Behaviour of isometric nonspherical aerosol particles in the aerodynamic particle sizer

J Aerosol Scl, Vol 21 No 5, pp 701 710, 1990 Printed m Great Britain

BEHAVIOUR PARTICLES

0021 8502,'90 $ 3 0 0 + 0 0 0 ,( 1990 Pergamon Press plc

OF ISOMETRIC NONSPHERICAL AEROSOL IN THE AERODYNAMIC PARTICLE SIZER Y. S.

CHENG,* B. T. CHEN and H. C YEH

Inhalation Toxicology Research Institute, Lovelace Bmme&caland Environmental Research Institute, P O Box 5890, Albuquerque, NM 87185, U S A (Received 5 October 1989, and m Jmal form 5 March 1990) Abstract--The theoretical response of an aerodynamic particle sxzeanalyzerTSI Model APS 3300 to

nonspherlcal, isometric particles, namely cube octahedra, octahedra and tetrahedra, was calculated with regard to the effects of non-Stokeslan flow and particle density For these three shapes calculated APS dmmeters are found to be smaller than those of spheres w~thsame aerodynamic size The deviation from the standard APS cahbratlon curve increases with dynamic shape factor However, some compensation can occur due to the opposing effect of particle density A simple lteratlve procedure is introduced to calculate correctmn factors for particles of known density, shape factor and non-Stokesmndrag The procedure is an extensmn of the idea of Wang and John [Wang, H. C and John, W (1987) Aerosol Scl. Technol 6, 191-198] and limited to isometric particles and other nonspherlcal particles mowng with a fixed orlentatmn xn the APS INTRODUCTION Aerodynamic diameter, dae, defined as the diameter of a unit density sphere having the same setthng velocity as the particle in question, is a very useful aerosol property in describing the dynamic behavtour of an aerosol particle. Particles of different shapes or density will behave similarly in the Stokes regime (Reynolds number <0.1) as long as they have the same aerodynamic diameter. F o r example, they will have the same deposition efficiency in the respiratory tract and in a fibrous filter, tf impaction and sedimentatton are the dominant deposition mechanisms. The response or calibration curve of inertial-type aerosol instruments, includmg tmpactors, centrifuges and cyclones, ts usually a function of the aerodynamic diameter alone. Since 1982/83 a new aerosol instrument for precise, quasi real-time measurement of da~, the Aerodynamic Particle Slzer (APS), has become available commercially. In the APS particles are accelerated through a nozzle and the times of flight of the particles travelling through two laser beams are measured and related to aerodynamtc diameter (Ananth and Wilson, 1988; Baron, 1987; Chen et al., 1985; Wang and John, 1989). Large parttcles travel more slowly than small particles because of their mertia. One would expect the calibration curve of the APS, which relates particle flight time to the d,e, to be a unique curve under a set of defined operating conditions (flow rate, ambtent pressure and temperature). However, at has been shown, both theoretically and experimentally, that spherical particles of the same dae, but different densities, respond differently m the APS (Ananth and Wilson, 1988; Baron, 1987; Chen et al., 1990; Wang and John, 1987, 1989). The reason for this phenomenon is that the d,e is defined in the Stokes region, where the relative velocity is very small (particle Reynolds number, Re<0.1), which is true for many situations m aerosol sampling and deposition. However, the Reynolds number in the sensing area of the APS ranges from 0.001 to 100 for particles of 0.1-15/am (Baron, 1987). As the particle Re increases to values greater than 0.1, Stokes law no longer applies and Newton's general equation for drag must be used instead. As a result the APS response ts a function of both dae and particle density, pp. For nonspherical particles the concept of the aerodynamic diameter is also vahd at Re<0.1, where Stokes law for the drag force mcludes a correction term called dynamtc shape factor (Fuchs, 1964). The behaviour of nonsphertcal particles in the non-Stokestan

* Author to whom correspondence should be addressed 701

702

",

~, ~. HF,"-,<, e~ a,

region is not fully understood. Only hmlted data on drag forces are a',adabte t~r nonspherlcal particles (Pettyjohn and Chnstiansen, 1948; Wadeil, 1934~ The APS has been used to determine the size &stnbutions of nonsphencal pamcles (Kasper and Wen, 1984: Griffiths et al.. 1984). Effects of shape and density on the APS response have been observed experimentally by Griffiths et al 11984), The APS results appeared smaller than aerodynamic diameters determined by an elu~riator However, no theoretical explanatmn was proposed This paper reports theoretical calculauon~ o| APS responses for some nonsphencal, ~somemc particles It shows the effect of non-Stokesmn flow of irregularly-shaped pamcles on the APS response and prowdes ways to properly interpret the data. MFTHODS

Equation of motion Figure 1 shows a schematic of the APS sampling and acceleration region. A particle travels through the inner and outer nozzles, being accelerated continuously. The time of flight of the particle travelling through the two laser beams located downstream of the outer nozzle is then recorded in a 1024-channel accumulator. The time of flight can be calculated by solving the following equation of motion, which relates the change of momentum to the drag force (Fd) experienced by the particle: dV

m~-=Fd,

(1)

where Vis the particle velocity, m IS the mass of the particle and t is the time. Values for drag force of nonspherical particles and the flow field in the APS are required to solve the equation.

Drag force of nonspherwal partzcles In the Stokes regime the drag force on a particle from the surrounding gas is linearly related to the drag coefficient (CD), the cross-sectional area (A), the air density (Pa), the relative velocity ( U - V ) , the viscosity (r/), the volume equivalent diameter (dr), and the

INNER NOZZLE

OUTER NOZZLE

I

2 LASER BEAMS

I SHEATH AIR FLOW

I;';,~,",' A E R O S O L FLOW

!

CENTERLINE

'

. . . . . . . . . . . . . . . . . .

.................

i ........................................... /

X-AXIS

/

/ [

2....11111.... i ...........

Fig 1 Schematic of the APS nozzle assembly

Behavmur of isometric nonsphencal aerosol particles in the aerodynamic particle slzer

703

dynamtc shape factor (x):

C DAp.(U Fd--

- -

V) 2

(2)

2

24 K CD-- Re C

(3)

Fd = 3~ldvx ~ It)-- v); .

.

.

.

(4)

whereas, in the non-Stokesian regime, an additmnal factor is needed: 3rr~/dv~ ( U - V)(1 + aRch),

Fd--

(5)

C

Re=P.dv(U- V),

(6)

q where C is the shp correction factor. The non-Stokesian correction factor (1 + a R e b) for spherical and some isometric particles has been obtained from oil tank experiments (Pettyjohn and Chrlstiansen, 1948; Wadell, 1933). Figure 2 shows the drag coefficient, CD, plotted as a function of Reynolds number (based on da=) for cube-octahedron-, octahedronand tetrahedron-shaped particles. The experimental data were reported by Pettyjohn and Chnstiansen (1948). A fitted equation of the following form and nonlinear least squares regression were used to describe the data (BDMP, Statistical Software, Umversity of Cahforma): 24x CD = R ~ ( 1 + aRch).

(7)

Parameters a and b are coefficients in the non-Stokesian correction factor (Table 1). In general, these fitted equations are valid when Reynolds numbers are less than or equal to 100, which is withm the range of APS operational con&tions. In this region the particles are mowng in a fixed orientation with maximum resistance without rotational motion. The correction term is greater than one, indicating that the drag force is greater than that predicted by Stokes law

DRAG FORCE OF ISOMETRIC P A R T I C L E S 10,00C

~'k

c=I,OOC 0 z I0C

~

~

o llt

O

10


~, I

Tetraheclron ~

"il~T~

/ /

Octahedron

// ~o "~+~+~+l+~+ " ~ l ~ ' ~ f ÷ +++ ~+ ++/ /

rr D 10

~

x" =~xxx/

Cube Octahedron ~

0

10 .3

I

10 2

i

10 +1

I

1

¢~0CI~0c

I

101

I

10 2

I

10 3

I

10 4

-,-~I

10 5

REYNOLDS NUMBER, Re

Fig 2 Drag coefficient as a function of Reynolds number for cube--octabedron-, octahedron- and tetrahedron-shaped particles Experimental data are from Pettyjohn and Chnstlansen (1948) The fitted curves are listed in Table 1

704

~ ,++ ++'F-iEN¢+e l

al

Table I Dynarm¢ s h a p e facto~ and n o n - S t o k e s l a n factor for ~somctr]c parucles P a r t M e shape Sphere Cube-octahedron Octahedron Tetrahedron

~

N o n - S t o k e s m n factor

10 I 029 1 065 1 182

! ~-0 I +0 t ~0 I +0

A p p l i c a b l e rarigc

1667Re 2 1850Re ° +,5oa 1874Re ° .663 1781 Re ° ,+9,+

0 < Re < 8~X~ 0
Flow field in the APS The velocity profile, especially the axial velocity (U) at the nozzles, is needed for calculating the APS response. Calculations based on incompressible, inviscid flow (Wang and John, 1987), or on a more complicated numerical solution of the Navier-Stokes equation (Ananth and Wilson, 1988), have been used. Both calculations predicted essentially the same results for the density effects (Wang and John, 1989). We expanded the simplified flow to a compressible, inviscid and ideal gas flow for this calculatior~ (Chen et al., 1985). The velocity at the axis ~s a function of distance x, where x = 0 at the exit of the outer nozzle: forx_< -1520/~m,

for - 1520/~m < x < -878 ktm,

4(2

U=Uo=-~5;

(8a}

U=[U2o+2RTn( UA ~l °5.

\ oAo/J

[

for x = - 878/~m = Xm,

=Wm= U g + M - - I n \Uodo/J

for -878/~m < x _< 200 tim,

U = U m + (U t - U m ) 2 0 0 - - - ; and ~m

forx>Z00/~m,

X--X

m

[uZ 2RT ( Po "]l °5 U=U,= L o+-~-lnkp--~_~fijj

+8b) (8c)

(8d)

(8e)

where Q is the total flow rate in the APS, M = 28.964 is the molecular weight of air, Po and AP are the ambient pressure and pressure drop across the APS nozzle assembly, d o Is the inner tube diameter, and Uo and Ut are the air velocity at the entrance of the inner tube and the terminal air velocity at the sensing volume. Figure 3 shows the velocity profiles under three operating conditions at the ambient pressure m Albuquerque (23~C and 625 torr). Terminal gas velocity, Ut, is derived based on the compressible, invmcid flow of 1deal gases: under normal operating conditions for the APS it can be simplified to"

[-2RT

Ut=[~_ln(

Po

"]l °

5

\Po-AP/ J

(8t)

because the entrance velocity, Uo, Is negligible. Equation (81) accurately predicted gas velocity under different ambient pressures and flow rates, as previously reported (Chen et al., 1985), and has been adopted for theoretical prediction of APS response (Wang and John, 1989).

APS response By substituting equation (5) into equation (1) we obtained: dV

dx

1 8 q ( U - V)K (1 + aRch). ppd 2 VC(dv)

(9)

Behavlour of isometric nonspher]cal aerosol particles in the aerodynamic pamcle slzer 30C

700

25(2

600

20(: ~~

1243tot,

"d

i

15C

1

705

I.LI

400 v 9 6 7 torr

' 71 5

torr

300 Q.

10C 200 5C Oz -1600

1O0

-1200

-800 -400 DISTANCE (/z m)

0

460

8

Fig 3 The axial velocityprofiles in the APS at 625 torr and 23~C for three operational condmons AP=1234, 967 and 71 5 torr

This one-dimensional equation was solved by using the 4th-order R u n g e - K u t t a method, with the initial conditions o f x = - 1520 #m, V= Uo and a constant increment ofAx = 1/~m. We obtained particle velocity, pressure, Reynolds number and the acceleration as a function of distance, x. The location and distance between the two laser beams depends on individual APS units (Ananth and Wilson, 1988). For our unit (No. 145) the distance between the two beams (L) was found to be 123.6/~m (Chen et al., 1985), and the location of the laser beam was between 400 and 523.6/~m, by best-fitting calculations to experimental data. The time (T) travelling between two detecting beams was calculated by 523 6

T= f

1vdx.

(10)

400

The channel number (N) was then calculated from T (nanosecond)" N = ( T + 2)/4.

(11)

The mean particle velocity and the acceleration at the detection volume were if=

L/T and dV

A = -dt-

-

18q(U t - lT)x -ppdv ~ - - - C(dO (1 +aRch).

(12)

(13)

The mean particle Reynolds number, Re, is calculated by using equation (6) with Ut and I7.

Cahbratmn of APS Polystyrene latex (PSL) particles, ranging from 0.3 to 7.0 #m, were used to calibrate the APS (unit No. 155). Particle densities were 1.05, and the particle sizes were determined from photomicrographs taken with a transmission electronmicroscope. The respective flow rates and corresponding pressure drops in the APS were 5.01 m i n - 1 (AP = 71.5 torr), 5.7 1m m (AP = 96.7 torr) and 6.3 1m i n - z (AP = 123.4 torr), under Albuquerque's ambient conditions [625 torr (83.3 kp) and 23°C].

706

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et al

RESUL'IS AND DISCUSSION

Calibratzon curves Figure 4 shows the calculated responses (channel number) for three cxpe~maenlat conditions (AP = 123.4, 96.7 and 71.5 tort, corresponding to sampling flow rates oi ~ 3.5 7 and 5 1min - ~, with 20% of the aerosol flow through the inner nozzle tubes). The oak ulated curves are in good agreement with the calibration data derived by using polystyrene latex particles of between 0.5 and 7.0 #m, wtth a density of 1.05. The corresponding Rcynotds number (Re) ranged from 0.0001 to 100 for particle sizes between 0.2 and 15 pro. clearly indicating that drag forces of larger particles were in the non-Stokeslan region Response of nonsphertcal particles Figure 5 shows the calculated responses for unit density particles of different shapes, including sphere (K= 1), cube-octahedron (x= 1.029), octahedron (x= 1.065) and tetrahedron (K= 1.182), as a function of the aerodynamic diameter, which is defined by the following equation: dae=[

RP C ( d v ) 1 0 5

dv

([4}

Lpo~ c(a.o)]

700r

6 o!

/

ooot ......... o,ot :

//

///

///

'oo t. < 450 0 400

f

350 /

/

3oot

/

/

,//,'(

.//y

100

10

PARTICLE SIZE ( / z m )

Fig 4. Comparison of calculated response of the APS and the experimental cahbratlon curves for polystyrene latex particles (density = 1 05) under three operating conditions (at 625 torr and 23 C!

1000 900 . . . . .

800

K=1.029

K*I oes K=I

182

K=I

rt~o =1

700 uJ

600

//

z

/

500 4OO

300 200

loo

1'o

1~0

(/zm) Fzg 5 Calculated APS responses for unit denszty particles of different shapes at' Po=625 torr, temperature = 23°C and AP = 96 7 torr AERODYNAMIC DIAMETER

Behavxour of isometric nonspherlcal aerosol particles m the aerodynamic particle sxzer

707

The aerodynamic diameter was calculated for Albuquerque's ambient pressure (625 torr) and temperature (23°C). These data were obtained for a pressure drop of 96.7 torr, but similar data were obtained for A P = 123.4 or 71.5 torr. Our results indicated that the measured APS diameters of nonspherical particles (x > 1) are smaller than those of spherical particles of the same aerodynamic dmmeters. For these unit density isometric particles, the deviation from the standard calibration curve increased with the dynamic shape factor (x). Figure 6 shows the mean acceleration (4) in the detection volume as a function of the mean particle velocity (17) for particles of different shapes. Regardless of particle shape, the mean acceleration is a function of particle velocity only. Not shown here are calculated mean accelerations for particles of different density and shape. These also follow the curve shown in Fig. 6. Our results confirmed the observations of Wang and John (1987) for spherical particles of different densities.

Effects of parttcle denstty Because particle density also affected the APS response, it was of interest to determine the combined effects of density and shape. Figure 7 shows the response curves for spherical partacles with densities 1, 2 and 4, obtained at a Ap of 96.7 torr. The response of dense spheres with the same d,e mcreases with particle density, but an opposite effect was obtained for changes in particle shape. Figure 8 shows response curves for tetrahedrons (x = 1.182) with 10 9

=o in

Eo 108 v z

_o

._J

0 + • - -

107

K=1029 K=1065 K=1182 K=I

(.9
_o 106

=<

5b

lo~

1~0

PARTICLE V E L O C I T Y

2bo

(m/sec)

Fig. 6 The mean particle acceleratzon between the detectzon beams of the APS, as a function of particle velocity, for unit density particles of &fferent shapes (Po = 625 torr, temperature = 2 3 ° C and A P = 9 6 7 torr) 1000 90C

i/

80C

/

700 0 z

//

60(~

Z 50C
/

300 200

1O0

1

110

100

AERODYNAMIC DIAMETER, ( / ~ m )

Fig 7 Calculated APS responses for spherical parUcles of &fferent densities at Po=625 torr, temperature = 2 3 ° C and A P = 96.7 torr

708

'~

3 ~. H I ' N ( , ~'l tit

120~

110( 1000 9OO

-

K=I la~ rt~7~-I ,'~ 1182 rhop=4

8O0 70O 600 50O 400 30O 20O

OOi 01

li0 AERODYNAMIC

1130 DIAMETER

(t.l m )

Fig 8 Calculated APS response for tetrahedron particles ( x = 1.182) of &fferent densities al Po = 625 torr, temperature = 23°C and AP = 96 7 torr

particle densities of 1, 2 and 4. The effects of density and shape tended to compensate for each other, and the resultant response was determined by a combination of density, shape factor and the non-Stokesian factor (1 +aReb).

Correction factor for shape and density effects The numerical calculations discussed in the previous sections provide information on the effects of particle shape and density on the response of APS. Our results show that the mean particle acceleration in the APS sensing zone depends only on the particle velocity. Based on this result, correction factors can be developed to estimate the deviation of APS response from the calibration curve of a particle of known shape and density. This concept was first proposed by Wang and John (1987, 1989) to correct for the density effect of spherical particles. The following analysis is an extension of their method that includes particle shape. The information needed to calculate the correction factors is: (1) APS calibration data for spherical particles (supplied by the manufacturer, including particle size, density and APS channel number); (2) the density and shape (dynamic shape factor and non-Stokeslan factor) of the particle of interest; and (3) APS operational condit]ons (flow rate, pressure drop, temperature and pressure). With this information one can calculate the volume equivalent diameter of a nonspherical particle having the same APS response. Because particles with the same APS response (same particle velocity) have the same mean acceleration in the sensing zone of the APS, we derived the following equation, based on equation (13): K ( l + a R e b) [ _

(l+aR.e~))

where subscript 1 denotes calibration data (usually for spherical particles) and subscript 2 denotes data for the nonspherical particle of interest. If spherical particles are used for calibration then a = 1/6 and b = 2/3 The mean Reynolds numbers are calculated as follows: R.e = pLdv (U, - ~ ) q

(I 6)

where U t is calculated from equation (8e), or from a simple form of (8t), and I7 is calculated from equation (12). Given the calibration data for a spherical particle of d, x and density pp ~, the volume equivalent diameter dr2 of a nonspherical particle of dynamic shape factor x and

Behavlour of mometnc nonsphencal aerosol particles in the aerodynamicparticle slzer

709

density Pp2, having the same time of flight, is calculated by solving equation (15) with an iterative Newton's method. In this procedure the slip correction factor and the gas density are evaluated at the APS sensing volume, with a reduced pressure of P - AP. For spherical particles of different densities equation (15) reduces to the equation [equation (5)] of Wang and John (1989). The aerodynamic diameter, d,~l for the calibration data and dae 2 for the nonspherIcal particle having the same time of flight m the APS, can be calculated from the following equations:

C(dvl)

d,cl ~ = d v 1 J P p l

d,e2~=dvz

/ pp2C(dvz)

for sphere

(17a)

for nonsphere.

(17b)

Note that the aerodynamic diameter can be evaluated at any chosen conditions; for example, at the ambient conditions. The correction factor then, is simply the ratio of dacz and dael :

CF- dae2

(18)

dac I "

Figure 9 shows correction factors for nonspherlcal particles of unit density and for spherical particles of density 2. When the correction factor is greater than 1 the APS underestimates the aerodynamic diameter of a nonspherical particle if the cahbration curve is not corrected. On the other hand, a more dense sphere gives a correction factor of less than 1. Also included in the figure is the correction factor for a tetrahedron of density 2. The correction factor is below 1 for smaller size particles, and increases to over 1 for particles greater than 2/~m. This is because the Reynolds number for smaller particles is smaller, so the particle behaves more like a dense sphere; however, for larger particles, the response is affected by both density and non-Stokesian drag. SUMMARY Theoretical analysis of the APS response of some isometric, nonspherlcal particles was presented. Our results indicate that both density and shape influence the APS response The following conclusions can be drawn from our analysis: (1) The APS response underestimates the size of a nonspherlcal particle. The difference between the indicated and true aerodynamic &ameters increases as the dynamic shape factor and the particle size increase.

15 - o 15

•

0 u-

12

7- 11

x

.,~

~×

a

....'',Lo6s

P~=,0

0

080

91

I I 4 ; 8 1'0 112 APS INDICATED DIAMETER, um

i 14 115

F~g 9 Calculated correction factor (true aerodynamicdiameter/indicatedaerodynamicdiameter), for nonsphemcal particles and spherical particles of different densmes

710

"Y ~ CHENGet al

(2) Particle shape has the opposite effect on the A P S response as particle denslty, and c o m b i n e d effects of shape and density will somewhat compensate each other for dense, nonspherical particles (pp > 1). However, for a light, nonspherical particle (pp < 1), both parameters have the same effect and will reduce the estimated dae from the APS further (3) Particles having the same mean velocity (APS channel number) are accelerating to the same extent; i e. they have the same acceleration at the A P S sensing volume, regardless of particle shape and density (4) Based on observation (3), a simple lterative procedure was derived to calculate the correction factor of A P S response for particles of k n o w n density, shape factor and nonStokesian drag expression. O u r results describe m o v e m e n t of nonspherical particles that travel in the A P S with fixed orientation. M o r e work is needed to allow an understanding of the possible orientation effects on particles which are less isometric particles, including elongated fibers and platelike particles. Acknowledgements- The authors are indebted to K. G. Hansen for techmcal help, many of our colleagues for reviewing the paper, Drs J L. Mauderly and C. H. Hobbs for support, T. A Coons for editing and the Word Processing Unit for typing. The research was supported by the U. S. Department of Energy. Office of Health and Environmental Research under Contract No DE-AC04-76EV01013

REFERENCES Ananth, G. and Wilson, J C (1988) Aerosol Sct. Technol 9, 189-200 Baron, P A (1987) Aerosol ScL Technol 5, 55-67 Chen, B. T, Cheng, Y. S and Yeh, H. C. (1985) Aerosol Scl Technol. 4, 89-97 Chen, B. T., Cheng, Y S and Yeh, H. C. (1990) Aerosol Scl. Technol. (m press). Fuchs, N. A (1964) The Mechanics of Aerosols. Pergamon Press, Oxford. Griffiths, W. D, Patrick, S. and Rood, A. P. (1984) J Aerosol Scz 15, 491-502 Kasper, G and Wen, H Y (1984) Aerosol ScL Technol 3, 405-409. Pettyjohn, E S and Chnstlansen, E. B. (1948) Chem, Engng Prooress 44, 157-172 Wang, H C and John, W. (1987) Aerosol Sc~ Technol 6, 191-198. Wang, H C and John, W (1989) Aerosol Sc~ Technol. 10, 501-505 Wadell, H (1934) J Frankhn Inst 217, 459-491