Annals of Occupational Hygiene Advance Access originally published online on April 24, 2007
Annals of Occupational Hygiene 2007 51(4):379-383; doi:10.1093/annhyg/mem012
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
A Simple Theoretical Model for the Air Flow and Particle Penetration in a Modified Horizontal Elutriator with Plates of Unequal Lengths
Department of Applied Environmental Science, Stockholm University, Stockholm, Sweden
* Author to whom correspondence should be addressed. E-mail: goran.liden{at}itm.su.se
 |
ABSTRACT |
|---|
 TOP ABSTRACT SIMPLE MODEL OF A...DISTRIBUTION OF AIR FLOWSPARTICLE SEPARATION DOWNSTREAM...DISCUSSIONREFERENCES |
|---|
A simple theoretical model is presented for the distribution of air speeds in a modified horizontal elutriator consisting of groups of plates with different lengths. Equations are presented for calculating the particle penetration both in channels of shortended plate lengths, as downstream the end of a shortended plate.
Keywords: aerosol • dust • elutriator • preseparator • sampling
Walton (1954) developed the theory for horizontal and vertical elutriation. Mercer (1973) reproduced the most important parts of the theory in his book Aerosol Technology in Hazard Evaluation. An aerosol separator based on horizontal elutriation consists of a stack of plates with identical dimensions and spacing, usually followed by a filter. Hamilton and Walton discussed the separation error caused by possible manufacture error consisting of unequal channel heights (1961). The horizontal elutriator has mainly been used for the sampling of respirable dust according to the British Medical Research Council (BMRC) and the Johannesburg sampling conventions (Orenstein, 1960; British Medical Research Council, 1961).
An ordinary horizontal elutriator consists of a stack of identical plates, mounted horizontally in a rectangular duct, usually with a filter downstream of the separation zone. The penetration, pen(Vs), of a particle size with sedimentation speed Vs may be written as
 | (1) |
The principal advantage of the horizontal elutriator as a pre-separator, over cyclones and impactors, is that its internal separation efficiency can be determined directly from theory, without any experimentally determined coefficients. The penetration may be modified by either changing the angle to the horizontal (Tillery and Buchan, 2002), unequal plate spacing (Hamilton and Walton, 1961) or unequal plate lengths (Myojo et al., 2007). (Other even more complicated ways to modify the horizontal elutriator in order to bend its inherent separation efficiency might be to have a range of outlet widths narrower than the inlet widths, or different inclination angles for different channels. However, no such experiments seem to have been published.) This short note will present a simplified theory of the air flow through a modified horizontal elutriator with plates of unequal lengths.
|
SIMPLE MODEL OF A HORIZONTAL ELUTRIATOR WITH UNEQUAL PLATE LENGTHS |
|---|
|
TOPABSTRACTSIMPLE MODEL OF A...DISTRIBUTION OF AIR FLOWSPARTICLE SEPARATION DOWNSTREAM...DISCUSSIONREFERENCES |
|---|
The model is based on the following assumptions: (1) All elutriator plates are assumed to begin at the same horizontal distance from the elutriator exit, they are all infinitesimally thin and the height of all channels are identical, h; (2) Longer plates are mounted below shorter plates (and thus plates with equal lengths are grouped together); (3) Plug flow exists inside each channel; (4) The space downstream each group of channels, is termed the corresponding segment, and the jets exited from the channels continue inside the segment with identical air speed as within the channels, without adjusting its speed to the air speed in the neighbouring segment above; (5) The pressure drop from the inlet of a channel to the end of the horizontal elutriator is the same for all routes, independent of which channel and which segments are passed,
P; (6) Entry and exit pressure drops of the channels and segments may be discarded in relation to the pressure drops in the channels and the segments themselves; (7) Pressure drops along the part of a segment that has no solid wall is assumed to be negligible; (8) The aspiration efficiency of all channels will be unity for particles penetrating the horizontal elutriator. See Fig. 1.
|
Schlichting (1979) showed that the boundary layer between two parallel infinite laminar planar streams at different speeds develops in the transversal direction as x0.5, where x is the distance from the junction. This is so slow that almost no air speed equilibration will occur between the segments of a modified horizontal elutriator. For enclosed streams, as within a modified horizontal elutriator, the boundary layer between the segments is expected to develop even slower. The model with identical air speed in both channel and segment is therefore plausible.
Let the horizontal elutriator consist of N channels, numbered from above, in groups of identical plate lengths from above. Let the length of the ceiling plates of the rth group of channels be termed Lr. The top channel belongs to the first group even though its ceiling length is identical to the nominal (maximum) length of the modified elutriator, L0. The segments are numbered in the same order as the groups of identical channel lengths. The length of segment r, Lsegr, is Lsegr = L0–Lr.
As an illustrating example, say that we have three groups of plate lengths, including the nominal plate length, with groups of plates having identical ceiling lengths beginning at channels 1, 4 and 8, and with the first two groups with plate lengths L1 = 0.3 L0 and L2 = 0.7 L0. The total number of channels is 20. This will result in three channel air speeds for the three groups of plate lengths, U1, U2 and U3. See Fig. 1.
|
DISTRIBUTION OF AIR FLOWS |
|---|
|
TOPABSTRACTSIMPLE MODEL OF A...DISTRIBUTION OF AIR FLOWSPARTICLE SEPARATION DOWNSTREAM...DISCUSSIONREFERENCES |
|---|
The distribution of velocities among the channels of the modified horizontal elutriator will be determined by the requirement that the pressure drop along all channel groups plus corresponding segment shall be identical.
In each channel in the rth group of the horizontal elutriator, the pressure drop, Pchanr, can be approximated by
 | (2) |
(Rechanr) is the friction factor, dh the hydrodynamic diameter of the channel (identical for all channels/groups) and 
is the dynamic viscosity. The hydrodynamic diameter for a flow of width W in a channel dh, or in a segment dhr, consisting of nr channels, is calculated as
 | (3) |
Psegr, depends upon how the flow is enclosed. Let each segment be divided into possibly three parts depending on how many plate surfaces enclose it, 0, 1 or 2. The pressure drop may then be approximated as follows: (1) For the part of the segment enclosed by both an upper and lower plate, the pressure drop is calculated similarly as in equation (2); (2) For the part of the segment enclosed either by an upper or lower plate, the pressure drop is approximated by half the corresponding value calculated similarly as in equation (2) and (3) For the part of the segment enclosed neither by an upper nor a lower plate, the pressure drop is approximated as zero. For the illustrating example in Fig. 1, this gives the following approximate pressure drops for the first and second segments
 | (4) |
Pr, is calculated as
 | (5) |
 | (6) |
|
PARTICLE SEPARATION DOWNSTREAM OF THE SHORTENED PLATES |
|---|
|
TOPABSTRACTSIMPLE MODEL OF A...DISTRIBUTION OF AIR FLOWSPARTICLE SEPARATION DOWNSTREAM...DISCUSSIONREFERENCES |
|---|
If the length of a group of channels is not very short compared to the nominal plate length, it will mainly be airborne particles in the lowest and the next to the lowest channels that may be separated in the segment downstream of the group.
For the channel at the bottom of a group r of channels, the following simplified model may be used to study the extra separation that occurs as the particles flow along the ‘roof’ of the next group, r+1, of channels. In the channel at the bottom of group r, let the penetration of a particle with sedimentation speed Vs, at horizontal position x (measured from the entry into the elutriator) be termed penr0(x, Vs). At the entry and at the exit of the first part of the segment downstream of the ceiling plate proper (that is for Lr<x<Lr+1), the penetration so far into the elutriator may be estimated as
 | (7) |
2.3, thus considerably decreasing the penetration through this individual channel. In contrasting to this, for the second group of channel lengths, i.e. the channel at the bottom of the second group (subscript r0 = 20), channel #7, the correction factor is only L3/L2 = L0/0.7L0 1.4. Thus, a calculation of the total penetration along the lowest channel of a group of channels with identical lengths may need to take account of the separation after the length of the channel plate proper.
For the channels (of a group of shortened plates) above the lowest of the channels of the group, the particles will have a much longer sedimentation distance until they are separated. The following simplified model may be used to study the effect on penetration. In the jth channel above the lowest channel in group r of channels of identical lengths, at the end of the channel plate, the particles are concentrated within the distance z
h penrj(LrVs), where z is measured from the floor of channel j. (See Fig. 1 for the enumeration with j of the channels within the groups of channels.) The sedimentation depth for all particles before reaching the floor in the first part of the downstream segment is h[j+penrj(Lr, Vs)] with a corresponding required sedimentation time, ts,
 | (8) |
 | (9) |
 | (10) |
Ur(h/Lr), which in combination with the requirement for zero losses in the first downstream segment gives
 | (11) |
In the example illustrating the model, the numerical value of the limiting factor in equation (11) for the first group of channels is 1.3, and thus, the tail of the total penetration curve through this channel (with subscript rj = 11, i.e. channel #2) will be somewhat sharpened for particles larger than the 50% cut size. In contrasting to this, the factor for the second group of channels is only 0.30, which implies no further separation in its downstream segment.
For groups of plates with lengths considerably longer than the length of the first part of the downstream segment, the particles generally will only sediment a short distance downwards in the segment, and therefore all particles not separated in the channel proper will escape further separation.
Figure 2 presents the penetration through channels 1, 2, 3, 5, 6 & 7 (i.e. subscript rj = 12, 11, 10, 22, 21 & 20, respectively) for both the velocity and penetration models presented here, and a simple model where the velocity in channel group r is proportional to L0/Lr, and only the penetration through the channel proper is determined. It can be seen that, for the first group of channels there is, as expected, significant influence on the penetration through channel #1, pen12, channel #2, pen11 and especially for channel #3, pen10. However, for the second group of channels there is only non-negligible influence on the penetration of channel #7, pen20.
|
The total penetration through all channels will then be calculated as the flow-weighted average over the penetration of the individual channels. For each group of channels with identical plate lengths, the penetrations of the channel at the bottom, the channel just above the channel at the bottom and all channels above these may be different.
|
DISCUSSION |
|---|
|
TOPABSTRACTSIMPLE MODEL OF A...DISTRIBUTION OF AIR FLOWSPARTICLE SEPARATION DOWNSTREAM...DISCUSSIONREFERENCES |
|---|
The simplified model presented above gives some guidelines to reduce the separation downstream of the channels proper. This may be achieved by two requirements: (1) The ratio of the additional length of a segment of the horizontal elutriator to the hydrodynamic diameter of the segment shall not be high; and (2) The additional length of the subsequent group of channels shall not be much longer than the plates of the present group of channels.
Due to the problems of determining whether additional separation occurs after the end of the channels proper (based on the simplified model above), it is recommended that this method for modifying the penetration curve of a horizontal elutriator to be used only when the number of plates or their spacing cannot be changed or that the manufacturing requirements for unequal plate spacing cannot be met, provided that the measured penetration can be modelled from first principles. Otherwise, a modification of the plate spacing with identical plate lengths is to be preferred, as the air flow is predictable and no simplified assumptions are needed to model the penetration. Using the approach with unequal plate spacing amounts to turning what Hamilton and Walton (1961) saw as a potential manufacturing problem into a design tool.
Received November 2, 2006; in final form February 5, 2007
|
REFERENCES |
|---|
|
TOPABSTRACTSIMPLE MODEL OF A...DISTRIBUTION OF AIR FLOWSPARTICLE SEPARATION DOWNSTREAM...DISCUSSIONREFERENCES |
|---|
British Medical Research Council. Recommendations of the MRC panels relating to selective sampling (Extracted from the minutes of a Joint Meeting of Panels 1, 2 and 3 held on 4th March, 1952). In: Inhaled particles and vapours—Davies CN, ed. (1961) Oxford, UK: Pergamon Press. 475.
Hamilton RJ, Walton WH. The selective sampling of respirable dust. In: Inhaled particles and vapours—Davies CN, ed. (1961) Oxford, UK: Pergamon Press. 465–75.
Mercer TT. Aerosol technology in hazard evaluation. (1973) New York (NY), USA: Academic Press.
Myojo T, Oyabu T, et al. Redesign of a static horizontal elutriator to perform according to the ISO 7708 Respirable Convention. Ann occup Hyg (2007) 51. in press.
Orenstein AJ. Recommendations adopted by the pneumoconiosis conference. In: The second pneumoconiosis conference—Orenstein AJ, ed. (1960) London, UK. J & A Churchill. 617–21.
Schlichting H. Boundary-layer theory. (1979) New York (NY), USA: McGraw-Hill.
Tillery M, Buchan R. Determination of large aerosol particle size by elutriation. Appl Occup Environ Hyg (2002) 17(10):717–22.[CrossRef][Medline]
Walton WH. Theory of size classification of airborne dust clouds by elutriation. Brit J Appl Phys (1954) 5(S3):S29–S37.[CrossRef]
CiteULike 
Connotea 
Del.icio.us What's this?
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||