The effects of finite reynolds number on the aspiration of particles into a bulky sampling head

J. Aerosol Sci.. Vol. 18. No 5, pp. 553-561, 1987.

0021-8502i87 $3.00+0.0(1 ~ 1987PergamonJournals Ltd.

Printed in Great Britain

THE EFFECTS OF FINITE REYNOLDS NUMBER ON THE ASPIRATION OF PARTICLES INTO A BULKY SAMPLING HEAD S. J. DUNNETTand D. B. INGHAM Department of Applied Mathematical Studies, The University of Leeds, Leeds LS2 9JT, West Yorkshire, U . K (Received 9 February 1987)

A b s t r a e t - - A study is made o f the aspiration of dust particles, suspended in an ideal fluid, into a bulky sampling head. The particles are assumed to be spherical and suspended withou~ sedimentation in the fluid flowing past the sampler. The effects o f the particle having a finite local Reynolds number are investigated. Numerical results arc obtained for two cases, namely when the sampler is smooth and clean and the particles arc solid so that no particles will adhere to the sampler and also when the sampler is sticky or the particles arc liquid so that all the particles which hit the sampler adhere to it. It is found that the effects on the aspiration o f particles ot:the increasing inertia forces arc dependent upon a parameter ,1, the particle Reynolds number. When this parameter is in the range of its largest value in most experimental situations the results obtained are found in some circumstances to differ by over 100~o from the corresponding results obtained by neglecting the finiteness of the Reynolds number.

INTRODUCTION Much research has been performed on dust sampling due to its importance to the health of many people, e.g. workers in coal mines. The air is usually sampled by sucking through an orifice and then collecting the aspirated dust on a filter which is then investigated to determine the concentration of airborne dust. In general the dust particles will have inertia and so will not be travelling with the same velocity as the fluid. Therefore the concentration of dust in the air entering the sampler will not be equal to the concentration of dust in the ambient fluid and the aspiration coefficient, A, which is defined as the ratio between these two concentrations, will not in general be unity. In order to determine the dust concentration in an environment it is therefore necessary to evaluate how this aspiration coefficient varies with different conditions. The dust samplers employed consist of a bulky sampling head containing one or more suction orifices, through which the dusty air is drawn. One attempt at modelling this problem has been to consider the walls of the sampler to be infinitely thin, this is known as 'thin walled' sampling. Past work in this area includes Badzioch (1959), Davies (1968), Belyeav and Levin (1974) and Addlesee (1980). In their work, the aspiration coefficient has been obtained experimentally and then generalized by empirical formulae. In reality the walls of the sampler wilt cause some obstruction to the flow, and more recently this case has received some attention, see for example, Vincent et al. (1979), Vincent and Mark (1982) and Ingham (1981 ). In Ingham's work an analytic approach was adopted in which the orifice in the sampler was assumed to be narrow and could therefore be modelled by a line sink. This approach limits the shape of the sampler as there are only a few cases for which analytic expressions are available for the flow and also only samplers with narrow orifices can be considered. In order to take into account the shape of the sampler and also the finite width of the orifice the problem has been studied adopting the Boundary Integral Equation method [see Dunnett and Ingham (1986)]. Using this method it is possible to consider samplers of any shape with suction orifices of finite size. This method uses an approximation of Greens theorem and involves more computer time and storage than the analytical approach. In all the past work mentioned it has always been assumed that the inertia forces of the particles are negligible compared with the viscous forces, i.e. the Reynolds number, which is defined as the ratio between the inertial and viscous terms in the equations of motion, is always small and therefore has been neglected. This is a reasonable assumption in many applications but as the effects of the inertia forces increase, assuming the Reynolds number to 553

554

S..I Dt.>,x~vr and D B. [',OH.~sl

be negligible increases the inaccuracy of the results. Some work has been pertormed ,,hich includes the Reynolds number, see lbr example Paw U (1983, 1984), where the rebound o~ particles from surfaces, including the Reynolds number effects, has been investigated. Fuchs (1964) investigated the error obtained in calculating the resistance of a medium by neglecting the Reynolds number as its value increases. Analysing experimental data it is found that at a Reynolds number as small as 0.5, the error exceeds 5 0,. The aim of this present work is to investigate the error incurred by neglecting the finiteness of the Reynolds number when considering the area of dust sampling. As it is necessary to model the sampling procedure as accurately as possible it is important to know whether including realistic values of the Reynolds number in the model makes a significant ditt~erence to the results obtained, The case of a cylindrical sampler is investigated which has a narrow suction orifice on the side facing the oncoming potential flow. Through the orifice an amount Q of fluid per unit time per unit length is drawn and it is assumed that the dust particles are spherical and suspended in the fluid without sedimentation. It was decided to adopt the analytical approach taken by Ingham and not the B.I.E. method as the aim of this work is to investigate the effects of including the finite Reynolds number in the model on the ,sampling of the particles for any shaped sampler with a suction orifice. As the aspiration coefficient is an important quantity in dust sampling its value ~s investigated for the cases when the Reynolds number is neglected and when it is included in the equations of motion and the results are compared. Two cases are considered, namely (a) when all the particles that hit the sampler between the two stagnation points (points of zero fluid velocity on the sampler} are eventually sampled e.g. in the case of a smooth clean sampler and solid particles which hit the sampler with insufficient kinetic energy and bounce in the wrong direction to escape the converging trajectories, so that all the particles which hit the sampler between the stagnation points either bounce off into the boundary layer or into the fluid converging towards the orifice, or (b) when particles adhere to the sampler surface so that the only particles to be sampled will be the ones that go directly into the orifice without first colliding with the walls e.g. in the case of a sticky sampler or liquid particles.

FORMULATION

The system considered is that of a cylindrical sampler, centre O, radius a, with a line sink at A along 0 = m See Fig. 1. The flow Uo is assumed uniform at large distances from the sampler and is moving in the positive x direction. The Reynolds number of the flow is assumed to be sufficiently large so that the flow may be assumed to be inviscid except near the boundary of the sampler. This should be a good approximation for x < 0. Using a complex variable method, the x and y components of the fluid velocity are shown to be U=--

x2 + v2

+ .12 + y2 +Uo

1

~ll

r

Fig 1. Coordinate system and notation.

Effects of finite Reynolds n u m b e r on aspiration of particles

xZ..~_y2

555

(x+a)2+y 2 - UoL(x~y2) 2 ,

(2)

see Davies (1967). The stagnation points S~ and $2 are defined as the points on the sampler where the fluid velocity is zero. Their position is given by cos

=

or

0 = 0 °,

(3)

see Ingham (1981). If the orifice has a width 26 then the mean velocity of sampling v,, will be given by vm = Q/(26). Therefore from equation (3) cos ~

= ~

(4)

where ~0 = (6vm)/(aUo) and hence if the stagnation points $I and $2 are assumed to be a distance 2s apart, s is given by a

s= ~

x/(4rt~0- tp2).

(5)

Considering a spherical particle of radius r, mass m, moving in this velocity field with resolutes of velocity (u, v) in the x and y directions, its equations of motion are, Fuchs (1964), du m-d-~- = 6n#r (1 + 0.167(Re) 3/2) (U - u) dv md-~ = 6rcpr(1 +O.167(Re)a/2)(V-v) dx

,

(6)

dy

u=-d- f,

V=dt

where p is the viscosity of the fluid and Re is the Reynolds number associated with the motion of the particles relative to the fluid defined as, Re = 2 r [ ( u -

u)2 + ( V - v ) 2 ] 1/2,

(7)

v

where v is the kinematic viscosity of the fluid. In much past work the Reynolds number has been assumed to be sufficiently small so that the Stokes drag formula could be used. Before performing any calculations all quantities were non-dimensionalized, lengths with respect to a, the radius of the sampler, and velocities with respect to Uo. The equations of motion (6) now become,

du'

(l +O.167(~((U'-u')2 +(V'-v')2)x/2)3/2)(U'-u ')

dt'

St

dr'

(l +O.167(e((U'-u')2 +(V'-v')2)l/2)3/2)(V'-v ')

-dt~ = u'

St dx' dt'

v'

,

(8)

dy' dr'

where primes denote the non-dimensional quantities,

2rUo

= --

V

(9)

556

S J. Dt;NN~ rq and D. B. INGH,~,',t

is a particle Reynolds number based on the flee stream speed and St is the Stokes numbe; given by St = (mU0)(6rt~ra), In order to evaluate the aspiration coefficient it is necessary to obtain the hmiting particle trajectories between which all particles are sampled. If the sampler is smooth and the pamcle,~ solid, then all the particles which hit the surface will bounce offit. Hence. assuming that those which hit the sampler between the two stagnation points do not have sutticient energy or do not bounce off in the correct direction to enter the diverging flow travelling around the sampler, the limiting trajectories in this case will be those that meet the sampler at the stagnation points. If the sampler is sticky or the particles liquid, then the particles will adhere to the sampler so the limiting trajectories will be the ones which meet the sampler at the corners of the inlet. A full discussion of the rebound of particles from surfaces can be found in Paw U (1983). For a given value of St and e the equations of motion were integrated starting from a large value o f x ' with various values ofy'. It was found that x' = - 1000 gave results to the accuracy required, of order 10 -4 , and so this was taken as the value at which conditions at infinity could be enforced. For each value of St and ~ the value o f y ' at x' = - 1000, say y'~, was found which gave the limiting trajectory. The aspiration coefficient is then given by 2 y'caUo

A= - -

Q

(I0)

When the Stokes number is small the particles tend to move with the fluid and A ~ 1, whereas when it is large the particles tend to move independently of the fluid, such that for a smooth sampler A --* (2sUo)/Q, and for a sticky sampler A --* ( 2 6 U o ) / Q as St --, zc. In most of the published experimental results, see, for example Vincent et al. (1979), results for A have been obtained for values of a up to order 10. As the errors in the calculated results obtained by neglecting the Reynolds number will be the greatest when the Reynolds number is largest, the case of • = 10 has been investigated in detail. In dust sampling, the parameter ~ will not usually rise greatly above the value 10 as the particle radius is generally small and the fluid in which the particles are suspended is not fast moving but it is physically possible that ~ will exceed this value, see, for example, Mark et al. (1982) where experiments have been performed for ~ = 20. In this work though, in order to obtain numerical results which relate to the majority of experimental results available in the area of dust sampling the case of ~ greater than 10 has not been investigated in detail. RESULTS The value of the aspiration coefficient has been obtained for a variety of values of the parameter St. Three values of a have been taken, the two limiting cases of a = 0 and :¢ = 10 and the intermediate value ~ = 5. In Fig. 2 the variation of A with St is shown for the case when the sampler is smooth and there is a fairly low rate of suction, i.e. ~o/rt = 0,01. It is seen that, as expected, for all values o f ~ A --* 1 as St --* 0 and A ~ s/(a~o) = 3.1831 as St ~ 3c. As increases, the value of A decreases. A case when the suction is increased is shown in Fig. 3. In this case (o/n = 1 and as before the sampler is assumed to be smooth and the particles solid. Once again the cases o f ~ = 0, ~ = 5 and ~ = 10 are shown. The same behaviour is noted for small and large St as before but in this case the value of A increases as ~ increases. It has been shown by Vincent and Mark (1982) that at a relatively low rate of suction the pattern o f the fluid flow into the sampling orifice exhibits the 'spring onion' shape (see Fig. 4). There are two distinct regions, the divergent part associated with the movement of the fluid around the sampler and the convergent part associated with the suction of the fluid into the sampling orifice. The two limiting streamlines meet the sampler at the stagnation points St and $2 and all the fluid between these two lines will enter the sampler. For a relatively large rate of suction, i.e. ~o > 1, however, the 'spring onion" shape is no longer present (see Vincent et al. (1982)), the flow o f fluid into the sampling orifice being wholly convergent. It was found that in plotting A against the particle Reynolds number, ~, (see Fig. 5) tor the constant Stokes number 1, that, depending on whether the flow exhibited the 'spring onion"

Effects of finite Reynolds n u m b e r on aspiration of particles

557

30 P h i / P i •0.01

/

/

~

/f

2.~

i//.~ I

,,.7./ /Z/ &

2.0 -

/Z/ /I/~/../

....

/~//

ALpha =0 _

J

Alpha =5

~,~

i

I

I IIIIII

I

I

2 25 4 56789i

I Ililll

[

2 3 4 56789t

x I0 - z

x I0 -I

I

I IIIIII

I

2 3 4 567891 x I0 °

I

ALpha•lO

I IIIlll

I

2 3 4 567891 x I0'

I

I IIIIII

2 3 4 56789i x I0 3

xlO 2

St Fig. 2.

The variation of A with St for the case o f a s m o o t h sampler with • = 0, 5 and lO and ~ / n = 0.01.

.2

.0

08

~

~

' . ~~

'

~

P h i / P i = I .0

-.'~.

-

ALpha -0

~ 1 A

0.6 --

~%.

....

ALpha-5

~'~

ALpho=lO

0.4

0.2

I

I

I Jlllll

I

2 3 4 567891 xI 0 - 2

I

I liltll

t

2 3 4 567891 x l O -I

I

L tlllll

I

2 3 4 56789~ x t0 °

I

I Illill

I

2 3 4 567891 x I01

I

I

IIII11

2 3 4 567891 xlO 2

St

Fig. 3. The variation of A with St for the caseof a smooth sampler with = = O,5 and ] 0 and ~o/n = 1.0.

xlO 3

558

5; ,I Dt;Nr~Ff-I and D B. INGHAM

Fig. 4. Example of potential flow at the inlet to a simple blunt sampler.

. . . .

I

0

~. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I

I

I

2

I

3

i

4

l

I

5

I

6

*

.

l

°

*

0

I

7

. . . . .

~

Phi/Pi

- 0.001

Phi/Pi

,, 0 . 0 1

~ . ~

Phi/Pi,'O.O05

.........

Phi/Pi

......

Phi IPi=O,l

-

Phi/Pi=05

.

-

.

I

8

m

.

*

m

.

.

.

.

.

-0.05

.

.

.

.

I

9

.

I

I0

Alpha Fig. 5. The variation of A with c{ for the case of a s m o o t h sampler with St = 1 and various values of (0/~.

effect or was wholly convergent, A either decreased or increased to unity as :e increased. The critical value of q}/n being approximately 0.1. Increasing the value o f ~ in the equations of motion equation (8) is, in effect, increasing the resistance o f the particles. As the Stokes number, St, is the ratio between the particle stop distance and the sampler radius a, this is equivalent to decreasing the Stokes number. For the case of a relatively small rate of suction the flow has a diverging and converging part. In this case when the Stokes number is relatively large the particles will have inertia and they will not follow the streamlines of the flow and so some particles outside the limiting streamlines will not be carried by the fluid around the sampler but hit it between the two stagnation points and therefore enter the converging flow and hence the orifice. As :t increases and the particles increasingly respond to the airflow, some of those which previously had sufficient inertia to hit the sampler between the two stagnation points will be carried by the diverging flow around the sampler and will never enter the converging flow and hence A decreasesl This is seen in Figs 2 and 5. This behaviour continues until ~ is so large that the particles have no significant inertia and travel with the fluid, so that A tends to unity.

Effects o f finite R e y n o l d s n u m b e r on a s p i r a t i o n o f particles

559

The same behaviour is noted for larger suction rates when the flow is wholly convergent, only in this case A increases to unity, as seen in Figs 3 and 5 for the larger values of ~0/rc. In Fig. 5 only values of ct up to those attained in most experimental results have been taken so the case of 0t > 10 is not shown and hence A has not attained its limiting value of unity. A value of the suction rate between the two previously mentioned was taken in order to investigate the case when the flow is changing from having a diverging and converging part to being wholly convergent. Taking q~/rc = 0.1 it was found that increasing ct had a negligible effect on A, the difference between the value for ct = 0 and ~ = 10 being at the most of order 10 -3 ' Considering the case when the sampler is sticky so that particles adhere to the surface for a relatively low rate of suction, ~0/~ = 0.01, so that the flow pattern exhibits the 'spring onion' shape. The case when the dimensionless inlet width Delta = 0.032 is shown in Fig. 6. When the Stokes number is large the particles move independently of the fluid and any decrease in its value causes the particles to follow more closely the fluid flow. As shown in Fig. 4, the fluid starts to deviate around the sampler and then converges near the orifice. As the particles become more affected by the fluid motion they will also follow this pattern. But at large St, as the Stokes number decreases, the particles begin to respond to the flow and the divergent part pulls them away from the orifice and some of the particles which previously entered will hit the walls of the sampler and adhere to them. Therefore a decrease in St at large values decreases the number of particles sampled, i.e. an increase in the Reynolds number causes a decrease in A. Further decrease o f the Stokes number causes the particles to be more influenced by the fluid flow. At some value of St a number of particles that did not move with the fluid into the orifice but adhered to the walls for higher St now enter. So a decrease in St leads to an increase in the number o f particles entering the sampler. Therefore, for small St, an increase in the Reynolds number leads to an increase in A. This behaviour is seen in Fig. 6. As can be seen there is a value o f St for which A when at = 10 is equal to A when ~ = 0. This is also true for ~t = 5.

I 2

08

A

',\\

j

0.6

. . . .

ALpha =0

04 Phi/Pi

=0.01 ALpha

• 5

DeL'ka = 0 . 0 3 2 02

~ . ~

o 10 -2

I I0-

ALpha • I0

I

I

I

I0 0

I01

10 2

St

Fig. 6. The v a r i a t i o n o f A with St for the case w h e n the particles a d h e r e with ~ = 0. 5 a n d 10. tp/n = 0.01 a n d D e l t a = 0.032.

I iO3

560

S J Dur, x~,-Fr and D B. INGHa.M 060

0.58

0.56

o5, r0.52 ~"

A

0.50

0.48

0.46 Detto =0.032

Phi/Pi'O.OI2

0 44

Phi/Pi

,0.014

0.42

Phi/Pi 0.40

I 5

I

io

I 15

I 20

I 25

I 30

35

I 40

"0016 I 45

I 50

ALpha

Fig. 7. The variation of A with c( for the case when the particles adhere with St = 1 and (p/n= 0.12, 0.014 and 0.016.

In Fig. 7, A has been plotted against a for different suction rates when the Stokes number is one, for the case when the particles adhere. It is seen that the minimum value attained by A is reached at a lower value o f the particle Reynolds number, a, as the suction rate increases until it occurs at ~, = 0 as seen for the case of q~/n = 0.016. This is to be expected because as the suction rate increases the convergence o f the fluid increases and the particles become affected by this flow at a higher value of the Stokes number. For smaller values of q~/n then, the ones showing the minimum in A occur at very large values of a, values which are not attained in any experimental situations. If the flow pattern does not exhibit the 'spring onion' effect, i.e. the rate of suction is such that the flow is wholly convergent, then in this case decreasing the Stokes number so that the particles are more influenced by the fluid flow means the particles increasingly converge towards the orifice. Hence A increases. This is shown in Fig. 8 when the case of q~/rt = t for the sticky sampler/liquid particles is investigated. As can be seen, as the value of the Reynolds number increases, i.e. c( increases, the aspiration coefficient increases. CONCLUSIONS It has been shown that the effects of the Reynolds number on the aspiration of dust particles is dependent upon a parameter c( = (2rUo)/V where r is the radius of the particle, Uo the free stream velocity and v is the kinematic viscosity of the fluid. The case of :t = 0 corresponds to neglecting the Reynolds number. It has been shown that the effects of neglecting this parameter on the results obtained depends on the rate of suction. If the rate of suction is relatively high then the effect will be to underestimate the value o f the aspiration coefficient and if it is relatively low then the effect will be to overestimate the value of A. The error obtained by neglecting the Reynolds number rises to over 100 oj;;in some cases when :( = 10. Therefore in order to obtain accurate numerical results to compare with the experimental results available at ~ > 10 obtained by, for example, Vincent et al. (1979), it is

Effects of finite Reynolds number on aspiration of particles

561

I 0

\

Alpha

I 0

-

ALpha

I5

--.--

Alpha

• IO

Phi/Pi=l.O

\

Delta

= 0.032

I I 04

\

''4 \

02

\

\

ol 10-2

I

I

I 0 -I

i0 o

....

I- . . . . . . I0 ;

I

I

I0 z

I0 ~

S't Fig. 8. The variation of A with St for the case when the particles adhere with :t = 0, 5 and 10, tp/rt = 1.0 and Delta = 0.032.

necessary in modelling the problem to include the finite Reynolds number. As ~t increases the error in assuming ct = 0 will increase. The maximum errors occur when the Stokes number is in the range 0.1 < St < 10 whereas for large and small values o f St the inclusion o f a finite Reynolds number in the equations o f m o t i o n has no significant effect on the results. REFERENCES Addlesee, A. J. (1980) J. Aerosol Sci. 11, 483. Badzioch, S. (1959) Br. J. appl. Phys. 10, 26. Belyeav, S. P. and Levin, L. M. (1974) J. Aerosol Sci. 5, 325. Davies. C. N. (1967) Br. J. appl. Phys. 18, 653. Davies, C. N. (1968) Br. J. appl. Phys. (J. Phys. D) Series 2, 921. Dunnett, S. J. and Ingham, D. B. (1986) J. Aerosol Sci. 17, 839. Fuchs, N. A. (1964) The Mechanics of Aerosols. Pergamon Press, Oxford. ingham. D. B. (1981) J. Aerosol Sci. 12, 541. Mark, D., Vincent, J. H. and Witherspoon, W. A. (1982) Aerosol Sci. Technol. 1,463. Paw, U. K. T. (1983) J. Colloid Interface ScL 93, 442. Paw. U. K. T. (1984) J. Aerosol Sci. 15, 657. Vincent, J. H, Hutson, D. and Mark, D. (1982) Atmos. Envir. 16, 1243. Vincent. J. H. and Mark. D (1982) Ann. occup. B y O. 26, 3. Vincent. J. H., Wood, J. D.. Birkett, J. L. and Gibson, H. (1979) Institute of Occupational Medicine (Edinburgh), Technical Memorandum TM/79/18