Table Of ContentLecture Notes in
Engineering
Edited by C. A. Brebbia and S. A. Orszag
38
S. J. Dunnett
D. B. Ingham
The Mathematics of
Blunt Body Sampling
Spri nger-Verlag
Berlin Heidelberg New York
London Paris Tokyo
Series Editors
C. A. Brebbia . S. A. Orszag
Consulting Editors
J. Argyris . K -J. Bathe' A. S. Cakmak . J. Connor· R. McCrory
C. S. Desai· K. -Po Holz . F. A. Leckie' G. Pinder· A. R. S. Pont
J. H. Seinfeld . P. Silvester' P. Spanos' W. Wunderlich . S. Yip
Authors
Sarah Jane Dunnett
Derek Binns Ingham
Dept. of Applied Mathematical Studies
The University of Leeds
Leeds, West Yorkshire LS 2 9JT
Great Britain
ISBN-13: 978-3-540-50147-3 e-ISBN-13: 978-3-642-83563-6
001: 10.1007/978-3-642-83563-6
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2161/3020-543210
III
ABSTRACT
The sampling of airborne particles in an environment is very
important in assessing the risk to health of workers. Blunt sampling
devices are usually employed to collect the particles and the rate
of aspiration will depend on the features of the fluid flow, which
are influenced by the sampler, and the size and shape of the
particles. Although much previous research has been performed on the
problem of blunt body sampling there is still a need for a greater
understanding of the factors which govern the performances of such
devices. In this book the case of sampling, by aerodynamically blunt
bodies, is investigated mathematically for different situations.
Existing approaches to the problem are described and an attempt is
made to extend and improve upon them. Also, a new approach is
described which includes the use of the Boundary Integral Equation
method. This method involves an approximation of Greens Integral
Formula and its advantages lie in its adaptability to model
different shaped samplers with sampling at arbitrary positions and
to include the finiteness of the orifice in the model. Initially
two-dimensional samplers are considered and the results obtained
indicate the usefulness of the method for modelling blunt sampling
devices. This method is then extended to three-dimensional samplers
and one sampler in particular, the human head, is investigated in
detail.
Also, using an analytical approach, the effects upon aspiration of
the particle shape and the finite particle Reynolds number are
investigated, in order to gain a better understanding of the
performances of particle samplers.
CONTENTS
Chapter 1 General Introduction 1
Chapter 2 A Review of Past Work on Particle Sampling 11
Chapter 3 A Comparison of Analytically Derived Expressions in
Blunt Body Sampling with those Derived Empirically 47
Chapter 4 A I30undary Integral Equation Analysis of Blunt Body
Sampling in Two Dimensions 67
Chapter 5 Mathematical Investigation of the Sampling Efficiency
of a Two-Dimensional Blunt Sampler 89
Chapter 6 The Effects of the Particle Reynolds Number on the
Aspiration of Particles into a Blunt Sampler 101
Chapter 7 TIle Aspiration of Non-Spherical Particles into a Bulky
Sampling Head 117
Chapter 8 An Empirical Model for the Aspiration Efficients of
Blunt Aerosol Samplers Orientated at an Angle to the
Oncoming Flow 139
Chapter 9 Use of the Boundary Element Method for Modelling TIlree
Dimensional Samplers 163
Chapter 10 The Human Head as a Blunt Aerosol Sampler 187
Chapter 11 Conclusions 217
References 223
NOMENCLATURE.
a Characteristic length-scale of blunt sampler.
Cross-sectional area of thin-walled sampler.
Cross-sectional area of streamsurface on the plane at
which the flow changes from diverging (or converging)
to converging for a blunt sampler.
a Cross-sectional area of orifice for a blunt sampler.
2
A Aspiration coefficient.
A, Aspiration coefficient relating to the flow at a distance
from the orifice of a blunt sampler.
Az Aspiration coefficient relating to the flow in the
vicinity of the orifice of a blunt sampler.
b Half length of rectangular sampler.
B Bluntness of the sampler.
c Dust concentration entering the sampler.
c, Dust concentration at the plane at which the flow changes
from diverging (or converging) to converging for a blunt
sampler.
C Parameter related to the Stokes number.
d Diameter of particle.
D Diameter of thin-walled sampler.
DELTA Dimensionless inlet width.
E Coefficient of distortion due to particle bounce, for a
r
thin-walled sampler.
Et Coefficient of distortion due to internal deposition for
a thin-walled sampler.
f Function of ~ determining s for a blunt sampler at an
angle to the flow.
VI
F Flux parameter.
G Rate of shear.
H Distance upstream from the thin-walled sampler at which
the flow is undisturbed by its presence.
k Empirical constant in expression for A for a thin-walled
sampler.
Empirical constants for a blunt sampler.
Length of thin-walled sampler.
m Mass of particle.
Fluid velocity.
Q Volume of fluid withdrawn through the orifice of a blunt
sampler per unit time.
R Velocity ratio (=Uo/vm).
Re Reynolds number of the particle (=d[(U_u)2+(V_v)2jl/2/v).
p
Re Reynolds number based on the sampler diameter and the
v
m
inlet velocity (=Dvmlv for a thin-walled sampler,
=av Iv for a blunt sampler).
m
s Characteristic dimension of the region on the blunt
sampler enclosed by the limiting streamsurface.
St Slokes number (=(d2p U )/(18~D) for a thin-walled
p 0
sampler, =(d2p U )/(18~a) for a blunt sampler).
p 0
St' Adjusted Slokes number for a sampling at an angle ~ to the
flow (=Stexp(O.022~».
Sli Slokes number for a blunt sampler determined by the inlet
diameter (=(d2p U )/(18~o».
p 0
Slokes numbers associated with the flow at a distance
from, and in the vicinity of the orifice, respectively
for a blunt sampler.
VII
t Time.
u,v,w Components of particle velocity in x,y,z directions
respectively.
U,V,W Components of fluid velocity in x,y,z directions
respect i vely.
U Freestream velocity.
o
U, Speed of the fluid and particles at the intermediate
plane for a blunt sampler.
v Mean velocity of sampling.
m
« Angle the direction of motion of a particle makes with
p
the horizontal.
~ Angle the sampler axis makes with the flow.
~c Vertex angle of conically shaped sampler.
~r Blockage ratio for a blunt sampler (=l-a/a in two
dimensions, =1_a2/a2 in three dimensions).
Fluid density.
7
Half the inlet width of a blunt sampler.
Angle the plane of a particle makes with the horizontal
plane.
Particle Reynolds number (=(Uod)/V).
Fluid viscosity.
Points at which the velocity towards the orifice is zero
for a three-dimensional blunt sampler.
Particle density.
Fraction of particles in defected fluid volume which, due
to their own inertia, enter the thin-walled sampler when
it faces the flow.
VIII
Fraction of particles in defected fluid volume which, due
~'
to their own inertia, enter the thin-walled sampler when it
is at an angle to the flow.
Values of ~ associated with the flow at a distance from,
and in the vicinity of the orifice, respectively for a
blunt sampler.
Ratio of the volume of the fluid sampled to that incident
on the sampler body ( for a blunt sampler, =(ov )/(aU )
m 0
in two dimensions, =(o2uo)/(a2vm) in three dimensions).
~ Velocity potential.
~£ Perturbation potential.
~ Stream function.
Coordinate Systems.
x,y,z Cartesian coordinates.
r,e,a Spherical polar coordinates.
CHAPTER 1.
GENERAL INTRODUCTION.
3
1 Introduction
Particle samplers are widely used in workplaces in order to
determine the concentration of airborne particles in the atmosphere.
They generally operate by drawing air, with the aid of a pump,
through one or more orifices in the sampler body and housed within
the sampler is a filter through which the air is subsequently drawn.
The airborne particles are collected on the filter and their
concentration is determined. Various samplers have been designed for
this purpose including "static" samplers, which are located in a
fixed position in a working environment and determine the dust
concentration averaged over a prescribed period of time at that one
point, and "personal" samplers which are mounted on a working person
near to the breathing zone. The ORB sampler, a static sampler
designed by Ogden and Birkett (1978) to have approximately the same
entry efficiency, for particles with aerodynamical diameter up to at
least 25~m, as a human head equally exposed to all wind directions
for wind speeds between 0 and 2.75m1s, is shown in Fig.l.l and
examples of personal samplers are shown in Fig. 1. 2a, b and c and
represent a single 4mm hole sampler, a seven hole sampler and a 25mm
open face filter holder respectively. These three samplers are some
of the most commonly used personal samplers for sampling the total
airborne concentrations of workplace dusts in Britain.
It is important that the particle sample collected on the filter
within the sampler is representative of the particle concentration
upstream of it, such a sample is known as a true sample. If changes
do occur they should be known or predicted as accurately as
possible.
The physical presence of the sampler and the action of sampling
