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Lecture 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 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9,1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin, Heidelberg 1988 The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. 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