Ann. occup. Hyg., Vol. 47, No. 2, pp. 151-156, 2003
© 2003 British Occupational Hygiene Society
Published by Oxford University Press
On the Inertial Range of Particles Under the Influence of Local Exhaust Hoods
Department of Environmental Sciences and Engineering, School of Public Health, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-7400, USA
Received 23 April 2002; in final form 4 November 2002
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONTHEORYMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUDING REMARKSAUTHOR’S NOTEREFERENCES |
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This paper presents results from numerical simulations conducted to estimate the inertial range (stopping distance) of large aerosol particles ejected away from local exhaust hoods. Potential flow theory is used to specify the air velocity along the centerline of a flanged circular hood. A high-order Gear method is used to integrate the particle equations of motion with drag coefficients in the transitional range. The results allow for a relative comparison of hood performance based on the energy cost needed to stop the particle within a specified distance. Hood size is shown to be a critical factor, while the capture velocity generated by the hood at the point of particle ejection is of secondary importance.
Keywords: local exhaust hoods; inertial range; stopping distance; aerosols
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONTHEORYMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUDING REMARKSAUTHOR’S NOTEREFERENCES |
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Local exhaust hoods are a primary engineering control for protecting workers from excessive exposure to toxic airborne contaminants. The current design procedure for these hoods relies on the concept of capture velocity outlined in several texts (Burgess et al., 1989; ACGIH, 2001). Table 3-1 in the Industrial Ventilation Manual (ACGIH, 2001) specifies a range of capture velocities as a function of initial contaminant momentum, or ‘condition of dispersion’ as it is identified in the table. Guidance as to the applicable portion of the range is provided based on mitigating conditions such as cross drafts, hood size, duration of contaminant generation, and toxicity. The table appears to have originated with the work of Kane (1946) and Brandt (1947).
It has long been known, however, that capture velocity is only part of the puzzle in designing effective, economic hoods for contaminant control. Even in the early days of studying the performance of these hoods it was clear that their ability to capture pollutants depended upon some integral of the air velocity over distance. In Industrial Health Engineering, Brandt states: ‘The capturing power of an air stream into an exhaust hood is a function of the product of the velocity and the distance through which it acts, not of the velocity alone. Obviously, a dust particle or a unit volume of contaminated air traveling away from an exhaust hood by virtue of the momentum imparted to it by a machine will not continue indefinitely in that direction against a counterflow of air even though it might readily continue through the small "defense depth" set up by a small hood.’ Later Brandt continues: ‘since large exhaust hoods accomplish considerable control by means of general ventilation in addition to unusually effective local exhaust control through "defense in depth" they will very frequently control the hazard satisfactorily with considerably less air flow than estimated by the equations given later in this chapter. The extent of our knowledge on this subject today does not permit an accurate solution of such problems.’ In Industrial Dust, Drinker and Hatch comment: ‘For a given required air velocity at the point of dust generation the velocity at the face of the hood should be as low as possible. This is contrary to the common idea that the hood opening should be made small in order to increase the suction.’
This paper provides a fundamental look at the ability of local exhaust hoods to capture contaminants generated with significant momentum and ties together some basic concepts governing their performance. The example selected is the penetration of large aerosol particles ejected at high velocity away from the hood along the centerline. Settling due to gravity is neglected and the flow field is one-dimensional along the centerline.
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THEORY |
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TOPABSTRACTINTRODUCTIONTHEORYMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUDING REMARKSAUTHOR’S NOTEREFERENCES |
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During the 1970s and 1980s, studies (Drkal, 1970; Tyaglo and Shepelev, 1970; Flynn and Fitzgerald, 1989) demonstrated that three-dimensional velocity fields into flanged unobstructed local exhaust inlets could be described accurately by potential flow solutions. Potential theory is concerned with solutions to Laplace’s equation, which is the form the conservation of mass equation takes when airflow is frictionless. Although the theory has been used widely over the years to describe hood flow fields, the interpretation of potential and its connection to contaminant control has been obscure.
The relationship between the air velocity vector and the potential function is:
(1)
For a one-dimensional flow along the x-axis (e.g. the hood centerline), this can be expressed as:
(2)
and subsequently:
(3)
Thus the integral of the air velocity over a given distance is the potential difference over that interval. The average air velocity in the interval is the potential drop divided by the distance. Thus, the ‘product of velocity and the distance through which it acts’ identified above by Brandt is in fact the potential drop indicated in equation (3).
Drkal (1970) obtained the analytic solution for the centerline velocity into an infinitely flanged circular exhaust hood with a uniform face velocity using potential flow theory as:
(4)
where Uh is the uniform hood face velocity, x is the centerline distance, and a is the hood radius. The corresponding potential function is:
(5)
If drag were strictly proportional to the air velocity, as in the Stokes range, one might anticipate that the potential drop would be adequate to define the stop distance for a given particle. However, for large particles the dependence of drag on velocity transitions from linear to quadratic and the analysis is complicated.
Serafini (1954) obtained an analytical solution for the equation of motion governing aerosol transport in a uniform (non-accelerating) airflow with the drag coefficient in the transitional range (i.e. particle Reynolds numbers between about 2 and 1000). These equations are given below starting with the following approximation for the drag coefficient:
(6)
where:
(7)
(8)
Here V is the particle velocity and U is the uniform air velocity both with respect to a fixed frame of reference. In the frame of reference moving with the uniform air velocity and incorporating the expression for the drag coefficient, Newton’s law of motion for the particle is:
(9)
Integration of this equation with initial conditions at time zero of W = W0 and x = 0, produces the following analytical forms for the relative velocity and displacement at a later time, t.
(10)
(11)
where
(12)
(13)
(14)
(15)
and
(16)
The expression for the inertial range (stopping distance) of large aerosol particles ejected into still air is
(17)
For the case when 
= 1/6, Hinds (1982) gives the following form of equation (17)
(18)
where the arctan argument is in radians. Unlike the spatially uniform air velocity field that Serafini used, local exhaust hoods generate accelerating airflow and an analytical solution appears unlikely. Thus, the equations of motion are integrated here numerically.
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MATERIALS AND METHODS |
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TOPABSTRACTINTRODUCTIONTHEORYMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUDING REMARKSAUTHOR’S NOTEREFERENCES |
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Neglecting gravity, the particle equations of motion with respect to a fixed frame of reference can be written as the following set of coupled first-order differential equations:
(19)
The air velocity on the hood centerline, (U), is given by equation (4) and for all work here = 0.158. Equations (18) and (19) are integrated using a double precision, fifth-order accurate, backward difference Gear-type algorithm, called DIVPAG, available in the Fortran IMSL library. The subroutine requires specification of an error tolerance, which keeps the global error proportional to its value. For all calculations conducted here the value of this tolerance was selected as 10–15.
Integration is continued until the particle velocity becomes negative (i.e. reverses direction). The stopping location (Xs) is estimated as the maximum of the last two particle locations on the centerline. The difference between the stopping location and the initial particle location (X0) is the stopping distance (s). The relative uncertainty in the stopping distance is estimated by repeating the calculation for finer time increments until the last two estimates of stopping distance differ by <0.0001.
An attempt was made to summarize the numerical work with a regression model based on dimensional analysis and the work of Serafini described above. The guiding principle in this effort was that as the particle ejection location (X0) moves away from the hood face the stopping distance (s) will approach the value, smax, defined by equation (17). Thus the ratio s/smax was selected as the dependent variable to model. The dependent variables were selected from a dimensional analysis, which suggested that as many as six dimensionless variables might be required.
The computer code was used to generate estimates of the inertial range for particles of 100 and 25 µm, with densities of 1.0 and 2.6 g/cm3. Five different hood diameters at 10 different face velocities were used. A particle ejection velocity of 4064 cm/s was employed with a centerline location of 5.08 cm. In addition, a 50 µm diameter particle with a density of 5.0 g/ cm3 was run at an ejection velocity of 2032 m/s and a centerline start location of 2.54 cm. These conditions resulted in 250 determinations of stopping distance. The particle densities are representative of water, silica, and iron oxide as the bulk materials. The ejection velocity is typical of a high-speed grinding operation, and hood diameters and flows were selected to be in the range of smaller local exhaust hoods with face velocities up to Mach 0.3 which is considered the limit for incompressible flow.
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RESULTS AND DISCUSSION |
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TOPABSTRACTINTRODUCTIONTHEORYMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUDING REMARKSAUTHOR’S NOTEREFERENCES |
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The data are presented in Figures 1–5 as plots of the inertial range or stopping distance as a function of the hood face velocity for each of the five different hood diameters. Figures 1 and 2 are for the 100 µm particle at densities of 1 and 2.6 g/cm3, respectively. Figures 3 and 4 are the corresponding plots for the 25 µm diameter particle. Figure 5 illustrates the data for the 50 µm particle with a density of 5.0 g/cm3. The plots demonstrate that for a given hood diameter, the stopping distance of the particle decreases (i.e. less penetration against the controlling airflow) as the hood face velocity increases. The rate at which this drop-off in stopping distance occurs is greater at larger hood sizes. For a given size particle of constant density, ejected along the centerline at a specified velocity and fixed centerline distance from the hood face, the penetration against the counter flow of air is minimized by selecting larger hoods for a fixed value of hood face velocity.
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Several different statistical models were evaluated to fit the numerical data, but none gave a compelling summary. Graphic plots of the numerical values, and regression analyses of appropriate logarithmic transforms indicated that the following correlation seemed to work as well as any with an R2 = 0.95.
(20)
Figure 6 illustrates how this relationship fits the 250 numerical values from which it is derived. The maximum error was 250%, and for 200 of the values the error was <25%. To test this summary model a separate numerical run was conducted with values of the dependent variable selected within the range used to fit the model. A 75 µm diameter particle with specific gravity of 1.8 was located at X = 0.1016 m with an initial ejection velocity of 20.32 m/s. All five hood diameters and the 10 different hood face velocities used in the previous runs were used for this validation test as well. Figure 7 shows the results—a clear bias to over-predict the numerical values, but within the data envelope of equation (20) shown in Figure 6. The statistical fit defined by equation (20) is not recommended when direct numerical integration is possible since it will provide superior results. However, it is offered as an indicator of the significant variables and a first-order estimate.
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An interesting application of the results is given with reference to Fig. 1. The objective is to limit the penetration of a unit density, 100 µm particle to no more than 0.15 m. A centerline location 0.0508 m in front of the hood face is selected with a particle ejection velocity of 40.64 m/s. The figure shows that at hood diameters below 0.1016 m the objective is essentially impossible. For larger hoods, Table 1 shows the face velocity required to accomplish the task and the associated capture velocities produced by the hoods at the particle ejection location. A relative energy cost factor is included assuming the largest size hood has a factor of unity. The cost factor is based on the fact that the energy required to stop the particle is a product of the hood static pressure and flow rate. This in turn is proportional to the product of the square of the hood diameter and the face velocity cubed. As seen from this example, considerable savings are associated with the larger hood (up to a factor of 10) despite its generating a much lower capture velocity (by as much as a factor of 2) at the particle generation point.
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CONCLUDING REMARKS |
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TOPABSTRACTINTRODUCTIONTHEORYMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUDING REMARKSAUTHOR’S NOTEREFERENCES |
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The work presented here is quite theoretical. Many important realistic considerations have been omitted, e.g. cross drafts, induced air flows by the particle generation process itself, obstructions to the hood, transport velocity considerations, settling, and many others. The effect of cross drafts in particular may reduce the importance of the hood size effect noted here since the ratio of the cross draft velocity to the hood face velocity is known to limit the reach of the hood. However, the results do quantify the fundamental interaction of a local exhaust hood and an inertial particle, and call attention to the importance of hood size in limiting their penetration. This suggests that successful control of large, heavy aerosol particles such as those generated during high-speed grinding of metals and concrete will be extremely difficult to achieve with small localized hoods. Even with very high airflow these hoods are unlikely to be successful unless well-designed enclosures or receiving hoods are employed. The work suggests that more emphasis needs to be placed on studying the effect of the size and shape of hoods, and that focus on capture velocity alone is unlikely to produce successful, efficient controls.
Acknowledgements—This work was partially supported by Grant No. 1 R01 OH07363–01 from the National Institute for Occupational Safety and Health (NIOSH). Its contents are the sole responsibility of the author and do not necessarily represent the official views of NIOSH.
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AUTHOR’S NOTE |
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TOPABSTRACTINTRODUCTIONTHEORYMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUDING REMARKSAUTHOR’S NOTEREFERENCES |
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Serafini’s equations appear in a slightly different form in The Mechanics of Aerosols by Fuchs. On page 79 of the 1964 translation of that text equation (18.10) is incorrect. The last term should be:
NOMENCLATURE
W = relative velocity of air and particle
V = particle velocity with respect to fixed frame
U = air velocity with respect to fixed frame
= density
µ = coefficient of viscosity
s = particle stopping distance
= dimensionless relaxation time in Serafini’s formulation
= fit factor; set at 0.15 in this work according to Serafini
CD = drag coefficient
Re = Reynolds number
dp = particle diameter
D = hood diameter
Subscripts
p = particle
f = fluid
0 = initial
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FOOTNOTES |
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* Tel: +1-919-966-1171; fax: +1-919-966-7911; e-mail: mike_flynn@unc.edu
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REFERENCES |
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TOPABSTRACTINTRODUCTIONTHEORYMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUDING REMARKSAUTHOR’S NOTEREFERENCES |
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ACGIH. (2001) Industrial ventilation—a manual of recommended practice. 24th edn. Cincinnati, OH: American Conference of Governmental Idustrial Hygienists.
Brandt AD. (1947) Industrial health engineering. New York: John Wiley & Sons.
Burgess WA, Ellenbecker MJ, Treitman R. (1989) Ventilation for control of the work environment. New York: John Wiley & Sons.
Drinker P, Hatch T. (1936) Industrial dust hygienic significance, measurement and control. New York: McGraw-Hill.
Drkal F. (1970) Stromungsverhaltnisse bei runden Saugoffnungen mit Flansch. Heizung, Luftung und Klimatechnik; 21: 271–3.
Flynn MR, Fitzgerald ML. (1989) A comparison of three-dimensional velocity models for flanged rectangular hoods. Appl Ind Hyg; 4: 210–16.
Fuchs NA. (1964) The mechanics of aerosols. Oxford: Pergamon Press.
Kane JM. (1946) Design of exhaust systems. Health Ventil; 42: 68.
Hinds WC. (1982) Aerosol technology. New York: John Wiley & Sons.
Serafini JS. (1954) Impingement of water droplets on wedges and double wedge airfoils at supersonic speeds. National Advisory Committee for Aeronautics Report No. 1159.
Tyaglo IG, Shepelev IA. (1970) Dvizhenie vozdushnog potoha k vytyazhnomu otversityu (Airflow near an exhaust opening). Vodosnabzhenie I sanitarnaya teknika; 5: 24–5.
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