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A mathematical investigation into the effects of the position of the sampling inlet upon the aspiration of a thin-walled sampler
J. Aerosol Sci., Vol. 23, Suppl. I, pp. $579-$582, 1992 Printed in Great Britain.
0021-8502/92 $5.00 + 0.00 Pergamon Press Ltd
A MATHEMATICAL INVESTIGATION INTO THE EFFECTS OF THE POSITION OF THE SAMPLING INLET UPON THE ASPIRATION OF A THIN-WALLED SAMPLER S.J.Dunnett Health and Safety Executive, Broad Lane, Sheffield $3 7HQ, U.K. (C) British Crown Copyright 1992
ABSTRACT Using a mathematical approach an investigation has been made into the effects upon sampling performance of the position of the sampling inlet for a thin-walled cylindrical tube. The airflow in the vicinity of the sampler is determined and the behaviour of the particles in the air are then investigated. Various situations are considered and the aspiration efficiency of the sampler determined. KEYWORDS Sampling, Thin-walled,
Mathematical modelling.
INTRODUCTION In many working environments there are airborne particles which are potential health hazards to the people working there. It is therefore important to be able to determine the concentration of such airborne particles in order to evaluate the health risks involved and many aerosol samplers have been developed for this purpose. Due to many factors, however, the particle sample collected by the sampler may not be a true representation of the particle concentration in the atmosphere. Therefore, in order to effectively use such sampling systems, it is necessary to have an understanding of the factors which affect their performances and much work has been undertaken in this area, see Vincent(1989). This has led to a greater understanding of sampler performance. However there are still many factors which affect performance which are not fully understood. In this work the effects upon aspiration of the position of the sampling inlet for a thin-walled cylindrical tube aligned with the flow are considered. The sampling inlet is taken to be the circular cross section of the tube where the flow is withdrawn at a constant velocity, see Fig.l. In general samplers operate by the application of suction within the sampling body at some distance, Xi, from the entrance of the sampler and hence in order to model the situation accurately the sampling inlet should be taken at some position within the tube, i.e. Xi>0. In this work aspiration is investigated for different values of Xl. In the model considered the developing boundary layer in the tube has been ignored, since for the Reynolds numbers of the flow considered $579
$580
S.J. DUNNETT
the layers will be extremely thin. The flow in the tube will become fully developed at a distance of approximately 0.06 x Reynolds number x Do from the tube entrance. In sampling situations the Reynolds number is typically O(104 ) and hence the flow is not fully developed until a distance of approximately 600Do from the entrance. In this work the assumption of constant velocity across the tube is made at distances considerably less than 600Do from the tube entrance where the boundary layer is extremely thin.
u~
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I
I
:/ leeS"
tI
If
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---~---~,. V
Ii
.-.rl--.4D-
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Fig.l. Notation used for cylindrical sampler. FORMULATION The situation of a thin-walled cylindrical tube is considered, although the method used could equally well be applied to thick walled samplers of arbitrary shape. In order to ensure that the sampler is thin-walled the ratio of the samplers outer diameter, Do, to inner diameter, Di, was taken to be Do/Di=l.0/0.95=l.05. From Belyaev and Levin (1974) a sampling tube is thin-walled if Do/Di
Aspiration of a thin-walled sampler
$581
from those that do not, were determined and hence the aspiration efficiency, A, of the sampler evaluated. Where A is defined as the ratio of the concentration of particles in the flow passing through the sampler entrance to that in the undisturbed flow far away from the sampler. Values of A were determined for different values of the velocity ratio v/U0 (where U0 is the freestream velocity), the Stokes number St, and for different positions of the sampling inlet along the sampling tube. The Stokes number, St, is given by St=(d29U0)/(18~Di), d is the particle diameter,q is the particle density and ~ Is the fluid viscosity. RESULTS AND DISCUSSION In Fig.2 the aspiration coefficient, A, is shown as a function of St for three different positions of the sampling inlet, (i) Xi=0, (ii) Xi=2Do, and (iii) Xi=10Do for v/U0=0.8 and 2. Also shown are the curves obtained from the semi-empirical formula of Belyaev and Levin (1974) for thin-walled samplers. This formula was obtained from experiments conducted for 0.18 ~v/U0 6 6.0 and 0.18 ~ St ~ 2.03 with Xi ~ 10Do, the results of which gave considerable scatter.
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~ " .8
0-----0
Empirical
formuls,
v/U o=0
8
D
"
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~
\
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X-----X
X~ =0 '
Jr-----Jr
X~.2O..
v / U o ,~0 " 8
~----~ ?--4
X,~lOOo
o
x,-2Do.
v/Uo=0.8 v/U,,-O.8
Emplrical formula V/Uo~2
.7
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o
,'Uo-2
.6
.5
___~_J__,_~_,
I0-2
,,J1
....
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J_ i I_LJ-.L.L[L__ * ~
i
,
~O
~
,
. . . . .i. ~0 !
,I
I I I I I WJ
iOa
st Fig.2. A as a function of the Stokes number When solving for the flow characteristics using the B.I.E. method it was found that inaccuracies were introduced when the sampling inlet was taken inside the cylindrical tube away from the entrance due to the inner corner formed where the sampling inlet joins the sampler walls. Hence when applying the method care was taken to ensure that the inaccuracies were kept to a minimum. It was found that, in all cases considered, the error in the calculated flux of fluid withdrawn across the sampler entrance was less than 10% with the majority of cases giving errors well AS
23 Supp|.
]--MM
$582
S.J. DUNNETT
below 10%. The greatest errors were found to occur for the largest values of Xi. As can be seen from Fig.2 the numerical results agree reasonably well with the empirical formula, always within 15%. This agreement is reasonable considering the simplicity of the mathematical model and the estimated possible errors of up to 10% in both the mathematical results and the experimental data from which the empirical formula was developed. Work is however currently in progress to improve the accuracy of the mathematical model. Also, as can be seen, there is no significant difference in the numerical values of A obtained for the different positions of the sampling inlet for both values of v/U0 shown. Other values of v/Uo and positions of sampling inlet were considered and in all cases it was found that the position of the inlet had no significant effect on A. CONCLUSIONS It has been shown that it is possible, by using a mathematical approach, to investigate the effects upon aspiration of varying the position of the sampling inlet. In sampling systems in use the position at which suction is applied will be at some distance from the samplers entrance. Also, when modelling aerosol samplers mathematically in order to investigate their sampling characteristics, it is necessary to apply the conditions of the sampling inlet at some position along the sampling tube. Hence it is important to understand the effects the position may have upon aspiration. It has been shown that, for the situations considered, the effects are not significant. The method adopted has advantages over experimental methods which could be applied to this problem as it can easily be adapted to consider other situations, for example, different flowrates, different size or shape sampl~rs, etc. It also has advantages over other numerical methods, e.g. finite difference methods, as once the problem is solved on the boundary of the sampler it is possible to determine the air velocity directly at any point of interest. REFERENCES
Belyaev,S.P. and Levin,L.M.(1974) Techniques for the collection of representative aerosol samples. J.Aerosol Sci., 5, 325-338. Dunnett,S.J. and Ingham,D.B.(1988) The mathematics of blunt body sampling, Springer-Verlag,Berlin. Vincent,J.H.(1989) Aerosol Sampling, John Wiley and Sons, Chichester.
0021-8502/92 $5.00 + 0.00 Pergamon Press Ltd
A MATHEMATICAL INVESTIGATION INTO THE EFFECTS OF THE POSITION OF THE SAMPLING INLET UPON THE ASPIRATION OF A THIN-WALLED SAMPLER S.J.Dunnett Health and Safety Executive, Broad Lane, Sheffield $3 7HQ, U.K. (C) British Crown Copyright 1992
ABSTRACT Using a mathematical approach an investigation has been made into the effects upon sampling performance of the position of the sampling inlet for a thin-walled cylindrical tube. The airflow in the vicinity of the sampler is determined and the behaviour of the particles in the air are then investigated. Various situations are considered and the aspiration efficiency of the sampler determined. KEYWORDS Sampling, Thin-walled,
Mathematical modelling.
INTRODUCTION In many working environments there are airborne particles which are potential health hazards to the people working there. It is therefore important to be able to determine the concentration of such airborne particles in order to evaluate the health risks involved and many aerosol samplers have been developed for this purpose. Due to many factors, however, the particle sample collected by the sampler may not be a true representation of the particle concentration in the atmosphere. Therefore, in order to effectively use such sampling systems, it is necessary to have an understanding of the factors which affect their performances and much work has been undertaken in this area, see Vincent(1989). This has led to a greater understanding of sampler performance. However there are still many factors which affect performance which are not fully understood. In this work the effects upon aspiration of the position of the sampling inlet for a thin-walled cylindrical tube aligned with the flow are considered. The sampling inlet is taken to be the circular cross section of the tube where the flow is withdrawn at a constant velocity, see Fig.l. In general samplers operate by the application of suction within the sampling body at some distance, Xi, from the entrance of the sampler and hence in order to model the situation accurately the sampling inlet should be taken at some position within the tube, i.e. Xi>0. In this work aspiration is investigated for different values of Xl. In the model considered the developing boundary layer in the tube has been ignored, since for the Reynolds numbers of the flow considered $579
$580
S.J. DUNNETT
the layers will be extremely thin. The flow in the tube will become fully developed at a distance of approximately 0.06 x Reynolds number x Do from the tube entrance. In sampling situations the Reynolds number is typically O(104 ) and hence the flow is not fully developed until a distance of approximately 600Do from the entrance. In this work the assumption of constant velocity across the tube is made at distances considerably less than 600Do from the tube entrance where the boundary layer is extremely thin.
u~
"'~----i I
I
I
I
:/ leeS"
tI
If
"
---~---~,. V
Ii
.-.rl--.4D-
I! I!
1! iI
• ,I
Fig.l. Notation used for cylindrical sampler. FORMULATION The situation of a thin-walled cylindrical tube is considered, although the method used could equally well be applied to thick walled samplers of arbitrary shape. In order to ensure that the sampler is thin-walled the ratio of the samplers outer diameter, Do, to inner diameter, Di, was taken to be Do/Di=l.0/0.95=l.05. From Belyaev and Levin (1974) a sampling tube is thin-walled if Do/Di
Aspiration of a thin-walled sampler
$581
from those that do not, were determined and hence the aspiration efficiency, A, of the sampler evaluated. Where A is defined as the ratio of the concentration of particles in the flow passing through the sampler entrance to that in the undisturbed flow far away from the sampler. Values of A were determined for different values of the velocity ratio v/U0 (where U0 is the freestream velocity), the Stokes number St, and for different positions of the sampling inlet along the sampling tube. The Stokes number, St, is given by St=(d29U0)/(18~Di), d is the particle diameter,q is the particle density and ~ Is the fluid viscosity. RESULTS AND DISCUSSION In Fig.2 the aspiration coefficient, A, is shown as a function of St for three different positions of the sampling inlet, (i) Xi=0, (ii) Xi=2Do, and (iii) Xi=10Do for v/U0=0.8 and 2. Also shown are the curves obtained from the semi-empirical formula of Belyaev and Levin (1974) for thin-walled samplers. This formula was obtained from experiments conducted for 0.18 ~v/U0 6 6.0 and 0.18 ~ St ~ 2.03 with Xi ~ 10Do, the results of which gave considerable scatter.
i.i
i.O A .9
"" p
"~ •
~ " .8
0-----0
Empirical
formuls,
v/U o=0
8
D
"
"A
~'?.\
"~"
~
\
~B\ \ ~ " .k\ ~
X-----X
X~ =0 '
Jr-----Jr
X~.2O..
v / U o ,~0 " 8
~----~ ?--4
X,~lOOo
o
x,-2Do.
v/Uo=0.8 v/U,,-O.8
Emplrical formula V/Uo~2
.7
"..L "~.'9..
o
,'Uo-2
.6
.5
___~_J__,_~_,
I0-2
,,J1
....
lO -I
J_ i I_LJ-.L.L[L__ * ~
i
,
~O
~
,
. . . . .i. ~0 !
,I
I I I I I WJ
iOa
st Fig.2. A as a function of the Stokes number When solving for the flow characteristics using the B.I.E. method it was found that inaccuracies were introduced when the sampling inlet was taken inside the cylindrical tube away from the entrance due to the inner corner formed where the sampling inlet joins the sampler walls. Hence when applying the method care was taken to ensure that the inaccuracies were kept to a minimum. It was found that, in all cases considered, the error in the calculated flux of fluid withdrawn across the sampler entrance was less than 10% with the majority of cases giving errors well AS
23 Supp|.
]--MM
$582
S.J. DUNNETT
below 10%. The greatest errors were found to occur for the largest values of Xi. As can be seen from Fig.2 the numerical results agree reasonably well with the empirical formula, always within 15%. This agreement is reasonable considering the simplicity of the mathematical model and the estimated possible errors of up to 10% in both the mathematical results and the experimental data from which the empirical formula was developed. Work is however currently in progress to improve the accuracy of the mathematical model. Also, as can be seen, there is no significant difference in the numerical values of A obtained for the different positions of the sampling inlet for both values of v/U0 shown. Other values of v/Uo and positions of sampling inlet were considered and in all cases it was found that the position of the inlet had no significant effect on A. CONCLUSIONS It has been shown that it is possible, by using a mathematical approach, to investigate the effects upon aspiration of varying the position of the sampling inlet. In sampling systems in use the position at which suction is applied will be at some distance from the samplers entrance. Also, when modelling aerosol samplers mathematically in order to investigate their sampling characteristics, it is necessary to apply the conditions of the sampling inlet at some position along the sampling tube. Hence it is important to understand the effects the position may have upon aspiration. It has been shown that, for the situations considered, the effects are not significant. The method adopted has advantages over experimental methods which could be applied to this problem as it can easily be adapted to consider other situations, for example, different flowrates, different size or shape sampl~rs, etc. It also has advantages over other numerical methods, e.g. finite difference methods, as once the problem is solved on the boundary of the sampler it is possible to determine the air velocity directly at any point of interest. REFERENCES
Belyaev,S.P. and Levin,L.M.(1974) Techniques for the collection of representative aerosol samples. J.Aerosol Sci., 5, 325-338. Dunnett,S.J. and Ingham,D.B.(1988) The mathematics of blunt body sampling, Springer-Verlag,Berlin. Vincent,J.H.(1989) Aerosol Sampling, John Wiley and Sons, Chichester.