NIOSH
EPA
United States
Department of Health
and Human Services
G6
•fy
' I Public Health Service
Centers for Disease Control
National Institute for Occupational Safety and Health
United States
Environmental Protection
Agency
Office of Environmental
Engineering and Technology
Washington, DC 20460
EPA-600/7-80-183
December 1980
Research and Development
Theoretical
Investigation of Inlet
Characteristics for
Personal Aerosol
Samplers
Interagency
Energy/Environment
R&D Program
Report
-------
THEORETICAL INVESTIGATION OF INLET CHARACTERISTICS
FOR PERSONAL AEROSOL SAMPLERS
Narayanan Rajendran
IIT Research Institute
Chicago, Illinois 60616
Contract No. 210-78-0092
U0S0 DEPARTMENT OF HEALTH AND HUMAN SERVICES
Public Health Service
Centers for Disease Control
National Institute for Occupational Safety and Health
Cincinnati, Ohio 45226
May 1981
-------
DISCLAIMER
Mention of company names or products does not
constitute endorsement by the National Institute for
Occupational Safety and Health.
NIOSH Project Officer: Jerome Smith
Principal Investigator: Narayanan Rajendran
-------
ABSTRACT
To keep the working environment as healthful as possible the permissible
exposure to dust particles has to be decreased. Lower permissible exposure
makes the representativeness of the sample more important and methods to
estimate the sampling errors are necessary. A theoretical model and a com-
puter program system has been developed to estimate such errors.
The computer program can handle various inlet geometries such as circular
tube and parallel plates. Samplers whose face is not perpendicular to the
ambient flow are simulated by a line sink. The model accounts for inertia!
and sedimentation effects on particle motion.
The results obtained by use of the computer program system agree very well
with the experimental data reported in the literature.
m
-------
CONTENTS
Abstract in
List of Figures v
List of Tables vil
Introduction I
Literature Survey . . 3
Theoretical Model 12
Method of Solution 33
Results and Discussion 50
References. 75
Appendixes
A. Derivation of Boundary Condition at Section II 78
B. Stream Function for Uniform Sink Strength 80
Distribution . 80
C. Stream Function for Triangular Sink Strength
Distribution 82
D. Computer Program Listing 84
E. Computer Program System User Manual 110
Glossary 125
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FIGURES
1.
2.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
Comparison of theoretical and experimental effects
of anisokinetic sampling
Comparison of experimental and theoretical data
taken from various authors
Flowfield in the case of sampling a stationary fluid
into a tube
Particle trajectories at U/U0 = 25, K = 0.3
Modeling strategy
Coordinate system for solving Equation (14)
Nomenclature of flow region for two parallel plates .
Line sink distribution along face plane of the sampler. . . 22
Triangular source/sink strength distribution ........ 24
Coordinate system for electrostatic force calculation ... 30
Rectangular mesh geometry ................. 34
Unequal grid size ..................... 36
Various inlet geometries .................. 37
Flow field boundary and grid point layout ......... 45
Boundary conditions for circular inlet ........... 46
C/CQ vs. Stokes 's number U/U0 =1.2 thin-walled
circular tube ...................... 51
C/Cg vs. Stokes 's number U/U0 =2.0 thin-walled
circular tube, a = 90° .................. 52
Sampling bias for thin-walled circular tube facing
the stream, velocity ratio =0.75 ............ 53
Sampling bias for thin-walled circular tube facing
the stream, velocity ratio = 0.375 ............ 54
Sampling bias for thin-walled circular tube facing
the stream, velocity ratio = 0.1875 ........... 55
Sampling bias for thin-walled circular tube facing
the stream, velocity ratio = 0.0938
0.375
0.375
0.375
0.375
0.375
Trajectory of particles, U
Trajectory of particles, U
Trajectory of particles,
Trajectory of particles, U
Trajectory of particles » U
U =
a
a
a
a =
a =
= 90°, K
= 90°, K
= 900, K
= 60°, K
= 0°, K =
= 0.01 . . .
=0.3 . . .
=3.0 . . .
= 0.01 . . .
0.01 . . .
56
57
58
59
60
61
C/C0 vs. orientation angle for U = 0.375 and lip
depth of 0.2 .................. ..... 63
C/C0 vs. Stoke 's number for a = 90°, square inlet,
ZLIP = 0 ......................... 64
C/C0 vs. Stoke 's number for a = 90°, square inlet,
ZLIP = 0 ......................... 65
-------
31. Stokes number vs. C/C0 for U = 0.375, square inlet,
ZLIP = 0 66
32. C/C0 vs. a for U = 0.375, square inlet, ZLIP = 0 67
33. Effect of bias on the distribution, a = 90°, square
inlet, ZLIP = 0 68
34. C/C0 vs. U/U0 for square inlet facing the stream 70
35. C/C0 vs. K for various thicknesses of wall with
circular inlet facing the stream, U/U0 = 0.375 71
36. Comparison of theoretical results with experimental
data of S. Badzioch (1959), circular tube facing
the stream 72
37. Parameter description for inlet geometry and
orientation 110
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TABLES
Permissible radii of tubes (cm) for sampling
aerosols in calm conditions 11
Typical values of Stokes number K, Peclet number Pe,
Froude number Fr for unit density spheres 31
Parameter description and physical signfficance 47
vfi
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INTRODUCTION
Almost all methods of aerosol characterization begin with aerosol sampling--
the capture and transport of aerosols to a characterizing instrument of some
kind. The most important aspect of sampling is the representativeness of the
sample. Representativeness exists when both the sample and the aerosol from
which it was drawn are identical with respect to concentration, particle size
distribution, and chemical composition.
As early as 1911, Brady and Touzalin1 predicted and experimentally verified
the possibility of obtaining nonrepresentative samples of participate matter
by a sampler. Since then, the experimental data of numerous investigators
have confirmed the impostance of sampling procedures. Few, however, have
made attempts to establish a theoretical basis for estimating sampling error.
To keep the working environment as safe as possible the permissible exposure
to particles is made less and less. Lower permissible exposure makes the
representativeness of the sample more important, and methods to estimate the
sampling errors are necessary.
The quality of air in an industrial environment is evaluated by use of sam-
pling methods. The recent emphasis on personal sampling has produced a trend
to wear small battery operated samplers. The usual approach is to attach a
filter in a two or three piece cassette to the workers breathing zone and
connecting the cassette to a belt mounted, battery operated pump with an
appropriate length of tubing. Personal respirable dust samplers consist of a
cyclone fitted to the sampling head that may be attached to the worker's
clothing near his breathing zone. The cyclone is designed to closely approx-
imate the AEC-ACGIH respirable dust curves. Some of the other commonly used
samplers are horizontal elutriator, open faced filter, closed face filter,
etc. The inlet geometries of the above mentioned samplers vary widely, from
a square inlet in a cyclone to a circular inlet in a closed face filter.
The goal of this research program is to develop a computer program system for
predicting the sampling errors for various inlet geometries encountered in
personal sampling.
The program involves the following tasks:
Task 1: Literature Survey
Task 2: Theoretical Simulation of Fluid Flow
Task 3: Theoretical Simulation of Particle Motion
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Fluid flow simulation required the modelling of the following:
(a) Sampling procedures and conditions for obtaining the best
results differ markedly for sampling from flowing and
stationary environments. Hence, a model for the environ-
ment from which the sample is drawn is very important for
meaningful results.
(b) The disturbance created by introducing a sampling inlet in-
to the above known environment establishes a new pattern of
fluid flow in the vicinity of the sampler inlet. The velo-
city of suction of the sample further changes the flu.id
dynamics of the sampling. Hence, a model for the fluid flow
in and around the sampling inlet has to be developed.
To estimate the performance of the sampler inlet, the effect of flow field on
particle transport has to be studied. Transport of particles depends on the
fluid flow, as well as such factors as particle size, density, and shape. So
a model that would enable us to calculate the transport of aerosols into the
probe has to be developed. This would complete the model for effectively pre-
dicting the errors that occur during the sampling.
-------
LITERATURE SURVEY
"Literature Search" was initiated upon commencement of this project with the
help of IITRI's Computer Search Center. The following data bases were
searched for pertinent works in "Inlet Characteristics of Aerosol Samplers."
(1) NTIS 1964-1978
(2) APTIC (Air Pollution Abstracts) 1966-1978
(3) Compendex (Engineering Index) 1970-1978
(4) CAC (Chemical Abstracts) 1969-1978
A total of 354 citations were obtained from the computer search. Both theo-
retical and experimental works that are directly connected to the program
were reviewed. Two substantially different types of aerosols are sampled in
practice—flowing and stationary aerosols.
(a) Sampling from flowing aerosols is encountered in the study of ducted
aerosols and atmospheric aerosols in the presence of wind. Lappel and
Shepperd2 were the first to present a theoretical approach and proposed an
equation for assessing the order of magnitude of anisokinetic errors.
DallaValle3 suggested the use of experimentally determined velocity contours
as a possible means for determining the errors.
There was a growing interest in this problem in the fifties and only the two
methods proposed by Watson1* and Badzioch5'6 appear to be practical. The
above methods allow the determination of deviations in measured concentration
of particulates. Watson" gave the following equation for estimating the
deviation.
where
f(K)
C
C
V
u =
K =
f(K)
unknown function of K and is evaluated by experimental data
measured concentration of particles (#/cc)
actual concentration of particles (#/cc)
stream velocity (cm/sec)
the mean air velocity at the sampling orifice
dimensionless inertial parameter called Stokes number
(1)
-------
Badzioch investigated the efficiency of collection of gas-borne particles by
an aspirated sampling nozzle. The efficiency was shown theoretically to de-
pend on (a) the ratio of the velocity of aspiration into the sampling nozzle
to the velocity of the undisturbed gas stream, and (b) the ratio of a length
representing the distance of disturbance upstream of the nozzle to the "range"
of a particle. The "range" is defined as the distance a particle would tra-
vel, before coming to rest, if projected into still gas with a velocity equal
to that of the gas stream. In the range of conditions investigated experi-
mentally, which included nozzles of 0.65 to 1.90 cm diameter aspirating from
turbulent gas streams, it is found that the length representing the upstream
disturbance is a function of the diameter of the nozzle.
He gave the following expression for estimation of sampling error:
^-=a f + (1-a) (2)
where
a = [1 - exp (-DA)]/(D/A)
X = KL = range of a particle
D = assumed distance from the plane of nozzle exit at which the
streamlines start either diverging or converging and is a
function of L. The value of D was deduced from experiments.
Both of the above models are semi-empirical, using experimentally derived
parameters to bridge gaps in theoretical deductions.
Levin was the first to consider the inertia! lag of particles with the par-
ticles flowing into a point sink in the field of uniform wind. His theore-
tical results are valid if
j- < 64K (3)
o
is fulfilled, where U is the suction velocity of the sample (cm/sec), U is
free stream velocity, and K is the Stokes number (U x/L). According to
Levin, if particle sedimentation is ignored, the aspiration coefficient due
to inertia A. = C/C , where C is the measured concentration (#/cm3) and CQ
is the actual concentration (#/cm3), is determined as follows:
IL Un
Ai = 1 - 3.2 y K + 0.44 ~K2 (4)
Where ¥/UQ values from 4.0 to 8.0 the experimental results were in good
agreement with Equation 4.
In the late fifties and early sixties there were a number of experimental
studies on the effect of probe shape on the sampling accuracy and directional
dependence of samplers8'9.
-------
Vitols10'11 calculated, by means of_ a computer, the flow field for an aspir-
ated nozzle for various values of U/U ignoring the turbulence and viscosity
of the fluid. Particle trajectories were computed for different values of
the Stokes number, K. Gravitational force on the particles was not consid-
ered. The graphs A. vs. U/U were obtained using the Stokes number as the
parameter. A typical curve is shown in Figure 1. The lowest value of K used
was large. For large values of K, the particle trajectories are nearly _
straight. Their motion is nearly independent of the flow lines and
Their motion is nearly
because the efficiency of capture
These theoretical error estimates
Badzioch which have a very large
Davies,12'13 when
be estimated from
this condition is
the
the
not met,
wind velocity U 5
sampling criteria
of the particles is 100% for large ft valSes.
compared well with the experimental data of
scatter in values of A... According to
U/5, the aspiration coefficient can
for stationary systems. Whenever
using Levin's solution for A.
sink that draws particles from
sampling tubes of small radii.
of the tube is:
the aspiration coefficient can be calculated by
Because Levin's solution is based on a point
a,uniform stream, it is applicable only to
According to Davies, the maximum diamter, D,
where F is the rate of suction of the sampler.
For larger sampling tubes Davies gave an empirical relation based on the
perimental data and A. is determined as follows:
(5)
ex-
- U
f(K)
(6)
where K is the particle Stokes number and
f(K) = 1 - T-l-r
(7)
Belyave and Levin11* showed that the observed aspiration coefficient A could
not be identified with purely inertia! aspiration A. that had been the main
subject of the earlier studies (1-13). Taking into account the rebound of
the particle and its deposition in the sampling tube, A will be a function
of A-, A, (the coefficient characterizing the particle concentration decrease
in a sample caused by deposition in the inlet channel) and A (the coeffi-
cient which depends on the particle rebound from the front eage of the sam-
pling nozzle and their subsequent aspiration into it).
All three coefficients, A., A,, and A , strongly depend on nozzle shape.
Since the character of this dependence for A and A. has yet to be estab-
lished, A and A,, were assumed Ib be united and gave an empirical relation
for A. as follow
«:
Ai =
1 +
" 1
lU
(8)
-------
IQ
C
fD
V> O
o> o
3 3
-O TJ
-*-• a
-j. -s
3 _j.
,8
ro
-s
CT>
3 -i.
O O
CD OJ
3
O
oo
•a
3-
o
en
Concentration sampled C
True concenfrotion Co
— i 01
O 3
i — i a.
3
n>
rt
cu
ro
a>
o
CO
O
Qi
3
CO
O
O
O
-5
O"
o.
(D
01
S ii.
Conceniration sampled C
True concentration Co
Isokinetic Sampling
o I
m •*
If
» 0
^T* *" U ^ _
1 -" ° " 2
1 r-l ^> O S
3
fD
O
-------
where 3 is a non-dimensional function, $ ->• 0 as K ->• 0, and $ •*• 1 as K •> «>.
All the previous works concluded that 3 is a function of K only. But in
Figure 2, which gives theoretical curves as well as experimental points, 3
shows a dependence on U/U and is given as follows:
1
1 + BK
and B, a non-dimensional function, is given by:
B = 2 + 0.617
JU_
II
(9)
(10)
(b) Sampling from calm air or stationary aerosols has been investigated much
less than sampling from flowing aerosols. Figures 3 and 4 demonstrate that
the character of the flow fields_at the entrance to a tube from a stationary
medium and from a stream 0 < U/U < 1 is quite different15.
Levin16 considered the sampling from stationary aerosols as a flow into a
point sink. He developed a relation for A., the inertia! aspiration coeffi-
cient, which is:
Ai + 1 - 0.8K + 0.08K2
(11)
where K = t(4 V 3/F)"2 is a parameter acting as a Stokes number, F is the flow
rate into the sink, and Vs is the sedimentation velocity of the particle.
In his well known theory of sampling, Davies
12 ,13 »15,?3,3 3
used the stopping dis-
tance of a particle to characterize inertia! effects and terminal settling
velocity to characterize sedimentation of particles. For inertia to be neg-
ligible, the stop distance should be small compared to the radius of the tube
D/2. For sedimentation to be negligible, the flow velocity in the probe
should be much larger than the terminal settling velocity. Then the complete
condition to obtain a true sample is:
1/3
«£«
(Trg-rj
(12)
where F is the rate of suction of the sample (cmVsec), T is the relaxation
time of particles (sec), and g is acceleration due to gravity (cm/sec2).
Using an arbitrary criteria of 1/5 he arrives at the following condition:
sffi|1/3 fr~
5 M
D
1*9*0
(13)
Table 1 shows the permissible tube sizes for sampling aerosols as a function
of suction rate and particle size. When the tube radius is greater than the
lower figure, sampling errors due to inertia are not significant. When the
radius is less than the upper limit, sedimentation is not significant. The
two criteria can be satisfied simultaneously for the unbracketed entries in
the table but not for bracketed entries. Satisfactory samples can be ob-
tained by using the tube size satisfying the lower limit condition and
orienting the sampler vertically to eliminate sedimentation.
-------
Agarwal36 studied the problem of aerosol sampling under calm air conditions by
solving the Navier-Stokes equations and equation of variable motion. The study
was continued to circular inlets. The sampling efficiency of an inlet was
found to depend upon two dimensional parameters, the Stoken Number, K, and the
relative velocity, Vs1. Using an arbitary criteria of 90x efficiency, he ar-
rives at the following condition:
2.K.VS1 £ 0.1
This criterion is less restrictive than the condition given by equation (J3)
and provides adequate accuracy.
Ter Kuile37 made the distinction between "representative sampling" and "com-
parable sampling". This results in two criteria.
1. The criterion for representative sampling which combines a modification of
DAVIES1 theory for representative sampling and LEVIN's sampling theory into
one criterion which is limited by three physical effects:
- impaction on the wall of the inlet;
- sedimentation on the wall of the inlet;
- dynamic escape from the sampling region.
2. The criterion for comparable sampling requires that the inlet is to point
vertically downwards with a filter near the inlet, resulting in a higher ef-
ficiency, sharper cut-off limits, better reproducibility and better compar-
ability.
Both these criteria were given in graphs in which the sampling efficiency is
a function of a dimensionless particle size number (k-n), and a tube size num-
ber ("inn). Theoretical limits of the region where the^efficiency is more than
90% forVepresentative sampling and more than 94% for comparable sampling are
plotted in these graphs. The tube size number only depends on parameters of
the sampling device, so that the design of this device determines the nature
of the physical limitation. As a result of this, sampling devices can be di-
vided into three classes for which different physical mechanisms limit the
aspiration efficiency of large particles.
8
-------
+ -Bodzioch (1959)
* -Watson (1954 )
o —Hams ond Hemeon(1954)
» -Sehmel ( 1967)
• -Zenker ( 1971)
— VOIDS nchu ok and Levin (1969!
Ruping (1968)
u/u.,- I
O-c1 0'«
Figure 2. Comparison of experimental and theoretical data taken from papers
of various authors (from reference [14]).
-------
Figure 3. Flow field in the case of sampling a stationary fluid into a tube
(from reference [17]).
10
-------
Figure 4. Particle trajectories at U/U0 = 25, K = 0.3 (from reference [17]),
11
-------
Fuchs presented a review of methods of sampling and methods to estimate the
bias. During this literature search the author came across a number of exper-
imental studies on the effect of probe shape, orientation and velocity of sam-
pling on entry efficiencies.
Glauberman9 studied the directional dependence of air samplers using uranium
oxide dusts. Two filtration type air samplers were tested. In turbulent air
no bias was found due to orientation. In a directional air stream, a sampler
head facing into the air stream collected more dust by a factor of two com-
pared to a sampler facing up or down. Schmel19 investigated particle sam-
pling errors from several sources. The sampling errors were significant for
particles as small as 1 micron in some cases. Pickett and Sansone22 studied
the effect of varying inlet geometry on collection characteristics of a
10-millimeter Nylon Cyclone. Samples of coal dust aerosol were collected
simultaneously with two filter holders: one designed to conform to Davies1
criteria, and one with smaller inlet dimensions corresponding to those of the
10 mm Nylon Cyclone. No differences in mass concentration or size distribu-
tion were obtained.
Breslin and Stein26 considered sampling inlets in calm air. Their results
showed that Davies1 criteria for inlet conditions for correct sampling are
overly restrictive.
Raynor30 studied the effect on the entrance efficiency of a filter holder
caused by variations in the following parameters—air speed, flow rate, angle
between the air flow and the filter holder, and particle size. Efficiencies
for various particles were determined over a range of angles from 60-120
degrees from horizontal for wind speeds of 100, 200, 400 and 700 cm/sec and
filter flow rates of 6.4, 12.7 and 25.4 liters/min. The entrance efficiency
varied with all parameters from less than 1% at highest wind speed and
lowest flow rate to over 100% at forward angles. Efficiency was lowest with
filter holder entrance at right angles to the air stream.
Table 1. Permissible radii of tubes (cm) for sampling aerosols
in calm conditions*
Particle
diameter, n
1
'2
5
10
20
50
100
MOO
r>oo
i
0.033 -
' 0.051 -
0 093 -'
0.15 -
(0.23 ~
(0/12 ~
(0 C3 ~
(0.80 •-
(i 21; ~
1.9
1.0
(Ml
0.21
0.10)
0.042)
0.02:1)
O.OM)
0 OOK)
10
0.071 -
0.11 -
0.20 -
0.31 -
(0.50 ~
(0.00 ~
(1.4 ~
(1.9 -
(2.7 ~
Rate of suction, F (crn'/scc)
100 1,000
G. 0
3.2
1.3
O.G5
0.33)
0.13)
0.071)
0.0.17)
0.025)
0.15 -19
0.23-10
0.4.3- 4.1
O.f.8- 2.1
(1.1 ~ 1.0)
(1.9 ~ 0.42)
(2.9 ~ 0.23)
(4.1 ~ 0.14)
(5.8 ~ O.OK)
0.33
0.51
0.93
1.5
2.3
(4.2
(G.3
(8.9
(12. G
-GO
-32
-13
- G.5
- 3.1
~ 1.33)
~ 0.71)
~ 0.37)
~ 0.25)
10
0.71
1.1
#.o
3.1
5.0
(9.0
(14.0
(19
(27
,000
-190
-100
- 41
- 21
- 10.3
~ 4.2)
~ 2.3)
~ 1 4)
~ 0.80)
100,000
1.
2.
4.
C.
11.
(19
(29
Ml
(58
5 -GOO
3-320
3-130
8- 05
0- 31
~ 13.3)
~ 7.1)
~ 3.7)
- 2.5)
* From Reference [12]
12
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THEORETICAL MODEL
An overall theoretical model to simulate a personal sampler requires modeling
of the following:
• Model for environmental flow patterns
• Model for flow pattern in-and around the sampler
• Model for motion of particles
The strategy used to obtain the overall model for personal sampling is given
in the form of a block diagram in Figure 5,
ENVIRONMENTAL FLOW PATTERNS: MODEL FOR INDUSTRIAL ENVIRONMENT
Industrial environmental flow patterns are so complex and individual that any
one particular model cannot describe the actual field for all types of work-
ing environments. Hence, it is necessary to make some simplifying assumptions
so that a model can be developed. Different types of flow patterns exist in
a variety of working environments.
(1) An environment essentially calm except for microscopic
fluctuations. This type of pattern can be found in
research laboratories, nuclear reactor plants, etc.
(2) A steady flow exists in a particular direction even
though the direction cannot be fixed. The effect of
varying directions can be studied by using it as a
parameter. This kind of flow pattern will exist in
industries which need high ventilation.
(3) In some working environments, the flow volume is so
large that the medium is essentially turbulent in
nature. In this type of environment the flow does not
realize the existence of an object (sampling head, for
instance) and barges into it. This kind of flow would
be predominant in environments which need very high
ventilation, such as mines.
Hence, the model uses the calm air type of sampling for environments of type
1 and a steady flow type for industrial environments of type 2. Type 3
environments could be realized by increasing the flow velocity of the uniform
flow of type 2 environments.
13
-------
Inlet
Geometry
Inviscid
Flow
Particle
Equations
Particle
Characteristics
Efficiency
of Capture
y
Efficiency
of Sampling
Figure 5. Modeling Strategy
-------
FLOW RATE AND INLET GEOMETRY: MODEL FOR FLOW PATTERN IN AND AROUND THE PROBE
The model for flow pattern in and around the probe is assumed to be given by
the inviscid flow pattern. In solving the inviscid problem above, it was
assumed the fluid was ideal, without any viscosity as it glides over the walls
of the boundary. In reality all fluids have viscosity, hence friction. So
the fluid in contact with the wall will be subjected to a no slip condition,
that is, the velocity at the wall is zero. This viscous effect extends only
to a very small distance from the wall and was not taken into account in this
study.
Sampler Facing the Stream
Equations in this section are formulated in the general form so that they can
be applied to any given geometry. Ideal fluid flow patterns and velocity
distributions can be determined by solving the Laplace equation for the stream
function, if;.16
V2 ty = 0
where V2 is the Laplace operator.
pv2 ^2
V2 = 7T-Z- + TT-Z- for two dimensions
3x 3y
where V2 is the second order central difference operator. The shape of the
body determines the boundary conditions on the stream function. General con-
ditions will be:
(1) 4> = constant at the body surface
(2) ijj = 0 at the axis of symmetry (15)
(3) 4* = ijj of the environmental flow pattern upstream and
downstream from the body
The velocity of flow can be calculated from fy as:
(i) for Cartesian coordinates (x,y,z) (see Figure 6)
U = li
x 8y
M (16)
M = ££
y 3x
(ii) for cylindrical coordinates (r,z) (see Figure 6)
ij - Ili
r " " r 3y
u .ia (17)
z r 3r
Hence, once if» is calculated from equation (14) with the use of conditions (15)
then the velocity distribution can be found from either (16) or (17).
15
-------
Figure 6. Coordinate system for solving Equation (14),
16
-------
The stream function i|» does not exist for three dimensional flows. That is,
there is no function fy such thajt a isoline is a stream line. But for a _
solenoidal vector field (where U satis fies_the 3^0 continuity equation, V • U
= 0), a so-called vector potential i|» = *x i + $ j + i|>2k does exist, such that
the velocity components are given by:
"H =
U
_ _ i?.
_
x 3y
-
w
= y _ _ i.
9x ay
and ty satisfies
V2 ^ = 0
Equation (18), when written in terms of the vector components, constitutes a
set of three equations to be solved for $x, 4> , and ^. They are:
V2 Tj). =0
A
V2 4, = 0 (20)
Equations (20) are similar to equation (14). But the boundary conditions are
not as simple as (15) and body shape needs to be specified to formulate them.
Due to the complexity involved in the boundary conditions and solution pro-
cedures the flow model covers only the two dimensional and axisymmetrical
cases.
The two-dimensional equation (14) for fy is useful where the sampling inlet is
either axisymmetrical or when the ratio of height to width of the inlet is
large enough to discard one dimension. Even if these conditions are not sat-
isfied for a particular inlet, equation (14) will still approximate the flow
in the core region of the sampler inlet.
Formulation of theProblem for Specific Inlet Geometries
Parallel Plate Inlet--
This type of inlet is often encountered in the horizontal elutriator.
Figure 7 shows the nomenclature for the fluid flow region. As can be seen
from the figure, some transverse distance Y0 from the center!ine of the
probe defined where the fluid stream line maintains a flow unperturbed by
the sampler. This assumption is necessary because a numerical solution for
the problem can be obtained only for a finite number of mesh points.*
The required boundary conditions now have to be specified at the centerline,
the probe surface, the constant ordinate YQ and section I and II as shown in
Figure 7.
17
-------
CO
"I
->•
Section I
W\
z v y v./ s. / \ /
>7^
A
~>,
Section II
Figure 7. Nomenclature for flow region--
two dimensional case.
-------
When Section I and II are sufficiently far up and downstream from the inlet
and the disturbed flow region, the velocities at these stations can be con-
sidered uniform and axial only. The axial component of velocity U and trans-
verse component U are related to the stream as follows:
Uz - f (21)
Uy •-£ (22)
The expression for ty at sections I and II can be obtained by integrating
equation (21):
* = Uy + Ci (23)
Where U is the appropriate velocity of the fluid and Ci is an orbitrary com-
tant of integration for which a value of zero can be assigned.
Then on the centerline of the probe (upstream and inside the probe) the value
of the stream function becomes:
ijj {_= ty at centerline = 0 (24)
By virtue of equation (23), the values for \i> at other boundaries follow:
fy at Section I =
ijj at y0 =
-------
= y*
= U*y*
Equation (14) together with the boundary conditions (29) pose a well defined
problem.
Circular Tube--
Circular inlets are the most commonly used geometry in aerosol sampling.
Although circular tubes are very rarely found in the personal samplers for
aerosols, the inlets to the closed face filters can be approximated by the
circular tube geometry. Open faced filters can be regarded as a wide and
very short circular tubes terminating in a filter.
Figure 8 gives the flow field boundary description and nomenclature. A cir-
cular tube of radius R, and wall .thickness W, is considered to be located at
Z = 0. the axial and radial velocity components, Uz and U are given as
follows:
H = 1 M
z " i i:
II - jL ££
ur " Y 3z
Following similar procedure used for parallel plates,
$ = h U r2
r2
(UR2 - UR2)
n
(R -
= ip at center! ine = 0
~ (R+W)2)
Nondimensionalizing velocities by h UT and distances by R boundary conditions
become :
= r*2 0 < r* <_ RQ*
ifijp* = U* r*2 0 i r* ^ 1 (32)
(R *2 .
(1+W*) < r < R *
= 0 - - o
20
-------
A
_yi\ '\- . '\• .." -.
xv V >% X > 1 > T X ' X ! 7 '• v'~
Xv I / . \! -- \ ' ' -' /\ I/V /\'-' ^-iv—
n/^\ / '^ ~ \;/--\'/ yy \< \j \
R
A
w
Figure 8. Nomenclature for flow region for circular inlet.
-------
Inviscid flow equations together with Equation (31) pose a well-deftned prpblem.
Sampler with General Orientation to the Stream--
In the two models discussed above, the stream function fy existed because the
problem was reduced to two dimensions. With the parallel plate sampler, the
width of the plates along the x-axis is assumed to be so great that the flow
field is unchanged. For the circular tube, the angular dimension can also be
factored out of the problem when the sampler's axis is parallel to the initial
flow. When the circular sampler face is oriented at an angle a f 90° towards
the oncoming stream, the axial symmetry is not preserved. In the three-
dimensional case, the stream function fy does not exist and Equation (19) is
not valid.
Even for the case of parallel plates, solution to Equation (14) for an arbi-
trary orientation requires a very large flow region to be solved. Hence other
approximate methods to simulate the flow are necessary. One such method is to
superpose simple potential flows to simulate the flow in question. In this
study, superposition of uniform flow over a line sink was used to simulate the
flow around a sampling head. In the past a number of investigators7'12'13'15'16
have used a point sink and a uniform flow to simulate the flow around the sam-
pling head. But the point sink is isotropic and the sampling head orientation
cannot be incorporated. This difficulty is overcome by the use of a line sink.
Uniform Strength Sink Distribution—
Let a line sink of length d be distributed along the face plane OA of the
sampler head. Let a be the angular orientation of the sampler head with the
oncoming streamj and m the sink strength per unit length. Then the stream
function ^ at any point P (Figure 9) can be divided into two parts, ^ . . and
"'uniform stream" The f°11owi"9 equation provides ^1nk and ^.
— r 2 2
., = -m tan"1^ • x - tan"1//.v • (x-d) + y g.n /x + ^
x Tx^dT /(x-d)2+y2J
(33)
cosa + Ujx sina (34)
Ux = velocity in the x direction
/ y o
UT cosa - m En /x *^ — (35)
/(x-d)2+y2
OP
Uj cosa - m Un
22
-------
Figure 9. Line sink distribution along face plane of the sampler.
23
-------
Uy = velocity in y direction
3x
= 1)i sina + m tan'1^- - tan"1-^- (36)
L x x~a_j
= Uj sina + m (6j-92)
If U is the anisokinetic velocity ratio and f the ratio of -*TT, then through
mass balance "
(37)
If U is greater than sina then m is positive and denotes a sink. If U is
smaller than sina then m is negative and denotes a source. The detailed
derivation of Equations (33), (38) and (36) is given in Appendix 8.
Triangular Strength Distribution--
A uniform sink/source strength distribution along the face plane of the sam-
pler does not take into accout the effect of the sampler wall. The effect of
sampler wall propagates towards the center of the sampler inlet and varies
with the distance from the centerline, In essence, the flow through the cen-
ter of the sampler is more than the flow closer to the walls. In order to
approximate this effect of retarded flow near the walls, the sink/source
strength distribution was made triangular (Figure 10). Using an approach
similar to the case of uniform strength, the stream function can be written
as the sum of two stream functions, \l> sink/source and fy uniform stream.
Then fv"om Figure 9
d
s = / fflr * 6 • d? (38)
0 ^
Where
and e = cot ~a (x-
(40)
24
-------
m r =m
0 5 C £ d/2
d/2 < ? < d
Figure 10. Triangular source/sink strength distribution.
25
-------
The details of integration of equation (38) with equation (39) and (40) are
given in Appendix C. The stream function due to source/sink YS is given by
VS = Cot -1 x/y [rox2/2-ray2/2]
+ Cot -1 x-d/2 [my2-m(x-d/2)23
y
-i- Cot -1 x-drm(d-x)2-my2"[
Jtny2+x2
+ (x-d/2)
+ my (d-x)/2
* (x-d/2)2")
+ (x-d)2 J
(41)
The velocities U and U in x and y direction due to the sink are given as
follows. * y
ux = ay
= mx/a x2-y2 - my cot-1 x/y
y Z 1 « t'JL
+ m (x-d/2) y2- (x-d/2)2 + 2my cof1 x-d/2
yz+ }x=d/2)2 y
+ m (x-d) (d-x)2-y2 - my cof1 x-d
2 (d-x)2+y2 y
+ m(d-x) infy2-!- (x-d/2)21+ my2 (d-x)
Ly2+ (x-d)2 J
(x-d/2)2
i+ (x-d)2J
and Uy = -
(42)
2 - y2 - mx cot "a x/y
2 + y^
- (x-d/2)2 + 2m (x-d/2) cot -1 x-d/2
y
(d-x) cot -1 x-d
(x-d/2)2
(d-x)2 + y2
26
-------
+ my/9 In
- myx
y2
- my (d-x)
2_ —
v "f" X
y + (x-d/2)2
+ my/,, li
x - (x-d/2) 1
+ x2 y2 + (x-d/2)2
fx-d/2)
_y2 + (x-d/21
ify2 + (x-d/2)2
y2 + (x-d)2
- (x-d) 1
I2 y2 +
(x-d)2_
(43)
The velocities in the flow field with the uniform stream will be
Ux - Ux + Uo cosa
Uy = Uy + Uo sina
(44)
The value of m in equations (41), (42) and (43) can be obtained by mass balance
as per Appendix C,
m = - 2(U-sina) Arrf2 (45)
where
U is the anisokinetic velocity ratio.
a the angular orientation of the sampler head.
f the fraction of the diameter over which the
sink/source is assumed to be distributed.
The equation (41) - (45) define the flow field completely. This simulation of
flow by superposition accounts for the angular orientation and the anisokinetic
sampling. The effect of inlet geometry js not taken into account.
27
-------
PARTICLE MOTION
AEROSOL MECHANICS OF RESPIRABLE PARTICLES
The general theory of the dynamics of spherical bodies suspended in a fluid as
a continuum restricts analysis to the condition where the Knudsen number of
the tody, Kn, approaches zero. In practice, these results can be applied to
particle behavior under conditions of Kn » 1, then the continuum theory can-
not be applied and free molecular theory takes over. For intermediate
Knudsen number particle dynamics, the continuum theory needs to be supplemen-
ted with a slip correction factor. All the above theories have been well
documented in the literature.13'11*
The various forces acting on a particle are as follows:
• inertia!
• gravitational
• diffusional
• electrostatic
Inertia!
In the course of movement, particles of aerosol may acquire motion relative to
the suspending gas, due to their inability to conform to the fluid flow in-
stantaneously. There is a certain amount of time lag in which the particle is
not affected by local velocity changes in the flow. This is characterized by
•T, the relaxation time of the particles. T is defined as the ratio of par-
ticle terminal settling velocity to the acceleration due to gravity. In the
same manner, if an aerosol at rest is accelerated to a velocity V/e in time T.
Hence, in a velocity gradient field, the particle overshoots the fluid when
decelerating or it undershoots it when accelerating.
Stokes number, K (the ratio of the stop distance of the particle to the char-
acteristic length of the system), is used to indicate the importance of
inertia! effects for a given set of conditions. The smaller the value of K,
the more negligible the inertia:
U T pd2U
v - ° - _
* " L " 18nL
0 (45)
28
-------
where U = the free stream velocity
T = the relaxation time of particles
L =- characteristic length (cm)
p = the density of the aerosol particle (gm/cc)
d = diameter of the particle (cm), and
n = the viscosity of the medium (gm cm/sec).
Gravitational
Gravitational force on particles is given by mg, where m is the mass of the
particles and g is the acceleration due to gravity. The terminal settling
velocity, Vs, is given by
Vs » zg (47)
where
T = the relaxation time of particles, and
g = acceleration due to gravity (cm/sec).
The importance of gravitational effect is indicated by the Froude number (Fr):
Vc VCT
Fr s = s {48)
Lg L
Diffusional
' *
Diffusion is the most dominant force on small particles (d<-2 ym). Particles
not under the influence of external forces diffuse in a random fashion called
Brownian motion. Diffusion also occurs because of velocity gradients, con-
centration gradients, and thermal gradients.
The characteristic numbers are the Schmidt number, Sc, and the Peclet number,
Pe.
Sc = I (49)
where
Y = kinematic viscosity (cm /sec), and
c - diffusivity of particles (cm /sec).
The Schmidt number indicates the ratio of momentum transfer to mass transfer.
Higher Sc values mean Brownian Diffusion is not as important as convective
diffusion. The combined effects of diffusion and fluid motion (convection)
on particle transport can be expressed as a function of the Peclet number, Pe:
Z9
-------
Pe = -2- (50)
where
V = the initial inviscid velocity (cm/sec), and
D = the diameter of the sampler inlet (cm).
In this study diffusion was not included.
Electrostatic Force
The electrostatic force considered in the model is only the coulombic force
between point charges of magnitude Qp and Qc located at the center of the
particle and the collector respectively. The coulombic force between a
charged particle and sampler head is given by
'»'
where
Qc = charge on the collector/unit length
Qp = charge on the particle
e0 = dielectric constant of air, and
R = distance between the center of particle
to the surface of collector.
If the linear dimension of the collector surface is L, the electrostatic force
can be obtained by integrating equation (51) over the length of the collector.
Using the coordinate system shown in Figure 11, the total force on the parti-
cle can be written as follows
L
F = J Fcdx (52)
o
Fx = x component of the force
K
and
(Sin 0! - Sin 62) (53)
Fy = y component of the force
= ~ (Cos 92 - Cos 9j)
where
K = Qc Qp
30
-------
JL Collector surface
Figure 11. Coordinate system for electrostatic force calculation.
Equation (51) is valid as long as the particle and the collector are not very
close. For closer distances, the equations (52) and (53) become infinite and
image forces have to be taken into account. This study does not take into
account tfve image forces.
Table 2 provides values of non-dimensional parameters such as Stokes number K,
Peclect number Pe, and Froude number Fr for various particles.
31
-------
Table 2. Typical values of Stokes number K, Peclet number Pe, Froude number Fr
for unit density spheres.*
^"""^How Velocity
Parti cle^--\( cm/sec)
Radius ^"\^
(M) \.
0.25
0.50
1.00
2.00
5.00
10.00
10
K
1.02(-05)
3.6 (-05)
1.3 (-04)
5.1 (-04)
3.1 (-03)
1.24(-02)
Pe
1.6 (07)
3.65(07)
7.85(07)
1.63(08)
4.17(08)
8.43(08)
50
K
5.1 (-05)
1.8 (-04)
6.5 (-04)
2.55(-03)
1.55(-02)
6.2 (-02)
Pe
8.0 (07)
1.83(08)
3.93(08)
8.15(08)
2.09(09)
4.22(09)
100
K
1.02(-04)
3.6 (-04)
1.3 (-03)
5.1 (-03)
3.1 (-02)
1.24(-01)
Pe
1.6 (08)
3.65(08)
7.85(08)
1.63(09)
4.17(09)
8.43(09)
Fr**
1.03(-09)
1.25(-08)
1.73(-07)
2.56(-06)
9.55(-05)
1.50(-03)
* Characteristic dimension 'L' of the sampler is assumed to be 1 cm.
** Froude number 'Fr1 is independent of flow velocities.
-------
EQUATION OF MOTION OF PARTICLES
When an aerosol particle travels in a moving medium the particle generally
tends to lag behind the flow of fluid. Assuming that the Stokes relation for
the particle drag may be used, then within the continuum approximation of the
fluid the equation of motion of the particle may be written as:1"
.3 dv
d p dt =
/TT —\ , IT j3 dU
(U-v) +7rdpnTr
3 2
+ "2 d Ap n
TT .3
12 d f
dt1
dU
dt1 <
t - t1
dv '
i?
55
dU
Ht
d\T)
" dtj
(54)
The first term on the left in equation (54) is the resultant force acting on
the particle. The first term on the right is the viscous resistance given by
the Stokes law. The second term is due to the pressure gradient in the gas
surrounding the particle, caused by the acceleration of the particle. The
third term denotes the force required to accelerate the apparent mass of the
particle relative to the ambient gas. The fourth term, known as the Basset
term, accounts for the deviation from the steady state in the gas flow pattern,
The last term is the force resulting from external potential. In general, the
second, third ,and fourth terms will be negligible, so equation (54) in simpli-
fied form will be
TT ,3 dv
6 d p dt
= 3irnd (U - v) + F
or
d_v
dt
U-v
em
(55)
(56)
and
where Fem is the external force per unit mass. Divide velocities by Uj an
time by L/Uj to obtain nondimensional form of equation (56). Denoting the
nondimensional quantities by stars, the equation of motion becomes:
where
d_V * U* - V* . F *
dt* K hem
K = Stokes number of particle =
(57)
em
em
(58)
33
-------
METHODS OF SOLUTION
In order to estimate the sampling error, it is necessary to solve the equation
of particle motion (equation 48) together with the fluid flow equations.
Analytical solutions to partial differential equations such as equations (57)
or (14) can be obtained for only very simple boundary configurations. Approx-
imate solutions, however, can be obtained by numerical methods by solving the
finite differenced equations of the governing differential equation,
FLUID FLOW
As indicated in the previous section, the model for the fluid flow has two
options. The first option is to solve the flow equation (19) with proper
boundary conditions to obtain the numerical solution to the exact problem.
The second option is to use the superposition of simple flows to approximate
the actual flow conditions. The first option requires numerical solution of
equation (14) and can be used only when the sampling head is facing the on-
coming stream. For general orientation of the sampling head, the second
option is used and requires numerical evaluation of the flow velocities given
by equations (41)-(45) for use in determining the trajectory of particles.
Sampler Facing the Stream
Thin-walled Plates--
Let the flow field be divided into a grid as shown in Figure 12. Then,
Laplace's equation (14) can be written in finite difference form and solved
by iteration.
Each iteration of the finite difference equation is analogous to solving the
time-dependent version of equation (14):
|i = ^ (59)
9t
We are not interested in the physical significance of this transient solution,
but a step in time At in the time-dependent ip is a convenient representation
for an iteration of the time-independent function. As the solution to
equation (59) approaches steady state, we have also converged to the desired
solution of Laplace's equation (14).
Now, we write equation (59) in discrete form for point I, 0 using FTCS
(Forward Time Centered Space) differencing
i K ' 1 i K *5 «
" <- c-^ .
+ SJL (60)
At Ax2 Ay
34
-------
-4V-
Ay
•O
1 .
-O- 0>
. J+l
J
- J-l
1-1
1+1
M,N
1,1
where
Figure 12. Rectangular mesh geometry.
= the stream junction fy at (I,J) at the k time step.
- 2
AZ
Az'
- 2
Ay
Ay'
For the difference equation (60) to be stable the condition is:
M.l
(61)
(62)
Since we wish to approach the solution as rapidly as possible, we take the
largest possible At from equation (62).
Defining the mesh aspect 8 = Az/Ay, then:
35
-------
2(1+3)
(63)
The convergence of equation (63) can be made faster by multiplying the
bracketed terms by a factor w such that l, ,.
An initial lower order solution has to be provided .either .by a previous'
problem or by arbitrary assumptions for ty. When 4^i j-*-r'i j, then the
solution is reached. This can be programmed in a digital computer and the
iteration will be stopped when
(error limit) (65)
The velocities are calculated as follows.
Uz(I,J) =
(66)
Uy(I,J) = -[^(J+1,J) - 4-(J-l,J)]/2.Az
Thin-walled Tube —
The time dependent problem for circular tube is given by
_
8t 3z2 r
Writing equation (67) in discreetized form for point I,J using FTCS differ-
encing
, k+1 . k . . .
^T i " ^T i ?2 , k j2 , k -, f , k
yIJ rIJ _ 5*\1) , 6 'J; 1 6tL /co%
. At = Az?- 4 Ar2 ' R Ar (68)
where
r ,
,
= stream function at (I,J) at kth time step
36
-------
R = Radial Coordinate at (I,J)
Ar
Ar2
Ar
2-Ar
The convergence of equation (68) is made faster by multiplying the right hand
side of the equation by a factor w such that Uw<2. The flow velocities are
calculated as follows
UZ(I,J) =
1
2«Ar
(69)
2.Az
Thick-Walled .Tube/Plate
Depending on the shape and thickness of the sampler wall the grid size in any
one direction may not be equal throughout the flow region (Figure 13).
J+l
J
J-l
Ar2
An
Az2
I 1+1
J-l
Figure 13. Unequal grid size.
37
-------
Writing the governing equation (67) in discreetized form for unequal grid size
and grouping the terms gives:
'I.J
1 fAri-Aral . 2 \
R [An-ArzJ ArvAr2/
\(AZi+Az2).Azj
Ar2
R (Ar!+Ar2)'Ari
; /- I . - Ari
'I.J+1 t R
(Ari+Ar2)
•ArJ
for circular tube,
2 2
Ari'Ar2
"
(Az1-fAz2).Az2
J
for plates
(70)
(71)
The stream function at the boundary points are given by either equation (29)
or by equation (32).
The various wall shapes that can be realized by use of wall thickness 'w1
and chamfer angle 'a' are shown in Figure 14. These shapes can be obtained
for both two-dimensional and axisymmetrical cases.
(c)
(a)
(b)
w = 0«
o, = 0-
w = finite
a = finite
,
'^ .'.-*••'"
>' "•• < s'-. '
'>' -V S : -;-•
w = finite
= 90°
Figure 14. Various inlet geometries
38
-------
Sampler with General Orientation
When the sampler is oriented at an angle 'a1 to the on-coming stream, the
approximate solution developed earlier is used. Equations (41)-(45) will be
evaluated for use in particle trajectory calculation.
PARTICLE MOTION
The equation of motion of particles as given by equation (48) is solved by
calculating the trajectories of particles. The method that is used to do
this is the predictor-corrector method together with an iterative convergence.
Even though the predictor-corrector method is a standard one and can be found
in any numerical methods book, we will give a short description of the method
here.
Prediction
If the particle is at position x=xo at time t=0, then at t=t+At, the particle
is predicted to be at
x = x + At
P o
x = x.
(72)
o
Correction
The above equation assumes that the velocity, v, of the particle remains
constant within the step. But actually it is a changing function:
x = x + At • v
- - (73)
X = X + X '
_o p_
2
Equation (73) is iterated until the corrected value x converges.
The above procedure is continued until the particle either touches the wall
of the sampling probe, in which case it is captured and lost from the sample,
or enters the probe inlet, in which case it could either be transported to
the sensing instrument or be lost by deposition to the walls. A limiting
trajectory which will just graze the inside of the probe wall can then be
calculated.
In order to calculate the velocity v for equation (73), the simplified equa-
tion for particle motion (48) is used. Equation (48) is the result of the
various forces influencing the particle. In this study, the particle motion
is determined by inertia! and gravitational forces. Electrostatic and dif-
fusion forces are neglected.
39
-------
Assumptions
The following assumptions are used in obtaining a solution to the particle
motion equation.
• The particles are uniformly distributed and, at a large
distance upstream from the probe inlet, move with the same
velocity as the free stream fluid.
• The particles are spherical and do not change in size due
to agglomeration, evaporation or condensation.
• The particles are considered to be sufficiently small in
comparison with probe size and they move as individual
particles with no hydro-dynamic interactions among themselves
or between themselves and the probe walls.
Solution Procedure
At a large distance upstream (5 radii) from the probe inlet the fluid flow is
uniform and the particles move with the stream. As the flow approaches the
sampling probe the disturbance due to the presence of the probe is felt and
the particles due to their inertia are not able to follow the fluid flow.
The equation (48) can be written in terms of the velocity components as
dVx Ux-VW-x
dt k-
(74)
dt
Now let the particle be at position (xo,y0) at time t=to. Then at time
t=t0+At, VX=VXP, Vy=VyP are the predicted values of velocities.
VXP -
VyP = V}
dv
dt
dt
x=x0
x=x0
y=yo
xP = X0 + VXP • At
yP = yo + vyp • At
At
At
(75)
40
-------
The corrected values are
. dVx
VyC =
Vs
dt
dV
X =
At
(76)
X =
At
xc = x0 + VXC • At
+ vyc • At
The equations (76) are iterated, replacing the Xp, and yp values by the
corrected values xc and yc until the successive values are within a pre
tolerance.
preselected
For the cases of very small Stokes number, it becomes necessary to adopt very
small time steps to compute the trajectory accurately in any flow region with
steep velocity changes (near the sampler head, for example). To overcome
this difficulty, the velocities of particle Vx and Vy can be approximated and
the following procedure is used.
Vx = Ux - K
Vy = Uy - K
—
dX
Ux + VGx
Ux + VGy
(77)
where
Vx
Vy
Ux
Uy
K
VGx
VGy
_ r
particle velocity in x direction
particle velocity in y direction
fluid velocity in x direction
fluid velocity in y direction
tokes number of the particle
sedimentation velocity in the x direction
sedimentation velocity in the y direction
41
-------
dx
-r
dt
dy
= Vx
vx
(78)
where
x , y
t =
coordinates of particles
time coordinate
From (78) we get
dx
Vx
(79)
Equations (77) and (78) are used to compute the trajectory of the particles.
Particle velocities Vx, Vy are computed from the equation (77). By using (79),
the new position of the particle is predicted as:
where
v = Y
TPN TPO
-
Vx
Ax
XPO
(80)
new position of particle in Y
old position of particle in Y
a preselected step size in x
old position of particle in x
YPN
YPO
Ax
XPO
If the ratio of Y^/Ypo is greater than 1.05, then the step size x is reduced
until this criteria is met. Then a corrected new position of the particle is
computed by calculating the velocities at the new position by equation (77)
and using an average value of the velocity ratios in equation (80). That is
YpN(corrected) = Y
pQ
Vx
old
Vx
new
AX
(81)
If we denote the corrected value of Ypfj as Yo then Equation (81) is iterated
until Yo converges. That is
Yon = Yo"'1 +
_
Vx
old
Vx
AX
42
-------
where superscript n represents iteration number. The procedure is stopped
when the successive values of YO are within a chosen error limit.
The above procedure is repeated until the particle passes the probe inlet, at
which point it either gets captured by the probe or escapes it.
Calculation of the Efficiency of the Sampling
Let C0 denote the number of particles/unit volume in the free stream and let
C be the actual number of particles sensed by the instrument, then
n --- . -
n " c0 c0 G!
where
n = the efficiency of the sampling, and
Cj = the concentration at the inlet to the probe
C0 u U
where
E = the efficiency of capture of inlet
_c_ CI-AC
Cl ". Cj
where
AC = the loss of particles in the probe.
In the case of a polydispersed system, the distribution can be divided into
n number of small groups. Each one can be treated in the same way as
described above. The effective n will be given as
neff = .* nifi
where
Hi = the efficiency of the ith group
fi = the fraction of ith group, and
n = the number of groups.
In most of the cases, the measured size distribution under non-ideal conditions
43
-------
are available and one likes to find the actual distribution. Denoting the
fraction of itn group by 'fi act', then;
"act
z ni
1=1
COMPUTER PROGRAM SYSTEM
The computer program system consists of two separate programs. Program
'FLOWFI1 solves for the flow field and program 'TRAJEC1 computes the particle
trajectories in the specified flow region. Since the model uses the stream
function equation (14) only when the sampling head faces the stream, program
1 FLOWFI' has to be run only with this option. For angular orientations the
flow field is approximated by a line sink/source in a uniform stream and the
flow field is incorporated in the 'TRAJEC1 program. A description of both
programs follows.
Program ' FLOWFI'
This program to solve for the values of Stokes/Lagrange stream function ty
and to calculate the fluid velocity components Ur, Uz has been written in
Fortran V for Univac 1108 digital computer. The program solves the flow
fluid in and around the sampling head with circular/parallel plate geometry.
The thickness of the sampling head W, chamfer angle a, and the velocity of
suction ratio U are treated as parameters. An explanation of various program
subroutines foll'ows.
The calling .sequence of the subroutines according to the user option of the
flow field is accomplished in the Main Program.
Subroutine FLBOUN, meaning FLow BOUNdary, specifies the boundary of the flow
field in the upstream (ZM), downstream (ZMA), and the ratial (RO) direction.
It also specifies the sampling velocity ratio (U), chamfer angle (ANG) and
the sampler wall thickness (W). The values of the upstream boundary ZM,
downstream boundary ZMA, and the radial boundary RO have to be chosen arbi-
trarily. Typical nondimensional distances are ZM=5, ZMA=5 and R0=5 for
sampling heads with 'W £0.2. For very thick-walled tubes, these boundary
values have to be increased so as to realize the uniform undisturbed flow
condition.
The above values are input to the program and have to be supplied by the user.
The subroutine calculates a value (ZOW) which is the axial coordinate of the
outer edge of the sampler for use in further calculations.
Subroutine GRID places a grid of specified grid spacing in axial (Z) and
radial (R) direction and calculates the coordinates of a given point. It also
calculates the maximum number of points in axial direction (IM) and maximum
number of points in radial direction (JM) . The total number of grid points
in the flow field would be IM x JM. The coordinates of any given point (I,J)
would be given by (Z(I),R(J)). ITW is an indicator for the sampler wall
44
-------
thickness. ITW:0 for W=0, ITW:1, for finite W and ANG-0 and ITW:-1 for finite
W and finite ANG. Calculation regions for various options are shown in
Figure 15. IGR, and JGR are number of grid points per unit distance in the
axial direction and radial direction.
Subroutine BCOND formulates the boundary conditions of the problem. The
boundary conditions are to be formulated at Section I, II, and III. The
sampler wall conditions are also stipulated. Figure 16 gives the values of
PSI at the boundary for circular tube.
Subroutine LAPLA solves the Stokes1 stream function equation formulated for
the problem earlier by using the boundary conditions provided by BCOND. The
solution for the flow field is provided at the grid points specified by GRID.
Successive over relaxation procedure is used to approach the solution. The
iteration procedure is stopped either when the successive values of PSI are
within the specified error tolerance (EPPS) or when the number of iterations
has exceeded the maximum number of iterations (ITERMA) specified by the user.
The relaxation factor (RELAX) has to be supplied by the user and the value
varies from 1.0 to 2.0 depending upon the cpnditions of the problem. Optimum
value has the be found by trial and error. The finite difference equation
formulated in the previous sections is used. This equation is solved at each
grid point of the flow field except the boundary points specified by BCOND.
The maximum error occurring at each iteration is printed out under convergence
rate.
Subroutine VELO calculates the axial and radial velocity components Uz, Ur,
respectively, from the stream function solution provided by the subroutine
LAPLA. Velocities are calculated by determining by second order accurate
finite difference expressions. Depending upon the position of the grid point,
either one of the following finite difference forms is chosen:
1) ESC (Equal Spaced, Centered)
2) ESB (Equal Spaced, Backward)
3) ESF (Equal Spaced, Forward)
4) UESB (Unequal Spaced, Backward)
5) UESF (Unequal Spaced, Forward)
6) UESC (Unequal Spaced, Centered)
Subroutine RESULT prints out the results obtained from various subroutines.
User can manipulate the output statements to suit his needs.
Subroutine STREAM calculates the locus of any specified stream line for use
in the plotting of stream lines. A lagrange interpolation is used to calcu-
late the locus. Function statement SAG is the lagrange interpolation formula
for three points with unequal intervals.
The program listing is provided in Appendix D.
45
-------
J=JM,R=RO-
J=JGRO,_
R=1'0
-4-
— -t
—1~
i
.. j „ _;.....
Z=ZM
1 = 1
-^- Flow direction
Sampler Wall
:Z
z=o
I = IBW
Z=W/TAN(ANG)
1=1 BWO
Z=ZMA
Figure 15. Flow field boundary and grid point layout.
-------
Section I
Section IV
Section III
Section II
Section PSI
I R(J)**2
II U*R(J)**2
III A*(R(J)**2-(1+W)**2)+U where A
IV R0**2
(RO**2-U)
(RO**2-(1+W)**2)
Figure 16. Boundary conditions for circular tube.
-------
Program 'TP.AJEC'
This program to solve for the limiting particle trajectory and to calculate
the true particle distribution from the measured distribution has been written
in Fortran V for Univac 1108 digital computer. For certain flow field options,
the flow has to be predetermined and is input to 'TRAJEC'. The particle
trajectories are determined by solving the equation of motion of particle with
Stokes1 drag. The program uses a predictor-corrector method with iterative
convergence. A description of various program subroutines follows.
Frogram 'INER1 is the Main Program and computes the particle trajectories.
The user has three options in choosing the flow field. These are controlled
by an integer 'NDIM1.
NDIM = 0 The flow field used is that of two parallel plate
inlets facing the stream.
NDIM = 1 The flow field used is that of a circular tube
inlet facing the stream.
NDIM = 2 The flow field used is obtained by superposition
of line sink/source with uniform stream.
Choosing the option 0 or 1 requires the output from 'FLOWFI'. However, for
option 2 the flow field is incorporated in the program 'TRAJEC1. Various
inlet geometries and physical conditions can be obtained by the parameters U,
W, ANG, NDIM, IDIM, ALPHA. Table 3 shows the physical conditions that can be
obtained by various combinations.
Table 3. Parameter desription and physical significance.
NDIM
0
1
2 '
W
0
Wo
Wo
0
Wo
Wo
-
_
_
ANG
0
90°
a
0
90°
a
-
_
-
ALPHA
900
90°
90°
90°
900
900
900
a
a
ZLIP
0
0
0
0
0
0
-
0
ZL
Physical Meaning
Two D thin-walled plates facing the stream.
Two D thick-walled plates facing the stream.
Two D thick-walled plates with sharp edge
facing the stream.
Thin-walled circular tube facing the stream.
Thick-walled circular tube facing the stream.
Thick-walled circular tube with sharp edge
facing the stream.
Approximate solution to sampler facing the
stream.
IDIM = 0 - Two • D
IDIM = 1 - circular tube
Sampler with an orientation angle a.
Sampler with an orientation angle a with a
lip depth 'ZL'.
48
-------
Program TRAJEC starts the trajectory calculation at position ZI, RI input by
the user. ZI is the upstream distance at which the particle is moving with
the stream and is usually at least 5 radii from the sampling head. RI is the
radial position to start the process to find RC, the radial position of limit-
ing trajectory at ZI. If the trajectory of the particle starting at position
RI enters the probe then the next trajectory is started from a position RI =
RI + OR, where DR is a preselected radial increment. If the trajectory from
RI escapes the sampler then the new RI is given by RI-OR. This process
continues until the trajectories from successive radial positions alternate
(i.e., one gets captured and another escapes). This process is repeated with
successive halving of OR until the radial positions RES (at which the particle
escapes) and RCA (at which the particle gets captured) are within a preselected
tolerance EPP.
To calculate the trajectory accurately it is necessary that the time or space
step is not very large. This is accomplished by taking the ratio RAT of
predicted radial position RP to initial radial position RO. If the ratio RAT
is greater than a prespecified value CAT then the time/space step is halved
and recalculation starts. The value of CAT used in the program is 1.05.
Even with a step size that satisfies the ratio test if the iterative conver-
gence fails within LIMIT iteration then the step size is halved and the
calculation procedure is restarted. The origin of the coordinate system lies
at the center of the sampling head. The Z axis is along the center!ine and
R axis along the face plane. The efficiency calculation is performed for 1ST
number of particles.
Subroutine FCT calculates the derivatives of the velocities at a given posi-
tion R, Z, When NDIM option of 2 is used, the fluid velocity can be calcu-
lated at any given point but for option 0 and 1 the velocities are available
only at the grid points and interpolation is needed. Subroutine INTERP
performs the interpolation and computes the fluid velocity at a given particle
location. A two-dimensional linear interpolation is used.
Subroutine QUTP prints the particle position at various time intervals. The
printing interval INT is chosen by the user. For example, when INT - 50, the
result is printed once in every 50 steps. Subroutine OUTP also tests for the
deposition, capture or escape, of particle from the sampler and is indicated
by KOOE. When KODE = 0, particle is escaped and when 1 particle is captured,
KODE = 2 indicates that particle has deposited on the sampler wall.
Subroutine BIAS calculates the actual size distribution from the measured
size distribution . It uses the efficiency calculated by the program INER.
The Stokes number of a given particle K is also calculated by BIAS, The
measured fraction PF, particle size P, relaxation time TAU are input to the
subroutine BIAS. The maximum number of particle intervals that can be used
is 10.
A computer listing of the program is provided in Appendix 0. The users
manual and an example problem is provided in Appendix E.
49
-------
LIMITATIONS OF THE MODEL
(1) The model uses inviscid flow pattern around the sampler to compute par-
ticle trajectories. Close to the sampler wall the effect of boundary
layer dominates the flow pattern and modifies the particle trajectories.
(2) The error in collection efficiency for particles with much smaller
Stokes number K (<0.001) is high when using the flow field option NDIM=2,
The main cause of the error is the deviation of the model flow lines to
the actual flow lines.
(3) The effect of the physical presence of sampler and inlet geometry are
not incorporated in the model for angular orientations of the sampler
(option NDIM=2).
50
-------
RESULTS AND DISCUSSION
Even though there are a number of theoretical and emperical models available
to estimate the sampling errors due to anisokinetic sampling they fail to ac-
count for the effect of sampler orientation. The major difficulty in incor-
porating the effect of orientation is solving the flow field. This is circum-
vented by approximating the flow field around the sampling head with a two-
dimensional line sink superposed on uniform flow. Such an assumption will ren-
der the exact flow field when the sampler inlet is a slit but for other inlet
geometries the assumption is valid in the core region of the inlets. Even
with such a simple model, one can gain a physical insight to the effect of sam-
ple orientation.
The computer program system was used to obtain the sampling bias for various
particles when sampled by a thin-walled circular tube. Sampler inlet geometry
of thin-walled circular tube approximates the flow around the vicinity of the
closed face filter used in personal sampling. The velocity ratios used are:
1.2; 2.0; 0.75; 0.375; 0.1875; and 0.0938.
The concentration ratios were calculated at least at 8 Stokes numbers K.
The results are given in Figures 17-22. The X axis represents the ratio of
measured concentration C/actual concentration C0. The Y axis represents
particle Stokes number K. For all the cases sedimentation and electrostatic
effect has been neglected.
Particles with negligible inertia (K->0) move with the fluid stream, thus
the concentration ratio C/Co approaches unity. But particles with infinite
inertia (K->00) continue in their original direction of motion and the con-
centration ratio C/C0 approaches the velocity ratio U0/U.
At velocity ratios close to one such as 1.2 or 0.75, the maximum error in con-
centration measurement is approximately 30%. But at extreme velocity ratios
the error becomes as high as 100% for super isokinetic velocity ratios (>1)
and infinite at subisokinetic velocity ratios (<1). Hence, sampling error
for subisokinetic velocity ratios are more important and significant than the
error for superisokinetic velocity ratios.
For the case of open-faced filter sampling, flow field option of NDIM=2 was
used. In calculating the limiting trajectory of particles, the filter
cassette was considered to have a lip depth of 0.2 x radius. The sampling
bias for various particles when sampled at an angle a to the oncoming stream
was calculated for a velocity ratio of 0.375.
Figures 23 through 27 show the trajectory of particles when sampled at a
velocity ratio of 0.375. Effect of Stokes number of the particle trajectory
is shown in Figures 23 through 25 at the orientation angle of 90°. Flat tra-
jectories are characteristic of large Stokes number K. Figures 26 and 27
51
-------
5.0 -
1.0
0.5
0.1
O.O
Figure 17. C/C0 vs. Stokes Number
U/U0 = 1.2
Thin walled circular tube
0.01
0.0
0.2
0.4
0.6
c/G0
0.3
1.0
1.2
52
-------
10.C
Figure 18. C/CQ vs. Stokes Number
U/U0 = 2.0
Circular tube
-------
en
-Pi
upper Limit for 'K' = °°
Stokes Number 'K
Figure 19. Sampling bias for a thin-walled circular tube
facing the stream velocity ratio = 0.75.
-------
tn
en
O
o
Upper Limit for K = °°
Stokes Number 'K1
Figure 20. Sampling bias for a thin-walled circular tube
facing the stream velocity ratio = 0.375.
-------
6.0
en
o
o
Upper Limit for 'K = °°
Stokes Number 'K
Figure 21. Sampling bias for a thin-walled circular tube
facing the stream velocity ratio = 0.1875.
-------
11.0 —
9,0
tn
Upper Limit for 'K' =
1.0
1.0
Stokes Number 'K'
Figure 22. Sampling bias for thin-walled circular tube
facing the stream velocity ratio = 0.0938.
-------
Radial Distance
-i
CD
co
ro
o
O
O
-h
O
o>
tn
x
_j*
OJ
OJ
3
O
n>
i
IN
O
7= QO
II II II
o <^> o
• o •
O o OJ
I—- -~J
(jn
-------
6S
Radial Distance
-s
ro
-s
Q)
ro
r>
o
a>
-i
o
ro
oo
o
x
_j.
QJ
O
_i.
in
cu
O
ro
I
ro
i
o
o
o
7* PO
II II tl
O ',O O
• o •
CO o CO
-------
+1 .0
U/U0
a
K
0.375
90°
3.0
OJ
O
fO
fO
ee
0.0
-1.0
-4.0
-3.0 -2.0
Axial Distance
-1.0
Figure 25. Trajectory of particles.
-------
-4.0
-3.5
-3.0
•2.5 -2.0
Axial Distance
-1.5
-1.0
-0.5
+0.5
Figure 26. Trajectory of particles.
-------
-1 .0
-0.8
-0.6.
-0.'
-0.;
0.1
-6.0
U/U0 = 0.375
a = 0°
K = 0.01
-4.0 -2.0 0.0
Figure 27. Trajectory of particles
-------
show the trajectory of a particle of Stokes number 0.01 at the orientation
angles of 60° and 0°, respectively.
Figure 28 shows the efficiency of sampling of various particles at different
orientation angles. The velocity ratio is 0.375. The theoretical limits of
efficiency for Stokes numbers of °° and 0 are shown in solid lines. The
experimental values of efficiency for a particle of Stokes number 0.553, as
obtained by G.S. Raynor30 is also shown. The values predicted by the current
model are higher than the experimental values. The reason for over estimation
is probably due to effects such as particle bounce-off, flow turbulence, etc.
Figure 28 predicts that there is an angular orientation a0 at which all the
particles are sampled isokinetically.
Sin a0 = U/UQ (83)
This holds only when the sampler wall is very thin and the sampler body
doesn't affect the flow.
Projected Area A1
Sampler Head
Area A
Isokinetic Sampling at an Angle to the Free Flow
The physical meaning of equation (83) is that the correct tilt of the sampler
ot0 can reduce the volume of air sampled to compensate for sampling velocities
U
-------
experimental
theory
0.
60.
90.
(Degree)
orientation of face plane with flow direction
Figure 28. C/C0 vs. orientation angle for U/U0 = 0.375 and lip depth is 0.2,
64
-------
4.0
3.0
en
en
a: ooi o.oi
U/U = 0.25^
U/UQ= 0.375
1.0
10.0
Figure 29. C/C0 vs. K for a = 90° square inlet, ZLIP = 0.
-------
cr»
CTl
0.01
1.0
10.0
Figure 30. C/C0 vs. K for a = 90° square inlet, ZLIP = 0.
-------
10.0
1.0
\
\
\
\
•
0.1
\
T
V
0.01
0.0
1.0
2.0
C/C0
Figure 31. K vs. C/C0 for U/U0 = 0.375 square inlet, ZLIP = 0.
67
-------
CO
0.0
30.0 60.0
orientation angle a >-
Figure 32. C/C0 vs. a for U/Ub = 0.375 square inlet, ZLIP = 0.
-------
10.0
CTl
10
20 30 40 50 60 7C 80 90 95 98 99
% Smaller Than
Figure 33. Effect of Bias on the Distribution a = 90°, square inlet, ZLIP = 0.
-------
with a line sink located at the face of the inlet (NDIM=2). The lip length
ZLIP is assumed to be 0 and a the orientation angle is 90°.
Figures 29 and 30 show the plot of the ratio C/C0 vs. K for various values of
the velocity ratio U. At lower Stokes numbers the value of C/C0 approaches 1
indicating that the measured concentration C is the same as the true concen-
tration CQ. This is true even in the case of extreme velocity ratios of
U=0.25 and 4.0. At higher Stokes number values, the concentration ratio
reaches the asymptotic limit of 1/U.
Figure 31 shows the plot of C/C0 vs. K for various orientation angles a.
When a equals 90° the sampler faces the oncoming stream and when a equals 0°
the face plane of the sampling head is tangential to the oncoming stream.
For low Stokes number values, the angular orientation seems to be immaterial.
The asymptotic value for the larger Stokes number is sin a/U. Figure 32
presents the plot of C/C0 vs. a for various values of K. The curves for K
equal to zero and K equal to °° are also known.
Figure 33 shows the effect of sampling bias on the cumulative size distribu-
tion for two velocity ratios. The measured distribution with the anisokine-
tic velocity ratio is assumed to be a lognormal distribution with a ag of 2.0.
The corrected "True" distribution for various values of U is shown.
Figure 34 shows the effect of sampling bias as a function of velocity ratio
for various Stokes numbers K. For smaller Stokes numbers such as 0.1, the
ratio C/C0 rema.ins close to 1 even at extreme velocity ratios. For large
Stokes numbers, K = 10 the curve approaches the asymptotic limit for K = °°,
i.e., C/C0 = U0/U.
The effect of wall thickness of the sampler on the sampling bias was deter-
mined for a circular tube. The flow field was obtained with option NDIM=1.
The sampler was assumed to be facing the stream. A velocity ratio of 0.375
was used. The maximum value of W the wall thickness used was 0.2 x radius.
Figure 35 presents the results. As W increases, the sampling bias decreases.
The reason for this is that the velocity ratio of 0.375 makes the stream
lines move away from the center of the sampler, whereas the effect of wall
thickness is to move the stream lines toward the center thereby nullifying
the effect of an anisokinetic velocity ratio to a certain extent.
The present model was used to compare with the experimental results of
Badzioch31 for a circular tube inlet. The comparison is shown in Figure 36.
Theory agrees very well with the experimental data. Here the Stokes number
of the particle used is 10.5 which is very much larger than usual values
encountered in personal sampling conditions. Experimental data for lower
Stokes numbers are not available at present.
For the physical conditions encountered in personal sampling the Stokes num-
ber of particles in the respirable range is much smaller than unity. Typi-
cally, for a lOym, particle sampled with a tube of 1cm radius at a velocity of
100 cm/sec the Stokes number K is on the order of 3.1 x 10'2. Particles with
70
-------
such a small Stokes number will follow the fluid stream lines even if the curv-
ature of the stream line is quite high. So the anisokinetic velocity ratio
or the inlet geometry is not expected to play a major role in influencing the
collection efficiency.
71
-------
Concentration Ratio C/C0
o
o
O
-s
00
X3
c
Oi
-s
n
3
tQ
ro
00
rt
n>
o>
-------
0.0
VI
0.0
0.1
0.2
1.0
s_
0>
o
•*->
0.1
0.01
0.001
0.0
0.5
1.0
1.5
C/C0 •
2.0
2.5
3.5
Figure 35. C/C0 vs. K for Various Thickness of Wall with
Circular Inlet Facing the Stream U/U0 = 0.375
73
-------
3.5
3.0
2.5
2.0
o
s.
c
o
tQ
cu ] 5
o ' •D
o
o
1.0
0.5 -
O.O
Experimental Data
S.Badzioch (1959) Zinc Spheres
K = 10.5
Theory
K ? 10.5
K = 0
I
0.0
0.5
1.0 1.5 2.0
Velocity Ratio U/U0 -
2.5
3.0
Figure 36. Comparison of Theoretical Results with Ex.perimental
Data of S. Badzioch (1959). Circular Tube Facing
the Stream.
74
-------
CONCLUSIONS
There exists a complete lack of purely theoretical investigations on the
effects of sampler orientation and the effects of sampler head. The present
research demonstrates the feasibility of such a study. The conclusions from
the results of this study are summarized below.
(1) The actual fluid flow patterns encountered around two-dimensional
or axisymmetrical inlet geometries can be obtained. The flow
around a square inlet presents a formidable problem and approxi-
mations are necessary to obtain flow patterns.
(2) Anisokinetic sampling errors are more important and significant
for subisokinetic velocity ratios (U/U0<1) than for superisokinetic
velocity ratios (U/U0>1).
(3) Theoretical error estimates may be regarded as upper limits. The
actual error is lower due to turbulence, particle bounce-off, and
variability of drag on the particles.
(4) For subisokinetic sampling (U/U0<1) the sampler wall thickness has
a counter effect on the sampling bias.
(5) Sampling bias for polydispersed aerosols can be obtained by use of
number of monodisperse aerosols.
(6) The experimental data available for circular inlets compare very
well with the theoretical collection efficiencies.
(7) For cases when the sampler is oriented at an angle to the stream,
the exact solution to the flow field is very difficult to obtain
and approximations are necessary. A line sink was used in the
present model.
(8) Sampling efficiencies obtained for angular orientations compare
well with the presently available experimental data. But expanded
experimental data are needed to ascertain the reliability of the
model.
(9) It is very difficult to arrive at a single or a multiple optimum
inlet geometries because of the variety of factors that influence
the particle collection. For the physical conditions encountered
in personal sampling, the inlet geometry is not expected to play a
major role in influencing the collection efficiency.
75
-------
(10) The major difficulty in evaluating various inlet geometries is
obtaining the flow field around the sampler. Experimentally deter-
mined flow field can be used with the particle motion part of the
program to accurately evaluate various inlets.
76
-------
REFERENCES
1. Brady, W. and L. A. Touzalin, J. Ind. Eng. Chem., 3 :662 (1911).
2. Lappel, C. E. and C. B. Shepherd, J. Ind. Eng. Chem., 32:605 (1940).
3. Dalla Valle, J. M. , Micrometrics, 2nd edition, Pitmann Publ. Corp.
New York. (1948).
4. Watson, H. H., Amer. Ind. Hyg. Assoc. Quart., _15 :21 (1954).
5. Badzioch, S., British J. Of. App. Physics. 10, 10:26 (1959).
6. Badzioch, S., J. Inst. Fuel. 3^, :106 (1960).
7. Levin, L. M., Izv. Adad. Nauk. Ser. Geoph, 7, :914 (1957).
8. Whiteley,.A. B., and L. E. Reed,, J. Inst. Fuel; 32:316
9. Glauberman, H., Amer. Ind. Hyg. Assoc. Quart, 23.235 (1962)
10. Vitols, V., Determination of Theoretical Collection Efficiency of
Aspirated Particulate Matter Sampling Probes Under Anisokinetic Flow,
Ph.D. Thesis, University of Michigan (1964).
11. Vitols, V., 0. Air. Poll. Con. Assoc.. 16:2, :79-84 (1966).
12. Oavles, C. N., Staub Reihalt. Luft (English). 28:6, :l-9, (1968).
13. Davies, C. N., Dust is Dangerous, Faber London, P. 21, (1954).
14. Belyave, S.P., and L. M. Levin, J. Aero. Sci.. §, :325, (1974).
15. Belyave, S.P., and L. M. Levin, J. Aero. Sci., 3, :127, (1972).
16. Levin, L. M., "Studies on coarsely dispersed aerosols", P. 3, Acad.
Sci. , U.S.S.R. (1961).
17. Fuchs, N. A., Atmos. Envir.. 9_, :697, (1975).
18. Gooddale, T. C., B. M. Carder, E. C. Evans, Ind. Hyg. Quart, 13.4,
:226 (1952).
77
-------
REFERENCES (continued)
19. Schmel, 6. A., Am. Ind. Hyg. Assoc. J. 31:758 (1970).
20. Carson, G. A., Ashrae Journal. :45-49, May (1974).
21. Bien, C. T., and Morton Corn, Amer. Ind. Hyg. Assoc. J., 32:7, :453,
(1971).
22. Pickett, W. E., and E. B. Sansone, Amer. Ind. Hyg. Assoc. J.. 34,
:421, (1973).
23. Davies, C. N.. Amer. Ind. Hyg. Assoc. J., 36:9, :714, (1975).
24. Strom. L., Atmos. Envir. 6 :133 (1972)
25. Myres, G. E., Amer. Ind. Hyg. Assoc. J.. 35:307 (1974).
26. Breslin, 0. A., and R. L. Stein, Amer. Ind. Hyg. Assoc. J. 36 :576,
(1975)
27. Lundgren, D.., and S. Calvert, Amer. Ind. Hyg. Assoc. J. 28: 208 (1967).
28. Roache, P. J., Computational Fluid Dynamics, Harmosa
Publication, N.M. , (1971). ~
29. Southwell, R.V., and G. Vaisey, Phil. Trans. Roy. Spc.,
A 240, :177, (1946).
30. Raynor, G.S., Amer. Ind. Hyg. Assoc. J. 31:294 (1970).
31. Fuchs, N. A., Mechanics of Aerosols, Macmillan Company, N.Y. , (1964).
32. Hidy, G.M. and J.R. Brock, Dynamics of Aerocolloidal Systems.
Pergammon Press (1970).
33. Davies, C. N., Aerosol Science, Academic Press, N.Y., (1966).
34. Milne-Thomson, L. M., Theoretical Hydrodynamics, 5th ed., Macmillan
Company, N.Y., (1968).
35. Schlichting, H., Boundary Layer Theory, 4th edition, McGraw Hill, N.Y.,
(1960).
78
-------
REFERENCES (continued)
36. Agarwal, J. K. Aerosol Sampling and Transport. Ph.D. Thesis,
University of Minnesota (1975).
37. Ter Kuile, VI. M., "Comparable Dust Sampling at the Work Place,"
Report F 1699, Instituut Vook Milieuhygiene en Gezondheidstechniek,
Postbus 214
79
-------
Appendix A. Derivation of Boundary Condition at Section II
Circular Tube
so
Using continuity of flow at Sections I and II
- 1-UR2 = llJ0R02 -Iu0 (R + W):
II = UiRp2 - UR2
0 Rn2 - (R + W)2
Ro2 - (R
denoting the stream function outside the probe at Section II iji0p then
1 2
on the probe wall ty is unique and ^op must be equal to ^jp, the stream
function inside the probe. So
and C is given by
C = -1 U0(R + W)2 + TJ- UR
(R + W) <_ r <_ RO
in non-dimensional form
(1 + W) £ r < RQ
80
-------
Two Parallel Plates
Following similar procedure as before
- UH
° Y0 - (H + W)
* (H
(H + W)
-------
Appendix B. Uniform Sink Strength Distribution
^sink = -/ m • dx: -6
o
Xi = x - y cot 9
dxi = y cosec26 • de
82
61
m * 9 ' cosec9 • de • y
= -my
- 62 • cot62 + £n
(x-d)
^ due to uniform stream
Ujy Cosa + Ujx sina
Ux = UT cosa - m in
/(x-d)2+y2
(B2)
u» = -
9X
= -Uj sina + m
-iy.
x2+y2
xy (x-d)y
x +y (x-d)2+y2
. ,
x (x-d)2+y2
- tan-3
= -Ui sina + m
tar
' -
(B3)
sina + m(6i-e2)
82
-------
The sink strength m is given as follows
H sina • Uj + irdm = H • U
U - sina
2ir • f
d
m - ^^r (B4)
A
where f =
83
-------
put
Appendix C. Triangular Sink Strength Distribution
= m
= m (d-O
d
6 = tan-
or 8 = cof
source =
d/2
/
o
cot
+ / m(d-C) • cof
d/2
(ci)
then
then
= -y • dx
y y
source = myx / cot-1x • dx + my2 /x * cot'-'x
dx
_! x-d
y y
+ my(d-x) / cot^x • dx - my2 / x
x-d d_
T" x"2
cot *x • dx
84
-------
Integrating and grouping the terms
source = cof*
+ cot"1-
+ cot
_ix-d
p-(d-x)2 my2!
2 -I
2 + (x-d)2
(C2)
^
2 d
The total sink strength = / m • £ • d5 + / m(d-C)
0
md2 md
8 8
md2
d_
2
The value of m is found by flow balance to the sampler. Let Q be the rate of
sampling for a square inlet (H x H). Then flow per unit width is
n- and
$= Uj sina • H - TT • 2*1
non-dimensional m = j^- = - iiJJ—-.g1""-)
UI
(C3)
where f is given by
velocity UX =
J A
5*
ource
^source
3x
85
-------
Appendix D. Computer Program Listing
(See following pages.)
86
-------
£ ********************************************************************
c
C 'FLOWFI' DESCRIPTION
C *********************
C
c ***************** ******* ************* ********************************
C THIS PROGRAM SOLVES FOR THE FLOW FIELD IN AND AROUND THE SAMPLING
C HE An.
C THfc PHVSRAL DIMENSIONS ARE NORMALIZED WITH RESPECT TO THF
C PROBfc RADIUS
c ALL THE VELOCITIES ARE NORMALIZED WITH RESPECT TO THE FREE
C STREAM VELOCITY
c
C METHfiu OF SOLUTION
r.
C SECOND ORDER FINITE DIFFERENCE APPROXIMATION
C SUCfkSSIVE OVER RELAXATION(SCR) TECHNIQUE IS USED
C
C THE FLU1U IS ASSUMED TO Hfc f-RICTIi)NLESS AND THE FLO* FIELD IS GOVERNED
C faY ptJUNTIAi. FLO.w EQUATIONS. THE FLUID Fl 0* MODEL COVERS THREE DIFFERENT
C SITUATIONS. 1} Twt) DIMENSIONAL FLOW BETWEEN PARALLEL PLATES
C 2)AxISVMMET«IC FLOw IN A CIRCULAR TUBE
C 3)LINE SINK wITH ARBITRARY ORIENTATION TO THt ONCOMING FLOH,
c THE USER CAN CHOOSE ANY OF THF ABOVE OPTION WITH pROPfcR VALUE OF AN
c INTEGER PARAMETER «NDIMI
C NDIM =0 Two DIMENSIONAL CASF
C NDlM *\ AX1SYMMETRIC CASE
C NDlM s2 LINE SINK
C
C «HOtFp VARIABLES DESCRIPTION
C *******************************
C
C REL AX*****RELAXATION PARAMETF.P (USF P INPUT ) f VALUF S BETWEEN 1,0 AND ^fQ
C
c ALPHA*****ANGLE OF ORIENTATION OF SAMPLER HEAD( INPUT) tustn ONLY WITH
c NDIM VALUE 2.
C IGR*******C.KID POINTS PER UNIT LENGTH IN AXIAL DIRECTION (INPUT)
C
C JGR*******tRJD POINTS PER UNIT LENGTH IN TRANSVERSE 01 RECT ION( INPUT )
C
c ZM *******FLQW FIELD BOUNDARY UPSTREAKINPUT) <»5,
c
C ZMA*******FL()« FIPLO BOUNDARY DOWNSTRE AM ( INPUT )
C
C HO *******Fluw F-HLD BOL'NDAWY RADIAL DIRECT ION ( INPUT )
C
C ITEPMA****MAXIMUM ALLOWABLE ITERATIONS FOR FLOW FIELD TO CONVERGE ( INPUT )
C
c in ******MAXIMUM NUMBER OF GRID POINTS IN AXIAL DIRECTION <101
C
c JH ******MAXIMUM NUMBI-R OF GRID POINTS IN RADIAL DIRECTION
-------
c -. . •
C IPUNCH****CUNTROl PARAMETER FOR GETTING PUNCHED OUTPUT OF
C PSI»UZ»URtZ»RtfcTC(lNPUT)
C *0 NO PUNCHED OUTPUT DESIRED
C M PUNCHED OUTPUT RESULTS
C
C NC«N******INTEGER PARAMETER INDICATING WHETHER SOLUTION CONVERGED
C OR NOT WITHIN ITER^A INTERATIONS
C =0 CONVERGENCE OBTAINED
C *i NO CONVERGENCE
C
C U *******SAFHLING VELOCITY RATIO (INPUT)
C
C UR********VfcLOCITY Of- FLUID IN TRANSVERSE DIRECT ICNC ARRAY Of IMXJM)
C
C UZ********VELOCITY OF FLUID IN AXIAL 0 IREC UON< ARRAY OF IMXJM)
C
C w*********THfc SAMPtER wALl THICKNtSS(INPUT) »USED ONLY WITH
C NOIM UP Tint* OF 0 OR 1
C
C ESC*******FUNCTIUf-' FOR EQUALLY SPACED CENTRAL flFf-fHFNCE
C
C ESF*******FUKCTU)Ki f-OR EQUALLY SPACED FORWARD MFFEHfNCt
C
C tS^*****1»*FliNCTION FOR EQUALLY SPACED BACKWARD DIFFERENCE
C
C UESC*******UNCTION FOP UNEfiiAlLY SPACED CENTRAL DIFFERENCE
C
C UESf******FHNCTIGf4 FOR UNEQUALLY SPACED FORWARD DIFfEHfNCE
C
C U£Sf»*****»FUNCTIUN FUR UNFGUALLY SPACED BACKWARD DIFFERENCE
C
C M*********FltTICIOUS SOURCE/SI** STRENGTHCFOR NDIM ?)
C
SUBROUTINE DESCRIPTION
#i)t^*»##|[***$»*»$ #4 ****************************
C
c MAI\******CALLS VARIOUS SUBRCUTINES
c
C FLBntN****FlxfS THE FLOW BOUNDARY
C
C G«IO******LAYS ft PKtSPfCI^TED G»IO ON THE FLO* FlftH ANU
C IALCUI.ATFS THt COORDINATES OF THF GRID JOINTS.
C
C fc)CONO*****tALCuLATf S BOUNDARY CONDITIONS FOR THE PROBLEM
C
C LAPL******I'OtVtS Tht FINITE CIFFERfNCE i-QUAUON BY SOR METHOD
C
C VELO****J»*CALCULATtS THE VEtCCIUES F«OM STREAM FUNCTION
C
C STH£M*****€ALCULATES THE CONTOUR OF A STREAM LINE
C
c RESULT****PHINTS THt RESULTS
c
C S!Nvfct****CALCULATF5 THE FLO^ FIELD WITH LINE SINK(NOIM*2)
C
C
£ ********************************************************************
******************************************************************
88
-------
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lR(I»J)sUESB(PSI(I.J)»PSIU»l»J)»PSI(I-2.J)»DZ»022)/(*RR)
lFU.E(,.(JbRO-l))Gl' TO 20
GO Tt) 21
10 L>Rf< = H( J+2)-H(J+l)
U7(I»J)=UtSMPSI(l*2tJ)tPSJ(I*l»J)«PSI(I«J) tDPtDRR)/RR
IMJ.fcU. ( JfcR+l ))GO TO 7
D7=zm-zu-n
UR(I«J)a-fcSh(PSI(I»J) .PSI(IM».J)»PSIU"2.J) .OZ)/KP
Gn TO 7
18 IF (I .hi.'. Ih*0)GU TO 22
26 URks«( J)-R( J-l )
D7Z=Z(J}-7(I-1)
UP( I. J)s-iits,C(PSI( I*lfJ)fPSI(liJ)fPSKI-itJ) »T)Z»DZZ)/FW
b?U»J)=UESC(PSI( J.J41) tPSl(ItJ)fPSI(If J-l) »DR.OKR
L-n TU 7
22 1F( J.GT .JGK.AND.J.l T.JGPO) GO TO 2?
IFU.EU'.JGK) (iO TO 2U
IF U.tu.JGknj GO TO ?b
UR=R( J+ 1 J-KCJJ
bn TO 26
24 tH(ItJ)s0.o
D7=Z(1)-Z( 1-1)
UpUtJ)=-tSH(PSI(I»J) tPSHI-l.J) »PSI(I-2*J) tDZ)/HH
OW=R(J)-R{J-1)
U?( 1 » J)=k Sb(PSi (I » J) »PSI
(,P TU 7
2b UR(I»J)sO.O
U7=Z(I)-Z( i-1)
UR( If J)=-f'stUVST( 1 » J) »PS
U7U »J)=tSFtPSl ( T , J + 2)»PSI ( I t JM ) »PSI ( I» J) iOK)/RR
Gil TU 7
23 ljR(ltJ)ro.y
U7CI» J)=0.0
7
5
DTMNSION ws(io»ioi 3
SA(.
-------
DIMENSION z(iM)tK
COMMON/BAT/ ALPHAfF.ISlNK
Ft|N(XtV)«SQRT(X*»2+Y**2)
. 141596/ieo.
CsCl)S(ALPHA)
SsSINCALPHA)
*z(U-S)/«./K
1F( ISINK.E0.1) M»(S»U)/2,/F**2
wPlTt(6f«) MtFtlSjNK
3 FORMAT(UM0.6)
« Fnf«WAT(2Fi0.6tla)
On 1 I*1»IM
Yr-7(I)
i^n 2 J=I»JM
X = kU)+F
XF=X-F
XF^SXF-F
iF(ISINK.tU.O) GO TO 13
lFUbSm.LE..01) GO TO 10
AsHJN(X»Y)
BsFUN(XFtY)
U = FUN(
HsB* XF
G=X>A
Uu2=-X/2./A*XF/H-XFF/2./D*ALOG(H/G)+ALOG(H/E)+
lY**2/2,*{2./M/B»l./G/A-l./t/D)
,/ Y/ A-Xf- **?/ Y /6* XFF **2/2./Y/D»Y/2.*( 2, /B-i,/ A- l./D)
GO TO U
10 JF(x.GL.O..ANU.X.LE.(2.*F)) GO TO 12
GO TU 11
12
GO TU 11
1 3 Asf-UN(X.Y)
Bs^U^l(XF^ tY)
lF(A.Lt..OD As. 01
lF(b.Lt..01) bs.01
UU/SM*tl./A-l./B)
UijH = *M*(X/A/Y»(XFP)/B/Y)
11 CONTINUE
U?( 1 1 J)B-UU«*S
UR( I» J)
2 CONTINUE
l CONTINUE
98
-------
PROGRAM 'TRAJEC'
0 A »
C
C TRAJEC SOLVES FUR THE LIMITING TRAJECTORY OF PARTICLES CAPTURED BY
C A SAMPLING Hf-Ad IN A PREDETERMINED FOO* FIELD
C
C PARTICLE TkAJECYORIF.S A»E OBTAINPO BY NLIMeRICALl.Y SOLVIKG THE EQUATION
C OF HUTJON OF PARTICLES *ITH STOKt'S DRAG,
C
C NUMpRICAL METHOD USFD..«.,...PKf-DICTOR»CORHECTOR WITH ITERATIVE CONVERGENCE
C
C WKw*rthMW,«fcWwN»w»MNW»*WMWkWfcfc^M^^t«w*fcri»MNfckWWI*W*fcHMW^
C PROG^AM TNAjer. VARIABLES DESCRIPTION
C ALPHA**»***UHItNT AT IQNi ANGLE OF THE SAMPLE RHP AD wITH RESPECT TO FREF STREAM
C IN UtGREES(USED OMY WITH THF OPTION NDIMs?) INPUT
C SO WHEN THE FACE PLA^t OF THE SAMPLfrW IS TANGENTIAL TO
C FKEfc STREAM
C
c s40 ^HEN THE. FACE PLANE FACES THE STREAM,
c
C ANG********TAPfc.RlNG ANGLF OF THF SAMPLER wALL(USEP ONI Y WITH THE OPTIONS
C NDIMsl OR 0) INPUT
C
C DTI********lNilIAL TIME INCREAM^NT FUR THfc TRAJECTORY C ALCUL AT ION ( DUH I NG
C THE EXECUTION OF THE PROGRAM THIS TIME INCREAMENT IS VARIED AS
c REQUIRED FOR CONVERGENCE CPITERIA. (USUALLY=O,D INPUT
C DR»******** INCREAMEN1 IN RADIAL DIRECTION, USED TO CHOUSfc A NE"'RIl
C BASED ON THE PREVIOUS TRAJECTORY CALCULATION (CAN Rt 0.1 OR 0.?)
C
C EPPS*******PRESET TOLERENCE TO STOP THF ITERATIVE PROCEDURE AND TRAJECTORY
C CALCULATION (INPUT)
C
C F**********FkALHnN OF DIAMETER OVER *HICH THE SINK IS DISTRIBUTED,
C (USED ONLY HITH OPTION NDlMs?) USUALLY FsO.Ol INPUT
C
C FR*#*******FROUDt NUMbFR CF THt PAkTlCLE (OUTPUT)
C
C IBtv********NUM«tR OF AXIAL GRID POINT AT WHICH THE INNER WALL OF THE
C SAMPLER IS LOCATED
C
C IHWfi*******NUMbER OF AXIAL GRID POINT AT WHICH THE OUTER WALL OF THE
c SAMPLER is LOCATED,
C IbWU=Irf* Ih OPTION NDIMr2 IS USED.
C
c iDiM*******siNK TYPE PARAMETER
C «0 TwO DIMf- WSIONAL
C si AXISYMMETRICAL
C
C IE*********ELEtT»05TATIC PARAMETER
C sOELECTROSTATIC EFFECT NOT TAKEN INTO ACCOUNT
c =1 * * * TAKEN INTO ACCOUNT
c
C 1M*********MAX1MUM NUMBER OF GRID POINTS IN AXJAI DIRECTION INPUT
C
99
-------
C lNT********INTfc«VAL FOR TRAJECTORY COORDINATES TO BE PRINTED OUT INPUT
C
C 1S*********SED1MENTATION PARAMETER
C BO SETTLING IS NOT TAKEN INTO ACCOUNT
C *\ * * TAKEN INTO ACCOUNT
L
C JSlNK******SJNK STRENGTH DISTRIBUTION PARAMETER.
C sO UNIFORM STRENGTH
C si TRIANGULAR STRENGTH DISTRIBUTION
C
C IST********NUMbkR OF SIZE INTERVALS IN PARTICLES DISTRIBUTION INPUT
C MAXIMUM OF 10
C
C LIMIT******MAXIMUM NUMBER OF TIMES TO PERFORM THE TRAJECTORY CALCULATION
C FUR ONE PARTICLE
C
C THfc FLUID IS ASSUMED TO RE FR ICT IDNt.ESS AND THE FLO* FIELD IS GOVERNED
C BY POTENTIAL FLU* fc GiUAT IONS . THfr FLUID FLOW MODEL COVERS THREE DIFFERENT
C SnuATIONS.l) TWO DIMENSIONAL FLO* BEWfeN PARALLEL PLATES
C <>)AxISYMMURlC FLO* IN A CIRCULAR TUBE
C 3)LINE SINK wHH ARBITRARY ORIENTATION TO THE ONCOMING FLOW,
C THE USER CAN CHOOSE ANY OF THE ABOVE- OPTION WITH PROPER VALUE OF AN
C iNTfGER PAHAMFUR »NDIMl
C NoIM =0 TwO DIMENSIONAL CASE
c NDIM sj AXISY^METRIC CASF
C NDlM *2 LINE SINK
c
C p**********pART ICLE RADIUS IN MICRONS (DIMENSION '1ST')
C
C PM********MEASURED FRACTION OF VARIOUS PARTICLE SIZES (DIMENSION '1ST')
C
C PL*********L!Nh AR DIMENSION CF THE SAMPLING HEAD
C
C OC*********tLEtTRUSfATIC CHARGF ON THE SAMPLER HEAD.
C
C RI*********INITIAL RADIAL POSITION OF THE PARTICLE
C
C RT*********RADIUS OF THf- TUBE ( S AHPL I NG) OR CHARF CT tP I ST 1C DIMENSION OF
C THE SAMPLER.
C
C STK********STOKe
c
c UR********VELOCITY OF FLUID IN TRANSVERSE DIRECTIONURRAY OF
r
C U *******SAMPLINr, VELOCITY RATIO (INPUT)
C
AXlAL COORDINATE (DIMENSION I«)
C
C ZI*********INIHAL AXIAL POSITION OF THf PARTICLE
100
-------
ZLIP*******LIP DEPTH Of- THE
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
r
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
L
C
C
C
J*HM AXIAL DISTANCE UPTO *HICH THF TRAJECTORY SHOULD BE
CONTlNUtO (USUALLY EQUAL TO 0.)
TRAJEC DATA INPUT
FORMAT(
INTERVAL
CARn
lfcso/1 ELECTROSTATIC EFFECT NOT INCLUDED/ INCLUDED
Iw=l/-1 UPPEW/intNhP ^ALl
lbso/i SEDIMENTATION EFFECT NOT INCLUDED/INCLUDED
(3F10.«>
U--VCLOCITY RATIO.SAKPLING VEL/FRF.F STREAM VEL
*--K,ALl TnICK\F.SS OF THE SAMPLER
AN&--IAPEKING ANGLE TN DEGREES OF SAMp| fcR WALL
; N[;JM FORMAT(la)
F L I** FIELD UPTIONI
= 0 TwU DIMENSIONAL FLOv^ HETwFFN PARALLEL PLATES
=i AxISYMMi1RICAL FLOw FOR CIRCULAR TUPE
= 2 FLOr; FltLD with LINE SINK
Os.1 REQUIRE THE Fl.C'W FI {- L D TO FE DETERMINED PY THE PROGRAM
I '
: 10 IM FURMATdaj OMIT IF
so T^t' DIMENSIONAL LINE- SINK
TRICH LINE SINK
sO OR J
CAHr> bt
i'F GRID POINTS Pf-R UNIT
RADIAL DISTANCE. sl./GPID SPACINGIN RAD.DIR*!
JGRO--RADIAL GRID POINT AT WHICH OUTER EDGE OF THR WALL IS LOCATED
JGRCzJGK U NDIMs? OP *- = U. FOR NPIMsOtl JGRiiJGRU ARE CALCULATED
t)Y IHt F-ROGRAC
LARD
7: I«w»lHrtO
IH«--AXIAL GRID POINT AT WHICH INNER EDGE OF SAMPLER WALL is LOCATEI;
I6WL--AXIAL GRID POUT AT WHICH OUTER EDGE OF SAMPLER WALL IS
101
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LOCATtO. IBwOsIH* IF MDIM=2 OH W«0 OR ANG«90, FOR MHM«0 OR li
IHfc AND Ib*0 ARE CALCULATED BY 'FLOWFP
CARf) Bl JM»JM FORMAT(2I«)
IM--MAXIMUM NUMBER OF POINTS IN AXIAL DiRtcTiON
ZI--1NITIAL AXIAt POSITION OF THE PARTICLE.
RI — INITIAL RADIAL POSITION OF PART ICLEsSQRT (U )
DH-- RADIAL INCREAMFNT (0.2)
) UINFtRF FORMAT(?F15.6)
U1NF--FREE STREAM VELOCITY USER INPUT
Kb— REYNOLD'S NUMBER BASED ON TUBE w
P FORMAT(F15.fo)
DEPTH OF SAMPLER.
DONE
ZLIP — LIP
CARn(M*+a): OP»OC»PL FORMAT(3FJO,«) OMIT IF IE*0
UP--ELECTHOSTATIC CHARGE ON THE PARTICLE
QC--EUCTKUSTATXC CHARGE ON THE SAMPLER,
PL--LINEAK DIMENSION OF THE SAMPLER.
» 1ST FDRMATU4)
» OF SIZE INTERVALS IN THE SIZE DISTRIBUTION
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FjlsO.
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J=o
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TsU.
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ir = i
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V 7 = U X
V p s U Y
CAlL FCT (Rti.ZI tf-VZtOVWtUXf bY»VZ» VP.DUX»DUY)
DIIXUSDUX
1) V Z 0 » D V 2
DT=DTa
r PREDICT NE* POSITION
9 lFTK.LT..I) GU TO <
Gn TO a.i
l.+STK*DUX)
105
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18 FpH'-'Al (//5Xf 'ESCAPED' )
21 FoKMAH tH,3(F 15.6»10X)
UU.e&.O) KSJ
lF(AbS(«CA-ktS).LT.EPP ) GC TO 2b
20 CONTINUE
2«> IF(1I*.EU,1) YUPSRCA/2.+RES/2.
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*PlTE(fe»«0) CAI.ICAT
aO PnH^AT(//t IHt lUXt 'MAX. POSITION RA T IDs ' »H 5, 6t // t 1 OX »-
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( ( VUP-YLIO/F «P
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26 FnRWAT(//«iXt ' VAL. » AT 1 0= ' » H 0 . fo t /SX t ' CONC . R AT I Os i i F 1 0 .6 »
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PF(lI)=PF(II)/t>F
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STOP
FCT
NE FCT(RUtZOit}V7»nvPtUXtUYtVXtVYtriJX»nOY)
S» It fOPf(JL» Ik»tPl ttSP
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IF (SI K.LI ..1) Gu TU 2
GO TU 1
ALPHA)/FR»F T
UV*=Uv«-SlN( ALPHA )/FR*F I
iF(n.f-U.O) GO TO 2
Ss(HU-J .)/MIN
L=ZU/FlN
^l -Z0)**2)
-1 ,)*(S-SS)
SliBROL't INE INTtRP(RCJ»ZUtUX»UYtDUX»OUY)
M)»Z( 101)
J01 »(jl)»UR( JOltbl)
JMtJGRtJGRDf !Bw»IH
CALl SINVEL(WO»ZOtUX»UYtOUX»OUY)
107
-------
iF(ZO.Gt.Zd) ,ANO.ZO.LT.ZtI + n)GO TO 2
1 CnMINUE
2 IF d.EG.IHw) I a !•!
On 3 JSI»JM
lF(RO.Gt.HtJj.ANB.RO.LT.R(J+l))Gn TO 4
3 CONTINUE
a CONTINUE
UxL=(UZ(Ui»J)-UZ(I»J))/(ZUf 1>»ZCI))* (ZO-Zd))
(Ufcd+l.J)-UR(I»J))/(Z(I*l)-Z(I))* (ZO-Zd) >
uvU*(UHll4l.J*l)--UH(I»J+J))/(Z(I*l).Z{I))*(ZO-Z(I))
If J)
I ? J+l )
UX*(UXli-UXL)/(R(J+1 )»R(J))*(RO»P(J))
UYS(UYU-UYLJ/(R(J+1)"R(J))»(RO-«(J))
UllY«(UYlJ-UYL)/(R(J*l)-R(J))
1*(RO-R(J))/(K(J+1),R(J))
Ux=UX+UXL
UY=LIY*UYL
Kf
SINVtL(ROtZatUX»UY,DUX.DUY)
Ff ALPHA»U» I DIM
THlL(XtYJs(X**2-Y**2)/(X**2+Y**2)
ARA(X,Y»Z)sALOG((X**2+Y**2)/(Z**2+Y**2))
ACH(X»Y)=l./(x**2+Y»*2)
RFAL M
SsSlN(ALPHA)
f=(U-S)/a./F
IF( ISIKK.F.U.1)
X=RD+F
IF (IUIK.EQ.O) GO TO 1
IFUSIKK.HJ.Q) GO TO 13
IF(ABS(Y) .LE..01) GO TO 10
Arf-UN(Xt Y)
HsFuN(XFtY)
DsFUN(XFF,Y)
fc s X F h + 0
GsX-^A
U|)Z3"X/2./AtXF/B-XFF/2./D+ALOG{H/G)+ALOG((-/E)
108
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111
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Appendix E. Computer Program System - User Manual
The computer program system consists of two separate programs. Program
'FLOWFI' solves for the flow field and program 'TRAJEC' computes the particle
trajectories in the specified flow region. Since the model uses the stream
function equation only when the sampling head faces the stream, program
'FLOWFI1 has to be run only with this option. For angular orientations the
flow field is approximated by a line sink/source in a uniform stream and the
flow field is incorporated in the 'TRAJEC1 program. User instructions for
both programs follow.
The fluid flow model covers various inlet geometries:
• Parallel plate inlet facing the stream
• Circular tube inlet facing the stream
• Slit inlet with arbitrary orientation to the stream
The user can choose any of the above options with proper value of an integer
parameter 'NDIM'.
NDIM = 0 Parallel plate inlet
NDIM, = 1 Circular tube
NDIM = 2 Arbitrary orientation - slit
For options 0 or 1, it is necessary to obtain the flow field data from pro-
gram 'FLOWFI' to do the trajectory calculations. However, for option 2 the
flow field is incorporated in the program 'TRAJEC1 and the use of 'FLOWFI'
is not required. Various inlet geometries and physical conditions are sim-
ulated by proper selection of parameters U, W, ANG, NDIM, ZLIP, ALPHA.
Figure 37 shows the physical meaning of these parameters.
N6
W
Figure 37. Parameter description for inlet geometry and orientation.
The 'FLOWFI1 program input data and their description follows. The recommended
values for some non-physical variables are also given.
112
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•FLCWFI' DATA INPUT
******************
CARD l: NQIMflPUNCH FORMATC )
NDIM,.....FLOw FIELD OPTION PARAMETER
so Two DIMFNSIONAI CASE
=1 AXISYMMEIRIC CASt
-^ LINE SINK
1PUNCH....CUN1ROL PARAMETER Fl)K GETTING PUNCHED OUTPUT OP
PSI tUT. »UR»ZtR»EWC
= 0 NO PUNCHED OUTPUT DESIRED.
= 1 PUNCHED our PUT DESIRED,
CARD t"> JSlNKfALPHA,F
CARD $
CARD
CARD
FORMAT (Ib»2F 10.
OMIT IF NOIM.NE.2
IS1NK ..... SINK CISTRIBUT IUN PARAMETER
=0 LINE SINK OF UNIFORM STRENGTH.
=1 LINE SINK OF TRIANGULAR STRENGTH DISTRIBUTION.
ALPHA ..... ANGULAR ORIENTATION OF THE FACE PLANE OE SAMPLER HEAD.
=90 DEGREES WHEN SAMPLER FACES THE STREAM
=0 DEGREES wH£N STREAM IS TANGENTIAL TO THE HEAD.
f" r ....... .FRACTION OF THE PROBE DIAMETER OVER WHICH
THE SINK IS ASSUMED TO BE DISTRIBUTED.
RECOMMENCED VALUE is 0,01.
ITERMA,R(- LAX
IS,F io.«)
IIERMA. . ..MAXIMUM ALLOWABLE ITERATIUNS FOR FLO* FIELD
TO CONVERGE
RELAX ..... RELAXATION PARAMETER USED IN THE PROCEDURE OF
SUCCESSIVE OVER RELAXAT ION
VALUES BETWEEN 1.0 AND
-------
c
C CAKQ bj II FORMAT( )
C
C II.. NUMBFK Of STREAM FUNCTION CONTOURS NEEDED.LE.10
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C CARD 7: STK FORMATC )
c
c STK.......STKEAH FUNCTION VALUES FOR THE CONTOUR,THIS ARHAV
C hJLL CONTAIN «II« VALUES
c
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114
-------
Program 'TRAJEC' solves for the limiting particle trajectory and calculates
the true distribution of particle size from the measured distribution.
Program 'TRAJEC1 starts the trajectory calculation at position ZI,RI input by
the user. ZI is the. upstream distance at which the particle is moving with
the stream and is usually at least 5 radii from the sampling head. RI is the
radial position to start the process to find RC, the radial position of
limiting trajectory at ZI. If the trajectory of the particle starting at
position RI enters the probe, then the next trajectory is started from a
position RI = RI + DR, where DR is a preselected radial increment. If the
trajectory from RI escapes the sampler, then the new RI is given by RI-DR.
This process continues until the trajectories from successive radial positions
alternate (i.e., one gets captured and another escapes). This process is
repeated with successive halving of DR until the radial positions RES (at
which the particle escapes) and RCA (at which the particle gets captured) are
within a preselected tolerance EPP.
To calculate the trajectory accurately, it is necessary that the time or space
step is not very large. This is accomplished by taking the ratio RAT of
predicted radial position RP to initial radial position RD. If the ratio RAT
is greater than a prespecified value CAT then the time/space step is halved
and recalculation starts. The value of CAT used in the program is 1.05.
Even with a step size that satisfies the ratio test, if the iterative conver-
gence fails within LIMIT iteration than the step size is halved and the cal-
culation procedure is restarted. The origin of the coordinate system lies
at the center of the sampling head. The Z axis is along the center!ine and
R axis along the face plane. The efficiency calculation is performed for 1ST
number of particles.
Subroutine BIAS calculates the actual size distribution from the measured
size distribution. It uses the efficiency calculated by the subroutine INER.
The Stokes number of a given particle K is also calculated by BIAS. The
measured fraction PF, particle size P, relaxation time TAU are input to the
subroutine BIAS. The maximum number of particle intervals that can be used
is 10.
The input data and their description follows. The recommended values for
non-physical variables are also given.
115
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CAkT; 11: --- It+I^/H 1(1) hOR>AT (6F 10.4)
AXIAL muRDINAUb OF GRID POINTS. FUK NUIKsutl ZU) IS LALLULAItb
HY 'Fiov.l-11. FOR NDI^=2 INPUT HY UStR. lHt«fc •-ILL. bfc
l*/b LATA CAKUS.
CARDS 11 + 1" /h--11 + If/H+ J*/b=N(SA> ) HI) HiKMAT (HF 1 0 . <4 )
kATIAL CnOKDlNATKS U(- GRIP PU1MS.KJK NOlM=Otl KC1) IS LAUULAItU
HY 'FLt^FJ'. hOH NDJ^s? IKPUT hY UStH. ThLRK ^ILL fit
JM/£ IATA CARHS.
CAkf; N--N+IM* JI-./H =M(SAY) L'Z(1»J) HJk M A T ( Hf- 1 0 . 4 ) UM 1 it- NUJM=2
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AXIAL POSITION Of- Tut HANTILLL.
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-------
For NDIM options of 0 or 1, the data cards 3 through 'MM' are punched out in
the same order by program 'FLOWFI1. However, for NDIM=2, the user has to
calculate the axial and radial coordinates as shown below.
The flow region of interest is given by upstream boundary (ZM), downstream
boundary (ZMA) and radial boundary (RO). Assuming the face of the sampler is
located at the origin, then
,„ . -ZM
JGR = + 1
where AZ, AR are the grid sizes in axial and radial direction. Coordinates of
the grid points are
Z(I) = ZM + AZ * (J-l) for I = l.IM
R(J) = AR * (J-l) for J = 1,JM
For typical values of ZM=-5, ZMA=5, R0=5 and AZ=AR=0.2, IM=51, IBW=26, JM=26
and JGR=6.
The axial and radial velocity data cards are omitted for option NDIM=2.
118
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.0007
.0007
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1.0000
1,0290
1.0219
1.0100
1.0022
.9977
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-------
SAMPLING BIAS DETERMINATION
THJAtCTORY CALCULATION
*ALL
TAptKlNG
1.500000
SSS .100000
90.000000DEG
HELD BOUNDARY
FLOW HELD FOR CIRCULAR TUBE
AXIALLY SYMMETRIC CASE
ZLlHs .000000
SEDIMENTATION PARAMETERS 0
ELECTROSTATIC FIELD PARAMETER* o
SEDIMENTATION TRAJECTORY PARA ITS«
US(HICRONS) FRACTION
i.b 0.1
7.5 0.2B
b 1 . 0
,000000
a. 9799^9
3.979998
a, 989997
b. 009997
b. 009997
O.i?
0,06
WALL PARAMETER
2
•«,999998
-a.019999
-3,019830
-2.01779«
•1.008228
-.489925
»,016*187
,001048
.001048
1.224754
1.224403
1.220267
1.198568
1.I6J534
1.103450
1.101434
1.101434
124
-------
,000000
,94000y
1.940000
2,939999
3.939998
4.469997
•a.999998
•4,059999
•3.059878
•2.057465
•1.043345
-.491433
.000842
1.024754
1.024754
1,024508
1,020901
,999490
.956265
.644296
,000000
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l.UOOOO
P. 140000
3.139999
4,069998
4,569997
4,929997
•4.999998
•4.P59998
•3.859999
-2.859684
•1.656431
-.913105
-.39P799
.013838
1.124754
.124754
.124753
.124218
.118748
.093251
1,045445
.909776
CApTuKtQ
.000000
,260000
1.200000
2.2599^9
3.2599V8
4,12999*
4.629Q97
U.9494V7
•4.999998
•4.739998
•3.739999
•2.739573
•1,735871
-,85369b
-.344664
.001965
1,174754
1.174754
1.174747
1.173951
1.167162
1.140491
1.092050
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CApTUWtD
.000000
.340000
1 .340000
2.339999
3.339998
4.169998
4.669997
4.989997
4,989997
•4.999998
•4,659998
•3.659995
•2.659484
•1.655U35
-.B13938
-.307929
,007761
,007781
1.19975'J
1.199/54
1,199737
1.198730
1.190942
1,163512
114391
1.052651
1.052651
1
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1,340000
2.3399V4
3.339998
4,169998
4.669997
a.969997
4,969997
•4,999996
•4,659998
•3,659995
•2,659475
•1.655329
-.813385
-.306744
.006864
,006864
1.187254
1.187254
1.187238
1.186239
1,178451
1.150797
1.100462
1.021694
1.02l69a
.000000
•4.999998
125
1.181004
-------
.381)000
1.360000
2.379999
3,37999ft
a.189996
4,689997
4.959997
4.959997
•«.619998
•3.619992
-2.619412
-1.61«929
-.792677
-.286671
.005292
.005292
.161004
.180984
.179879
,17154ft
.143184
.090367
.006424
.006424
MAX. POSITION RATIO=
VALUt Of- 1CAT IS= 0
1.500000
RATIOS 1,500000
CONC, RATIOS ,92491ft
STo*ts NUMBER* .ooeooo
CRITICAL ORU.s 1.177870
.000000
2b.b
51.0
0.093
0.261
0.2?3
0.340
0.070
126
-------
GLOSSARY
A-J Aspiration coefficient due to inertia
B Non-dimensional function (Equation 8)
C Measured concentration (#/cm3)
C0 Actual concentration (#/cm3)
D Length of line sink (cm)
d Diameter of particle
E Efficiency of capture
f Fraction of diameter with line sink
F Pate of suction of sample (cm3/sec)
Fe External force (gnrcm/sec2)
Fr Froude number (Vst/L)
f.j Fraction of itn particle group
g Acceleration due to gravity (cm/sec2)
H Half width of channel (cm)
I Grid point along Z direction
J Grid point along y or r direction
K Stoke's number (Uft/L)
L Characteristic length (cm)
m Sink/Source strength
n Number of particle groups
Pe Peclet number (UjL/E)
Qp Electrostatic charge on particle (statcoulums)
Qc Electrostatic charge on collector (statcoulums)
R Radius of the tube
R0 Radial boundary of flow field
r Cylindrical coordinate
Sc Schmidt number
t. Time scale
127
-------
u Suction velocity of sample
DI Free stream velocity
U Flow velocity vector
Ux, Uy Flow velocity along x + y direction
UY, Uz Flow velocity along y + z direction
v" Particle velocity vector
Vs Terminal settling velocity of particle
W Width of sampler wall
x,y,z Cartesian coordinates
Ay, Az, Ay Grid size intervals in y, z, y directions
ty Stream function
ty{_ Stream function at center line
a Orientation of sampler face plane to the direction of flow
T Relaxation time of particles
n Efficiency of sampling
OUSGPO: 1981 — 757-074/1075
128
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