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A theoretical study of aerosol sampling by an idealized spherical sampler in calm air
Aerosol Science 34 (2003) 1135 – 1150 www.elsevier.com/locate/jaerosci
A theoretical study of aerosol sampling by an idealized spherical sampler in calm air R.S. Galeev, S.K. Zaripov∗ Institute of Mathematics and Mechanics, Kazan State University, Universitetskaya St., 17, Kazan, Republic of Tatarstan 420008, Russia Received 21 January 2003; accepted 3 April 2003
Abstract The performance of an idealized spherical sampler operating in calm air for an inlet arbitrarily oriented relative to the gravity force is studied theoretically. Under potential .ow assumption the air velocity /eld is obtained by using a model of a /nite-size sink on a sphere. The particle motion equations are solved to /nd the limiting trajectory surface and to calculate the aspiration e3ciency. The singular points of the motion equations as a function of settling velocity of particles and the sampler orientation angle are investigated. The connection between the pattern of typical zones of particle trajectories around the sampler and the location of the singular points is illustrated. The e4ects of partial sampling from zones without particles and of particle screening are discussed. The results of parametrical investigations of the dependence of the aspiration e3ciency on the Stokes number and their analysis are presented. In the case of vertically upwards orientation of the sampler the proposed mathematical model gives fair agreement with experimental data from the work by Su and Vincent (Abstracts of sixth international aerosol conference, Taipei, Taiwan, 2002a, pp. 639 – 640). ? 2003 Elsevier Ltd. All rights reserved. Keywords: Spherical sampler; Calm air; Aspiration e3ciency
1. Introduction Aerosol sampling has important practical applications in industrial hygiene, ambient atmosphere and living rooms. Due to the deviation of the aerosol particle paths from the air.ow streamlines in the vicinity of the sampler ori/ce under real sampling conditions the concentration of sampled particles inside the measuring device can be considerably di4erent from the particle concentration ∗
Corresponding author. E-mail address: [email protected] (S.K. Zaripov).
0021-8502/03/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0021-8502(03)00091-0
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far from sampler. To correct the aerosol measurement distortions, a parameter named aspiration e3ciency is introduced in sampling theory. It is de/ned as the ratio of a measured particle concentration to the concentration in real environment. A review of many experimental and theoretical investigations into the aspiration e3ciency for various sampling devices is given by Vincent (1989). Presently there is increasing interest in indoor aerosol measurements. This is due to the essential in.uence of indoor aerosol on human inhalation and hence human health. For indoor aerosol measurements a blunt sampler with spherical head is attractive. Particle movement near the sampling ori/ce is close to the situation near a human head. Understanding the aspiration process in a spherical sampler will help us to increase the knowledge of the human inhalation. The performance of blunt samplers operating in moving air was studied in many previous experimental and theoretical works. Semi-empirical models for blunt samplers were developed by Vincent (1984), Vincent (1987), Tsai and Vincent (1993) and Tsai, Vincent, and Mark (1996). Results of intensive mathematical study of blunt body sampling were summarized in the book of Dunnett and Ingham (1988). Two-dimensional model of aspiration into the sampler with head of blunt shape was developed by Chung and Dunn-Rankin (1992). In the recent paper of Dunnett and Vincent (2000) a mathematical approach was adopted to calculate the aerosol .ow into an idealized spherical sampler and to investigate aspiration e3ciency for various orientations of the sampler relative to the wind direction, based on a simple approximate model of gas .ow—a sink on a sphere. In living and industrial rooms the air is usually calm or moves with low velocity. At these conditions the gravity becomes important, and the aspiration e3ciency depends on the orientation of the sampler relative to gravity. The science of calm air sampling into blunt samplers has received far less attention by comparison with the case of moving air. The gap remains to complete a full understanding of the aspiration process, especially, on blunt sampler orientation in.uence. The results of experimental studies of the aspiration e3ciency for thin-walled and thick-walled tubes operating in still air were presented by Kaslow and Emrich (1973, 1974), Yoshida, Uragami, Hasuda, and Iinoya (1978), Belyaev and Kustov (1980), Grinshpun, Lipatov, and Semonyuk (1989), Su and Vincent (2002b). Mathematical models for thin-walled tube sampling in still air were developed by Agarwal and Liu (1980), Dunnett (1992). The aspiration e3ciency for the two-dimensional blunt sampler operating in calm air at horizontal and vertically upwards and downwards orientations was studied by Chung and Dunn-Rankin (1993). Grinshpun, Willeke, and Kalatoor (1993) constructed the formulae /tting the available experimental data and applicable for sampling through the thin-walled tubes from moving air as well as from calm air. The experiments connected with the study of spherical sampler performance in still air have been made early by Su and Vincent (2002a). In the present paper we propose a mathematical model of sampling into the spherical sampler with a circular ori/ce operating in calm air. The /nite-size sink on the sphere is modelled as a number of single point sinks and the gas velocity /eld induced by the /nite-size sink is expressed in analytical form. The equations of motion of aerosol particles are numerically solved to /nd the limiting particle trajectory surface and to calculate the aspiration e3ciency. The results of parametrical investigations of the aspiration e3ciency and a comparison with experimental data are presented. We also analyse singular points of the particle motion equations and their connection with typical trajectory zones around the sampler.
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2. Formulation Let us consider aerosol sampling into an idealized spherical sampler with a single circular ori/ce that is oriented at the various angles relative to the gravity direction (Fig. 1). Denote by the angle between negative direction of the x-axis and the line that connects the centre of the ori/ce with origin. The angle changes from −=2 to +=2 in the plane z = 0. The ori/ce diameter is smaller than the sampling head diameter D. Far from the sampler the gas is assumed to be stagnant and the particles settle gravitationally parallel to the y-axis. The limiting particle trajectory surface separates the trajectories of particles that are sampled from those that are not. The aspiration e3ciency A is de/ned as the ratio of the concentration c of particles in the air .ux passing through the sampler inlet to that c0 in undisturbed air far from the sampler. By using the relation of continuity of particle .uxes the aspiration e3ciency for the calm air sampling is written in the form A = c=c0 = VS S=Q = 4VS S=US 2 ;
(1)
where US is the mean velocity across the plane of the sampling ori/ce, VS =g is the particle settling velocity, Q = 2 US =4 is the sampling .ow rate, = p d2 =18 is the particle relaxation time, p is the particle density, d is the particle diameter, is the dynamic viscosity of the .uid, g is the gravity acceleration, and S is the horizontal area above the sampler enclosed by the limiting particle trajectory surface.
Fig. 1. Representation of the sampler head orientation and limiting particle trajectories.
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The determination of the area S reduces to calculating the limiting particle trajectories by solving the equations of particle motion in a given velocity /eld. The motion of the air is induced by the action of air suction through an ori/ce in the surface of the sampler head. The air.ow for the typical sampling velocities can be considered at /rst approximation as a potential .ow of an incompressible .uid and viscous e4ects can be neglected. Let the single point sink with strength Q be located in the point (−1; 0; 0) in the non-dimensional coordinate system (x; y; z). The sphere radius D=2 and the sampling velocity Us are the scales of the length and velocity. In the axially symmetric .ow of an ideal .uid the stream function non-dimensioned with respect to the quantity US D2=4 is given by H 2 ( cos + 1) ; 4 1 1 = 2 + 1 + 2 cos ; 0
=−
H = =D;
(2)
where 2 = x2 + y2 ; is the angle between the positive direction of the x-axis and the radius-vector of an arbitrary point on the sphere in the meridional plane z = 0. In accordance with the Butler theorem (Milne-Thomson, 1960) the stream function of the .ow induced by sink (2) in the presence of sphere can be written as 1 = 0 (; ) − 0 ; : (3) It follows from (2) and (3), that =
H 2 (2 − 1) : 4 1
(4)
The radial and tangential components of air velocity by means of the formulae u = −
2
@ 1 ; sin @
u = −
1 @ sin @
(5)
can be expressed in the form H 2 (2 − 1) ; u = − 4 31
1 H2 2 (2 − 1)( + cos ) : u = − 3 4 1 sin 1
(6)
The Cartesian gas velocity components ux ; uy ; uz resulting from (6) are 2x + 1 2x + 1 H 2 1 r 2 + 3 + 4x H2 2 H2 2 − y z ; uy = ux = − 3 ; uz = − 3 ; 4 r 4 r2 4 r2 r13 r1 r1 r 2 = x2 + y2 + z 2 ;
r12 = r 2 + 2x + 1;
r2 = r 2 r1 (x + 1) + rr12 x:
(7)
Formulae (7) can be used to express the velocity /eld for the case when a point sink is located at an arbitrary point (x0 ; y0 ; z0 ) (x0 ¡ 0) on the sphere. A new coordinate system (x ; y ; z ) is introduced
R.S. Galeev, S.K. Zaripov / Aerosol Science 34 (2003) 1135 – 1150
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for which the sink location is (−1; 0; 0). This system is generated by rotating the coordinate system (x; y; z) relative to the y- and z-axis by the two angles 0 = arcsin(y0 );
0 = arcsin(z0 );
(8)
respectively. In this case the coordinates and velocity components are transformed by formulae rM = A (0 ; 0 ) rM ; u( M r) M = AT (0 ; 0 )uM (rM ); cos 0 cos 0 −sin 0 cos 0 sin 0 cos 0 A(0 ; 0 ) = sin 0 sin 0 −cos 0 sin 0 where
x rM = y; z
x
rM = y ; z
ux
uM = uy ; uz
(9) sin 0 0
;
(10)
cos 0
ux
uM = uy ; uz
A(0 ; 0 ) is the rotation matrix, AT (0 ; 0 ) is the transposed matrix. In the vicinity of the inlet ori/ce the above-described single point sink model gives a velocity /eld di4erent from the one of a /nite-size ori/ce as in real sampling devices. In order to obtain a more accurate velocity /eld of the carrier .uid we employ the following model. For H 1 the .ow through the circular inlet ori/ce will be represented as a .ow through the spherical segment of radius H with the same boundary as the boundary of the suction ori/ce. The gas .ow is modelled as a .ow induced by a number of single point sinks uniformly distributed within the spherical segment with the centre in the point (−1; 0; 0). The model approaches the one of the /nite-size sink in the surface of the sampler head with increase in the number of single point sinks. The coordinates of the single point sinks are de/ned by 2(i − 1) 2(i − 1) Hj Hj yi; j = cos ; zi; j = sin ; xi; j = 1 − yi;2 j − zi;2 j ; J I −1 J I −1 1 6 j 6 J;
1 6 i 6 I (j); I (j) = 6j:
(11)
The number of sinks can be found by formula N = 1 + 3J (J + 1). For example the case N = 91 is shown in Fig. 2. If the inlet ori/ce is oriented under the angle relative to the x-axis the location of the point sinks (11) will be determined by the two angles i; j = arcsin(yi; j ) + ;
i; j = arcsin(zi; j ):
(12)
The transformations (9) and (10) are used to express the velocity /eld induced by the arbitrary single sink in point (11). The velocity /eld from a number of sinks is given by sum uM =
J I ( j) 1
T A (i; j ; i; j )uM (rMi; j ); N j=1 i=1
(13)
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Fig. 2. Example of distribution of point sinks within the /nite-size sink at N = 91.
where rMi; j = A(i; j ; i; j )r. M Formulae (7), (11)–(13) determine completely the velocity /eld of the carrier .uid induced by suction through the /nite-size sink at arbitrary orientation relative to the gravity force direction. The accuracy of the simulation of the /nite-size suction ori/ce increases with the number of the simulated point sinks. The equations of particle motion are written under the assumption that Stokes’ law for drag is valid and that any external forces except gravity are negligible u y − vy dvx ux − vx dvy vs dvz u z − vz = ; = − ; = ; dt St dt St St dt St dy dz dx = vy ; = vz ; = vx ; dt dt dt
(14)
where vx ; vy ; vz are the Cartesian components of particle velocity non-dimensioned with respect to US ; vs = VS =US , t is the time, St = US =(D=2) is the Stokes number. Solving Eqs. (14) with the initial conditions for t = 0, vx = 0;
vy = −vs ;
vz = 0;
x = x0 ;
y = y0 ; z = z0
(15)
allows tracing the particle trajectories. To /nd the limiting trajectories we assume that the particle will be sampled if it reaches the spherical segment of radius H with the same centre as the inlet ori/ce centre. Within the considered model the aspiration e3ciency is determined by the four parameters St; vs ; ; H .
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3. Singular points The calm air sampling behaviour for aerosol particles of a spherical sampler is determined to a considerable degree by existence of singular points of the system of Eqs. (14) that can appear in the domain of the solution as a function of the values of the stationary settling velocity vs and angle . For simplicity consider the singular points for the single point sink model (N = 1) at the /xed value H = 0:1. In these points which are also called the “saddle” points (Vincent, 1989) the particle settling velocity directed downwards is balanced by the upward gas .ow created by the suction. In the general case the singular point is the point where the particle has zero velocity. Hence the coordinates of singular points can be found from the condition that the right-hand side of Eqs. (14) are equal to zero vx = 0;
vy = 0;
vz = 0;
ux = 0;
uy = v s ;
uz = 0:
(16)
The gas velocity components ux ; uy ; uz for the case = 0 can be expressed by the ones ux0 ; uy0 ; uz0 of .ow induced by the point sink at = 0: ux = ux0 cos + uy0 sin ;
uy = −ux0 sin + uy0 cos ;
uz = uz0 :
(17)
The Cartesian coordinates of the system turned on the angle are determined by the formulae x0 = x cos − y sin ;
y0 = x sin + y cos ;
z0 = z:
(18)
Taking into account (16) and (17) we obtain ux0 cos + uy0 sin = 0;
−ux0 sin + uy0 cos − vs = 0;
uz0 = 0:
(19)
For = ± =2 it follows from (19) that ux0 = −vs sin ;
uy0 = vs cos ;
z = 0:
(20)
In this case all singular points are in the symmetry plane z =0. Substituting the expressions ux0 ; uy0 from (7) into (20) and taking into account (18) yields the system of non-linear algebraic equations for coordinates x; y of singular points H 2 1 r 2 + 3 + 4(x cos − y sin ) = −vs sin ; (21) − 4 r r13 H2 (x sin + y cos ) 4 r 2 = x2 + y2 ;
2(x cos − y sin ) + 1 2 − 3 r2 r1
= vs cos ;
r12 = r 2 + 2(x cos − y sin ) + 1;
r2 = r 2 r1 (x cos − y sin + 1) + rr12 (x cos − y sin ): The results of numerical solution of (21) are shown in Fig. 3. The closed curves for various values of vs are obtained by variation of . The marks correspond to certain values of the angle .
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R.S. Galeev, S.K. Zaripov / Aerosol Science 34 (2003) 1135 – 1150
1 0 -1
VS =0.0001 VS =0.0002 VS =0.001 VS =0.01 VS =0.1 α=π/2 α=π/4 α=0 α=−π/4 α=−π/2
Y
-2 -3 -4 -5 -6 -7 -1
0
1
2
3
4
X Fig. 3. The curves of the singular point positions for the various values of vs and the angle . The fat circle is the sphere projection on the plane z = 0.
The curves cross the sphere in the points x = ± 1, y = 0 for all values of the settling velocity. Part of the curves inside the sphere has no physical meaning. The distances of the singular points from the sphere increase with decreasing vs . The curve of singular points outside the sphere is widened, acquiring a “waisted” shape (vs = 0:0002) and being divided into two almost circular curves at the value vs = 0:0001. One of them crosses the sphere but the second curve is located on some distance away from the sphere. The lower positions of singular points at the /xed values of the settling velocity correspond to the downward-facing orientation of the sampler (Fig. 3, = −=2). The singular points approach the sphere with the increase of . The critical values of the settling velocity vs and the angle correspond to the location of singular points on the sphere (x = ± 1; y = 0) and can be obtained from (21) in the form vs1 = H 2 sin [sin−1 (=2) − 1]=4;
x = −1; y = 0;
vs2 = H 2 sin [cos−1 (=2) − 1]=4;
x = 1; y = 0:
(22)
Functions (22) at 0 ¡ ¡ =2 are shown in Fig. 4. Three regions can be discerned in the plane (vs ; ): I (vs ¿ vs1 )—singular points outside the sphere are absent, II (vs2 ¡ vs 6 vs1 )—a unique singular point exists in the symmetry plane z = 0 outside the sphere, III (vs 6 vs2 )—two singular points are in the plane z = 0. Two types of singular points appear in the last case. In the far point
R.S. Galeev, S.K. Zaripov / Aerosol Science 34 (2003) 1135 – 1150
1143
y
x 0
P2
10
-1
10
I
vS1 vS2
-2
VS
10
-3
10
II
-4
10
III
-5
10
0
π /4 α
π/2
Fig. 4. The functions vs1 ; vs2 and the regions I–III in the plane (vs ; ).
P1
Fig. 5. The trajectories in the plane z = 0 for St = 0:1, vs = 0:0001, H = 0:1 and = =4.
the particle trajectories merge and bifurcate into two trajectories, one of which leads to the sink and the second trajectory by which the settling particles follow. In the other singular point that is close to the sphere the trajectories are doubled and .ow around the sphere by all sides. Fig. 5 illustrates the trajectories passing through such two typical singular points that are denoted by P1 and P2 for the case St = 0:1, vs = 0:0001 and = =4. In the region 6 0 we have the unique singular point in the plane z = 0 as in the region II for all settling velocities vs . The singular points for ¡ 0 and reasonably large vs can be located in the sphere shadow. The particle trajectories do not pass through such singular points and the aspiration e3ciency equals to zero in this case. In the axially symmetric case = =2, an in/nite number of singular points can form a circle with the centre on the y-axis. This circle is contracted and moves down with decrease of vs . It is illustrated in Fig. 3 by formation of the “waist” at vs = 0:0002. The circle is shrunk into one point at vs ≈ 0:00014. Further decrease of vs gives the two singular points on the y-axis where the particle trajectories merge or are doubled as in the above described case. The illustrations of particle behaviour near the singular points for various orientation of the sampler are given below. The coordinates of singular points as a function of vs and within the /nite-size ori/ce model are close to the considered case of the single point sink model. Note also that the similar analysis of singular points would be interesting for the thin-walled tube sampler operating in calm air. 4. Calculation results To determine the limiting particle trajectory the Cauchy problem for system of equations of particle motion is usually solved repeatedly. It is also possible to /nd the limiting trajectories by means of the approach based on the boundary value problem for particle motion equations (Galeev & Zaripov,
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R.S. Galeev, S.K. Zaripov / Aerosol Science 34 (2003) 1135 – 1150
2003). Let t1 be the time at which the particle reaches the sampler inlet edge. On the left end of the interval [0; t1 ], the boundary conditions for the particle velocity components are de/ned as vx = 0;
vy = −vs ;
vz = 0:
(23)
On the right-hand side, the particle coordinates are x = x1 ;
y = y1 ;
z = z1 ;
(24)
where (x1 ; y1 ; z1 ) is a point on the inlet boundary. The boundary value problem (14), (23) and (24) is solved using the /nite-di4erence method with deferred corrections of Pereyra (1978). Use of the boundary value problem is more e4ective but it is limited by the case of absence of the singular points. To compare the results and check the accuracy of calculations both methods were used when it was possible. The trajectories and shape of cross-sectional area of the tube of limiting trajectories signi/cantly di4er at various orientations of the inlet ori/ce relative to the gravity force direction. The behaviour of aerosol particles close to the spherical sampler in calm air is shown in Fig. 6 as a function of , where the trajectories in the plane z = 0 (a), projections of limiting particle trajectories on the symmetry plane z =0 (b) and cross-sectional area of the limiting particle trajectory surface (area S) (c) are presented. Fig. 6(1) – (5) correspond to the angles ==2; =4; 0; −=4; −=2, respectively and were obtained at H = 0:1, vs = 0:001 and St = 1. The corresponding values of the aspiration e3ciency are also presented in Fig. 6(1c) – (5c). The fat lines in Fig. 6(1a) – (5a) designate the limiting trajectories. In the general case the region around the sampler can be divided into four typical zones: zone (i) of trajectories of sampled particles, zone (ii) of trajectories of particles that deposit on the sphere, zone (iii) of trajectories of particles that pass by the head and zone (iv) without particles. Analysis of these zones near the cylindrical sampler head for inertialess particles was given by Davies (1967) and Dunnett (2002). In the axially symmetric case = =2, vs = 0:001 is the limiting value for singular points to exist (see Fig. 4); at larger values of the settling velocity they are absent. The cross-sectional area of the tube of limiting particle trajectories for upwards facing ori/ce orientation is the circular area and the singular points form the circle on the sphere: x2 + y2 = 1, y = 0 (Fig. 6(1)); they are denoted by P in the plane z = 0. In these points the particle trajectory bifurcates into two trajectories one of which creeps close to the sphere and reaches the sampler inlet. The other trajectory branch goes down and bounds the zone without particles. The symmetry is lost with changing the angle , i.e. for vs = 0:001 we have only one singular point left. Fig. 6(2) illustrates this for the case = =4. The dashed line divides the zones (ii) and (iii). The limiting particle trajectories merge in the point P (Fig. 6(2a, b)). Zone (iv) becomes connected with suction ori/ce and the air from zone without particles is additionally sampled. Because of this the aspiration coe3cient can be less than unity even for inertialess particles. The area of initial points of trajectories of sampled particles is displaced but is kept close to the circular area at = =2. The boundary of the area S is divided into the two parts that correspond to the initial points of limiting trajectories passing through the singular point and reaching directly the inlet opening. For horizontal orientation of suction ori/ce ( = 0) the “sickle-shaped” area S is formed by concave and convex curves (Fig. 6(3)). All limiting trajectories with initial positions on the convex curve go through the point P. A portion of the falling particles impacts the spherical head of the sampler. The area of initial positions of the particles deposited on the sphere is subtracted from the area S. For the next angle = −=4 the cross-sectional area of the limiting particle trajectory surface becomes doubly connected. The area of initial positions
R.S. Galeev, S.K. Zaripov / Aerosol Science 34 (2003) 1135 – 1150 y
1145
y
z
A=1.001
i 2 0 iii
iii
P
-4
-2
x
P
x 4
2 -2
x P
P
iv
-4
1b
1a
1c
y
y
z A=0.989
i
2 x
0 -4
ii
iii
P
P
-4
iv
2b
2a
2 -2
x
x
iii
-2
y
2c y
z
A=0.909
2 ii
i
x
0 x
x
-4
-2
2 -2
iii
P
iv
iii P
-4
3a
3c
3b y
y
ii
z A=0.895
i
i
2 x
x
-4
-2
2 -2
P iii
x
0
iv
iii
P
-4
4a i
4c
4b y
y
z
ii i
A=0.908
2 x
x
iv
0 -4
-2
2
x 4
-2 iii
P
iii
5a
P
5b
-4
5c
Fig. 6. (1) – (5). The trajectories in the plane z = 0 (a), projections of limiting particle trajectories on the symmetry plane z = 0 (b) and the cross-sectional area of the limiting particle trajectory surface (area S) (c) for St = 1, vs = 0:001, H = 0:1 and various .
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R.S. Galeev, S.K. Zaripov / Aerosol Science 34 (2003) 1135 – 1150
2
y
1 ii
1.0
i
0.5
3 x
ii
0
-1.5
-1.0
-0.5
-0.5
P 4
iii
P -1.0
iv
6a
6b
6c
Fig. 6. continued. (6) The same as in Fig. 6. (1) – (5) for horizontal orientation of the sampler for St = 50 and vs = 0:01.
of deposited particles is completely inside the area S. The particles are sampled from all sides, and because of this, the zone (iv) adjacent to the sphere becomes closed. When the sampler is oriented vertically downwards ( = −=2) the cross-sectional area of the tube of the limiting particle trajectories is the axially symmetric annular area. The unique singular point is located on the y-axis (x = 0; z = 0). Note one interesting e4ect at the sampling from calm air that can be observed for the particles at high Stokes numbers. The interaction between inertial and gravitational forces can lead to appearance of complex trajectories in the vicinity of the suction ori/ce. Davies (1968) was the /rst to discuss the “orbiting” particle trajectories near the sampling ori/ce at the calm air sampling. We have obtained in our calculations such complex trajectories for large St but their behaviour di4ers from the one of “orbiting” trajectories around the point sink as described by Davies. This di4erence is due to the presence of the spherical sampling head and the trajectories are oscillated near the singular point. It is di3cult to present the calculated complex trajectories in space for clear illustration. We tried to do it in Fig. 6(6) where the particle trajectories and the area S as in Fig. 6(1) – (5) at St = 50 and vs = 0:01 for horizontal orientation of the sampler are presented. The fat lines in Fig. 6(6a) designate the limiting trajectories. In the plane z = 0 we have the two limiting trajectories 1 and 2 that hit the inlet edge and con/ne the sampled particles zone. Trajectory 3 also reaches the inlet ori/ce because the particle inertia decreases but the gravitational force is not enough for particle to fall down. Therefore, a new zone of sampled particles appears. This zone is limited by the trajectory 3 and the next limiting trajectory 4 that goes through the singular point P after “orbiting” travel. The particles that move between the limiting trajectories 2 and 3 will deposit on the sphere. This leads to the formation of a doubly connected area S inside which the area of initial positions of trajectories of particles deposited on the sphere below the inlet ori/ce appears. To obtain this doubly connected area very accurate calculations are needed. The particle following the limiting trajectory and passing through the singular point will move to inlet or settle down with equal probability. It is expected that a peak of the particle concentration should be observed in the corresponding positions on the bottom of the experimental chamber including the sampling into blunt sampler from calm air.
R.S. Galeev, S.K. Zaripov / Aerosol Science 34 (2003) 1135 – 1150 VS =0.1, N=91 VS =0.01, N=91 VS =0.001, N=91
1.4 1.2
VS =0.1, N=1 VS =0.01, N=1
1.0
0.8
1.0 0.8
A
A
0.6
0.6
α=π/2 α=π/4 α=0 α=−π/4
0.4 0.4
0.2
0.2
0.001
1147
0.01
0.1
1
10
100
1000
St
Fig. 7. The St-dependence of A at the various values of vs for vertically upwards facing sampler. Black dots are the experimental points from work of Su and Vincent (2002a).
0.001
0.01
0.1
1
10
100
1000
St
Fig. 8. The St-dependence of A at the various values angle for vs = 0:01(a), 0.001(b).
Parametrical calculations of the aspiration e3ciency A with variable the Stokes number St, the settling velocity vs and the sampler orientation angle were made for the single point sink and /nite-size suction ori/ce models. The area S was found by means of calculating the limiting particle trajectories and is used to obtain the aspiration e3ciency A by formula (1). The numerical results and their analysis are given below. The St-dependencies of the aspiration e3ciency for = =2 and di4erent values of vs and N are shown in Fig. 7. The aspiration coe3cient calculated by means of the /nite-size ori/ce model stops to be sensitive to the number of the simulated point sinks beginning from N = 91. Because of this, the calculations results for N = 91 and single point sink model (N = 1) are presented for comparison. We also see that there is a .at part in aspiration e3ciency curve with A = 1 + vs (Grinshpun et al., 1993) for small values of the Stokes number. In this case the deviation of the aspiration e3ciency from unity is caused by gravity. The length of the constant part of the St-dependence of the aspiration e3ciency increases with the decrease of the settling velocity. For intermediate values of the Stokes numbers there exists a transition region where the in.uence of particle inertia on sampling becomes noticeable. With further increase of St the aspiration coe3cient drops to the value A = vs . Note that at 0 6 6 =2 in two limiting cases of inertialess particles and particles with very high inertia the aspiration e3ciency for upward-facing orientation of the sampler is given by formulae A=1+vs sin and A=vs sin , respectively. The experimental points from work of Su and Vincent (2002a) are denoted by black dots in Fig. 7. The Stokes number in their work is based on the suction ori/ce diameter as the length scale and St values were recalculated accordingly. Fair agreement of the St-dependence of the aspiration e3ciency calculated according to our models with experimental data is observed. The results of the /nite-size sink model are closer to the experimental points than those of the single point sink model for vs =0:1 where we have larger di4erences between the two models. This di4erence decreases for small values of the settling velocity. At vs =0:01 both models give reasonable agreement with experiment. In the case vs =0:001 the aspiration e3ciency calculated by two models does not distinguish practically.
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0.8
A
0.6 α=−π/4 α=−3π/8 α=−π/2
0.4
0.2
0
0.002
0.004
0.006
0.008
0.010
VS
Fig. 9. The variation of A for inertialess particles at various .
The numerical results of the aspiration e3ciency as a function of the Stokes number for vs = 0:01 and N = 91 for various angles of the sampler orientation relative to the gravity force direction are shown in Fig. 8. The transitional part of the St-dependence curve of the aspiration e3ciency for vs = 0:01 becomes steeper with decreasing absolute values of . There is a region of St values where A for the horizontal orientation is larger than for a vertically upward sampler position. There are the particles that impact on the spherical head around the sampler ori/ce and are not sampled. For the horizontal sampler orientation the spherical surface limits the access of aerosol particles into the inlet opening only from one side. Because of this, the area of sampled particles far from the sampler for = 0 can be larger than the one for = =2 in some range of the Stokes number or sizes of particles. It is also seen that for = 0 and −=4 the e4ect of the additional suction of air without particles and screening of settled particles leads to a decrease of the aspiration e3ciency below unity for small Stokes numbers. The area S of initial positions of sampled particles far from the sampler becomes large at very small values of the settling velocity and the initial area of deposited particles will be comparatively small. Because of this, the in.uence of screening of the settled particles by the sphere and orientation of sampler on the aspiration e3ciency will not be signi/cant. Calculations of the St-dependence of A for the various angles at vs = 0:001 show that all curves are very close. The aspiration e3ciency becomes less than unity for very large values of the Stokes number (see the corresponding curve in Fig. 7). Taking into account the screening of falling particles by the sphere the aspiration e3ciency of inertialess particles (St = 0) in the case of a vertically downwards facing sampler ( = −=2) can be expressed by formula A = 1 + vs (1 − H −2 ):
(25)
From (25) under the condition A = 0 the formula for the settling velocity beginning from which the particles are not sampled can be written: vs = H 2 =(1 − H 2 ). The dependence of the aspiration e3ciency on the settling velocity for inertialess particles at the some angles obtained by numerical calculations is presented in Fig. 9. The calculated values of the aspiration coe3cient at the angle = −=2 completely coincide with formula (25) (solid line). At the two other angles = −3=8,
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1149
−=4 the values of A are on the theoretical straight line for the values vs for which the area S of initial positions of sampled particle trajectories is the doubly connected area. At larger values of vs the aspiration e3ciency continues to decrease but more slowly.
5. Conclusions A mathematical model for sampling into a spherical sampler from calm air has been developed. The /nite-size sink on the sphere is presented as the sum of a number of single point sinks uniformly distributed within the spherical segment with the same boundary as one of the inlet ori/ce. The gas velocity /eld induced by the single point sink is expressed in analytical form. The equations of motion of aerosol particles have been solved numerically to /nd the limiting particle trajectory surface and to calculate the aspiration e3ciency. The behaviour of the trajectories of aerosol particles in the vicinity of the sampler is determined by the settling velocity and the Stokes number as well as the angle of orientation of the suction ori/ce relative to the direction of gravity. In the general case, the region around the sampler can be divided into four zones: zone of trajectories of sampled particles, zone of trajectories of particles that deposit on the sphere, zone of trajectories of particles that pass by the head and zone without particles. The patterns of these zones depend on the existence and location of the singular points of the motion equations of aerosol particles. The coordinates of the singular points as functions of the settling velocity and of the sampler orientation angle have been investigated. The regions in the plane (vs ; ) where one or two singular points exist and where there are no singular points have been determined. When the sampling region includes the zone without the particles the aspiration e3ciency even for inertialess particles may fall below unity. The dependence of the aspiration e3ciency on the Stokes number at various values of the settling velocity and the angle of the sampler orientation has been studied. It was shown that in certain range of the Stokes numbers the aspiration e3ciency for horizontal orientation could be larger than for a vertically upwards-facing sampler. The formula for the aspiration e3ciency of inertialess particles in the downwards-facing sampler case has been given. This formula is applicable for angles when the cross-sectional area of the limiting particle trajectory surface is a doubly connected area. Fair agreement between the numerical results from our model and experimental data for upwardsfacing orientation of the sampler ori/ce was observed. To check other theoretical results of the paper and to validate the mathematical model, an experimental study of the aspiration e3ciency of blunt samplers in calm air is important, including the e4ects of sampler orientation. The distribution of concentration of particles deposited on the bottom of the experimental chamber can indicate the location of zones without particles and singular points. The latter ones are of special importance, because a peak of the particle concentration should be observed below these points.
Acknowledgements Authors gratefully acknowledge the support of this work by Russian Foundation for Basic Research under grant no. 02-01-00836 and by Academy of Science of Republic of Tatarstan.
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References Agarwal, J. K., & Liu, B. Y. H. (1980). A criterion for accurate aerosol sampling in calm air. American Industrial Hygiene Association Journal, 41, 191–197. Belyaev, S. P., & Kustov, V. T. (1980). Sampling from calm air. Trudy IEM, 25, 102–108 (in Russian). Chung, I. P., & Dunn-Rankin, D. (1992). Numerical simulations of two-dimensional blunt body sampling in viscous .ow. Journal of Aerosol Science, 23, 217–232. Chung, I. P., & Dunn-Rankin, D. (1993). The e4ects of bluntness and orientation on two-dimensional samplers in calm air. Aerosol Science and Technology, 19, 371–380. Davies, C. N. (1967). Movement of dust particles near a horizontal cylinder containing a sampling ori/ce. British Journal of Applied Physics, 18, 653–656. Davies, C. N. (1968). The entry of aerosols into sampling tubes and heades. British Journal of Applied Physics, 25(1), 921–932. Dunnett, S. J. (1992). A mathematical study of the sampling characteristics of a thin-walled sampler operating in calm air. Aerosol Science and Technology, 17, 93–104. Dunnett, S. J. (2002). Particle motion in the vicinity of a bulky sampling head operating in calm air. Aerosol Science and Technology, 36, 308–317. Dunnett, S. J., & Ingham, D. B. (1988). The mathematics of blunt body sampling. Lecture Notes in Engineering. London: Springer. Dunnett, S. J., & Vincent, J. H. (2000). A mathematical study of aerosol sampling by an idealised blunt sampler oriented at an angle to the wind: The role of gravity. Journal of Aerosol Science, 31, 1187–1203. Galeev, R. S., & Zaripov, S. K. (2003). Deposition of aerosol particles on a sphere: The role of gravity. Aerosol Science and Technology, 37, 325–329. Grinshpun, S. A., Lipatov, G. N., & Semonyuk, T. I. (1989). A study of sampling of aerosol particles from calm air into thin-walled cylindrical probes. Journal of Aerosol Science, 20, 1561–1564. Grinshpun, S. A., Willeke, K., & Kalatoor, S. (1993). A general equation for aerosol aspiration by thin-walled sampling probes in calm and moving air. Atmospheric Environment, 26A, 1459–1470. Kaslow, D. E., & Emrich, R. J. (1973). Aspirating ;ow pattern and particle inertia e
A theoretical study of aerosol sampling by an idealized spherical sampler in calm air R.S. Galeev, S.K. Zaripov∗ Institute of Mathematics and Mechanics, Kazan State University, Universitetskaya St., 17, Kazan, Republic of Tatarstan 420008, Russia Received 21 January 2003; accepted 3 April 2003
Abstract The performance of an idealized spherical sampler operating in calm air for an inlet arbitrarily oriented relative to the gravity force is studied theoretically. Under potential .ow assumption the air velocity /eld is obtained by using a model of a /nite-size sink on a sphere. The particle motion equations are solved to /nd the limiting trajectory surface and to calculate the aspiration e3ciency. The singular points of the motion equations as a function of settling velocity of particles and the sampler orientation angle are investigated. The connection between the pattern of typical zones of particle trajectories around the sampler and the location of the singular points is illustrated. The e4ects of partial sampling from zones without particles and of particle screening are discussed. The results of parametrical investigations of the dependence of the aspiration e3ciency on the Stokes number and their analysis are presented. In the case of vertically upwards orientation of the sampler the proposed mathematical model gives fair agreement with experimental data from the work by Su and Vincent (Abstracts of sixth international aerosol conference, Taipei, Taiwan, 2002a, pp. 639 – 640). ? 2003 Elsevier Ltd. All rights reserved. Keywords: Spherical sampler; Calm air; Aspiration e3ciency
1. Introduction Aerosol sampling has important practical applications in industrial hygiene, ambient atmosphere and living rooms. Due to the deviation of the aerosol particle paths from the air.ow streamlines in the vicinity of the sampler ori/ce under real sampling conditions the concentration of sampled particles inside the measuring device can be considerably di4erent from the particle concentration ∗
Corresponding author. E-mail address: [email protected] (S.K. Zaripov).
0021-8502/03/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0021-8502(03)00091-0
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far from sampler. To correct the aerosol measurement distortions, a parameter named aspiration e3ciency is introduced in sampling theory. It is de/ned as the ratio of a measured particle concentration to the concentration in real environment. A review of many experimental and theoretical investigations into the aspiration e3ciency for various sampling devices is given by Vincent (1989). Presently there is increasing interest in indoor aerosol measurements. This is due to the essential in.uence of indoor aerosol on human inhalation and hence human health. For indoor aerosol measurements a blunt sampler with spherical head is attractive. Particle movement near the sampling ori/ce is close to the situation near a human head. Understanding the aspiration process in a spherical sampler will help us to increase the knowledge of the human inhalation. The performance of blunt samplers operating in moving air was studied in many previous experimental and theoretical works. Semi-empirical models for blunt samplers were developed by Vincent (1984), Vincent (1987), Tsai and Vincent (1993) and Tsai, Vincent, and Mark (1996). Results of intensive mathematical study of blunt body sampling were summarized in the book of Dunnett and Ingham (1988). Two-dimensional model of aspiration into the sampler with head of blunt shape was developed by Chung and Dunn-Rankin (1992). In the recent paper of Dunnett and Vincent (2000) a mathematical approach was adopted to calculate the aerosol .ow into an idealized spherical sampler and to investigate aspiration e3ciency for various orientations of the sampler relative to the wind direction, based on a simple approximate model of gas .ow—a sink on a sphere. In living and industrial rooms the air is usually calm or moves with low velocity. At these conditions the gravity becomes important, and the aspiration e3ciency depends on the orientation of the sampler relative to gravity. The science of calm air sampling into blunt samplers has received far less attention by comparison with the case of moving air. The gap remains to complete a full understanding of the aspiration process, especially, on blunt sampler orientation in.uence. The results of experimental studies of the aspiration e3ciency for thin-walled and thick-walled tubes operating in still air were presented by Kaslow and Emrich (1973, 1974), Yoshida, Uragami, Hasuda, and Iinoya (1978), Belyaev and Kustov (1980), Grinshpun, Lipatov, and Semonyuk (1989), Su and Vincent (2002b). Mathematical models for thin-walled tube sampling in still air were developed by Agarwal and Liu (1980), Dunnett (1992). The aspiration e3ciency for the two-dimensional blunt sampler operating in calm air at horizontal and vertically upwards and downwards orientations was studied by Chung and Dunn-Rankin (1993). Grinshpun, Willeke, and Kalatoor (1993) constructed the formulae /tting the available experimental data and applicable for sampling through the thin-walled tubes from moving air as well as from calm air. The experiments connected with the study of spherical sampler performance in still air have been made early by Su and Vincent (2002a). In the present paper we propose a mathematical model of sampling into the spherical sampler with a circular ori/ce operating in calm air. The /nite-size sink on the sphere is modelled as a number of single point sinks and the gas velocity /eld induced by the /nite-size sink is expressed in analytical form. The equations of motion of aerosol particles are numerically solved to /nd the limiting particle trajectory surface and to calculate the aspiration e3ciency. The results of parametrical investigations of the aspiration e3ciency and a comparison with experimental data are presented. We also analyse singular points of the particle motion equations and their connection with typical trajectory zones around the sampler.
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2. Formulation Let us consider aerosol sampling into an idealized spherical sampler with a single circular ori/ce that is oriented at the various angles relative to the gravity direction (Fig. 1). Denote by the angle between negative direction of the x-axis and the line that connects the centre of the ori/ce with origin. The angle changes from −=2 to +=2 in the plane z = 0. The ori/ce diameter is smaller than the sampling head diameter D. Far from the sampler the gas is assumed to be stagnant and the particles settle gravitationally parallel to the y-axis. The limiting particle trajectory surface separates the trajectories of particles that are sampled from those that are not. The aspiration e3ciency A is de/ned as the ratio of the concentration c of particles in the air .ux passing through the sampler inlet to that c0 in undisturbed air far from the sampler. By using the relation of continuity of particle .uxes the aspiration e3ciency for the calm air sampling is written in the form A = c=c0 = VS S=Q = 4VS S=US 2 ;
(1)
where US is the mean velocity across the plane of the sampling ori/ce, VS =g is the particle settling velocity, Q = 2 US =4 is the sampling .ow rate, = p d2 =18 is the particle relaxation time, p is the particle density, d is the particle diameter, is the dynamic viscosity of the .uid, g is the gravity acceleration, and S is the horizontal area above the sampler enclosed by the limiting particle trajectory surface.
Fig. 1. Representation of the sampler head orientation and limiting particle trajectories.
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The determination of the area S reduces to calculating the limiting particle trajectories by solving the equations of particle motion in a given velocity /eld. The motion of the air is induced by the action of air suction through an ori/ce in the surface of the sampler head. The air.ow for the typical sampling velocities can be considered at /rst approximation as a potential .ow of an incompressible .uid and viscous e4ects can be neglected. Let the single point sink with strength Q be located in the point (−1; 0; 0) in the non-dimensional coordinate system (x; y; z). The sphere radius D=2 and the sampling velocity Us are the scales of the length and velocity. In the axially symmetric .ow of an ideal .uid the stream function non-dimensioned with respect to the quantity US D2=4 is given by H 2 ( cos + 1) ; 4 1 1 = 2 + 1 + 2 cos ; 0
=−
H = =D;
(2)
where 2 = x2 + y2 ; is the angle between the positive direction of the x-axis and the radius-vector of an arbitrary point on the sphere in the meridional plane z = 0. In accordance with the Butler theorem (Milne-Thomson, 1960) the stream function of the .ow induced by sink (2) in the presence of sphere can be written as 1 = 0 (; ) − 0 ; : (3) It follows from (2) and (3), that =
H 2 (2 − 1) : 4 1
(4)
The radial and tangential components of air velocity by means of the formulae u = −
2
@ 1 ; sin @
u = −
1 @ sin @
(5)
can be expressed in the form H 2 (2 − 1) ; u = − 4 31
1 H2 2 (2 − 1)( + cos ) : u = − 3 4 1 sin 1
(6)
The Cartesian gas velocity components ux ; uy ; uz resulting from (6) are 2x + 1 2x + 1 H 2 1 r 2 + 3 + 4x H2 2 H2 2 − y z ; uy = ux = − 3 ; uz = − 3 ; 4 r 4 r2 4 r2 r13 r1 r1 r 2 = x2 + y2 + z 2 ;
r12 = r 2 + 2x + 1;
r2 = r 2 r1 (x + 1) + rr12 x:
(7)
Formulae (7) can be used to express the velocity /eld for the case when a point sink is located at an arbitrary point (x0 ; y0 ; z0 ) (x0 ¡ 0) on the sphere. A new coordinate system (x ; y ; z ) is introduced
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1139
for which the sink location is (−1; 0; 0). This system is generated by rotating the coordinate system (x; y; z) relative to the y- and z-axis by the two angles 0 = arcsin(y0 );
0 = arcsin(z0 );
(8)
respectively. In this case the coordinates and velocity components are transformed by formulae rM = A (0 ; 0 ) rM ; u( M r) M = AT (0 ; 0 )uM (rM ); cos 0 cos 0 −sin 0 cos 0 sin 0 cos 0 A(0 ; 0 ) = sin 0 sin 0 −cos 0 sin 0 where
x rM = y; z
x
rM = y ; z
ux
uM = uy ; uz
(9) sin 0 0
;
(10)
cos 0
ux
uM = uy ; uz
A(0 ; 0 ) is the rotation matrix, AT (0 ; 0 ) is the transposed matrix. In the vicinity of the inlet ori/ce the above-described single point sink model gives a velocity /eld di4erent from the one of a /nite-size ori/ce as in real sampling devices. In order to obtain a more accurate velocity /eld of the carrier .uid we employ the following model. For H 1 the .ow through the circular inlet ori/ce will be represented as a .ow through the spherical segment of radius H with the same boundary as the boundary of the suction ori/ce. The gas .ow is modelled as a .ow induced by a number of single point sinks uniformly distributed within the spherical segment with the centre in the point (−1; 0; 0). The model approaches the one of the /nite-size sink in the surface of the sampler head with increase in the number of single point sinks. The coordinates of the single point sinks are de/ned by 2(i − 1) 2(i − 1) Hj Hj yi; j = cos ; zi; j = sin ; xi; j = 1 − yi;2 j − zi;2 j ; J I −1 J I −1 1 6 j 6 J;
1 6 i 6 I (j); I (j) = 6j:
(11)
The number of sinks can be found by formula N = 1 + 3J (J + 1). For example the case N = 91 is shown in Fig. 2. If the inlet ori/ce is oriented under the angle relative to the x-axis the location of the point sinks (11) will be determined by the two angles i; j = arcsin(yi; j ) + ;
i; j = arcsin(zi; j ):
(12)
The transformations (9) and (10) are used to express the velocity /eld induced by the arbitrary single sink in point (11). The velocity /eld from a number of sinks is given by sum uM =
J I ( j) 1
T A (i; j ; i; j )uM (rMi; j ); N j=1 i=1
(13)
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Fig. 2. Example of distribution of point sinks within the /nite-size sink at N = 91.
where rMi; j = A(i; j ; i; j )r. M Formulae (7), (11)–(13) determine completely the velocity /eld of the carrier .uid induced by suction through the /nite-size sink at arbitrary orientation relative to the gravity force direction. The accuracy of the simulation of the /nite-size suction ori/ce increases with the number of the simulated point sinks. The equations of particle motion are written under the assumption that Stokes’ law for drag is valid and that any external forces except gravity are negligible u y − vy dvx ux − vx dvy vs dvz u z − vz = ; = − ; = ; dt St dt St St dt St dy dz dx = vy ; = vz ; = vx ; dt dt dt
(14)
where vx ; vy ; vz are the Cartesian components of particle velocity non-dimensioned with respect to US ; vs = VS =US , t is the time, St = US =(D=2) is the Stokes number. Solving Eqs. (14) with the initial conditions for t = 0, vx = 0;
vy = −vs ;
vz = 0;
x = x0 ;
y = y0 ; z = z0
(15)
allows tracing the particle trajectories. To /nd the limiting trajectories we assume that the particle will be sampled if it reaches the spherical segment of radius H with the same centre as the inlet ori/ce centre. Within the considered model the aspiration e3ciency is determined by the four parameters St; vs ; ; H .
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3. Singular points The calm air sampling behaviour for aerosol particles of a spherical sampler is determined to a considerable degree by existence of singular points of the system of Eqs. (14) that can appear in the domain of the solution as a function of the values of the stationary settling velocity vs and angle . For simplicity consider the singular points for the single point sink model (N = 1) at the /xed value H = 0:1. In these points which are also called the “saddle” points (Vincent, 1989) the particle settling velocity directed downwards is balanced by the upward gas .ow created by the suction. In the general case the singular point is the point where the particle has zero velocity. Hence the coordinates of singular points can be found from the condition that the right-hand side of Eqs. (14) are equal to zero vx = 0;
vy = 0;
vz = 0;
ux = 0;
uy = v s ;
uz = 0:
(16)
The gas velocity components ux ; uy ; uz for the case = 0 can be expressed by the ones ux0 ; uy0 ; uz0 of .ow induced by the point sink at = 0: ux = ux0 cos + uy0 sin ;
uy = −ux0 sin + uy0 cos ;
uz = uz0 :
(17)
The Cartesian coordinates of the system turned on the angle are determined by the formulae x0 = x cos − y sin ;
y0 = x sin + y cos ;
z0 = z:
(18)
Taking into account (16) and (17) we obtain ux0 cos + uy0 sin = 0;
−ux0 sin + uy0 cos − vs = 0;
uz0 = 0:
(19)
For = ± =2 it follows from (19) that ux0 = −vs sin ;
uy0 = vs cos ;
z = 0:
(20)
In this case all singular points are in the symmetry plane z =0. Substituting the expressions ux0 ; uy0 from (7) into (20) and taking into account (18) yields the system of non-linear algebraic equations for coordinates x; y of singular points H 2 1 r 2 + 3 + 4(x cos − y sin ) = −vs sin ; (21) − 4 r r13 H2 (x sin + y cos ) 4 r 2 = x2 + y2 ;
2(x cos − y sin ) + 1 2 − 3 r2 r1
= vs cos ;
r12 = r 2 + 2(x cos − y sin ) + 1;
r2 = r 2 r1 (x cos − y sin + 1) + rr12 (x cos − y sin ): The results of numerical solution of (21) are shown in Fig. 3. The closed curves for various values of vs are obtained by variation of . The marks correspond to certain values of the angle .
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1 0 -1
VS =0.0001 VS =0.0002 VS =0.001 VS =0.01 VS =0.1 α=π/2 α=π/4 α=0 α=−π/4 α=−π/2
Y
-2 -3 -4 -5 -6 -7 -1
0
1
2
3
4
X Fig. 3. The curves of the singular point positions for the various values of vs and the angle . The fat circle is the sphere projection on the plane z = 0.
The curves cross the sphere in the points x = ± 1, y = 0 for all values of the settling velocity. Part of the curves inside the sphere has no physical meaning. The distances of the singular points from the sphere increase with decreasing vs . The curve of singular points outside the sphere is widened, acquiring a “waisted” shape (vs = 0:0002) and being divided into two almost circular curves at the value vs = 0:0001. One of them crosses the sphere but the second curve is located on some distance away from the sphere. The lower positions of singular points at the /xed values of the settling velocity correspond to the downward-facing orientation of the sampler (Fig. 3, = −=2). The singular points approach the sphere with the increase of . The critical values of the settling velocity vs and the angle correspond to the location of singular points on the sphere (x = ± 1; y = 0) and can be obtained from (21) in the form vs1 = H 2 sin [sin−1 (=2) − 1]=4;
x = −1; y = 0;
vs2 = H 2 sin [cos−1 (=2) − 1]=4;
x = 1; y = 0:
(22)
Functions (22) at 0 ¡ ¡ =2 are shown in Fig. 4. Three regions can be discerned in the plane (vs ; ): I (vs ¿ vs1 )—singular points outside the sphere are absent, II (vs2 ¡ vs 6 vs1 )—a unique singular point exists in the symmetry plane z = 0 outside the sphere, III (vs 6 vs2 )—two singular points are in the plane z = 0. Two types of singular points appear in the last case. In the far point
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y
x 0
P2
10
-1
10
I
vS1 vS2
-2
VS
10
-3
10
II
-4
10
III
-5
10
0
π /4 α
π/2
Fig. 4. The functions vs1 ; vs2 and the regions I–III in the plane (vs ; ).
P1
Fig. 5. The trajectories in the plane z = 0 for St = 0:1, vs = 0:0001, H = 0:1 and = =4.
the particle trajectories merge and bifurcate into two trajectories, one of which leads to the sink and the second trajectory by which the settling particles follow. In the other singular point that is close to the sphere the trajectories are doubled and .ow around the sphere by all sides. Fig. 5 illustrates the trajectories passing through such two typical singular points that are denoted by P1 and P2 for the case St = 0:1, vs = 0:0001 and = =4. In the region 6 0 we have the unique singular point in the plane z = 0 as in the region II for all settling velocities vs . The singular points for ¡ 0 and reasonably large vs can be located in the sphere shadow. The particle trajectories do not pass through such singular points and the aspiration e3ciency equals to zero in this case. In the axially symmetric case = =2, an in/nite number of singular points can form a circle with the centre on the y-axis. This circle is contracted and moves down with decrease of vs . It is illustrated in Fig. 3 by formation of the “waist” at vs = 0:0002. The circle is shrunk into one point at vs ≈ 0:00014. Further decrease of vs gives the two singular points on the y-axis where the particle trajectories merge or are doubled as in the above described case. The illustrations of particle behaviour near the singular points for various orientation of the sampler are given below. The coordinates of singular points as a function of vs and within the /nite-size ori/ce model are close to the considered case of the single point sink model. Note also that the similar analysis of singular points would be interesting for the thin-walled tube sampler operating in calm air. 4. Calculation results To determine the limiting particle trajectory the Cauchy problem for system of equations of particle motion is usually solved repeatedly. It is also possible to /nd the limiting trajectories by means of the approach based on the boundary value problem for particle motion equations (Galeev & Zaripov,
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2003). Let t1 be the time at which the particle reaches the sampler inlet edge. On the left end of the interval [0; t1 ], the boundary conditions for the particle velocity components are de/ned as vx = 0;
vy = −vs ;
vz = 0:
(23)
On the right-hand side, the particle coordinates are x = x1 ;
y = y1 ;
z = z1 ;
(24)
where (x1 ; y1 ; z1 ) is a point on the inlet boundary. The boundary value problem (14), (23) and (24) is solved using the /nite-di4erence method with deferred corrections of Pereyra (1978). Use of the boundary value problem is more e4ective but it is limited by the case of absence of the singular points. To compare the results and check the accuracy of calculations both methods were used when it was possible. The trajectories and shape of cross-sectional area of the tube of limiting trajectories signi/cantly di4er at various orientations of the inlet ori/ce relative to the gravity force direction. The behaviour of aerosol particles close to the spherical sampler in calm air is shown in Fig. 6 as a function of , where the trajectories in the plane z = 0 (a), projections of limiting particle trajectories on the symmetry plane z =0 (b) and cross-sectional area of the limiting particle trajectory surface (area S) (c) are presented. Fig. 6(1) – (5) correspond to the angles ==2; =4; 0; −=4; −=2, respectively and were obtained at H = 0:1, vs = 0:001 and St = 1. The corresponding values of the aspiration e3ciency are also presented in Fig. 6(1c) – (5c). The fat lines in Fig. 6(1a) – (5a) designate the limiting trajectories. In the general case the region around the sampler can be divided into four typical zones: zone (i) of trajectories of sampled particles, zone (ii) of trajectories of particles that deposit on the sphere, zone (iii) of trajectories of particles that pass by the head and zone (iv) without particles. Analysis of these zones near the cylindrical sampler head for inertialess particles was given by Davies (1967) and Dunnett (2002). In the axially symmetric case = =2, vs = 0:001 is the limiting value for singular points to exist (see Fig. 4); at larger values of the settling velocity they are absent. The cross-sectional area of the tube of limiting particle trajectories for upwards facing ori/ce orientation is the circular area and the singular points form the circle on the sphere: x2 + y2 = 1, y = 0 (Fig. 6(1)); they are denoted by P in the plane z = 0. In these points the particle trajectory bifurcates into two trajectories one of which creeps close to the sphere and reaches the sampler inlet. The other trajectory branch goes down and bounds the zone without particles. The symmetry is lost with changing the angle , i.e. for vs = 0:001 we have only one singular point left. Fig. 6(2) illustrates this for the case = =4. The dashed line divides the zones (ii) and (iii). The limiting particle trajectories merge in the point P (Fig. 6(2a, b)). Zone (iv) becomes connected with suction ori/ce and the air from zone without particles is additionally sampled. Because of this the aspiration coe3cient can be less than unity even for inertialess particles. The area of initial points of trajectories of sampled particles is displaced but is kept close to the circular area at = =2. The boundary of the area S is divided into the two parts that correspond to the initial points of limiting trajectories passing through the singular point and reaching directly the inlet opening. For horizontal orientation of suction ori/ce ( = 0) the “sickle-shaped” area S is formed by concave and convex curves (Fig. 6(3)). All limiting trajectories with initial positions on the convex curve go through the point P. A portion of the falling particles impacts the spherical head of the sampler. The area of initial positions of the particles deposited on the sphere is subtracted from the area S. For the next angle = −=4 the cross-sectional area of the limiting particle trajectory surface becomes doubly connected. The area of initial positions
R.S. Galeev, S.K. Zaripov / Aerosol Science 34 (2003) 1135 – 1150 y
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y
z
A=1.001
i 2 0 iii
iii
P
-4
-2
x
P
x 4
2 -2
x P
P
iv
-4
1b
1a
1c
y
y
z A=0.989
i
2 x
0 -4
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iii
P
P
-4
iv
2b
2a
2 -2
x
x
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-2
y
2c y
z
A=0.909
2 ii
i
x
0 x
x
-4
-2
2 -2
iii
P
iv
iii P
-4
3a
3c
3b y
y
ii
z A=0.895
i
i
2 x
x
-4
-2
2 -2
P iii
x
0
iv
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P
-4
4a i
4c
4b y
y
z
ii i
A=0.908
2 x
x
iv
0 -4
-2
2
x 4
-2 iii
P
iii
5a
P
5b
-4
5c
Fig. 6. (1) – (5). The trajectories in the plane z = 0 (a), projections of limiting particle trajectories on the symmetry plane z = 0 (b) and the cross-sectional area of the limiting particle trajectory surface (area S) (c) for St = 1, vs = 0:001, H = 0:1 and various .
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R.S. Galeev, S.K. Zaripov / Aerosol Science 34 (2003) 1135 – 1150
2
y
1 ii
1.0
i
0.5
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0
-1.5
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Fig. 6. continued. (6) The same as in Fig. 6. (1) – (5) for horizontal orientation of the sampler for St = 50 and vs = 0:01.
of deposited particles is completely inside the area S. The particles are sampled from all sides, and because of this, the zone (iv) adjacent to the sphere becomes closed. When the sampler is oriented vertically downwards ( = −=2) the cross-sectional area of the tube of the limiting particle trajectories is the axially symmetric annular area. The unique singular point is located on the y-axis (x = 0; z = 0). Note one interesting e4ect at the sampling from calm air that can be observed for the particles at high Stokes numbers. The interaction between inertial and gravitational forces can lead to appearance of complex trajectories in the vicinity of the suction ori/ce. Davies (1968) was the /rst to discuss the “orbiting” particle trajectories near the sampling ori/ce at the calm air sampling. We have obtained in our calculations such complex trajectories for large St but their behaviour di4ers from the one of “orbiting” trajectories around the point sink as described by Davies. This di4erence is due to the presence of the spherical sampling head and the trajectories are oscillated near the singular point. It is di3cult to present the calculated complex trajectories in space for clear illustration. We tried to do it in Fig. 6(6) where the particle trajectories and the area S as in Fig. 6(1) – (5) at St = 50 and vs = 0:01 for horizontal orientation of the sampler are presented. The fat lines in Fig. 6(6a) designate the limiting trajectories. In the plane z = 0 we have the two limiting trajectories 1 and 2 that hit the inlet edge and con/ne the sampled particles zone. Trajectory 3 also reaches the inlet ori/ce because the particle inertia decreases but the gravitational force is not enough for particle to fall down. Therefore, a new zone of sampled particles appears. This zone is limited by the trajectory 3 and the next limiting trajectory 4 that goes through the singular point P after “orbiting” travel. The particles that move between the limiting trajectories 2 and 3 will deposit on the sphere. This leads to the formation of a doubly connected area S inside which the area of initial positions of trajectories of particles deposited on the sphere below the inlet ori/ce appears. To obtain this doubly connected area very accurate calculations are needed. The particle following the limiting trajectory and passing through the singular point will move to inlet or settle down with equal probability. It is expected that a peak of the particle concentration should be observed in the corresponding positions on the bottom of the experimental chamber including the sampling into blunt sampler from calm air.
R.S. Galeev, S.K. Zaripov / Aerosol Science 34 (2003) 1135 – 1150 VS =0.1, N=91 VS =0.01, N=91 VS =0.001, N=91
1.4 1.2
VS =0.1, N=1 VS =0.01, N=1
1.0
0.8
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α=π/2 α=π/4 α=0 α=−π/4
0.4 0.4
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0.2
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0.1
1
10
100
1000
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Fig. 7. The St-dependence of A at the various values of vs for vertically upwards facing sampler. Black dots are the experimental points from work of Su and Vincent (2002a).
0.001
0.01
0.1
1
10
100
1000
St
Fig. 8. The St-dependence of A at the various values angle for vs = 0:01(a), 0.001(b).
Parametrical calculations of the aspiration e3ciency A with variable the Stokes number St, the settling velocity vs and the sampler orientation angle were made for the single point sink and /nite-size suction ori/ce models. The area S was found by means of calculating the limiting particle trajectories and is used to obtain the aspiration e3ciency A by formula (1). The numerical results and their analysis are given below. The St-dependencies of the aspiration e3ciency for = =2 and di4erent values of vs and N are shown in Fig. 7. The aspiration coe3cient calculated by means of the /nite-size ori/ce model stops to be sensitive to the number of the simulated point sinks beginning from N = 91. Because of this, the calculations results for N = 91 and single point sink model (N = 1) are presented for comparison. We also see that there is a .at part in aspiration e3ciency curve with A = 1 + vs (Grinshpun et al., 1993) for small values of the Stokes number. In this case the deviation of the aspiration e3ciency from unity is caused by gravity. The length of the constant part of the St-dependence of the aspiration e3ciency increases with the decrease of the settling velocity. For intermediate values of the Stokes numbers there exists a transition region where the in.uence of particle inertia on sampling becomes noticeable. With further increase of St the aspiration coe3cient drops to the value A = vs . Note that at 0 6 6 =2 in two limiting cases of inertialess particles and particles with very high inertia the aspiration e3ciency for upward-facing orientation of the sampler is given by formulae A=1+vs sin and A=vs sin , respectively. The experimental points from work of Su and Vincent (2002a) are denoted by black dots in Fig. 7. The Stokes number in their work is based on the suction ori/ce diameter as the length scale and St values were recalculated accordingly. Fair agreement of the St-dependence of the aspiration e3ciency calculated according to our models with experimental data is observed. The results of the /nite-size sink model are closer to the experimental points than those of the single point sink model for vs =0:1 where we have larger di4erences between the two models. This di4erence decreases for small values of the settling velocity. At vs =0:01 both models give reasonable agreement with experiment. In the case vs =0:001 the aspiration e3ciency calculated by two models does not distinguish practically.
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0.8
A
0.6 α=−π/4 α=−3π/8 α=−π/2
0.4
0.2
0
0.002
0.004
0.006
0.008
0.010
VS
Fig. 9. The variation of A for inertialess particles at various .
The numerical results of the aspiration e3ciency as a function of the Stokes number for vs = 0:01 and N = 91 for various angles of the sampler orientation relative to the gravity force direction are shown in Fig. 8. The transitional part of the St-dependence curve of the aspiration e3ciency for vs = 0:01 becomes steeper with decreasing absolute values of . There is a region of St values where A for the horizontal orientation is larger than for a vertically upward sampler position. There are the particles that impact on the spherical head around the sampler ori/ce and are not sampled. For the horizontal sampler orientation the spherical surface limits the access of aerosol particles into the inlet opening only from one side. Because of this, the area of sampled particles far from the sampler for = 0 can be larger than the one for = =2 in some range of the Stokes number or sizes of particles. It is also seen that for = 0 and −=4 the e4ect of the additional suction of air without particles and screening of settled particles leads to a decrease of the aspiration e3ciency below unity for small Stokes numbers. The area S of initial positions of sampled particles far from the sampler becomes large at very small values of the settling velocity and the initial area of deposited particles will be comparatively small. Because of this, the in.uence of screening of the settled particles by the sphere and orientation of sampler on the aspiration e3ciency will not be signi/cant. Calculations of the St-dependence of A for the various angles at vs = 0:001 show that all curves are very close. The aspiration e3ciency becomes less than unity for very large values of the Stokes number (see the corresponding curve in Fig. 7). Taking into account the screening of falling particles by the sphere the aspiration e3ciency of inertialess particles (St = 0) in the case of a vertically downwards facing sampler ( = −=2) can be expressed by formula A = 1 + vs (1 − H −2 ):
(25)
From (25) under the condition A = 0 the formula for the settling velocity beginning from which the particles are not sampled can be written: vs = H 2 =(1 − H 2 ). The dependence of the aspiration e3ciency on the settling velocity for inertialess particles at the some angles obtained by numerical calculations is presented in Fig. 9. The calculated values of the aspiration coe3cient at the angle = −=2 completely coincide with formula (25) (solid line). At the two other angles = −3=8,
R.S. Galeev, S.K. Zaripov / Aerosol Science 34 (2003) 1135 – 1150
1149
−=4 the values of A are on the theoretical straight line for the values vs for which the area S of initial positions of sampled particle trajectories is the doubly connected area. At larger values of vs the aspiration e3ciency continues to decrease but more slowly.
5. Conclusions A mathematical model for sampling into a spherical sampler from calm air has been developed. The /nite-size sink on the sphere is presented as the sum of a number of single point sinks uniformly distributed within the spherical segment with the same boundary as one of the inlet ori/ce. The gas velocity /eld induced by the single point sink is expressed in analytical form. The equations of motion of aerosol particles have been solved numerically to /nd the limiting particle trajectory surface and to calculate the aspiration e3ciency. The behaviour of the trajectories of aerosol particles in the vicinity of the sampler is determined by the settling velocity and the Stokes number as well as the angle of orientation of the suction ori/ce relative to the direction of gravity. In the general case, the region around the sampler can be divided into four zones: zone of trajectories of sampled particles, zone of trajectories of particles that deposit on the sphere, zone of trajectories of particles that pass by the head and zone without particles. The patterns of these zones depend on the existence and location of the singular points of the motion equations of aerosol particles. The coordinates of the singular points as functions of the settling velocity and of the sampler orientation angle have been investigated. The regions in the plane (vs ; ) where one or two singular points exist and where there are no singular points have been determined. When the sampling region includes the zone without the particles the aspiration e3ciency even for inertialess particles may fall below unity. The dependence of the aspiration e3ciency on the Stokes number at various values of the settling velocity and the angle of the sampler orientation has been studied. It was shown that in certain range of the Stokes numbers the aspiration e3ciency for horizontal orientation could be larger than for a vertically upwards-facing sampler. The formula for the aspiration e3ciency of inertialess particles in the downwards-facing sampler case has been given. This formula is applicable for angles when the cross-sectional area of the limiting particle trajectory surface is a doubly connected area. Fair agreement between the numerical results from our model and experimental data for upwardsfacing orientation of the sampler ori/ce was observed. To check other theoretical results of the paper and to validate the mathematical model, an experimental study of the aspiration e3ciency of blunt samplers in calm air is important, including the e4ects of sampler orientation. The distribution of concentration of particles deposited on the bottom of the experimental chamber can indicate the location of zones without particles and singular points. The latter ones are of special importance, because a peak of the particle concentration should be observed below these points.
Acknowledgements Authors gratefully acknowledge the support of this work by Russian Foundation for Basic Research under grant no. 02-01-00836 and by Academy of Science of Republic of Tatarstan.
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References Agarwal, J. K., & Liu, B. Y. H. (1980). A criterion for accurate aerosol sampling in calm air. American Industrial Hygiene Association Journal, 41, 191–197. Belyaev, S. P., & Kustov, V. T. (1980). Sampling from calm air. Trudy IEM, 25, 102–108 (in Russian). Chung, I. P., & Dunn-Rankin, D. (1992). Numerical simulations of two-dimensional blunt body sampling in viscous .ow. Journal of Aerosol Science, 23, 217–232. Chung, I. P., & Dunn-Rankin, D. (1993). The e4ects of bluntness and orientation on two-dimensional samplers in calm air. Aerosol Science and Technology, 19, 371–380. Davies, C. N. (1967). Movement of dust particles near a horizontal cylinder containing a sampling ori/ce. British Journal of Applied Physics, 18, 653–656. Davies, C. N. (1968). The entry of aerosols into sampling tubes and heades. British Journal of Applied Physics, 25(1), 921–932. Dunnett, S. J. (1992). A mathematical study of the sampling characteristics of a thin-walled sampler operating in calm air. Aerosol Science and Technology, 17, 93–104. Dunnett, S. J. (2002). Particle motion in the vicinity of a bulky sampling head operating in calm air. Aerosol Science and Technology, 36, 308–317. Dunnett, S. J., & Ingham, D. B. (1988). The mathematics of blunt body sampling. Lecture Notes in Engineering. London: Springer. Dunnett, S. J., & Vincent, J. H. (2000). A mathematical study of aerosol sampling by an idealised blunt sampler oriented at an angle to the wind: The role of gravity. Journal of Aerosol Science, 31, 1187–1203. Galeev, R. S., & Zaripov, S. K. (2003). Deposition of aerosol particles on a sphere: The role of gravity. Aerosol Science and Technology, 37, 325–329. Grinshpun, S. A., Lipatov, G. N., & Semonyuk, T. I. (1989). A study of sampling of aerosol particles from calm air into thin-walled cylindrical probes. Journal of Aerosol Science, 20, 1561–1564. Grinshpun, S. A., Willeke, K., & Kalatoor, S. (1993). A general equation for aerosol aspiration by thin-walled sampling probes in calm and moving air. Atmospheric Environment, 26A, 1459–1470. Kaslow, D. E., & Emrich, R. J. (1973). Aspirating ;ow pattern and particle inertia e