally to particles sedimenting in static fluids with Re_p < 0.1 (Clift et al., 1978). This high C_D suggests that it is probably affected by the horizontal gas flow, which is perpendicular to the direction of sedimentation, as turbulence in the gas could increase the drag on the particle (note high Re_c in Table 1).

Thus, a correlation for C_D may be suggested, valid in the ranges 0.02£ Re_pL£ 0.1 and 500£ Re_c £ 3,000, in which the Reynolds number of the conduit is taken into account. The following was found to fit the data:

(9)

A graphic representation of the above correlation is given in Figure 8.

Figure 8
: C_DL as a function of the Reynolds numbers of the conduit and particle.

The Aerodynamic Diameter

Results from the comparison of aerodynamic diameters of particles (d_pA) with diameters observed on clean slides (d_L) are shown in Figure 9. The following expression relating these quantities was found by fitting a linear equation to the data:

d_pA = 0.3055d_L (10)

Figure 9:
Relation between the aerodynamic diameter and d_L.

In Figure 10 the values for the particle diameter derived from Equations (8) and (10) are compared graphically. Note that the values obtained from Equation (8) are about twice those from Equation (10). At this stage, the cause of this discrepancy, which directly affects the calculated value of C_D, remains unknown. As each of the two estimates involves a reliable measuring technique, the reason for the difference must be the object of further detailed study.

Figure 10:
Comparison of diameters obtained from slides and APS

Velocity Profile in the Conduit

In Figure 11, velocities at the seven y-positions are plotted with fitted curves for positions 2, 3 and 4 (see Figure 4) along the z-axis. Air flow was adjusted to keep the velocity at position 1 close to 0.26 m/s. Note that the shape of the central region of the profile evolves from a flatter form at position 2 (z = 26 cm) to a more parabolic one at position 4 (z = 130 cm), signifying that the flow is developing along the conduit.

Figure 11:
Velocity profiles in the ADDC.

This variation clearly influences the value calculated for C_D, as Equation 2 is only valid for profiles that do not alter along the conduit. A more rigorous analysis using numerical methods should be adopted as this work develops. For the moment, the most acceptable approach would be to incorporate the effects of the variable profile into an empirical adjustment of C_D.

CONCLUSIONS

The custom-made apparatus, designed to determine the drag coefficient of particles under the gas flow conditions described, proved to be simple to operate and gave promising results.

The C_D values obtained are higher than those calculated from published correlations, which do not take into consideration gas flow in the direction perpendicular to the downward motion of the particles. It was clear that the Reynolds number of the conduit had an important influence, but initial results were inconclusive.

The establishment of a correction factor, relating the diameter of a particle suspended in the air to that of a similar particle on a clean glass slide, deserves more attention. Measurement of the aerodynamic diameter of the particle would give an improved factor, but more data will have to be collected.

The velocity profiles obtained showed that air velocity, U₀, varies along the length of the ADDC. Thus, a mean value will probably have to be used in calculating C_D.

NOMENCLATURE

m Viscosity of the gas, kg.s^-1.m^-1

r_f Density of the gas, kg.m^-3

r_p Density of the particle, kg.m^-3

C_D Drag coefficient

C_DL Drag coefficient from clean slide data

D_h Hydraulic diameter of the conduit, m

d_L Diameter of the particle on clean slide, m

d_p Diameter of the particle in air, m

d_pA Aerodynamic diameter of the particle, m

d_pL Diameter of the particle in air, calculated from clean slide data, m

F_L Diameter correction factor

g Acceleration due to gravity, m.s^-2

K Constant

m_p Mass of the particle, kg

n Constant

Re_c Reynolds number of the conduit

Re_p Reynolds number of the particle

t_q Particle falling time, s

U₀ Velocity of the gas, m.s^-1

v_t Terminal velocity of the particle, m.s^-1

v_y Velocity of the particle in the y-direction, m.s^-1

v_z Velocity of the particle in the z-direction, m.s^-1

Y Depth of fall of the particle, m

Z Horizontal distance traveled by the particle, m

ACKNOWLEDGMENTS

The authors wish to express their thanks to FAPESP and PRONEX-FINEP for the financial assistance received, without which this research could not have been done.

• Clift, R., Grace, J.R. and Weber, M.E., Bubbles, Drops, and Particles. Academic Press (1978).
• Liu, B.Y.H., Pui, D.Y.H. and Wang, X., Drop Size Measurement of Liquid Aerosols. Atmospheric Environment, Vol. 16, No. 3, pp. 563-567 (1982).
• Marra Jr., W.D., Personal communication (1998).
• Olan-Figueroa, E., Mcfarland, A.R. and Ortiz, C.A., Flattening Coefficients for DOP and Oleic Acid Droplets Deposited on Treated Glass Slides. American Industrial Hygiene Association Journal, 43, pp. 395-399, June (1982).

Publication Dates

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

History

• Accepted
  10 Sept 1999
• Received
  30 July 1999