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The effects of rebound upon the sampling efficiency of a thin-walled sampler
J. Aerosol Sci., Vol. 19, No. 7, pp. 997 - i000, 1988 Printed in Great Britain
0021-8502/88 $3.00. + 0.00 Pergamon Press plc
THE EFFECTS OF REBOUND UPON THE SAMPLING EFFICIENCY OF A THIN-WALLED SAMPLER.
S.J. Dunnett, University of Leeds, Leeds,U.K.
During the process of particle sampling it is possible that secondary aspiration will occur due to particles bouncing off the outer surface of the sampler into the flow converging towards the orifice and hence entering the inlet. If the sampler is designed to determine the concentration of particles in an atmosphere then it is necessary to be able to determine the extent to which particle rebound contributes to the sample collected. Although much research has been performed in the area of particle sampling the effects of secondary aspiration have not been thoroughly investigated. In the case of a thin-walled sampler facing the flow many semi-empirical formulae have been developed for the aspiration coefficient, A, which is defined as the ratio of the concentration of particles in the fluid entering the sampler to that in the undisturbed flow. Such work includes Belyeav and Levin (1974) and Davies and Subari (1982). In the work presented here a mathematical approach is adopted to the problem which accurately predicts the fluid motion and hence the motion of the particles in the flow. This enables the extent of oversampling caused by particle rebound to be determined for various situations. The idealised case of a thin-walled sampler has been investigated in order to compare with the previous work performed into the effects of particle rebound. Hence the sampling system considered is that of a two-dimensional tube of length L and width D, which is facing a uniform potential flow U O. The sampling velocity is denoted as v and the system is investigated for various values of the velocity ratio V/Uo=k, say.
The method adopted to solve the flow problem was a numerical method known as the Linear Boundary Integral Equation (L.B.I.E.) method which has been described earlier by Dunnett and Ingham (iSSS).Once the flow pattern has been determined the paths of the particles in the flow can be traced and their behaviour near the sampler investigated. In tracing the particle paths it is assumed that the particles are spherical and small enough to cause no disturbance to the flow. It is also assumed that the Reynolds number associated with the motion of the particles relative to the flow is small enough for the Stokes drag to be applied and also that all external forces, e.g. gravity and electrostatic forces, are negligible.
An important flow characteristic to determine is the position of the stagnation points as these mark the positions where the streamlines dividing the sampled and unsampled fluid meet the sampler. In this study they were found to be influenced by the ratio of L/D and also k. In order to investigate the effects of k for one particular value of L/D, the ratio L/D=14 was considered in detail. It was found that for k~2 the stagnation points occurred on the edges of the inlet, but for larger values of the velocity ratio the stagnation points were positioned on the walls of the sampler and hence fluid is drawn back around the sampler body in order to enter the inlet. Therefore for this particular sized sampler with k~2 it is possible that secondary aspiration will occur due to particles not adhering to the sampler surface on impact and subsequently being carried by the converging flow into the tube. As k increases the distance from the inlet of the position of the stagnation points also increases and hence it is likely that if the particles do not
997
998
S.J.
DUNNETT
adhere, the c o n t r i b u t i o n to the sample collected from secondary a s p i r a t i o n will increase as k increases. When varying L/D for one particular value of k, k=lO, it was found that the stagnation points at the front of the body were positioned on the walls of the sampler for all values of L/D considered in the range 2
In order to investigate the accuracy of the model taken values of the aspiration coefficient were determined for the situation when the only particles that enter the sampler are those that enter directly, i.e. there no s e c o n d a r y aspiration. The aspiration coefficient is denoted as A i. In
is
Fig. l. the v a r i a t i o n St=(d2~U0)/(18wD),
of A with k is shown for k up to 34 and St=0.22 where i d is the aerodynamic particle diameter, ~ the particle
density and ~ the fluid viscosity. This dimensionless quantity is a measure of the inertia of the particle. Also shown is the curve obtained from the formula of Belyeav and Levin as their experimental study involved no secondary aspiration effects. The value of L/D=I4 has been taken as it corresponds to a value taken in the experimental study. As can be seen agreement between the numerical results and the empirical formula is good for small values of k but deteriates as k increases. The empirical formula developed by Belyaev and Levin was based on experimental results obtained by a sampler whose size corresponds to that taken in the numerical model and for values of the velocity ratio, k, less than S. In this range of k, Fig. l shews that agreement between numerical and experimental work is good. Recently Lipatov et al (1988) obtained experimental results which agree well with the empirical formula for k up to 35. In their work no information is given as to the size of the sampler used and hence it was decided to investigate the effects of the size of the sampling body upon the coefficient A i for these larger values of k. Values
of A
were obtained
for a sampler
with dimensions
such that L/D=28
and
for St=0.22 and these are included in the figure. As can be seen these results correspond more closely to the empirical formula and hence the e f f i c i e n c y of the sampler also has a dependence on its size. Therefore empirical formulae developed for a particular sampler may not hold for other sized samplers and hence can not be widely applied to any body. Other values of St were investigated and the same behaviour was noticed.
Another extreme situation, when all the particles which hit the sampler between the two front s t a g n a t i o n points and the inlet enter the sampler, was investigated and results compared with the empirical formulea of Davies and Subari who included contributions from secondary a s p i r a t i o n in their experiments. The same behaviour was found to occur with agreement reasonable for small k but d e t e r i a t i n g as k increased. Also, increasing the value of L/D improved the agreement.
The two situations s t u d i e d are extreme situations, e.g. in the second case it has b e e n assumed that all the particles that hit the sampler surface between the two front s t a g n a t i o n points and the inlet bounce off and are carried by the c o n v e r g i n g flow into the sampling body. Whether this situation occurs or not will d e p e n d upon many factors, e.g. size and nature of the particles, the surface of the sampler, suction rate, etc. Hence it was decided to investigate the effects upon rebounding particles of some of these factors, and an extreme situation, when all the particles that impact upon the sampler surface are a s s u m e d to bounce off it without a n y loss of e n e r g y , was considered. The ratio L/D was taken to be 14 as this corresponds to a value taken in previous experimental work. For this situation the effects of the size of the Stokes number, St, and the rate of suction, upon the subsequent motion of the r e b o u n d i n g particles were investigated. It was found that within
Sampling efficiency of a thin-walled sampler
999
the range of St taken in past experimental studies for k~3 there is no contribution to the sample collected from secondary aspiration and A., i the coefficient investigated by Belyeav and Levln, is equivalent to A. For kAlO then all the rebounding particles hitting the surface between the inlet and the stagnation points are sampled. In Fig. 2. the variation of A with k for the three situations considered is shown for St=0.22. As before A i is the coefficient when no secondary aspiration occurs, A is the coefficient when all e the particles hitting between the two front stagnation points and the inlet are sampled and A is the coefficient obtained when the particles are assumed to be perfectly elastic. As can be seen for k~2 the values of all three coefficients are the same because in this case the limiting streamlines dividing the sampled and unsampled fluid meet the tube at the edge of the inlet and hence there is no secondary aspiration. For larger values of the velocity ratio the stagnation points move around the sampler body and hence the values of A. and A differ. As expected from the earlier investigation I
e
into the motion of the elastic particles the values of A and A. are equivalent I
for small values of k but as the velocity ratio increases the convergent flow has more effect upon the motion of the particles. Until, for some value of k, some of the rebounding particles no longer bounce into the diverging flow but are carried by the converging flow into the inlet. This effect increases until all the particles hitting the sampler between the stagnation point and the inlet are eventually sampled and hence A is equivalent to A . By investigating e
different values of St it was found that A differs from A i at a smaller value of k as St decreases. This is due to the fact that as St decreases the particles are more affected by the flow and hence rebounding particles are more likely to be carried by the convergent flow into the inlet, therefore A differs from A and tends to A at a lower value of k as St decreases. i •
It therefore appears from this work that when using samplers to determine the concentration of particles in an atmosphere the particle sample collected inside the sampler will be distorted by secondary aspiration when the sampling velocity is large compared with the freestream velocity. Such conditions do occur in experimental investigations, e.g. Gibson and Ogden (1977), and hence rebound must be accounted for in such studies. Acknowledgement. The author wishes to thank D.B. Ingham for his help and encouragement. References. Belyaev,S.P. and Levin, L.M.(1974) Techniques for collection of representative aerosol samples. J. Aerosol Sci. 5, 325-338. Davies,C.N. and Subari,M.(1982) Asplration above wind velocity of aerosols with thin-w~lled nozzles facing and at right angles to the wind direction. J. Aerosol Sci. 13, 59-71. Dunnett,S.J.and Ingham, D.B.(1986) A mathematical theory to two-dimensional blunt body sampling. J. Aerosol Sci. 17, 838-853. Gibson, H. and Ogden, T.L.(1977) Some entry efficiencies for sharp-edged samplers in calm air. J.Aerosol Sci. 8, 361-365. Lipatov, G.N., Grinspun, S.A., Shinaryov, G.L. and Sutugin, A.G.(1988) Aspiration of coarse aerosol by a thin-walled sampler, J.Aerosol Sci. 17, 763-769.
AS
19:7-P
I000
S.J.
DUNNETT
1.0
ST=O. 22
0.9-
NUMERICAL
0.8-
.............
BELYEAV AND LEVIN
+
L/D=2B
0.7O.,L.
0.5-
+
0.4
~3
0,2
0.1
0.O 0
]
]
I
S
tO
t5
Flg. I The v a r i a t i o n formula hO
for St=0.22.
k
of A i w l t h k obtained
I
I
I
20
2s
30
numerically
and from the empirical
L/D=14.
0.8. 0.7. 0.6. k 0.5. 5T=O. 22 0.4. ........... 0.3
.
.
.
.
.
.
.
.
.
^i A Ae
0.20.1 0.0
I
I
I
I
I
I
2
~
4
S
F i g . 2 The v a r i a t i o n L/D=14.
of
A., 1
A and A w i t h •
k
I
I
I
I
6
?
8
9
k obtained
numerically
l0
for
St=0.22
and
0021-8502/88 $3.00. + 0.00 Pergamon Press plc
THE EFFECTS OF REBOUND UPON THE SAMPLING EFFICIENCY OF A THIN-WALLED SAMPLER.
S.J. Dunnett, University of Leeds, Leeds,U.K.
During the process of particle sampling it is possible that secondary aspiration will occur due to particles bouncing off the outer surface of the sampler into the flow converging towards the orifice and hence entering the inlet. If the sampler is designed to determine the concentration of particles in an atmosphere then it is necessary to be able to determine the extent to which particle rebound contributes to the sample collected. Although much research has been performed in the area of particle sampling the effects of secondary aspiration have not been thoroughly investigated. In the case of a thin-walled sampler facing the flow many semi-empirical formulae have been developed for the aspiration coefficient, A, which is defined as the ratio of the concentration of particles in the fluid entering the sampler to that in the undisturbed flow. Such work includes Belyeav and Levin (1974) and Davies and Subari (1982). In the work presented here a mathematical approach is adopted to the problem which accurately predicts the fluid motion and hence the motion of the particles in the flow. This enables the extent of oversampling caused by particle rebound to be determined for various situations. The idealised case of a thin-walled sampler has been investigated in order to compare with the previous work performed into the effects of particle rebound. Hence the sampling system considered is that of a two-dimensional tube of length L and width D, which is facing a uniform potential flow U O. The sampling velocity is denoted as v and the system is investigated for various values of the velocity ratio V/Uo=k, say.
The method adopted to solve the flow problem was a numerical method known as the Linear Boundary Integral Equation (L.B.I.E.) method which has been described earlier by Dunnett and Ingham (iSSS).Once the flow pattern has been determined the paths of the particles in the flow can be traced and their behaviour near the sampler investigated. In tracing the particle paths it is assumed that the particles are spherical and small enough to cause no disturbance to the flow. It is also assumed that the Reynolds number associated with the motion of the particles relative to the flow is small enough for the Stokes drag to be applied and also that all external forces, e.g. gravity and electrostatic forces, are negligible.
An important flow characteristic to determine is the position of the stagnation points as these mark the positions where the streamlines dividing the sampled and unsampled fluid meet the sampler. In this study they were found to be influenced by the ratio of L/D and also k. In order to investigate the effects of k for one particular value of L/D, the ratio L/D=14 was considered in detail. It was found that for k~2 the stagnation points occurred on the edges of the inlet, but for larger values of the velocity ratio the stagnation points were positioned on the walls of the sampler and hence fluid is drawn back around the sampler body in order to enter the inlet. Therefore for this particular sized sampler with k~2 it is possible that secondary aspiration will occur due to particles not adhering to the sampler surface on impact and subsequently being carried by the converging flow into the tube. As k increases the distance from the inlet of the position of the stagnation points also increases and hence it is likely that if the particles do not
997
998
S.J.
DUNNETT
adhere, the c o n t r i b u t i o n to the sample collected from secondary a s p i r a t i o n will increase as k increases. When varying L/D for one particular value of k, k=lO, it was found that the stagnation points at the front of the body were positioned on the walls of the sampler for all values of L/D considered in the range 2
In order to investigate the accuracy of the model taken values of the aspiration coefficient were determined for the situation when the only particles that enter the sampler are those that enter directly, i.e. there no s e c o n d a r y aspiration. The aspiration coefficient is denoted as A i. In
is
Fig. l. the v a r i a t i o n St=(d2~U0)/(18wD),
of A with k is shown for k up to 34 and St=0.22 where i d is the aerodynamic particle diameter, ~ the particle
density and ~ the fluid viscosity. This dimensionless quantity is a measure of the inertia of the particle. Also shown is the curve obtained from the formula of Belyeav and Levin as their experimental study involved no secondary aspiration effects. The value of L/D=I4 has been taken as it corresponds to a value taken in the experimental study. As can be seen agreement between the numerical results and the empirical formula is good for small values of k but deteriates as k increases. The empirical formula developed by Belyaev and Levin was based on experimental results obtained by a sampler whose size corresponds to that taken in the numerical model and for values of the velocity ratio, k, less than S. In this range of k, Fig. l shews that agreement between numerical and experimental work is good. Recently Lipatov et al (1988) obtained experimental results which agree well with the empirical formula for k up to 35. In their work no information is given as to the size of the sampler used and hence it was decided to investigate the effects of the size of the sampling body upon the coefficient A i for these larger values of k. Values
of A
were obtained
for a sampler
with dimensions
such that L/D=28
and
for St=0.22 and these are included in the figure. As can be seen these results correspond more closely to the empirical formula and hence the e f f i c i e n c y of the sampler also has a dependence on its size. Therefore empirical formulae developed for a particular sampler may not hold for other sized samplers and hence can not be widely applied to any body. Other values of St were investigated and the same behaviour was noticed.
Another extreme situation, when all the particles which hit the sampler between the two front s t a g n a t i o n points and the inlet enter the sampler, was investigated and results compared with the empirical formulea of Davies and Subari who included contributions from secondary a s p i r a t i o n in their experiments. The same behaviour was found to occur with agreement reasonable for small k but d e t e r i a t i n g as k increased. Also, increasing the value of L/D improved the agreement.
The two situations s t u d i e d are extreme situations, e.g. in the second case it has b e e n assumed that all the particles that hit the sampler surface between the two front s t a g n a t i o n points and the inlet bounce off and are carried by the c o n v e r g i n g flow into the sampling body. Whether this situation occurs or not will d e p e n d upon many factors, e.g. size and nature of the particles, the surface of the sampler, suction rate, etc. Hence it was decided to investigate the effects upon rebounding particles of some of these factors, and an extreme situation, when all the particles that impact upon the sampler surface are a s s u m e d to bounce off it without a n y loss of e n e r g y , was considered. The ratio L/D was taken to be 14 as this corresponds to a value taken in previous experimental work. For this situation the effects of the size of the Stokes number, St, and the rate of suction, upon the subsequent motion of the r e b o u n d i n g particles were investigated. It was found that within
Sampling efficiency of a thin-walled sampler
999
the range of St taken in past experimental studies for k~3 there is no contribution to the sample collected from secondary aspiration and A., i the coefficient investigated by Belyeav and Levln, is equivalent to A. For kAlO then all the rebounding particles hitting the surface between the inlet and the stagnation points are sampled. In Fig. 2. the variation of A with k for the three situations considered is shown for St=0.22. As before A i is the coefficient when no secondary aspiration occurs, A is the coefficient when all e the particles hitting between the two front stagnation points and the inlet are sampled and A is the coefficient obtained when the particles are assumed to be perfectly elastic. As can be seen for k~2 the values of all three coefficients are the same because in this case the limiting streamlines dividing the sampled and unsampled fluid meet the tube at the edge of the inlet and hence there is no secondary aspiration. For larger values of the velocity ratio the stagnation points move around the sampler body and hence the values of A. and A differ. As expected from the earlier investigation I
e
into the motion of the elastic particles the values of A and A. are equivalent I
for small values of k but as the velocity ratio increases the convergent flow has more effect upon the motion of the particles. Until, for some value of k, some of the rebounding particles no longer bounce into the diverging flow but are carried by the converging flow into the inlet. This effect increases until all the particles hitting the sampler between the stagnation point and the inlet are eventually sampled and hence A is equivalent to A . By investigating e
different values of St it was found that A differs from A i at a smaller value of k as St decreases. This is due to the fact that as St decreases the particles are more affected by the flow and hence rebounding particles are more likely to be carried by the convergent flow into the inlet, therefore A differs from A and tends to A at a lower value of k as St decreases. i •
It therefore appears from this work that when using samplers to determine the concentration of particles in an atmosphere the particle sample collected inside the sampler will be distorted by secondary aspiration when the sampling velocity is large compared with the freestream velocity. Such conditions do occur in experimental investigations, e.g. Gibson and Ogden (1977), and hence rebound must be accounted for in such studies. Acknowledgement. The author wishes to thank D.B. Ingham for his help and encouragement. References. Belyaev,S.P. and Levin, L.M.(1974) Techniques for collection of representative aerosol samples. J. Aerosol Sci. 5, 325-338. Davies,C.N. and Subari,M.(1982) Asplration above wind velocity of aerosols with thin-w~lled nozzles facing and at right angles to the wind direction. J. Aerosol Sci. 13, 59-71. Dunnett,S.J.and Ingham, D.B.(1986) A mathematical theory to two-dimensional blunt body sampling. J. Aerosol Sci. 17, 838-853. Gibson, H. and Ogden, T.L.(1977) Some entry efficiencies for sharp-edged samplers in calm air. J.Aerosol Sci. 8, 361-365. Lipatov, G.N., Grinspun, S.A., Shinaryov, G.L. and Sutugin, A.G.(1988) Aspiration of coarse aerosol by a thin-walled sampler, J.Aerosol Sci. 17, 763-769.
AS
19:7-P
I000
S.J.
DUNNETT
1.0
ST=O. 22
0.9-
NUMERICAL
0.8-
.............
BELYEAV AND LEVIN
+
L/D=2B
0.7O.,L.
0.5-
+
0.4
~3
0,2
0.1
0.O 0
]
]
I
S
tO
t5
Flg. I The v a r i a t i o n formula hO
for St=0.22.
k
of A i w l t h k obtained
I
I
I
20
2s
30
numerically
and from the empirical
L/D=14.
0.8. 0.7. 0.6. k 0.5. 5T=O. 22 0.4. ........... 0.3
.
.
.
.
.
.
.
.
.
^i A Ae
0.20.1 0.0
I
I
I
I
I
I
2
~
4
S
F i g . 2 The v a r i a t i o n L/D=14.
of
A., 1
A and A w i t h •
k
I
I
I
I
6
?
8
9
k obtained
numerically
l0
for
St=0.22
and