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A SIMPLE MODEL OF AXIAL FLOW CYCLONE PERFORMANCE UNDER LAMINAR FLOW CONDITIONS
PII:
J. Aerosol Sci. Vol. 31, No. 2, pp. 151}167, 2000 Crown Copyright ( 1999 Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain S0021-8502(99)00035-X 0021-8502/99/$ - see front matter
A SIMPLE MODEL OF AXIAL FLOW CYCLONE PERFORMANCE UNDER LAMINAR FLOW CONDITIONS Andrew D. Maynard Health and Safety Laboratory, Broad Lane, She$eld, S3 7HQ, UK (First received 21 April 1998; and in xnal form 19 March 1999 ) Abstract=Axial #ow cyclones are particularly attractive in both gas cleaning and aerosol sampling due to the axial alignment between the inlet and outlet. In order to aid the development of suitable instruments for aerosol sampling, a mathematical model of axial #ow cyclone performance under laminar #ow conditions has been developed, using a number of simplifying assumptions. Aerosol penetration is considered separately through the vane section of the cyclone and its body, and the two models combined. Relating all dimensions to vane pitch, penetration as a function of Stokes' number in the cyclone vanes is predicted to be dependent on vane outer radius, vane inner radius and the number of vane turns. Penetration in the cyclone body is predicted to depend on body length alone. Encouraging agreement is seen between published experimental data for aerosol sampling cyclones and predicted d values. Crown Copyright ( 1999 Published by Elsevier 50 !% Science Ltd. All rights reserved
N OME NC LAT U RE C # d d 50 !% d !% d $ g ¸ ¸* n N P Pen Pen " Pen 7 Q r r* r* 0 r* "* Re Re 1 o f r* i r .!9 r* .!9 r .*/ r* .*/ o 1 r* 70 St St 50 St " St 7 t q ; ! ; 5 < < ! < 3
Cunningham slip correction factor particle diameter aerodynamic diameter at which 50% penetration occurs aerodynamic diameter droplet diameter gas viscosity e!ective length of cyclone body relative e!ective length of cyclone body number of vane turns number of equally spaced vanes pitch of vanes penetration penetration through cyclone body penetration through vanes #ow rate particle radial position in the cyclone relative particle radial position in the cyclone relative particle radial position at t"0 relative particle radial position at inlet to body #ow channel Reynolds number particle Reynolds number gas density relative particle radial position at cyclone inlet that de"nes the boundary between penetration and the deposition inside radius of cyclone body relative inside radius of cyclone body radius of vane spindle relative radius of vane spindle density relative particle radial position at outlet of vanes Stokes' number Stokes' number at which 50% penetration occurs Stokes' number in cyclone body Stokes' number in cyclone vanes time particle relaxation time mean axial gas velocity in cyclone body tangential gas velocity in the cyclone body mean gas velocity in vanes axial gas velocity in vanes radial gas velocity in vanes 151
152
A. D. Maynard < 5 w w* z*
tangential gas velocity in vanes vane thickness relative vane thickness relative particle axial position
I N T RO DU CT I O N
The use of cyclones to separate out particles or droplets entrained in gases is well established. All cyclone separators are based on centrifugal separation of particles in an induced vortex within the gas #ow. The method by which the vortex is induced, and the operating conditions of the separator primarily de"ne di!erent groups of devices. They may further be grouped according to their main function, whether it is to remove as much particulate as possible from the gas #ow (gas cleaning), or whether it is to remove particles according to a speci"c penetration probability curve (size selective sampling). Cyclone separators are predominantly used as industrial gas cleaning devices. As a means of removing particulates from a gas #ow they have the advantage over "lters of allowing continuous material removal without clogging, while exhibiting relatively low-pressure drops as a function of #ow rate. However, at small particle sizes particle penetration through cyclones increases, leading to relatively poor performance compared to "lters. The simplest devices create a vortex within a cylindrical volume by introducing the gas at a tangent to the cylinder wall. Cleaned gas is removed along the axis of the vortex. These are termed tangential #ow cyclones, and they are in widespread use throughout industry. Their success lies in part in the simplicity of their design. However, the need for the inlet to be tangential to the outlet restricts their use to situations where the change in #ow direction is not a serious problem. For applications where a particulate separator needs to be positioned in a straight section of a #ow system an alternative cyclone con"guration is possible. The axial #ow cyclone induces a vortex within the #ow using a series of vanes in line with the #ow direction. In this con"guration both the inlet and outlet are in line, allowing the separator to be positioned in straight #ow system sections, while retaining the low-pressure drop and no-clogging characteristics of cyclone separators. This con"guration has the added advantage of enabling units to be placed in parallel, thus increasing throughput, or in series, thus increasing e$ciency. Cyclone separators are also used as aerosol samplers to remove particles above a speci"c size from a gas #ow, while allowing smaller particles to remain entrained for further analysis. Historically the cyclone has been the instrument of choice for personal respirable aerosol sampling, with the Higgins and Dewell cyclone being developed in Europe, and the Dorr}Oliver cyclone in the USA. These devices have penetration characteristics close to the international respirable convention (ISO, 1995), which in turn de"nes the probability of a particle with a given aerodynamic diameter of penetrating to the alveolar region of the respiratory system upon inhalation. Both of these devices are tangential cyclones, and operate by allowing the respirable fraction of an aerosol through to be collected on a "lter, while removing the larger super-respirable particles from the #ow. When used for personal aerosol sampling they have many positive attributes: pressure drop at operating #ow rates is relatively low, enabling long operating times with a portable pump; most particles removed from the #ow are transported to a large collection chamber (grit pot), thus avoiding the problems associated with deposit build-up; they may be designed as compact devices when used at reasonable #ow rates for portable pumps, while achieving separation characteristics comparable with health-related sampling conventions. Because of this there is still considerable interest in using cyclones as aerosol samplers in preference to alternatives such as impactors when carrying out health-related sampling. The main drawback of the tangential #ow cyclone in any aerosol sampling application is the physical necessity of having an inlet that is at right angles to the outlet. Although not a serious problem when carrying out personal sampling, this poses problems when attempting to cascade cyclones with other devices. Cascade tangential #ow cyclones are used in speci"c situations (such as stack sampling), although these systems have a complex
Axial #ow cyclone performance
153
geometry and in general this approach is not suitable for many potential applications. An obvious solution is to follow the route taken in gas cleaning applications and develop axial #ow cyclones for aerosol sampling. These would allow the opportunity of cascading the cyclones with other devices, allowing for instance easily con"gurable cascade cyclones for size distribution analysis, or size separators to be placed in line with other devices such as well characterised sampling heads and aerosol pre-separators. Despite the attractiveness of axial #ow cyclones for aerosol sampling, very little has previously been published on the subject. One of the earliest applications of axial #ow cyclones was within the Czechoslovakian mining industry (Weiss et al., 1987). Two devices were developed, the DP20 and DP50 samplers, to give a respirable penetration curve at 20 and 50 l min~1 respectively. These devices are still in use within the Czech Republic, although have not been used widely elsewhere. Another early application of an axial #ow cyclone separator was the Wedding dichotomous sampling head (Wedding et al., 1982), designed to act as a pre-separator for particles smaller than 10 km. This design has now been superseded by impactors for the PM10 and PM2.5 ambient particulate sampling standards in the U.S.A. In 1984 Liu and Rubow developed a static cascade axial #ow cyclone system for determining aerosol size distributions (Liu and Rubow, 1984). Despite encouraging initial results from this system there is no evidence of it being pursued. However, the idea of using axial #ow cyclones as personal aerosol samplers was taken up by Vaughan (Vaughan, 1988). Vaughan measured the penetration characteristics of a number of cyclone con"gurations. Although performance appeared unpredictable under certain conditions he demonstrated the feasibility of developing a compact respirable axial #ow cyclone sampler. One of the barriers to the development of the axial #ow cyclone as an aerosol sampler appears to be the lack of information on performance as a function of various design parameters, together with a lack of incentive following the successful development of the Higgins and Dewell, and Dorr}Oliver cyclones. While there is a clear advantage in developing axial #ow cyclones for use in certain aerosol sampling applications, this is unlikely to proceed without an investigation of key factors a!ecting sampler performance. As a preliminary to carrying out a systematic experimental study, a simple mathematical model has been developed. This enables an initial estimate of the key parameters in cyclone operation and design, together with possible values for practical operation. The model indicates which dimensions in the cyclone are likely to have a dominant in#uence on cyclone behaviour, de"ned as the particle aerodynamic diameter at which penetration is 50% ( d ). It also allows an estimate of realistic values to assign these dimensions when 50 !% developing prototypes for experimental testing. Where possible the model has been validated against published experimental data.
M AT HE MA TIC AL M O DE L O F A XIA L F LO W C YCL ON E PE NE TR ATI ON
An axial #ow cyclone may be idealised by considering a cylinder of internal radius r with a vane system at the inlet for imparting rotational motion to the incoming gas, and .!9 an axial outlet at the opposite end, with a radius less than r (Fig. 1). By ensuring that the .!9 outlet protrudes along the axis of the cyclone, a zone of still air is created around it, thus e!ectively forming a receptacle for deposited particles. In the model a helicoidal vane system is used to allow some simpli"cation of air velocities through this region. This is de"ned in terms of the pitch P of the helicoidal vanes, the radius of the central spindle r , .*/ the outer radius of the vanes r and n; the number of complete turns of each vane about .!9 the central spindle. Initially, it was assumed that the vanes were in"nitely thin, and that a single vane was used. When developing the model it was assumed particle separation within the vanes and the cyclone body would dominate penetration, with separation at the outlet leading to secondary e!ects. The model developed therefore deals solely with separation in these two regions. Particle motion in each region is considered under much simpli"ed laminar #ow conditions (channel Reynolds number Re(2000).
154
A. D. Maynard
Fig. 1. Schematic of the simpli"ed axial #ow cyclone, showing key model parameters.
P EN ET RAT I O N TH RO U GH CY CLO N E V AN ES
To a "rst approximation, the radial position of a particle within the vanes of the cyclone may be estimated from its radial velocity. If the relaxation time of the particle is assumed to be small, radial velocity < may be calculated by balancing the lack of centripetal force at 3 a given radius r by the drag on the particle. If particle motion is in the Stokes regime (particle Reynolds number Re (1), then 1 q < " <2 , (1) 3 r 5 where q is the particle relaxation time and < is its tangential velocity. Relaxation time q is 5 given as d2o C q" 1 # , (2) 18g where d is the particle diameter, o the particle density and g the gas viscosity. C is the 1 # Cunningham slip correction factor. Assuming plug #ow, tangential air velocity through a helicoidal channel is given by 2nr< <" , (3) 5 (P2#4n2r2)1@2
Axial #ow cyclone performance
155
where < is the mean gas speed through the channel. The relationship between particle radial position within the vanes and time t from entering the vanes may therefore be obtained by replacing < in equation (1) by the expression in equation (3), and integrating. The resulting 5 expression is simpli"ed to a dimensionless form by relating all dimensions to helicoid pitch P. Relative dimensions, de"ned as the absolute dimension divided by P, are denoted by a superscript *. The relationship between t and r is therefore given as
C AB
where
D
r* #2n2(r*2!r*2 ) , 0 r* 0
t"A Ln
P A" 4n2< St
and
(4)
q< St " , 7 P
(5) 7 where r* is the initial radial position of a particle as it enters the cyclone, and St is the 7 0 particle Stokes' number in the vanes, with vane pitch P as the characteristic length. Using equation (4) to calculate particle position as a function of time is not trivial, and does not lead to a simple analytical solution. However for instances where r*'0.16 the log term in 0 equation (4) is always smaller than the remaining bracketed term for r*'r* , leading to 0 cyclone behaviour being dominated by the remaining term. Although there will always be theoretical systems where the in#uence of the log term is signi"cant, it may be ignored to a "rst approximation for practical systems, giving
A
r*(t)"
B
2< St 1@2 7 t#r*2 . 0 P
(6)
This is identical to the solution obtained assuming plug #ow and a tangential velocity that is independent of r*. Relative particle position z* along the axis of the cyclone may be obtained by integrating axial velocity < with time ! 1 z*" < dt, (7) P !
P
< may be expressed as ! 1 <" < ! J1#4n2r*2
(8)
in the helicoidal channel. Substituting equations (6) and (8) into equaion (7) and integrating gives
CA
B
D
t 1@2 1 !(1#4n2r*2)1@2 . z*(t)" 1#4n2r*2#8n2 St2 (9) 7 q 0 0 4n2St 7 From equations (6) and (9) the initial particle radial position r* that leads to total deposition 0 in a helicoidal channel of axial length l (where l"nP) may be calculated. Denoting this as r*, l r*2"4n2n2St2 !2n St (1#4n2r*2 )1@2#r*2 (10) 7 7 .!9 .!9 l With the cylindrical geometry of the cyclone, penetration through the vanes, Pen , may be 7 calculated by comparing the total cross-sectional area to that containing aerosol which does not deposit (de"ned by the boundary radius r*). Pen is therefore expressed as 7 l r*2!r*2 0 . Pen " l (11) 7 r*2 !r*2 .!9 0 Substituting equation (10) into equation (11) therefore gives 4n2n2St2!2nSt (1#4n2r*2 )1@2 .!9 7 7 Pen "1# . 7 r*2 !r*2 .!9 0
(12)
156
A. D. Maynard
Setting Pen to 0.5 gives the particle Stokes number for 50% penetration through the 7 cyclone vanes: 2nJ1#4n2r*2 !(4n2(1#4n2r*2 )!8n2n2(r*2 !r*2 ))1@2 .*/ .!9 .!9 .!9 . St (vanes)" 50 8n2n2
(13)
Note that only the smaller of the two possible solutions to St gives a physically feasible result. 50 P EN ET RAT I O N TH RO U GH T HE CYC LO N E B OD Y
Nieuwstadt and Dirkzwager have developed a simple mathematical model looking at droplet motion within the body of an axial #ow hydrocyclone assuming no deposition in the turning vanes (Nieuwstadt and Dirkzwager, 1995). Tangential gas velocity in the cyclone body is assumed to vary as k ;" , 5 2nr
(14)
where k is a constant. Nieuwstadt and Dirkwager's model uses a vane system designed to give a tangential #ow velocity at the trailing edge of the vanes of twice the mean axial #ow velocity ; , giving ! 2r ; k"4nr ; and therefore ; " .!9 ! , (15) .!9 ! 5 r By balancing the lack of centripetal force at radius r against Stokes' drag they formulated a simple expression for particle transit time ¹ before it intercepted the outer wall of the hydrocyclone: o 18g n2 1 ¹"(r4 !r4) , (16) 0 o !of o d2 k2 .!9 1 1 $ r is the droplet radial position as it enters the cyclone body, and d the diameter of the 0 $ droplet. This model has been used as the starting point for estimating aerosol penetration through the body of an axial #ow cyclone. The same radial velocity pro"le as used by Nieuwstadt and Dirkwager has been assumed (equation 14). However, the constant term has been modi"ed to allow for di!erent vane con"gurations. To a "rst approximation the tangential gas velocity in the cyclone body may be given as k ; "Pr* ; 5 .!9 ! 2nr
(17)
with 4n2 k" (18) (1#4n2r*2 )1@2 .!9 Rewriting (16) for relatively high density particles in a gas using the expression in equation (18) for k gives n2P ¹"(r*4 !r*4) (19) .!9 0 r*4 k2; St .!9 ! " when written in terms of particle Stokes' number St , calculated using ; . " ! As ¹ is simply the ratio of the distance along the axis a particle will travel before depositing and the axial #ow velocity ; , it is a simple step to estimate the initial radial ! position that de"nes the boundary between deposition and no deposition in a cyclone body of relative length ¸*:
A
B
1@2 k2 r*2"r*2 1! St . ¸* .!9 0 " n2
(20)
Axial #ow cyclone performance
157
Equating the cyclone body inlet area de"ned by r* to the total cross sectional area gives 0 body penetration Pen as " 1@2 k2 Pen " 1!St . (21) ¸* " " n2
A
B
At 50% penetration r*2"r*2 /2, giving .!9 0
3 n2 St " . "50 4 k2 ¸*
(22)
CO MB IN ED MO D EL
In the laminar #ow regime assumed for the cyclone, no turbulent aerosol mixing will take place between the vane system and the cyclone body. Calculation of overall penetration is therefore only possible by modelling particle motion through the complete cyclone. By making simplistic assumptions about the nature of particle motion between the two cyclone sections, this is relatively easy using the two models developed above. In their model of body penetration, Nieuwstadt and Dirkwager assumed that just prior to entering the cyclone body the aerosol was uniformly distributed across the body inlet. In other words, there was no inertial separation of particles in the transition between the vane system and the body. By making the same assumption about the axial #ow cyclone, particle radial position as it leaves the vanes may be related to its radial position as it enters the cyclone body. Through conservation of mass, the aerosol mass #ux leaving the vanes through the annulus de"ned by radii r* and r* is identical to that passing through the circular area in 70 .*/ the cyclone body de"ned by r* (Fig. 2). Therefore, "* r*2 .!9 r*2" (23) (r*2!r*2 ), "* .*/ r*2 !r*2 70 .*/ .!9 where r* now de"nes the radial position of a particle entering the cyclone body for a given "* vane outlet radial position r* . The assumption that this transition takes place over an 70 insigni"cantly small length, with no inertial separation, is physically unfeasible. However, it is su$cient to allow both penetration models to be combined to allow an approximate determination of overall penetration.
Fig. 2. Idealised axial #ow lines in the transition region between the vane system and the cyclone body. It is assumed that the length of the transition region is insigni"cant compared to the length of the cyclone, and that particles follow the streamlines in this region (i.e. no inertial separation).
158
A. D. Maynard
Equation (20) gives the initial radial position of a particle entering the cyclone body that de"nes the boundary between deposition and penetration. Equating r* in equation (20) to 0 r* in equation (23) therefore gives the radial position of a particle leaving the vanes that will "* just be deposited within the cyclone. Likewise equating r* with r* in equation (10) gives .!9 "0 the initial radial position of a particle entering the cyclone that de"nes the boundary between penetration and deposition (r*). Particle Stokes numbers in the cyclone vanes and i body are related by the ratio of the mean air speed in the respective sections: St "! St 7 "
(24)
< !" . ; !
(25)
where
Thus the de"ning inlet radius r* may be expressed as * r*2"r*2 .*/ i #(r*2 !r*2 ) J1!16n2St ¸* .*/ " .!9 #4n2n2!2 St2 " !2n! St (1#4n2r*2 #4n2 J1!16n2St ¸* )1@2 " .*/ "
(26)
and overall cyclone penetration as Pen"(1!16n2St ¸*)1@2 " 4n2n2!2 St2 " # r*2 !r*2 .!9 .*/ 2n! St " (1#4n2r*2 #4n2 J1!16n2St ¸* )1@2. ! .*/ " r*2 !r*2 .*/ .!9
(27)
Because of the form of this expression for cyclone penetration, it is not possible to de"ne an analytical expression for St . However determination of St using iterative techniques is 50 50 a trivial matter using relatively simple computer codes. PR ED ICT ED CY CLO NE P ER FO RM AN CE A S A F UN CT I O N OF S t
Figures 3}5 show predicted cyclone performance for three speci"c design regions. In Fig. 3 the number of inlet vane turns n is signi"cantly smaller than 1, and the cyclone has a small relative body diameter. The plot shows penetration dominated by deposition within the cyclone body, with penetration through the vanes being close to 100%. This is the ideal mode of operation for industrial gas cleaning (or liquid cleaning in the case of hydrocyclones) applications, where deposition in the vanes is likely to lead to a build-up of deposits, and most industrial axial #ow cyclones are built with n(1 and a relatively small body diameter. It is also the preferred mode of operation for aerosol sampling where a build-up of deposits in the vanes increases the likelihood of particle carryover in the outlet #ow. Figure 4 demonstrates the e!ect of increasing the number of vane turns to greater than 1 and increasing the relative body diameter. By selecting the various design parameters appropriately particle deposition in the vanes becomes dominant, with body losses only contributing to secondary e!ects in cyclone penetration. This design region lies closer to that used by Vaughan. Figure 5 demonstrates the design parameters necessary for comparable contributions from both the vane section and the cyclone body.
Axial #ow cyclone performance
159
Fig. 3. Modelled axial #ow cyclone penetration. r* "0.2; r* "0.02; n"0.2; ¸*"3. .!9 .*/
Fig. 4. Modelled axial #ow cyclone penetration. r* "1; r* "0.5; n"2; ¸*"4. .!9 .*/
In Figs 6}9 the in#uence of ¸*, r* , r* and n on St is demonstrated. In each case St 50 50 .!9 .*/ is calculated using the mean axial body velocity ;. These are the four key parameters that determine St according to the combined model. Figure 8 shows the clear delineation of 50 operating regimes, with vane deposition dominating above n+1, and body deposition dominating below this region. However, although it may be generalised that cyclones with n(1 will be dominated by deposition within the body, and cyclones with n'1 will be dominated by deposition within the vanes, the model predicts a broad crossover region. It is therefore likely that losses within the vanes of an axial #ow cyclone will be signi"cant until the number of vane turns is signi"cantly smaller than 1.
160
A. D. Maynard
Fig. 5. Modelled axial #ow cyclone penetration. r* "0.2; r* "0.02; n"1.5; ¸*"2. .!9 .*/
Fig. 6. Plotting St as a function of r* . r* "0.02, n"1.5, ¸*"2. St referenced to mean axial 50 .!9 .*/ 50 body gas velocity ;.
E XTE N SI ON OF TH E MO D EL TO CYC LO NES W IT H M U LT IPL E V AN ES O F FI NI T E W ID TH
For simplicity the model cyclone initially had a single helicoidal vane that was in"nitely thin. Determination of penetration as a function of St using this model was valid as increasing the number of vanes and the vane thickness will primarily a!ect the mean gas velocity in the vanes, thus a!ecting St and not penetration as a function of St in the separate cyclone sections. In the combined model however there is a dependence on the gas velocity
Axial #ow cyclone performance
Fig. 7. Plotting St
50
161
as a function of r* . r* "0.2, n"1.5, ¸*"2. St referenced to mean axial .*/ .!9 50 body gas velocity ;.
Fig. 8. Plotting St as a function of n. r* "0.2, r* "0.02, ¸*"2. St referenced to mean axial 50 .!9 .*/ 50 body gas velocity ;.
in the cyclone vanes in the expression for ! (25). More importantly, when relating modelled penetration to aerodynamic diameter it is necessary to know <. The simplest approach to incorporating the e!ect of multiple vanes of "nite thickness into the model is to develop an appropriate expression for !. If the helicoidal vanes have a relative thickness w*, then the relative vertical channel height h* de"ned by multiple vane turns is given by h*"1!w* .
(28)
162
A. D. Maynard
Fig. 9. Plotting St as a function of ¸*. r* "0.2, r* "0.02, n"1.5. St referenced to mean 50 .!9 .*/ 50 axial body gas velocity ;.
Fig. 10. Plotting St as a function of Nw* assuming (a) in"nitely thin vanes and (b) vanes of "nite 50 thickness w*. r* "0.2, r* "0.02, n"1.5, ¸*"2. St referenced to mean axial body gas .!9 .*/ 50 velocity ;.
Generalising for a system with N vanes equally spaced about the cyclone axis, the e!ective relative channel height h* may be expressed as %&& 1!Nw* h* " . (29) %&& N The relative cross-sectional area perpendicular to the gas #ow of a channel de"ned by adjacent vanes is therefore given by 2nr* (1!Nw*) .!9 Area*(channel)" (r* !r* ). .*/ .!9 N J1#4n2r*2 .!9
(30)
Axial #ow cyclone performance
163
Assuming a #ow rate Q through the cyclone, ! is given by Q Area*(body) !" . Area*(channel)N Q
(31)
Thus substituting the cyclone body relative area and equation (30) into equation (31) gives r* J1#4n2r*2 .!9 . .!9 (32) !" 2(1!Nw*) (r* !r* ) .*/ .!9 Figure 10 demonstrates the in#uence of the factor Nw* over St for a cyclone dominated 50 by aerosol separation in the vanes. COMP ARI SO N B ET WE EN THE MO D EL AND EXP ER IME NTAL D ATA
In the published literature there are two sets of experimental data from axial #ow cyclones designed for aerosol sampling at relatively low Re. Of these Vaughan gives the most comprehensive account of axial #ow cyclone performance (Vaughan, 1988). Cyclones were constructed with two vane systems and three body lengths (Table 1), and the penetration as a function of particle size measured at four #ow rates. Table 2 compares the experimental d values obtained by Vaughan at four #ow rates, 50 !% with those estimated using the axial #ow cyclone model given in equation (27), with the
Table 1. Dimensions of the vane systems used by Vaughan (Vaughan, 1988). These systems were used with cyclone bodies of e!ective length 12.5 mm (S), 17.5 mm (M) and 22.5 mm (L). A single helicoid was used in each vane system. The dimensions are attributed to di!erent helix arrangements than in the published paper following discussions with Vaughan (Vaughan, 1998) Vane system
P/mm
w/mm
r /mm .!9
r /mm .*/
n
Helix 0 Helix 1
5 5
3.3 3.3
5 5
2.5 3.8
1.5 1.5
Table 2. Comparison between d values experimentally measured by Vaughan (Vaughan, 1988), and those 50 !% calculated using the model in equations (27) and (32). Data marked with s indicate that they originate from a penetration curve that is kinked. Experimental measurements using body M* were conducted with a liquid aerosol, all others used solid particles. The experimental data are attributed to di!erent helix arrangements than in the published paper following discussions with Vaughan (Vaughan, 1998) Helix 0
Helix 1
Body
Q/l min~1
Experiment d /km 50 !%
Theory d /km 50 !%
Vane Re
Experiment d /km 50 !%
Theory d /km 50 !%
Vane Re
L
1.240 2.281 3.235 3.750
4.85s 2.7 1.8 1.6
3.6 2.6 2.2 2.0
496 912 1293 1499
3.1s 1.96 1.45 1.32
1.8 1.3 1.1 1.0
715 1316 1867 2399
M
1.240 2.281 3.235 3.750
4.32s 2.4 1.7 *
3.6 2.6 2.2 2.0
496 912 1293 1499
2.35s 1.65 1.27 *
1.8 1.3 1.1 1.0
715 1316 1867 2399
M*
1.240 2.281 3.235 3.750
6.5s 3.42 2.47 1.93
3.6 2.6 2.2 2.0
496 912 1293 1499
4.75s 2.84 1.88 1.41
1.8 1.3 1.1 1.0
715 1316 1867 2399
S
1.240 2.281 3.235 3.750
4.0s 2.08 1.54 1.35
3.6 2.6 2.2 2.0
496 912 1293 1499
2.25s 1.83 1.35 *
1.8 1.3 1.1 1.0
715 1316 1867 2399
164
A. D. Maynard
Fig. 11. Comparing measured cyclone penetration and modelled penetration for helix 0 and body ¸ at 2.281 l min~1 (Vaughan, 1988).
expression for ! given in equation (32). At the lowest #ow rate, Vaughan found the penetration curve was not a smooth transitional function, but showed a step or &&kink'', indicating some form of complex #ow behaviour was taking place. Good agreement between experiment and theory was therefore not expected at these points, and indeed was not seen. For solid aerosols, the agreement between theory and experiment is good, considering the simplistic assumptions made in the model. In most cases agreement is well within 25%. Poorer agreement is seen for the liquid aerosol, although even here the estimated points agree to well within an order of magnitude. Although helix 0 has a #ow Reynolds number in the vanes greater than 2000 at 3.75 l min~1 (calculated using the square root of the vane channel area), and is therefore likely to be operating outside the laminar #ow regime, there is no indication of this in#uencing cyclone performance signi"cantly. However there is evidence for the length of the cyclone body having a more signi"cant e!ect on performance than the model predicts. Figure 11 compares the measured penetration with the modelled penetration for one cyclone con"guration from Vaughan. Although the model predicts the d point reason50 !% ably well, the shapes of the penetration curves di!er considerably. Similar comparisons are seen for all other sets of measurements where no kink appeared in the experimentally determined penetration. Liu and Rubow constructed a cascade axial #ow cyclone and published data on the measured penetration of each stage (Liu and Rubow, 1984). Unfortunately, no information was published on either the number of vane turns in the vane section of the cyclones, or the e!ective body length. In modelling the stages educated estimates have therefore been made of these dimensions. n has been assumed to be 1 for each stage as this is indicated in the drawings published, and gives reasonable agreement when used with the model. ¸ has been taken as the body length published, which is not equivalent to the e!ective body length used in the model. However, it can be shown that with the vane systems used, the model is insensitive to variations in ¸. Table 3 gives the dimensions of each stage of the cascade cyclone. Vane thickness w was not given, although "gures in the paper indicate it to the small relative to P. It has therefore been considered negligible when modelling cyclone performance. Table 4 presents the measured d values of each cyclone stage when operated at 30 l min~1, together with 50 !% values estimated from the model given in equation (27), together with equation (32). As shown in Table 4, all stages bar stage 1 are operating at Re within the cyclone vane systems greater than 2000, and are therefore unlikely to be operating under laminar #ow conditions.
Axial #ow cyclone performance
165
Table 3. Dimensions of the cascade axial #ow cyclones used by Liu and Rubow (Liu and Rubow 1984). The number of vane turns n has been estimated at 1. The body length ¸ is the length given by Liu and Rubow, and does not correspond exactly with the e!ective body length (Fig. 1). The #ow rate through each stage was 30 l min~1 Stage
N
P/mm
r /mm .!9
r /mm .*/
n (est)
¸/mm
1 2 3 4 5
2 2 2 2 1
33.9 25.4 12.7 10.2 5.1
25.2 25.2 15.2 10.4 10.4
15.4 18.7 9.2 7.2 9.1
1 1 1 1 1
51 51 32 32 32
Table 4. Comparison between d values experimentally mea50 !%Rubow, 1984), and those estisured by Liu and Rubow (Liu and mated using the cyclone model given by equations (27) and (32). In calculating the expected d values it was assumed that the vanes 50 !%in"nitely thin were
Stage
d /km 50 !% Experiment
Theory
Re (vane system)
1 2 3 4 5
12.2 7.9 3.6 2.05 1.05
11.6 6.9 4.4 2.1 0.6
1958 2793 4291 6723 14,849
Table 5. Dimensions of the Czech Republic axial #ow cyclones Cyclone
N
P/mm
r /mm .!9
r /mm .*/
n
¸/mm
DP20 DP50
4 4
22 28
7.5 12
4.5 4.5
0.5 0.36
25 12
Table 6. Comparison between d experimentally measured values and those 50 !% estimated using the cyclone model given by equations (27) and (32)
Cyclone DP20 DP50
Flow Rate (l min~1)
d /km 50 !% Experiment
Theory
Vane Re
8.1 15.0 10.4 15.2
4.8 3.0 9.6 7.3
5.9 4.3 13.8 11.4
1359 2532 1232 1800
Despite this, there is good agreement between experimental and modelled values for stages 1}4, with the largest variation of 22% occurring at stage 3. This may be slightly optimistic, as with the vane systems used variations in n a!ect the modelled d signi"cantly. For 50 !% instance, assuming n to be 0.5 for stages 1}4 (which would have been the smallest feasible value using two vanes) increases the modelled d for stage 1 to 16 km. However, even 50 !% allowing for variations in n between 0.5 and 2, the modelled d values are well within an 50 !% order of magnitude of the experimental data. For stage 5, the model underestimates d signi"cantly. However the high Re in the 50 !% vanes of this stage will lead to turbulent #ow, therefore placing the cyclone's operation outside the limits of the model. As well as published data on small axial #ow cyclone performance, limited measurements have been made on the DP20 and DP50 axial #ow cyclones used for aerosol sampling in the Czech Republic (Weiss et al., 1987). These are compact devices, capable of sampling health-related aerosol fractions at #ow rates between 8 and 50 l min~1; dimensions of the cyclones are given in Table 5. Table 6 presents the results of limited measurements made of
166
A. D. Maynard
Fig. 12. Comparison between experimental and modelled
d values. 50 !%
the internal penetration e$ciency (i.e. penetration e$ciency through the cyclone vanes and body, excluding inlet aspiration e!ects) using the TSI Aerodynamic Particle Sizer 3310 based technique described by Maynard and Kenny (Maynard and Kenny, 1995). The #ow rates were selected to achieve low Re within the cyclone vanes, and were not representative of operating #ow rates under normal conditions. Both of these cyclones di!er from those of Vaughan, and Liu and Rubow, in that the number of vane turns is less than one in each case. However, equation (27) predicts that separation within the cyclone vanes is still likely to be the dominant factor in determining overall penetration. The modelled d value for DP20 at 8 l min~1 (laminar #ow) is within 50 !% 25% of the experimentally measured value. The agreement is not as close at 15 l min~1, although the Re in the vanes indicates that the cyclone may be operating outside the laminar #ow regime. DP50 shows relatively poor agreement with the model (although d values are well within an order of magnitude), despite operating at low Re. It is not 50 !% clear what the cause of this discrepancy may be, although it may be signi"cant that this cyclone has the lowest number of vane turns out of all cyclones examined. Another possible explanation is that the penetration e$ciency of the cyclone begins to be in#uenced by factors such as the aspiration e$ciency of the vane inlets as the internal penetration e$ciency d increases to around 10 km and above. 50 !% Figure 12 plots experimentally measured d values for all cyclones against those 50 !% estimated from the model. Overall the agreement between experiment and theory is encouraging, considering the simplicity of the model developed. The assumption that the log term in equation (4) is insigni"cant is valid for most cyclone con"gurations. The DP20 and DP50 cyclones have small values of r* and r* (r* "0.16 for the DP50 cyclone), .!9 .*/ .*/ and this is likely to have led to some divergence of the model arising from the assumption. Data for these cyclones are lower than modelled values, but still in reasonable agreement with the model. Although more experimental data are required to further validate the model, existing data indicate it is suitable for giving an initial estimate of axial #ow cyclone performance. However, isolation of the modelled contribution to overall penetration from
Axial #ow cyclone performance
167
the vane system and the body indicates that for each of the experimental datasets considered, overall penetration is dominated by particle separation within the cyclone vanes (e.g. Fig. 4). Modelled performance where body separation dominates, characterised by Fig. 3, has been largely unexplored. Further experimental work is therefore required in this area to fully explore the applicability of the model. CO NC LU SIO N S
The aim here has been to develop a mathematical model of axial #ow cyclone behaviour under laminar #ow conditions (low Re) that is accurate enough to allow an initial estimate of the performance of new cyclone designs. Although a number of simplifying assumptions have been made concerning air #ow and particle motion within such a cyclone, there is encouraging agreement between experimental data and modelled d points. Fair agree50 !% ment is also seen at relatively high #ow Reynolds numbers in the cyclone vanes, where the #ow pro"le is likely to diverge further from the oversimpli"cation of laminar #ow than at low Re. However, comparisons have only been possible for cyclones where aerosol separation in the vanes is likely to dominate overall penetration e$ciency. Thus for devices where separation below the vanes dominates, the model is largely unvalidated. The next step will be to construct a series of cyclones based on predictions made by the model, and measure their performance as a function of both particle aerodynamic diameter and Stokes' number. Most importantly, validation of the model of penetration as a function of St for a wide range of cyclone dimensions will help determine whether the key design parameters identi"ed may in reality be used to determine cyclone performance. As it stands, the model predicts the development of compact, high #ow rate sampling devices following workplace and environmental sampling conventions*necessary criteria for devices either used for monitoring personal aerosol exposure, or used over a relatively short sampling duration. Combining further validation of the model with empirical data on cyclone performance will lay the groundwork for developing axial #ow cyclones suited to a range of size-selective aerosol sampling applications REF ER E NCE S ISO (1995) Air quality*Particle size fraction de"nitions for health-related sampling. Geneva, International Standards Organisation: ISO Standard 7708. Liu, B. Y. H. and Rubow, K. L. (1984). A new axial #ow cascade cyclone for size characterisation of airborne particulate matter. In: Aerosols (Edited by Liu, B. Y. H., Pui, D. Y. and Fissan, H. J.), pp. 115}118. Elsevier, Amsterdam. Maynard, A. D. and Kenny, L. C. (1995). Performance assessment of three personal cyclone models, using an aerodynamic particle sizer. J. Aerosol Sci. 26(4), 671}684. Nieuwstadt, F. T. M. and Dirkzwager, M. (1995) A #uid mechanical model for an axial cyclone separator. Ind. Engng Chem. Res. 34, 3399}3404. Vaughan (1998) Private communication. Vaughan, N. P. (1988). Construction and testing of an axial #ow cyclone preseparator. J. Aerosol Sci. 19(3), 295}305. Wedding, J. B., Weigand, M. A. and Carney, T. C. (1982) A 10 lm cutpoint inlet for the dichotomous sampler. Environ. Sci. ¹echnol. 16(9), 602}606. Weiss, Z., Martinec, P. and Vitek, J. (1987).
J. Aerosol Sci. Vol. 31, No. 2, pp. 151}167, 2000 Crown Copyright ( 1999 Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain S0021-8502(99)00035-X 0021-8502/99/$ - see front matter
A SIMPLE MODEL OF AXIAL FLOW CYCLONE PERFORMANCE UNDER LAMINAR FLOW CONDITIONS Andrew D. Maynard Health and Safety Laboratory, Broad Lane, She$eld, S3 7HQ, UK (First received 21 April 1998; and in xnal form 19 March 1999 ) Abstract=Axial #ow cyclones are particularly attractive in both gas cleaning and aerosol sampling due to the axial alignment between the inlet and outlet. In order to aid the development of suitable instruments for aerosol sampling, a mathematical model of axial #ow cyclone performance under laminar #ow conditions has been developed, using a number of simplifying assumptions. Aerosol penetration is considered separately through the vane section of the cyclone and its body, and the two models combined. Relating all dimensions to vane pitch, penetration as a function of Stokes' number in the cyclone vanes is predicted to be dependent on vane outer radius, vane inner radius and the number of vane turns. Penetration in the cyclone body is predicted to depend on body length alone. Encouraging agreement is seen between published experimental data for aerosol sampling cyclones and predicted d values. Crown Copyright ( 1999 Published by Elsevier 50 !% Science Ltd. All rights reserved
N OME NC LAT U RE C # d d 50 !% d !% d $ g ¸ ¸* n N P Pen Pen " Pen 7 Q r r* r* 0 r* "* Re Re 1 o f r* i r .!9 r* .!9 r .*/ r* .*/ o 1 r* 70 St St 50 St " St 7 t q ; ! ; 5 < < ! < 3
Cunningham slip correction factor particle diameter aerodynamic diameter at which 50% penetration occurs aerodynamic diameter droplet diameter gas viscosity e!ective length of cyclone body relative e!ective length of cyclone body number of vane turns number of equally spaced vanes pitch of vanes penetration penetration through cyclone body penetration through vanes #ow rate particle radial position in the cyclone relative particle radial position in the cyclone relative particle radial position at t"0 relative particle radial position at inlet to body #ow channel Reynolds number particle Reynolds number gas density relative particle radial position at cyclone inlet that de"nes the boundary between penetration and the deposition inside radius of cyclone body relative inside radius of cyclone body radius of vane spindle relative radius of vane spindle density relative particle radial position at outlet of vanes Stokes' number Stokes' number at which 50% penetration occurs Stokes' number in cyclone body Stokes' number in cyclone vanes time particle relaxation time mean axial gas velocity in cyclone body tangential gas velocity in the cyclone body mean gas velocity in vanes axial gas velocity in vanes radial gas velocity in vanes 151
152
A. D. Maynard < 5 w w* z*
tangential gas velocity in vanes vane thickness relative vane thickness relative particle axial position
I N T RO DU CT I O N
The use of cyclones to separate out particles or droplets entrained in gases is well established. All cyclone separators are based on centrifugal separation of particles in an induced vortex within the gas #ow. The method by which the vortex is induced, and the operating conditions of the separator primarily de"ne di!erent groups of devices. They may further be grouped according to their main function, whether it is to remove as much particulate as possible from the gas #ow (gas cleaning), or whether it is to remove particles according to a speci"c penetration probability curve (size selective sampling). Cyclone separators are predominantly used as industrial gas cleaning devices. As a means of removing particulates from a gas #ow they have the advantage over "lters of allowing continuous material removal without clogging, while exhibiting relatively low-pressure drops as a function of #ow rate. However, at small particle sizes particle penetration through cyclones increases, leading to relatively poor performance compared to "lters. The simplest devices create a vortex within a cylindrical volume by introducing the gas at a tangent to the cylinder wall. Cleaned gas is removed along the axis of the vortex. These are termed tangential #ow cyclones, and they are in widespread use throughout industry. Their success lies in part in the simplicity of their design. However, the need for the inlet to be tangential to the outlet restricts their use to situations where the change in #ow direction is not a serious problem. For applications where a particulate separator needs to be positioned in a straight section of a #ow system an alternative cyclone con"guration is possible. The axial #ow cyclone induces a vortex within the #ow using a series of vanes in line with the #ow direction. In this con"guration both the inlet and outlet are in line, allowing the separator to be positioned in straight #ow system sections, while retaining the low-pressure drop and no-clogging characteristics of cyclone separators. This con"guration has the added advantage of enabling units to be placed in parallel, thus increasing throughput, or in series, thus increasing e$ciency. Cyclone separators are also used as aerosol samplers to remove particles above a speci"c size from a gas #ow, while allowing smaller particles to remain entrained for further analysis. Historically the cyclone has been the instrument of choice for personal respirable aerosol sampling, with the Higgins and Dewell cyclone being developed in Europe, and the Dorr}Oliver cyclone in the USA. These devices have penetration characteristics close to the international respirable convention (ISO, 1995), which in turn de"nes the probability of a particle with a given aerodynamic diameter of penetrating to the alveolar region of the respiratory system upon inhalation. Both of these devices are tangential cyclones, and operate by allowing the respirable fraction of an aerosol through to be collected on a "lter, while removing the larger super-respirable particles from the #ow. When used for personal aerosol sampling they have many positive attributes: pressure drop at operating #ow rates is relatively low, enabling long operating times with a portable pump; most particles removed from the #ow are transported to a large collection chamber (grit pot), thus avoiding the problems associated with deposit build-up; they may be designed as compact devices when used at reasonable #ow rates for portable pumps, while achieving separation characteristics comparable with health-related sampling conventions. Because of this there is still considerable interest in using cyclones as aerosol samplers in preference to alternatives such as impactors when carrying out health-related sampling. The main drawback of the tangential #ow cyclone in any aerosol sampling application is the physical necessity of having an inlet that is at right angles to the outlet. Although not a serious problem when carrying out personal sampling, this poses problems when attempting to cascade cyclones with other devices. Cascade tangential #ow cyclones are used in speci"c situations (such as stack sampling), although these systems have a complex
Axial #ow cyclone performance
153
geometry and in general this approach is not suitable for many potential applications. An obvious solution is to follow the route taken in gas cleaning applications and develop axial #ow cyclones for aerosol sampling. These would allow the opportunity of cascading the cyclones with other devices, allowing for instance easily con"gurable cascade cyclones for size distribution analysis, or size separators to be placed in line with other devices such as well characterised sampling heads and aerosol pre-separators. Despite the attractiveness of axial #ow cyclones for aerosol sampling, very little has previously been published on the subject. One of the earliest applications of axial #ow cyclones was within the Czechoslovakian mining industry (Weiss et al., 1987). Two devices were developed, the DP20 and DP50 samplers, to give a respirable penetration curve at 20 and 50 l min~1 respectively. These devices are still in use within the Czech Republic, although have not been used widely elsewhere. Another early application of an axial #ow cyclone separator was the Wedding dichotomous sampling head (Wedding et al., 1982), designed to act as a pre-separator for particles smaller than 10 km. This design has now been superseded by impactors for the PM10 and PM2.5 ambient particulate sampling standards in the U.S.A. In 1984 Liu and Rubow developed a static cascade axial #ow cyclone system for determining aerosol size distributions (Liu and Rubow, 1984). Despite encouraging initial results from this system there is no evidence of it being pursued. However, the idea of using axial #ow cyclones as personal aerosol samplers was taken up by Vaughan (Vaughan, 1988). Vaughan measured the penetration characteristics of a number of cyclone con"gurations. Although performance appeared unpredictable under certain conditions he demonstrated the feasibility of developing a compact respirable axial #ow cyclone sampler. One of the barriers to the development of the axial #ow cyclone as an aerosol sampler appears to be the lack of information on performance as a function of various design parameters, together with a lack of incentive following the successful development of the Higgins and Dewell, and Dorr}Oliver cyclones. While there is a clear advantage in developing axial #ow cyclones for use in certain aerosol sampling applications, this is unlikely to proceed without an investigation of key factors a!ecting sampler performance. As a preliminary to carrying out a systematic experimental study, a simple mathematical model has been developed. This enables an initial estimate of the key parameters in cyclone operation and design, together with possible values for practical operation. The model indicates which dimensions in the cyclone are likely to have a dominant in#uence on cyclone behaviour, de"ned as the particle aerodynamic diameter at which penetration is 50% ( d ). It also allows an estimate of realistic values to assign these dimensions when 50 !% developing prototypes for experimental testing. Where possible the model has been validated against published experimental data.
M AT HE MA TIC AL M O DE L O F A XIA L F LO W C YCL ON E PE NE TR ATI ON
An axial #ow cyclone may be idealised by considering a cylinder of internal radius r with a vane system at the inlet for imparting rotational motion to the incoming gas, and .!9 an axial outlet at the opposite end, with a radius less than r (Fig. 1). By ensuring that the .!9 outlet protrudes along the axis of the cyclone, a zone of still air is created around it, thus e!ectively forming a receptacle for deposited particles. In the model a helicoidal vane system is used to allow some simpli"cation of air velocities through this region. This is de"ned in terms of the pitch P of the helicoidal vanes, the radius of the central spindle r , .*/ the outer radius of the vanes r and n; the number of complete turns of each vane about .!9 the central spindle. Initially, it was assumed that the vanes were in"nitely thin, and that a single vane was used. When developing the model it was assumed particle separation within the vanes and the cyclone body would dominate penetration, with separation at the outlet leading to secondary e!ects. The model developed therefore deals solely with separation in these two regions. Particle motion in each region is considered under much simpli"ed laminar #ow conditions (channel Reynolds number Re(2000).
154
A. D. Maynard
Fig. 1. Schematic of the simpli"ed axial #ow cyclone, showing key model parameters.
P EN ET RAT I O N TH RO U GH CY CLO N E V AN ES
To a "rst approximation, the radial position of a particle within the vanes of the cyclone may be estimated from its radial velocity. If the relaxation time of the particle is assumed to be small, radial velocity < may be calculated by balancing the lack of centripetal force at 3 a given radius r by the drag on the particle. If particle motion is in the Stokes regime (particle Reynolds number Re (1), then 1 q < " <2 , (1) 3 r 5 where q is the particle relaxation time and < is its tangential velocity. Relaxation time q is 5 given as d2o C q" 1 # , (2) 18g where d is the particle diameter, o the particle density and g the gas viscosity. C is the 1 # Cunningham slip correction factor. Assuming plug #ow, tangential air velocity through a helicoidal channel is given by 2nr< <" , (3) 5 (P2#4n2r2)1@2
Axial #ow cyclone performance
155
where < is the mean gas speed through the channel. The relationship between particle radial position within the vanes and time t from entering the vanes may therefore be obtained by replacing < in equation (1) by the expression in equation (3), and integrating. The resulting 5 expression is simpli"ed to a dimensionless form by relating all dimensions to helicoid pitch P. Relative dimensions, de"ned as the absolute dimension divided by P, are denoted by a superscript *. The relationship between t and r is therefore given as
C AB
where
D
r* #2n2(r*2!r*2 ) , 0 r* 0
t"A Ln
P A" 4n2< St
and
(4)
q< St " , 7 P
(5) 7 where r* is the initial radial position of a particle as it enters the cyclone, and St is the 7 0 particle Stokes' number in the vanes, with vane pitch P as the characteristic length. Using equation (4) to calculate particle position as a function of time is not trivial, and does not lead to a simple analytical solution. However for instances where r*'0.16 the log term in 0 equation (4) is always smaller than the remaining bracketed term for r*'r* , leading to 0 cyclone behaviour being dominated by the remaining term. Although there will always be theoretical systems where the in#uence of the log term is signi"cant, it may be ignored to a "rst approximation for practical systems, giving
A
r*(t)"
B
2< St 1@2 7 t#r*2 . 0 P
(6)
This is identical to the solution obtained assuming plug #ow and a tangential velocity that is independent of r*. Relative particle position z* along the axis of the cyclone may be obtained by integrating axial velocity < with time ! 1 z*" < dt, (7) P !
P
< may be expressed as ! 1 <" < ! J1#4n2r*2
(8)
in the helicoidal channel. Substituting equations (6) and (8) into equaion (7) and integrating gives
CA
B
D
t 1@2 1 !(1#4n2r*2)1@2 . z*(t)" 1#4n2r*2#8n2 St2 (9) 7 q 0 0 4n2St 7 From equations (6) and (9) the initial particle radial position r* that leads to total deposition 0 in a helicoidal channel of axial length l (where l"nP) may be calculated. Denoting this as r*, l r*2"4n2n2St2 !2n St (1#4n2r*2 )1@2#r*2 (10) 7 7 .!9 .!9 l With the cylindrical geometry of the cyclone, penetration through the vanes, Pen , may be 7 calculated by comparing the total cross-sectional area to that containing aerosol which does not deposit (de"ned by the boundary radius r*). Pen is therefore expressed as 7 l r*2!r*2 0 . Pen " l (11) 7 r*2 !r*2 .!9 0 Substituting equation (10) into equation (11) therefore gives 4n2n2St2!2nSt (1#4n2r*2 )1@2 .!9 7 7 Pen "1# . 7 r*2 !r*2 .!9 0
(12)
156
A. D. Maynard
Setting Pen to 0.5 gives the particle Stokes number for 50% penetration through the 7 cyclone vanes: 2nJ1#4n2r*2 !(4n2(1#4n2r*2 )!8n2n2(r*2 !r*2 ))1@2 .*/ .!9 .!9 .!9 . St (vanes)" 50 8n2n2
(13)
Note that only the smaller of the two possible solutions to St gives a physically feasible result. 50 P EN ET RAT I O N TH RO U GH T HE CYC LO N E B OD Y
Nieuwstadt and Dirkzwager have developed a simple mathematical model looking at droplet motion within the body of an axial #ow hydrocyclone assuming no deposition in the turning vanes (Nieuwstadt and Dirkzwager, 1995). Tangential gas velocity in the cyclone body is assumed to vary as k ;" , 5 2nr
(14)
where k is a constant. Nieuwstadt and Dirkwager's model uses a vane system designed to give a tangential #ow velocity at the trailing edge of the vanes of twice the mean axial #ow velocity ; , giving ! 2r ; k"4nr ; and therefore ; " .!9 ! , (15) .!9 ! 5 r By balancing the lack of centripetal force at radius r against Stokes' drag they formulated a simple expression for particle transit time ¹ before it intercepted the outer wall of the hydrocyclone: o 18g n2 1 ¹"(r4 !r4) , (16) 0 o !of o d2 k2 .!9 1 1 $ r is the droplet radial position as it enters the cyclone body, and d the diameter of the 0 $ droplet. This model has been used as the starting point for estimating aerosol penetration through the body of an axial #ow cyclone. The same radial velocity pro"le as used by Nieuwstadt and Dirkwager has been assumed (equation 14). However, the constant term has been modi"ed to allow for di!erent vane con"gurations. To a "rst approximation the tangential gas velocity in the cyclone body may be given as k ; "Pr* ; 5 .!9 ! 2nr
(17)
with 4n2 k" (18) (1#4n2r*2 )1@2 .!9 Rewriting (16) for relatively high density particles in a gas using the expression in equation (18) for k gives n2P ¹"(r*4 !r*4) (19) .!9 0 r*4 k2; St .!9 ! " when written in terms of particle Stokes' number St , calculated using ; . " ! As ¹ is simply the ratio of the distance along the axis a particle will travel before depositing and the axial #ow velocity ; , it is a simple step to estimate the initial radial ! position that de"nes the boundary between deposition and no deposition in a cyclone body of relative length ¸*:
A
B
1@2 k2 r*2"r*2 1! St . ¸* .!9 0 " n2
(20)
Axial #ow cyclone performance
157
Equating the cyclone body inlet area de"ned by r* to the total cross sectional area gives 0 body penetration Pen as " 1@2 k2 Pen " 1!St . (21) ¸* " " n2
A
B
At 50% penetration r*2"r*2 /2, giving .!9 0
3 n2 St " . "50 4 k2 ¸*
(22)
CO MB IN ED MO D EL
In the laminar #ow regime assumed for the cyclone, no turbulent aerosol mixing will take place between the vane system and the cyclone body. Calculation of overall penetration is therefore only possible by modelling particle motion through the complete cyclone. By making simplistic assumptions about the nature of particle motion between the two cyclone sections, this is relatively easy using the two models developed above. In their model of body penetration, Nieuwstadt and Dirkwager assumed that just prior to entering the cyclone body the aerosol was uniformly distributed across the body inlet. In other words, there was no inertial separation of particles in the transition between the vane system and the body. By making the same assumption about the axial #ow cyclone, particle radial position as it leaves the vanes may be related to its radial position as it enters the cyclone body. Through conservation of mass, the aerosol mass #ux leaving the vanes through the annulus de"ned by radii r* and r* is identical to that passing through the circular area in 70 .*/ the cyclone body de"ned by r* (Fig. 2). Therefore, "* r*2 .!9 r*2" (23) (r*2!r*2 ), "* .*/ r*2 !r*2 70 .*/ .!9 where r* now de"nes the radial position of a particle entering the cyclone body for a given "* vane outlet radial position r* . The assumption that this transition takes place over an 70 insigni"cantly small length, with no inertial separation, is physically unfeasible. However, it is su$cient to allow both penetration models to be combined to allow an approximate determination of overall penetration.
Fig. 2. Idealised axial #ow lines in the transition region between the vane system and the cyclone body. It is assumed that the length of the transition region is insigni"cant compared to the length of the cyclone, and that particles follow the streamlines in this region (i.e. no inertial separation).
158
A. D. Maynard
Equation (20) gives the initial radial position of a particle entering the cyclone body that de"nes the boundary between deposition and penetration. Equating r* in equation (20) to 0 r* in equation (23) therefore gives the radial position of a particle leaving the vanes that will "* just be deposited within the cyclone. Likewise equating r* with r* in equation (10) gives .!9 "0 the initial radial position of a particle entering the cyclone that de"nes the boundary between penetration and deposition (r*). Particle Stokes numbers in the cyclone vanes and i body are related by the ratio of the mean air speed in the respective sections: St "! St 7 "
(24)
< !" . ; !
(25)
where
Thus the de"ning inlet radius r* may be expressed as * r*2"r*2 .*/ i #(r*2 !r*2 ) J1!16n2St ¸* .*/ " .!9 #4n2n2!2 St2 " !2n! St (1#4n2r*2 #4n2 J1!16n2St ¸* )1@2 " .*/ "
(26)
and overall cyclone penetration as Pen"(1!16n2St ¸*)1@2 " 4n2n2!2 St2 " # r*2 !r*2 .!9 .*/ 2n! St " (1#4n2r*2 #4n2 J1!16n2St ¸* )1@2. ! .*/ " r*2 !r*2 .*/ .!9
(27)
Because of the form of this expression for cyclone penetration, it is not possible to de"ne an analytical expression for St . However determination of St using iterative techniques is 50 50 a trivial matter using relatively simple computer codes. PR ED ICT ED CY CLO NE P ER FO RM AN CE A S A F UN CT I O N OF S t
Figures 3}5 show predicted cyclone performance for three speci"c design regions. In Fig. 3 the number of inlet vane turns n is signi"cantly smaller than 1, and the cyclone has a small relative body diameter. The plot shows penetration dominated by deposition within the cyclone body, with penetration through the vanes being close to 100%. This is the ideal mode of operation for industrial gas cleaning (or liquid cleaning in the case of hydrocyclones) applications, where deposition in the vanes is likely to lead to a build-up of deposits, and most industrial axial #ow cyclones are built with n(1 and a relatively small body diameter. It is also the preferred mode of operation for aerosol sampling where a build-up of deposits in the vanes increases the likelihood of particle carryover in the outlet #ow. Figure 4 demonstrates the e!ect of increasing the number of vane turns to greater than 1 and increasing the relative body diameter. By selecting the various design parameters appropriately particle deposition in the vanes becomes dominant, with body losses only contributing to secondary e!ects in cyclone penetration. This design region lies closer to that used by Vaughan. Figure 5 demonstrates the design parameters necessary for comparable contributions from both the vane section and the cyclone body.
Axial #ow cyclone performance
159
Fig. 3. Modelled axial #ow cyclone penetration. r* "0.2; r* "0.02; n"0.2; ¸*"3. .!9 .*/
Fig. 4. Modelled axial #ow cyclone penetration. r* "1; r* "0.5; n"2; ¸*"4. .!9 .*/
In Figs 6}9 the in#uence of ¸*, r* , r* and n on St is demonstrated. In each case St 50 50 .!9 .*/ is calculated using the mean axial body velocity ;. These are the four key parameters that determine St according to the combined model. Figure 8 shows the clear delineation of 50 operating regimes, with vane deposition dominating above n+1, and body deposition dominating below this region. However, although it may be generalised that cyclones with n(1 will be dominated by deposition within the body, and cyclones with n'1 will be dominated by deposition within the vanes, the model predicts a broad crossover region. It is therefore likely that losses within the vanes of an axial #ow cyclone will be signi"cant until the number of vane turns is signi"cantly smaller than 1.
160
A. D. Maynard
Fig. 5. Modelled axial #ow cyclone penetration. r* "0.2; r* "0.02; n"1.5; ¸*"2. .!9 .*/
Fig. 6. Plotting St as a function of r* . r* "0.02, n"1.5, ¸*"2. St referenced to mean axial 50 .!9 .*/ 50 body gas velocity ;.
E XTE N SI ON OF TH E MO D EL TO CYC LO NES W IT H M U LT IPL E V AN ES O F FI NI T E W ID TH
For simplicity the model cyclone initially had a single helicoidal vane that was in"nitely thin. Determination of penetration as a function of St using this model was valid as increasing the number of vanes and the vane thickness will primarily a!ect the mean gas velocity in the vanes, thus a!ecting St and not penetration as a function of St in the separate cyclone sections. In the combined model however there is a dependence on the gas velocity
Axial #ow cyclone performance
Fig. 7. Plotting St
50
161
as a function of r* . r* "0.2, n"1.5, ¸*"2. St referenced to mean axial .*/ .!9 50 body gas velocity ;.
Fig. 8. Plotting St as a function of n. r* "0.2, r* "0.02, ¸*"2. St referenced to mean axial 50 .!9 .*/ 50 body gas velocity ;.
in the cyclone vanes in the expression for ! (25). More importantly, when relating modelled penetration to aerodynamic diameter it is necessary to know <. The simplest approach to incorporating the e!ect of multiple vanes of "nite thickness into the model is to develop an appropriate expression for !. If the helicoidal vanes have a relative thickness w*, then the relative vertical channel height h* de"ned by multiple vane turns is given by h*"1!w* .
(28)
162
A. D. Maynard
Fig. 9. Plotting St as a function of ¸*. r* "0.2, r* "0.02, n"1.5. St referenced to mean 50 .!9 .*/ 50 axial body gas velocity ;.
Fig. 10. Plotting St as a function of Nw* assuming (a) in"nitely thin vanes and (b) vanes of "nite 50 thickness w*. r* "0.2, r* "0.02, n"1.5, ¸*"2. St referenced to mean axial body gas .!9 .*/ 50 velocity ;.
Generalising for a system with N vanes equally spaced about the cyclone axis, the e!ective relative channel height h* may be expressed as %&& 1!Nw* h* " . (29) %&& N The relative cross-sectional area perpendicular to the gas #ow of a channel de"ned by adjacent vanes is therefore given by 2nr* (1!Nw*) .!9 Area*(channel)" (r* !r* ). .*/ .!9 N J1#4n2r*2 .!9
(30)
Axial #ow cyclone performance
163
Assuming a #ow rate Q through the cyclone, ! is given by Q Area*(body) !" . Area*(channel)N Q
(31)
Thus substituting the cyclone body relative area and equation (30) into equation (31) gives r* J1#4n2r*2 .!9 . .!9 (32) !" 2(1!Nw*) (r* !r* ) .*/ .!9 Figure 10 demonstrates the in#uence of the factor Nw* over St for a cyclone dominated 50 by aerosol separation in the vanes. COMP ARI SO N B ET WE EN THE MO D EL AND EXP ER IME NTAL D ATA
In the published literature there are two sets of experimental data from axial #ow cyclones designed for aerosol sampling at relatively low Re. Of these Vaughan gives the most comprehensive account of axial #ow cyclone performance (Vaughan, 1988). Cyclones were constructed with two vane systems and three body lengths (Table 1), and the penetration as a function of particle size measured at four #ow rates. Table 2 compares the experimental d values obtained by Vaughan at four #ow rates, 50 !% with those estimated using the axial #ow cyclone model given in equation (27), with the
Table 1. Dimensions of the vane systems used by Vaughan (Vaughan, 1988). These systems were used with cyclone bodies of e!ective length 12.5 mm (S), 17.5 mm (M) and 22.5 mm (L). A single helicoid was used in each vane system. The dimensions are attributed to di!erent helix arrangements than in the published paper following discussions with Vaughan (Vaughan, 1998) Vane system
P/mm
w/mm
r /mm .!9
r /mm .*/
n
Helix 0 Helix 1
5 5
3.3 3.3
5 5
2.5 3.8
1.5 1.5
Table 2. Comparison between d values experimentally measured by Vaughan (Vaughan, 1988), and those 50 !% calculated using the model in equations (27) and (32). Data marked with s indicate that they originate from a penetration curve that is kinked. Experimental measurements using body M* were conducted with a liquid aerosol, all others used solid particles. The experimental data are attributed to di!erent helix arrangements than in the published paper following discussions with Vaughan (Vaughan, 1998) Helix 0
Helix 1
Body
Q/l min~1
Experiment d /km 50 !%
Theory d /km 50 !%
Vane Re
Experiment d /km 50 !%
Theory d /km 50 !%
Vane Re
L
1.240 2.281 3.235 3.750
4.85s 2.7 1.8 1.6
3.6 2.6 2.2 2.0
496 912 1293 1499
3.1s 1.96 1.45 1.32
1.8 1.3 1.1 1.0
715 1316 1867 2399
M
1.240 2.281 3.235 3.750
4.32s 2.4 1.7 *
3.6 2.6 2.2 2.0
496 912 1293 1499
2.35s 1.65 1.27 *
1.8 1.3 1.1 1.0
715 1316 1867 2399
M*
1.240 2.281 3.235 3.750
6.5s 3.42 2.47 1.93
3.6 2.6 2.2 2.0
496 912 1293 1499
4.75s 2.84 1.88 1.41
1.8 1.3 1.1 1.0
715 1316 1867 2399
S
1.240 2.281 3.235 3.750
4.0s 2.08 1.54 1.35
3.6 2.6 2.2 2.0
496 912 1293 1499
2.25s 1.83 1.35 *
1.8 1.3 1.1 1.0
715 1316 1867 2399
164
A. D. Maynard
Fig. 11. Comparing measured cyclone penetration and modelled penetration for helix 0 and body ¸ at 2.281 l min~1 (Vaughan, 1988).
expression for ! given in equation (32). At the lowest #ow rate, Vaughan found the penetration curve was not a smooth transitional function, but showed a step or &&kink'', indicating some form of complex #ow behaviour was taking place. Good agreement between experiment and theory was therefore not expected at these points, and indeed was not seen. For solid aerosols, the agreement between theory and experiment is good, considering the simplistic assumptions made in the model. In most cases agreement is well within 25%. Poorer agreement is seen for the liquid aerosol, although even here the estimated points agree to well within an order of magnitude. Although helix 0 has a #ow Reynolds number in the vanes greater than 2000 at 3.75 l min~1 (calculated using the square root of the vane channel area), and is therefore likely to be operating outside the laminar #ow regime, there is no indication of this in#uencing cyclone performance signi"cantly. However there is evidence for the length of the cyclone body having a more signi"cant e!ect on performance than the model predicts. Figure 11 compares the measured penetration with the modelled penetration for one cyclone con"guration from Vaughan. Although the model predicts the d point reason50 !% ably well, the shapes of the penetration curves di!er considerably. Similar comparisons are seen for all other sets of measurements where no kink appeared in the experimentally determined penetration. Liu and Rubow constructed a cascade axial #ow cyclone and published data on the measured penetration of each stage (Liu and Rubow, 1984). Unfortunately, no information was published on either the number of vane turns in the vane section of the cyclones, or the e!ective body length. In modelling the stages educated estimates have therefore been made of these dimensions. n has been assumed to be 1 for each stage as this is indicated in the drawings published, and gives reasonable agreement when used with the model. ¸ has been taken as the body length published, which is not equivalent to the e!ective body length used in the model. However, it can be shown that with the vane systems used, the model is insensitive to variations in ¸. Table 3 gives the dimensions of each stage of the cascade cyclone. Vane thickness w was not given, although "gures in the paper indicate it to the small relative to P. It has therefore been considered negligible when modelling cyclone performance. Table 4 presents the measured d values of each cyclone stage when operated at 30 l min~1, together with 50 !% values estimated from the model given in equation (27), together with equation (32). As shown in Table 4, all stages bar stage 1 are operating at Re within the cyclone vane systems greater than 2000, and are therefore unlikely to be operating under laminar #ow conditions.
Axial #ow cyclone performance
165
Table 3. Dimensions of the cascade axial #ow cyclones used by Liu and Rubow (Liu and Rubow 1984). The number of vane turns n has been estimated at 1. The body length ¸ is the length given by Liu and Rubow, and does not correspond exactly with the e!ective body length (Fig. 1). The #ow rate through each stage was 30 l min~1 Stage
N
P/mm
r /mm .!9
r /mm .*/
n (est)
¸/mm
1 2 3 4 5
2 2 2 2 1
33.9 25.4 12.7 10.2 5.1
25.2 25.2 15.2 10.4 10.4
15.4 18.7 9.2 7.2 9.1
1 1 1 1 1
51 51 32 32 32
Table 4. Comparison between d values experimentally mea50 !%Rubow, 1984), and those estisured by Liu and Rubow (Liu and mated using the cyclone model given by equations (27) and (32). In calculating the expected d values it was assumed that the vanes 50 !%in"nitely thin were
Stage
d /km 50 !% Experiment
Theory
Re (vane system)
1 2 3 4 5
12.2 7.9 3.6 2.05 1.05
11.6 6.9 4.4 2.1 0.6
1958 2793 4291 6723 14,849
Table 5. Dimensions of the Czech Republic axial #ow cyclones Cyclone
N
P/mm
r /mm .!9
r /mm .*/
n
¸/mm
DP20 DP50
4 4
22 28
7.5 12
4.5 4.5
0.5 0.36
25 12
Table 6. Comparison between d experimentally measured values and those 50 !% estimated using the cyclone model given by equations (27) and (32)
Cyclone DP20 DP50
Flow Rate (l min~1)
d /km 50 !% Experiment
Theory
Vane Re
8.1 15.0 10.4 15.2
4.8 3.0 9.6 7.3
5.9 4.3 13.8 11.4
1359 2532 1232 1800
Despite this, there is good agreement between experimental and modelled values for stages 1}4, with the largest variation of 22% occurring at stage 3. This may be slightly optimistic, as with the vane systems used variations in n a!ect the modelled d signi"cantly. For 50 !% instance, assuming n to be 0.5 for stages 1}4 (which would have been the smallest feasible value using two vanes) increases the modelled d for stage 1 to 16 km. However, even 50 !% allowing for variations in n between 0.5 and 2, the modelled d values are well within an 50 !% order of magnitude of the experimental data. For stage 5, the model underestimates d signi"cantly. However the high Re in the 50 !% vanes of this stage will lead to turbulent #ow, therefore placing the cyclone's operation outside the limits of the model. As well as published data on small axial #ow cyclone performance, limited measurements have been made on the DP20 and DP50 axial #ow cyclones used for aerosol sampling in the Czech Republic (Weiss et al., 1987). These are compact devices, capable of sampling health-related aerosol fractions at #ow rates between 8 and 50 l min~1; dimensions of the cyclones are given in Table 5. Table 6 presents the results of limited measurements made of
166
A. D. Maynard
Fig. 12. Comparison between experimental and modelled
d values. 50 !%
the internal penetration e$ciency (i.e. penetration e$ciency through the cyclone vanes and body, excluding inlet aspiration e!ects) using the TSI Aerodynamic Particle Sizer 3310 based technique described by Maynard and Kenny (Maynard and Kenny, 1995). The #ow rates were selected to achieve low Re within the cyclone vanes, and were not representative of operating #ow rates under normal conditions. Both of these cyclones di!er from those of Vaughan, and Liu and Rubow, in that the number of vane turns is less than one in each case. However, equation (27) predicts that separation within the cyclone vanes is still likely to be the dominant factor in determining overall penetration. The modelled d value for DP20 at 8 l min~1 (laminar #ow) is within 50 !% 25% of the experimentally measured value. The agreement is not as close at 15 l min~1, although the Re in the vanes indicates that the cyclone may be operating outside the laminar #ow regime. DP50 shows relatively poor agreement with the model (although d values are well within an order of magnitude), despite operating at low Re. It is not 50 !% clear what the cause of this discrepancy may be, although it may be signi"cant that this cyclone has the lowest number of vane turns out of all cyclones examined. Another possible explanation is that the penetration e$ciency of the cyclone begins to be in#uenced by factors such as the aspiration e$ciency of the vane inlets as the internal penetration e$ciency d increases to around 10 km and above. 50 !% Figure 12 plots experimentally measured d values for all cyclones against those 50 !% estimated from the model. Overall the agreement between experiment and theory is encouraging, considering the simplicity of the model developed. The assumption that the log term in equation (4) is insigni"cant is valid for most cyclone con"gurations. The DP20 and DP50 cyclones have small values of r* and r* (r* "0.16 for the DP50 cyclone), .!9 .*/ .*/ and this is likely to have led to some divergence of the model arising from the assumption. Data for these cyclones are lower than modelled values, but still in reasonable agreement with the model. Although more experimental data are required to further validate the model, existing data indicate it is suitable for giving an initial estimate of axial #ow cyclone performance. However, isolation of the modelled contribution to overall penetration from
Axial #ow cyclone performance
167
the vane system and the body indicates that for each of the experimental datasets considered, overall penetration is dominated by particle separation within the cyclone vanes (e.g. Fig. 4). Modelled performance where body separation dominates, characterised by Fig. 3, has been largely unexplored. Further experimental work is therefore required in this area to fully explore the applicability of the model. CO NC LU SIO N S
The aim here has been to develop a mathematical model of axial #ow cyclone behaviour under laminar #ow conditions (low Re) that is accurate enough to allow an initial estimate of the performance of new cyclone designs. Although a number of simplifying assumptions have been made concerning air #ow and particle motion within such a cyclone, there is encouraging agreement between experimental data and modelled d points. Fair agree50 !% ment is also seen at relatively high #ow Reynolds numbers in the cyclone vanes, where the #ow pro"le is likely to diverge further from the oversimpli"cation of laminar #ow than at low Re. However, comparisons have only been possible for cyclones where aerosol separation in the vanes is likely to dominate overall penetration e$ciency. Thus for devices where separation below the vanes dominates, the model is largely unvalidated. The next step will be to construct a series of cyclones based on predictions made by the model, and measure their performance as a function of both particle aerodynamic diameter and Stokes' number. Most importantly, validation of the model of penetration as a function of St for a wide range of cyclone dimensions will help determine whether the key design parameters identi"ed may in reality be used to determine cyclone performance. As it stands, the model predicts the development of compact, high #ow rate sampling devices following workplace and environmental sampling conventions*necessary criteria for devices either used for monitoring personal aerosol exposure, or used over a relatively short sampling duration. Combining further validation of the model with empirical data on cyclone performance will lay the groundwork for developing axial #ow cyclones suited to a range of size-selective aerosol sampling applications REF ER E NCE S ISO (1995) Air quality*Particle size fraction de"nitions for health-related sampling. Geneva, International Standards Organisation: ISO Standard 7708. Liu, B. Y. H. and Rubow, K. L. (1984). A new axial #ow cascade cyclone for size characterisation of airborne particulate matter. In: Aerosols (Edited by Liu, B. Y. H., Pui, D. Y. and Fissan, H. J.), pp. 115}118. Elsevier, Amsterdam. Maynard, A. D. and Kenny, L. C. (1995). Performance assessment of three personal cyclone models, using an aerodynamic particle sizer. J. Aerosol Sci. 26(4), 671}684. Nieuwstadt, F. T. M. and Dirkzwager, M. (1995) A #uid mechanical model for an axial cyclone separator. Ind. Engng Chem. Res. 34, 3399}3404. Vaughan (1998) Private communication. Vaughan, N. P. (1988). Construction and testing of an axial #ow cyclone preseparator. J. Aerosol Sci. 19(3), 295}305. Wedding, J. B., Weigand, M. A. and Carney, T. C. (1982) A 10 lm cutpoint inlet for the dichotomous sampler. Environ. Sci. ¹echnol. 16(9), 602}606. Weiss, Z., Martinec, P. and Vitek, J. (1987).