Human head model as an aerosol sampler: Calculation of aspiration efficiencies for coarse particles using an idealized human head model facing the wind

Pergamon

J. At,roaol 8c~ VoL 26, No. 2, pp. 25~-27Z 1995 Copyright C 1995 ~ ~ Lid Printed in Great Britain. All fil~t~ nmav~ 0021-8502/95 $9.50+ O.O0

0021-8502(94)0010%3

H U M A N HEAD MODEL AS AN AEROSOL SAMPLER: CALCULATION OF ASPIRATION EFFICIENCIES FOR COARSE PARTICLES USING AN IDEALIZED H U M A N HEAD MODEL FACING THE WIND Serap Erdal* and Nurtan A. Esmen t

*EA Engineering, Science, and Technology, 8520 154th Avenue NE, Redmond, WA 98052, U.S.A. tEsmen Research and Engineering, 9800 McKnight Road, Suite B210, Pittsburgh, PA 15237, U.S.A. (First received 22 November 1993; and in finai form 2 August 1994) AbstractqTheoretical quantification of the entry of coarse particles to the human mouth or nose under different inhalation rates and ambient wind velocities is an important topic in understanding inhalability, in extrapolating empirical results and designing inhalable aerosol samplers. Using an idealized human head model facing the wind and potential flow approximation for the velocity fields, coarse particle aspiration efficiencies were calculated for a set of inhalation and external fluid motion parameters. The inhalation parameters are orifice size, the frequency and the tidal volume of breathing which depend upon the level of exertion assumed as well as interpersonal variabilities. The external fluid motion parameters are introduced by the fluid motion external to the head and the orientation of the orifice axis with respect to the flow axis. The total velocity field was modeled by superimposing the velocity field generated by breathing onto the external velocity field generated by an ambient wind. In this three-dimensional velocity field around the idealized human head figure, the particle equations of motion for coarse particles result in a set of coupled ordinary differential equations. The numerical solution of this set can be used to obtain particle trajectories and particle aspiration efficiency. The aspiration efficiency calculation for 25-200 lan particle aerodynamic equivalent diameters, for tidal volumes of 750, 1450, 2250 cm s and for 0.5-9 m/s ambient wind speed conform to the main trends observed in the experimental studies. Thus they may be used to extrapolate the experimental findings. Most importantly, the calculations did not suggest an evidence of zero aspiration efficiency (cutoff) for aerodynamic equivalent diam~ers up to 200 pan for the range of inhalation and external fluid motion parameters examined.

NOMENCLATURE radius of mouth, c m

A(d., Uo, UO b C~ du f(d.®, Uo, UO g K~ 1 P~,P.P. Re

Uo UI

Vxb, V,b, V. Vxo, V,o,V,o u~,, v,,,,,v . ,

u~, v~, u~ ~o, V,o,V~o VTs X-t. Xb y+ Z+ Z!

aspiration efficiency radius of human head, cm capture space box particle aerodynamic equivalent diameter,/an frequency of favorable outcomes gravitational constant, cm/s 2 drag correction factor to extend Stokes' law distance between the center of circular mouth opening and the plane of z--0, cm components of particle relative velocity, cm/s particle Reynolds number ambient air speed, cm/s Inhalation velocity amplitude, cm/s components of the velocity field generated by ambient wind around the semi-infinite cylindrical body, cm/s components of the velocity field generated by ambient wind around the the hemi-spherical cap, cm/s components of the velocity field generated by breathing motion, cm/s components of the velocity field generated by combined inhalation and external velocity fields, cm/s particle velocity components when t --- 0, cm/s particle terminal settling velocity, cm/s dimensionless space variable (x/a) in the direction of flow horizontal coordinate of the center of circular mouth opening, cm dimensionless space variable (y/a) orthogonal to flow in the horizontal direction dimensionless space variable (y/a) orthogonal to flow in the vertical direction vertical coordinate of the center of circular mouth opening, on velocity potential 253

254

S. Erdal and N. A. Esmen amplitute of the sinusoidalwave stream function particle relaxationtime,sec

1. INTRODUCTION If the objective of air sampling is to assess the nature and the magnitude of a potential health risk due to inhalation of airborne particles, in general, the particles collected by a sampler should be representative of what is inhaled by a human subject under given receptor and ambient conditions. Vincent and Mark (1984) suggested that biologically relevant definition of total dust sampling should represent total amount of particles which enter the nose and/or mouth during the act of breathing. Therefore, the inhalability, I(d), of the human head acting as a dust sampler is governed by the same factors as the entry efficiency, E(d), of the dust sampling instrument which one might use to separate the inhalable fraction of airborne dust. While fine particles such as those considered to be in the respirable sub-fraction tend to follow closely the motion of the air, the coarse particles do not exactly follow the flow streamlines due to their appreciable momentum and terminal settling velocities. Therefore, the concentration of coarse particles entering the nose and/or mouth, or entering a sampling device, and ambient concentration are not necessarily equal. Vincent and Mark (1981) showed that l(d) varies markedly with particle size and may differ significantly from unity. In its totality, the theoretical treatment of inhalation of coarse particles is very complicated. For example, the treatment of inhalation at the wake requires a level of theoretical understanding of flow separation around human head, hitherto unavailable. These difficulties not withstanding, the recognition of the fact that the human head is an imperfect dust sampler and it ought to be possible to design a sampler that mimics its performance has led to the definition of inhalability criteria, sampler inlet and aspiration efficiency studies and to some extent evolution of a new approach to particle sampler design. The experimental aspiration efficiency is defined by the ratio of the concentration of the aerosol collected just inside the head to the concentration of aerosol collected by thin walled isokinetic reference probes. This ratio is called aspiration ratio. capture efficiency, entry efficiency or inhalability. Intuitively, the aspiration efficiency is influenced by the fluid mechanical considerations determined by inhalation and external fluid motion parameters. The inhalation parameters are the size and shape of the mouth opening, the frequency and the tidal volume of breathing all of which change depending upon the level of exertion assumed as well as interpersonal variabilities. The external fluid motion parameters are introduced by the fluid motion external to the head and mouth system. The orientation of the mouth orifice axis with respect to the flow axis at infinity, fluid velocity field are the parameters which govern the external motion. The complexity of the fluid flow around a blunt body, particle fluid interactions for coarse particles and the relative instability of coarse aerosols introduce both experimental and theoretical difficulties. While there are a number of experimental studies currently in progress, the bulk of the data on aspiration efficiency studies was produced mainly by researchers in the Institute of Occupational Medicine (IOM) in Edinburgh, U.K. and Bergbauforschung in Essen, Germany. In these studies, using sculpted heads and tailor mannequins in wind tunnels, the aspiration efficiency of the human nose and/or mouth has been determined for the particle aerodynamic equivalent diameters up to 100/~m for different inhalation rates and ambient wind velocities (Ogden and Birkett, 1977, 1978; Armbruster and Brauer, 1982; Vincent and Armbruster, 1981; Vincent and Mark, 1982; Vincent et al., 1982, 1990). The results from these studies exhibit a number of important common characteristics. (1) In general, the inhalability decreases from unity as particle aerodynamic equivalent diameter, dae, increases from zero to 40 #m and levels out at about 0.5 for the larger sizes. (2) Within the range of experimental conditions examined, at least for particles up to 100/2m dae, there is no evidence of zero aspiration efficiency, i.e. particle size cutoff. (3) Although the measured aspiration efficiency for the different orientations of the head relative to the wind demonstrate a strong dependency on flow parameters, the inhalability data for the orientation

Human head model as an aerosol sampler

255

averaged cases appear to be relatively insensitive to variations in wind speed up to 4 m/s. For the higher wind speeds, there is an apparent dip in the inhalability of "smaller" sized particles and pronounced increase in the inhalability of"larger" sized particles. Also, for the orientation averaged case, the inhalability does not appear to be dependent on inhalation parameters except for the upturn in the inhalability at high ambient wind speeds, which seem to be more marked at the lower breathing that corresponds to respiratory level at rest. (4) The effects associated with mouth versus nose breathing and facial structural features seem to be relatively unimportant• The particle trajectory calculation procedure for inhalation by an idealized human head may be used to estimate the aspiration efficiency with certain limitations. The theoretical framework suggested in the idealized head model simplifies the flow field by keeping the head and mouth orientation within a range of angles with respect to ambient fluid flow (i.e. facing the wind) such that the boundary layer effects are minimized, thus the potential flow theory would be applicable, In addition, due to the limits imposed upon orientation, this approach does not consider the turbulent wake effects, flow separation and the complexities of back flow in the presence of aspiration in a wake. These limitations to the application of the theory notwithstanding, the calculation process proposed by this approach provides a vehicle to examine the behavior of aspiration efficiency as a function of particle and flow parameters from a fundamental, theoretical point of view. The simplest application of the theory is for the case when the velocity field far from the idealized head is assumed to be parallel to the mouth axis. In this case, the symmetry considerations simplify the calculations required by the model to an extent that a large number of changes in the parameters which influence the aspiration efficiency may be investigated. 2. D E S C R I P T I O N

OF THE

MATHEMATICAL

MODEL

The idealized human head model consists of two parts; one part is a semi-infinite circular cylindrical body and the other part is a hemi-spherical cap placed on the top of the circular cylindrical body. The model is taken to be a three-dimensional object with its origin at the point O as shown in Fig. 1. z A

Z=O •

J

•

I

S

Fig. 1. Schematic presentation of the human head model.

X

256

S. Erdal and N. A. Esmen

The coordinates x, y, z are chosen such that x-axis goes away from the face in the horizontal direction, y-axis points to the sideways either in the positive ( y > 0 ) or negative ( y < 0 ) direction and z-axis is a vertical one pointing upward in the opposite direction to the gravitational force. It should be emphasized at this point that the three-dimensional human head figure is symmetric only with respect to the y-axis. In other words, the results of any analysis done using y > 0 will be the same as using y < 0. In Fig. l, the plane z = 0 where the cylindrical body and the hemi-spherical cap intersect represents as arbitrary plane chosen for the purpose of mathematical simplicity so that cylindrical body and hemi-spherical cap can be conveniently treated separately. The human mouth is assumed to be a circular orifice of radius, a, placed on the cylindrical body at a location, representative of a human mouth with respect to the top of a head. As shown in Fig. 1, this location is designated by I which is the distance between the center of the circular opening of the mouth and the plane of z = 0. The radius of human head is presented by b. The constants of a, b and I were chosen to be 1, 7.5, and 9.5 cm respectively and these values of a, b and l were used throughout the calculations. For any condition which represents an external flow perpendicular to the axis of the cylinder, the x-axis of the model can always be taken to be parallel to the flow. Thus, the relationship between the external flow and the mouth is determined by the angle between the x-axis and the direction orthogonal to the mouth on the xy-plane (normal axis of the mouth). For the purposes of this paper, the normal axis of the mouth was assumed to be parallel to the x-axis. The representation of the fluid flow may be simplified considerably if the flow around the head model may be taken to obey the potential flow equations. The Reynolds' number of the model for ambient bulk velocities in excess of 0.5 m/s is in excess of 5000. Clearly either for a cylinder or for a sphere, the velocity and pressure distribution in a +30 ° arc from the front side stagnation point would be extremely well represented by potential flow theory. Therefore, it may be suggested that the potential flow is a good approximation of the flow field and the velocity field may be obtained by solving the equation of continuity and equation of motion with the assumptions that the fluid is ideal (p = constant,/~ = 0) and that the flow is irrotational [(V × U ] = 0) (potential flow). For an ideal fluid, the two-dimensional equation of continuity for steady flow is given by Bird et al. (1960): 8U~ , BUy

v. u=0,--, ~

,~

+ c?R~-v= v .

(1)

Based on the potential theory, the component velocities of a flow field may be expressed in terms of a single-valued functions of stream function qJ and velocity potential ~b (Lamb, 1945): 8T

v~-

8y '

8q~ Ux = - - , 8x

8T

u,.= 8~' 0~b Uy= ~--. c3'

(2) (3)

The physical significance of the stream function is that the streamlines are the curves actually traced out by the particles of the fluid in steady flow. In the model considered here, the flow field is a coupled velocity field composed of two different potential fields. The first one is the velocity field around the cylindrical body and the hemi-spherical cap generated by the ambient air flow. The second one is the velocity field generated by the inhalation pattern appropriate to the tidal volume of inhalation. For the sake of simplicity, we assumed a sinusoidal breathing pattern. The ambient velocity field around the head is superimposed on the velocity field accounting for the effect of inhalation and exhalation to create the resultant coupled velocity field which simulates the dynamic velocity field in front of a walking or a stationary receptor and consists the basis for all of the calculations carried out to characterize the coarse particle behavior under specific conditions. Using the potential flow formulation we seek to define the path of the fluid particles in the vicinity of the circular cylindrical body portion (z < 0) of the model. It is assumed that an

Human head model as an aerosol sampler

257

infinitely long circular cylinder of radius r is an infinite mass of fluid is placed with its axis normal to a flow of free stream speed Uo. The free stream velocity, Uo, would exist everywhere if the cylinder were absent and that still exists far away from the cylinder. The stream function for the ideal flow around such a circular cylinder in polar (cylindrical) coordinates is given by Lamb (1945):

b2 ~F = Uo (r - r ] sin 0.

(4)

The cylindrical coordinates of r, 0, z are related to the rectangular coordinates x, y, z by x=rcos0, r = ~

y=rsin0, 2 and

z=z,

O=arctan(y/x).

The stream function can now he rewritten in terms of x, y and z in rectangular coordinates by substituting the value of r and sin 0 into equation (4) and rearranging as follows: • =Uoy(1

x2¥y2). b2

(5)

The ambient air flow which generates the velocity field around the cylindrical body has the velocity components of Uxb, Uyb and U,b in the x-, y- and z-direction, respectively. Here, free stream is flowing in the x-direction. The velocity components in the x- and y-direction are defined by the direct application of equation (2). Since there is no fluid flow in the z-direction, the velocity component of Uzb=0, and

- a-y '

'b =-~X"

(6)

By introducing dimensionless variables given below, x + = x/a,

y + = y/a,

(7)

z + = z/a

the final set of equations which enable one to determine the component velocities of air flow around the cylindrical body portion (z < O) of the modelare For z < 0:

.l

Uxb = - o o

b+2 2y+2b+2 1 X+2+y+2 + (X+2+y+2)2j, F 2x + +b+2 n

U~b =O ,

(8a)

(8b) (8c)

with b + = b/a.

The equations (8a)-(8c) satisfy the equation of continuity, equation (1). Thus, OU~b ~Uyb + O U , b = O a x + t- Oy + Oz + .

The ambient air flow around the hemi-spherieal cap has the velocity components of U~, U,c and U,c. Although a three-dimensional stream function is necessary to define the flow field around this portion (z > 0) of the model, there is a stream function analogous to the two-dimensional stream function given by equation (5) in all eases where the motion of a fluid takes place in a series of planes passing through a common line, and is the same in each such plane. Based on the formulation given by Lamb (1945), with the parameters shown in Fig. 2, if two points of A and P are taken in any plane through the axis of

S. Erdal and N. A. Esmen

258

z

I

I

It

I

r ll

0

//

...¢/ //

z / w

A x..N. /L.-.-"'~a \\\

/"

N "N

___;__

"

. /i

/

//I

,Y

.

X Fig. 2. Schematic representation of spherical coordinate system and xw-plane. The ranges of variables are 0~
symmetry such that A is arbitrary, but fixed, and P is variable, then the flux across the annular surface generated by any line AP is a function of the position of P. This function can be expressed in terms of stream function. By taking the axis of x to coincide with that of symmetry, one may say that ue is a function of x and w, where x is the abscissa of P, and w = ~ / y 2 - t - 2 2 is its distance from the axis. The two-dimensional stream function for the ideal flow around the hemi-spherical cap is given as a function of x and w by equation (9). The curves of ue = constant are evidently streamlines of such flow. ue= Uow(1

x w2).

,9,

The component velocities, Ux¢ and Uwc, are Ux¢ =

due ~W

Uw~ = - ~ - .

(10)

GX

'

It may be shown that Ux¢ = - Uo 1

2(y2 +22)b2 X2-t- y2 + z 2 + ( x 2 + y 2 + z 2 ) 2

vF2xwb

l

J

(lla) '

(llb)

Equations (11a) and (1 lb) satisfy the equation of continuity as

ou= ouwo ouwc

0x + - C

=°

Using Fig. 2, it is easy to show that Uy¢ = U,,¢ cos a,

(12)

U:~ = U ~ sin ct.

(13)

Human head model as an aerosol sampler

259

The final set of equations which are used in the calculation of the component air veloeities around the hemi-spherical cap portion (z >0) of the model can be written in terms of dimensionless variables of equation (7). For z > 0 :

[

Ux¢=-Uo_l

b÷2

2(y.2÷z--)b+2 ]

x+2+y+2+z+2 + (x+2+y÷2+z+2)2_],

v,o= Uor [

2x+y+b+22)2~,.~ 2x+z+b+2

(14a) (14b)

]

Vzo=Vo[i + + y-7 + ;+ 2)2j. The velocity components of the fluid flow around the cylindrical body (z < (3) and the hemi-spherical cap (z > 0) in each direction must be equal to each other when z = 0, since the flow must be continuous across this plane. One can prove that the following set of relationships will be obtained if z = 0 is inserted into the equations (14a)-(14c). Forz=0:

The breathing is simulated in the mathematical model by assuming that the breathing follows a sinusoidal wave pattern consisting of two stages. Although in reality there is a pause between the two cycles, the pause is neglected in the model and exhalation is represented by the second half of the sinusoidal function. The amplitude of the sinusoidal wave, ~,, is given by

~=ct + 2c; where c=~/2. (15) As shown in Fig. 1, xb=x--b, y=y, z,=z+l are the coordinates of the center of the circular opening (mouth). The velocity field generated by inhalation resembles the aspiration of air into the circular orifice in an infinite plane. Based on the development of potential flow equation given by Lamb (1945), using an arbitrary plane xw with w2=y2+z 2, the potential flow is described by X2

W2

sinh2~

t- ~ = a

2.

(16)

Since cosh2~b = 1 + sinhZ~b, equation (16) may be rewritten as x2(1 + sinh2tb) + (y2 + z~) sinh2~b = a 2 sinh2~b (1 + sinh2~b).

(17)

a2sinh*~b _ (x 2 + y2 + z 2 _ a 2) sinh2 q~_ x 2 = 0.

(18)

With arranging

The roots of equation 08) are sinh2~ = (x 2 + y2 + z~ - a 2) + ~/(x 2 + y2 + z 2 _ a2)2 + 4a2x~ 2a 2 t.

(19)

I

Let the dimensionless variable A be the right-hand side of equation (19). Thus, = sinh-1 ,,/~.

(20)

By introducing dimensionless variables x~, y+ and zz+ defined as,

x~ =xda,

y+ =y/a,

z(~=zt/a,

(21)

the dimensionless velocity components of fluid flow generated by breathing motion may be

260

S. Erdal and N. A. Esmen

calculated from the set of equations presented below. U+_

Oq~_ 1 c3x 2 ~

dA A dx~-'

(22a)

U~--

c~q5 ~Y

1 dA 2x/A 2 + A dy + '

(22b)

i

v;-

dz

2 ~

dA A dzl+ "

(22c)

These velocity components U~+, U~- and U~+ have to be corrected so that tidal volume of inhalation controlled by inhalation (or aspiration) velocity U~ and time-dependent sinusoidal wave pattern of breathing can be incorporated into the mathematical model. The corrected velocity components Uxb~, Uybr, Uzbr are Uxbr=

Ut 2 ~ A

dA dx~ sinqJt,

(23a)

Urbr--

Ui 2 ~ A

dA dy + sin Ot,

(23b)

Uzbr=

Ul 2 ~ A

dA dzt+ sin~Ot.

(23c)

The set of equations (23a)-(23c) are the final set of equations which are used to calculate the velocity components of the fluid flow field induced by the breathing motion. Here, (dA/dx~ ), (dA/dy+), (dA/dz~ ) are given by dA dx• =x] I

x/~

dA Xb+2 q--y+ 2 q.- Z/+ 2 -- 11 =y+ [1 + dy + dA

--=z? dz?

,

(24a)

,

(24b)

[1-4X;2+y+2+Z~-2--11 7./-; J'

(24c)

here, N=(x~ -2+y+2 + z / e _ 1)2 + 4 x ; 2 . These individual velocity fields may now be superimposed to provide the overall velocity field associated with the idealized human head model acting as an aerosol sampler. The velocity components of the coupled velocity field are For z<0:

U2 s = Uxb~+ U~b,

(25a)

U7 s= Uybr+ Urb,

(25b)

U [ s= U~br+ U~b.

(25C)

U~+Sm - U~br+ Ux¢,

(26a)

U~-s= Uybr+ U~c,

(26b)

U+~s= U~br+ U~c.

(26c)

Ux+S =U2 s,

(27a)

For z>0:

When z = O:

Human head model as an aerosol sampler

261

U ; s = u~-S,

(27b)

U+ s = U7 s.

(27c)

For convenience, the velocity components of the overall velocity field are represented by U s , U s , U s with the understanding of the proper definitions given above. 3. C A L C U L A T I O N OF P A R T I C L E TRAJECTORIES When the particle motion obeys the Stokes' law (Re ~ 1), the equation for the curvilinear motion of particles in vector form is given as (Fuchs, 1964) dV ( U - V ) --= --+--, dt z

F m

(28)

where m

6n#r here, m is the particle mass, V and U are the velocity vectors of the particle and the medium, F is the external force vector, r is the particle radius assuming that particles are rigid spheres,/z is the absolute viscosity, and z is the relaxation time for the aerosol particle. Although, this equation applies to spherical particles in the strictest sense, if r is taken to be the aerodynamic equivalent radius, then the equation can be extended to all nearly isometric particles. The equation of particle motion is more complicated at large Re where inertial forces cannot be neglected in comparison to viscous forces. The generalization of the equation of particle motion to include coarse particles may be achieved by the inclusion of the drag coefficient factor, K,, into the equation (28): dV ( U - V ) . F -&-= z /%+m"

(29)

If a threc-dimensional system is considered such that the only external force is the gravitational force, g, acting in the negative z-direction, then components of equation (29) with the introduction of a relative velocity vector, P = U - V , will be:

dPx_dUx .Px K,, dt

dt

dPy_dUy PYK,, dt

dt

dP~ dU~ = - dt dt

- -

(30a)

z (30b)

z P~ K, z

+

g.

(30c)

For K,, a sixth-order alternating sign polynomial as Reynolds' number dependent correction of Stokes' solution was recently reported for Re up to 200 by Erdal and Esmen (1990). K r = 1 +9.5977436 x 10 -2 R e - 2.4461587 x 10 -3 Re 2 +3.9659374 x 10 -5 Re 3 -

3.1005466 x 10-7 Re 4 + 1.1330414 x 10-9 Re 5 - 1.5574985 x 10-12 Re 6.

(31)

Here, Re number with pf as density of fluid is Re -

2rPpf

,

(32)

and 2 + Py2 + Pz.

(33)

With the expansion of the first terms on the right-hand side of equations (30a)-(30c),

262

S. Erdal and N. A. Esmen

dUx/dt, dUy/dt, dUJdt, by the application of the chain rule, and the insertion of Vx=dx/dt= U~-P~ (for y- and z-direction as well), another set of equations for dUx/dt, duff&, dUz/dt may be obtained. dUx dU~

dt

dU~

dU~

(34a)

dx (Ux-Px)+~y (Uy-PY)+-~z (Uz-P~)'

dUrdt- d U Ydx(U,,-P~)+-~y(Uy-PO+ d Uy _~ff (U~-P~),

(34b)

dUz dUz ~ dt - dx (Ux-Px)+ (Ur-Py)+

(34c)

(Uz-P~).

The final set of equations describing the curvilinear motion of particles can now be obtained by placing equations (34a)-(34c) into equations (30a)-(30c) for each dimension.

dPx (dUx u=+d.~ U +dUx

( dr,, dP'=.dU'u~,+ d~y Uy+ dt

\dx

/dU:, K,\

)

dz

]

-

\dy

r]

/dU~\ p P,

-\~-x]

/dUx\

-\~-z]

P:,

(35b)

With the airflow field defined, i.e. the velocity of the fluid flow is known at all points in the vicinity of the head, the calculation of the actual particle trajectory as the particle proceeds through the three-dimensional overall velocity field may now be performed by inserting the coupled velocity components, dU s, dU s, dU s in terms of dimensionless variables of x +, y+, z ÷ into the equations (35a)-(35c), and solving differential equations of particle motion along with equations describing the displacements of the particle in x +-, y +- and z ÷- directions.

dP~ l~<(dUS s dUS s dUS

) / d U s aK,\p

:a

/dUS\

-)

.(dUS\

}

'

(36a)

dPr

l J'(dUs

/dUSt aK,\p _{dUSr~

dUrs s dUrs s \

/dUSt\

) (36b)

s

s

"~ /dU s aK,\

/dUS\

dPz a t\~x ~<(dUs+ -="s4-dU:---~ dy -'tss-~dU'-~dz - Us ) - t ~ z ~ + ~ ) P : - t ~ x +

/dUS\

)

)Px-t~y+ )Pr~+g' (36c)

dx += 1 (uS _ Px),

(36d)

dy+ -~ = (uSy-Py),

(36e)

dz + 1 s =-(U~-P~), dt a

(36f)

dt

a

where For z<0:

dx + = \dx~Jb, + \d--~Jb'

(37a)

Human head modelas an aerosol sampler

(dVx dy+ = td-~-;b + \ d y + ] b '

dU s [dU~'~

[dU~

dU s fdU~'~

{dU~'~

dUS~ {dU,'X

{dU~'~

dU s [dUx'~

fdU~

263 (37b) (37c)

For z>0: (38a)

(38c)

Equations (36a)-(36f) form a set of six ordinary differential equations coupled by the correction factor, K,, which is a function of the magnitude of the total relative velocity, P. Beginning from any point where magnitudes of fluid and particle velocities are known, solution of this equation set defines a trajectory of a certain size particle for a given free stream velocity and inhalation velocity. 4. C O M P U T A T I O N METHODS AND D E F I N I T I O N S Two different numerical approximation methods were used to solve the system of equations (36a)-(36f) for particle position and relative velocity as a function of time. These methods are (1) The fourth-order Runge-Kutta followed by Adams-Moulton (Gerald and Wheatley, 1989). (2) Backward differentiation formula method of the differential equation solver, DI VPAG, in the IMSL software package (IMSL, 1989). DIVPAG is a predictor-corrector type solver and finds approximations to the solution of a system of first-order ordinary differential equations of the form y' =f(x, y) based on initial conditions (IMSL, 1989). In the numerical analysis of a mathematical problem without an analytical solution, the numerical results of a particular method used to solve the problem may be validated by using another independent numerical method. While there is small chance that two numerical methods will agree only on the number of validation cases chosen, and nowhere else, such an occurrence is not likely. The two different methods were, therefore, used to provide an independent validation of the results presented here. With, h as the step size of the Adams-Moulton method, it is found out that the percentage difference between the numerical results of the two methods is in the order of 10-2 when h = 10-4 and in the order of 10 -a when h= 10 -5. DIVPAG does not have a fixed step size and has the advantage of optimizing the step size at each calculation step based on an error test. Therefore, the more efficient and quicker DIVPAG was used as the method of choice for calculating particle trajectories. With a known free stream velocity Uo and inhalation velocity Ui, the solution of the problem defines a trajectory of a particle of a given size (d,e) for certain initial conditions of position Xo, + Yo, + Zo+ and particle velocity components ~o, ~o, ~o- In carrying out the calculations, the terminal settling velocity for each particle size was calculated using the numerical techniques discussed above and the results at the precision level required for the numerical methods are presented in Table 1. The calculation procedure is simple, beginning from one of the initial points presented where ~o = Uxo, ~o = 0 and ~o = VTs, the particle path was calculated by solving equations (36a)-(36f) for given Uo, U~ and particle size (d,e) by the method of DIVPAG. The calculation is started when independent variable, time, is equal to 0 (t = 0) and solution for the dependent variables of relative velocity components,

264

S. Erdal and N. A. Esmen Table 1. Terminal settling velocity (I/rs) for each particle size (d,c) d., (#m)

Vrs (cm/s)

25 50 75 100 125 150 175 200

1.845059190438658 7.235726859943728 15.5223069880543 25.59793079544202 36.43792472232983 47.45081551401448 58.42940784888413 69.38118147255223

Px, Pr and P~, and the displacements of the particle in the each direction, x, y and z, is obtained at every 0.1 s. After each solution at this small time segment, the computer program performs an inhalation test. This test checks the following conditions and assigns an outcome to a specific category; (a) If z < - 3 5 , then the particle is assumed to be settled. (b) If x < 3.0, then the particle is assumed to go around the head with the air flow. (c) If z < 0 and (xS+y2)~<(7.51) 2, then another condition is checked whether the particle is in the mouth or not. If it is found to be in the mouth, then it is categorized as inhaled, if it is not, then it is regarded as hit the face. (d) If z/> 0 and (x2+ y 2 + z 2) ~<(7.51) 2, then it is assumed that the particle hit the cap portion of the head. (e) While the particle is moving towards the head, if t becomes ~>2 s, then the calculation is halted at that point, since next 2 s will be spent on exhalation. Thus, all of the calculations were based on one single breath and inhalation cycle which lasts 2 s. In each cycle, the trajectory computation was continued until one of the conditions of the inhalation test outlined above was met.

-4

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The Distonce in X-Direction [cm] Fig. 3. The results of particle trajectory calculations for the particle sizes of 25-200/am, the wind speed of Uo = 0.5 m/s and the inhalation velocity of U~ = 362.5 em/s when the particle starts its motion at Xo = 32.5 cm, Yo = 0 and Zo = - 5 cm.

Human head model as an aerosol sampler

265

-4.

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Fig. 4. The results of particle trajectory calculations for the particle sizes of 25--200 p_rn. the wind speed of Uo = 4 m/s and the inhalation velocity of U~ = 362.5 cm/s when the particle starts its motion at Xo = 32.5 era, Yo = 0 and Zo = - 5 cm.

Two examples of particle trajectories calculated for particle aerodynamic equivalent diameters between 25 and 200 #m, in conjunction with ambient wind speeds of 0.5 and 4 m/s. In these calculations, initial particle coordinate of Xo= 32.5, Yo=0 and Zo= - 5 cm was used. The results of these calculations are shown in Figs 3 and 4. With the development of the numerical methods for the calculation of particle trajectories, the aspiration efficiency of the model head may be calculated. Consider a monodisperse aerosol uniformly distributed in space such that all particles have reached their terminal settling velocity. With a specified set of inhalation and external fluid motion parameters for a head in this space, there is a subspace enclosed by a surface (capture surface) connected to the mouth such that all particles found in that subspace (capture space) at time zero are inhaled within one inhalation cycle and none found outside that subspace is inhaled in that cycle. Clearly, the volume of the subspace defined by the capture surface is a function of the inhalation and external fluid motion parameters and the particle size. Furthermore, for a specified set of inhalation and external fluid motion parameters, the size specific aspiration efficiency, A(dae, Uo, U, . . . . . ), may be defined as the ratio of the volume of capture space for d=dae to the volume of capture space for d=0. In order to determine the capture volumes necessary for the computation of aspiration efficiency, a Monte Carlo method of integration was used. In a Monte Carlo integration to determine a volume, only the maxima and minima in each coordinate direction is required. Therefore, it was unnecessary to determine the capture surface in great detail. An idealized head is represented by a semiinfinite cylinder and a hemi-spherical cap. The idealized head model used has its cylindrical axis coincident to the vertical (z-axis). Its mouth axis is orthogonal to the yz-plane (in the wind direction) with its origin located at (7.5, 0, 9.5). Consequently, the xz-plane is the plane of symmetry and the particle size dependent maxima and minima of the capture surface boundary for x and z directions occur in xz-plane, for the y-direction occur in the plane AS 26;Z-G

266

s. Erdal and N. A. Esmen

which contains the line segment between point (7.5, 0, 9.5) and the maximum in the x-direction and the line (7.5, 0, y). The computations necessary to determine the bounds of the capture surface for a given set of conditions may be performed in the xz- and xy- planes separately using a simple systematic search process that tests the trajectory of a particle located at the nodes of a grid drawn on each plane for its entry into the mouth. The simplest computational scheme is to contrast a grid with mesh size Ax = 0.5 cm and Ay or Az =0.25 cm in the appropriate plane and solve the coupled equations (36a)-(36f) starting at each grid point until the trajectory is completed by a test which determines whether the particle is settled, gone around the head, hit the face or head or inhaled. The boundary of the capture surface is determined by a repetitive systematic search. Arbitrarily, the x coordinate was chosen to be the index coordinate for the plane considered. For a fixed x=x~, the trajectory of a particle is calculated starting from the first complementary coordinate (i.e. y or z) where the particle is inhaled and continued for grid points of the complementary coordinate indexed along x = xi until a miss occurred. The process is the same for either complementary coordinate; except that, due to symmetry, only the maximum of the capture surface on the y coordinate has to be determined. In order to determine the size and flow parameter dependent maxima and minima, this procedure was carried out for all inhalation and external fluid motion parameters studied in combination for all the particle aerodynamic equivalent diameters studied. For a specified ambient wind speed, inhalation velocity and particle size, the volume of the capture space may be calculated by a Monte Carlo method. By definition, the capture space is the space which defines the loci of all those particles at time zero which will enter into the mouth. If this space is enclosed in an oblong box (capture space box, Cb), then the fraction of the randomly selected points from this box which belong to the capture space multiplied by the volume of the capture space box is the volume of the capture space. Although time consuming, the computation algorithm to determine the required fraction is simple. A random coordinate which is a member of the capture space box is picked as the initial position of a particle and if the particle trajectory calculation shows that the particle is inhaled then, by definition, the point picked belongs to the capture space. If such a point is picked then the outcome is said to be favourable. The statistical basis for the unbiased estimate of the frequency of favorable outcomes and the calculation of the coefficient of variation of the estimate is the inverse sampling method developed by Haldene (1945). The favorable outcome frequency, f(da~, Uo, UO, for m favorable outcomes in a total random sample of n, is

f(dae, Uo, UO = m - 1 n-l' Coefficient of variation = x ~

m-l'

. m > 1,

(39)

f(d~e, Uo, U~) <<.0.5

(40)

The volume of the capture space is the volume of the capture space box multiplied by the frequency of favorable outcomes. For any given set of inhalation and external fluid motion parameters, a capture space volume for any d~e ~<200/~, including d~e = 0 may be calculated by this Monte Carlo method. For d~e = 0, the aspiration efficiency is unity by definition; thus, the aspiration efficiency for any d = dae may be calculated by A (da~, Uo, U,) =

f(dae, Uo, UI) x Volume [Cb (dae, Uo, U~)] f(0, Uo, U0 x Volume [Cb (0, Uo, U0]

(41)

The minima and maxima of the capture space may be used to determine an efficient capture space box. It must be recognized that the proper determination of position of the capture space box in relation to the capture space is important. Theoretically, the careful determination of the location of capture space box is not necessary so long as the box chosen is certain to contain the capture space. However, if the volume of the capture space is very small compared to the capture space box volume, then the number of random points necessary to obtain 402 favorable outcomes may be very large with attendant excessive

Human head model as an aerosol sampler

267

computer time costs. The computation time can be substantially reduced by the judicious selection of the enclosing box. The capture volume is more or less ellipsoidal; therefore, if the enclosing box volume is determined by the capture surface maxima and minima plus a small amount, then the enclosing box volume would be approximately 2.5 times the enclosed volume. For the calculations reported here, an oblong box 0.5 cm larger than the maxima and the minima in each coordinate, oriented along the capture space was constructed to act as the enclosing volume for the capture space. 5. RESULTS AND DISCUSSION Capture surface maxima and minima, capture space boxes and aspiration efficiencies were calculated for nine particle sizes between 0 and 200# in steps of 25 #m for six ambient wind velocities and three tidal volumes. The tidal volumes 750, 1450 and 2250 cm 3 were chosen to represent at rest, moderate and heavy work load. With the assumed orifice size, these values correspond to 187.5, 362.5 and 562.5 cm/s amplitude for the inhalation. The breathing pattern was assumed to be sinosoidal with a period of 4 s (15 inhalations/rain). Since the half cycle is the determining factor in the aspiration efficiency, the assumption of a pause between inhalation and exhalation is not necessary. In addition, for each work load and breathing frequency, appropriate adjustments can be made to take the variation in breathing into account. However, we felt that due to the limitations of the model mentioned above, such a refinement would not be warranted. In general, the maximum in x dimension of the capture space boundary is approximately the distance of travel at the ambient wind speed and the maxima in y and z dimensions are influenced by size, tidal volume and ambient velocity. The change in the relative maximum in z direction, Zm,x,is weakly tidal volume but strongly ambient wind speed and particle size dependent. For example, for tidal volume 1450 cm 3, as particle size increases from 25 to 200/~m, Zm,x decreases by 80% for 0.5 m/s and by 43% for 9 m/s in contrast, for ambient wind speed 1 m/s and for the same increase in particle size the decrease in Zm,x is 77 and 79.6% for tidal volumes 750 and 2250 cm a respectively. The behavior of the maxima in the y direction is somewhat more complicated. While y=,~ exhibits similar behavior to Zmax with respect to ambient wind speed and particle size, the dependency of y=~ on tidal volume is more pronounced, especially for smaller sizes. For example, for ambient speed of I m/s as the tidal volume increases from 750 to 2250 cm a the increases in Ymaxfor 25 and 200/~m are 68 and 50%, respectively. The shape of the capture surface may be readily visualized by examining selected results of calculations used in the computation of the maxima and minima of capture surface. The calculated boundaries of the capture spaces on the xz- and xy-planes for all particle sizes with ambient wind velocity of 1 m/s and tidal volume of 1450 cm a are shown in Figs 5 and 6. As intuitively expected, the capture space in the xz-plane is skewed vertically upwards as a function of particle size, The capture space exhibits the characteristic shape which was experimentally observed and described as a spring onion by Vincent and Mark (1982). This characteristic shape, as well as the effect of tidal volume on the capture space are shown in Figs 7 and 8 for the three tidal volumes studied with one ambient wind speed and particle size (d,e =25 #m, Uo = 1 m/s) The aspiration efficiency calculations for all particle sizes and ambient velocities studied with tidal volume 1450 cm 3 are shown in Fig. 9. As particle size increases, the aspiration efficiency decreases for the lowest ambient wind speeds, 0.5 and 1 m/s. For 2 and 4 m/s ambient wind speeds, a plateau is attained for the aspiration efficiency at about 14 and 32% for 150 and 75 #In, respectively. The higher wind speeds, 6 and 9 m/s, each, exhibit an associated minimum occurring at a different particle size and efficiency. These observations are tempting to conjecture that for each ambient wind speed there is a critical particle size such that above which aspiration efficiency increases with increasing particle size. While for particles with sufficient inertia, as the ambient wind speed approaches and exceeds exhalation velocity amplitude, a distinct minimum in the aspiration efficiency is expected, the unknown effects of head orientation with respect to the appearance of the minima and the

268

S. Erdal and N. A. Esmen 140 130

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Fig. 5. The boundaries of capture spaces in xz-plane for particle aerodynamic diameters 0-200 #m with ambient wind speed 1 m/s and tidal volume 1450 cm 3,

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The Distance in X-Direction [cm]

Fig. 6. The boundaries of capture spaces in xy-plane for particle aerodynamic diameters 0-200 #m with ambient wind speed 1 m/s and tidal volume 1450 cm 3.

Human head model as an aerosol sampler

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A = 187.5 ~/g~

269

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60

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140

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Fig. 7. The boundaries of capture spaces in xz-plane for tidal volumes 750-2250 cm 3 with particle aerodynamic diameter 25/zm and ambient wind speed I m/s. 7.75

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270

S. Erdal and N. A. Esmen 110 100 90 Uo

80

•

o

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60

so I U, = 562.5 cm/sec I "~. 40

30

20

0

2s

so 7s too ~.s 1so Particle Aerodynamic Diameter [~.m]

17s

20o

Fig. 9. Aspiration efficiency as a function of particle aerodynamic size for the particle size range of 0-200 #m, the wind speed range of Uo =0.5-9 m/s and the tidal volume of 1450 cm 3.

complexities of the fluid mechanics involved beyond the simplest cases preclude a facile generalization. The influence of the tidal volume on the aspiration efficiency for ambient wind speeds 0.5, 1 and 2 m/s are shown in Figs 10-12, respectively. These results correspond to what one might expect intuitively. At calm conditions, the aspiration efficiency of the smaller particles are affected. In contrast, as the wind speed increases the effect of the tidal volume starts to shift to larger sizes and the aspiration efficiency shows an increase as the tidal volume decreases. These observations pertaining to the theoretically calculated results lead to several important conclusions. The results of the aspiration efficiency calculations are in concordance with the experimentally observed trends. Significantly, the main trend obtained in the experimental results for low to moderate wind speeds, i.e. the decrease and leveling off of the aspiration efficiency is clearly present. While this observation is a corollary to theoretical observations for inertia dominated particle transport near a blunt dust sampler (Vincent and Mark, 1982), by the theoretical calculations, the empirical observation of a lack of "cutoff" is extrapolated to the aerodynamic equivalent diameter of 200 #m. Intuitively and by simple logical deduction one would expect that, for any ambient wind speed there is a "cutoff" size simply due to the gravity. However, theoretical and empirical estimates of the cutoff sizes are not yet available. The theoretical demonstration of the increase in the aspiration efficiency by increasing particle size at high ambient wind speeds is in concordance with the experimental observations. While, this trend is expected to have its maximum manifestation at head orientations which present the mouth as a clear target to the oncoming particles, the theoretical observations suggest a very significant increase in aspiration efficiency for particles larger than 100 #m at ambient wind speeds significantly in excess of the inhalation amplitude. The extension of the calculation method presented here to flow not parallel to the mouth axis is

Human head model as an aerosol sampler

271

110

90

•~ .

=

.

~ 6o c

5O

3O 2O 10 0 0

25

50

75

100

125

150

175

200

Particle Aerodynomic Diameter INto] Fig. 10. Aspiration efficiency as a function of particle aerodynamic size for the particle size range of 25-200 ~tm, the tidal volume range of 750-2250 cm 3 and the wind velocity of Uo =0.5 m/s. t10

100

90

80

Ui = 187.5 cm/sec 362.5 cmLsec 562.5 cr'n~sec

i

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50

40

30

20

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0

0

25

50 75 100 t25 155 Particle Aerodynomic Diameter [/~m]

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20o

Fig. 11. Aspiration efficiency as a function of particle aerodynamic size for the particle size range of 25-200/~m, the tidal volume range of 750-2250 cm 3 and the wind velocity of Uo = 1 m/s.

272

S. Erdal and N. A. Esmen 110

100 ,

90

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125

150

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Particle Aerodynamic Diameter [,u,m] Fig. 12. Aspiration efficiency as a function of particle aerodynamic size for the particle size range of 25-200/~m, the tidal volume range of 750-2250 cm 3 and the wind velocity of U0 =2 m/s.

a straightforward proposition, provided that the angle is not large enough to place the mouth in wake. If an expression for the aspiration of particles into an orifice in the wake of a cylinder can be developed, the Monte Carlo method used here can be extended to consider the entire space around the head. Of course, this suggests the possibility that rotationally averaged aspiration efficiency of very coarse particles may be predicted by the inertia dominated behavior inherent in the theoretical approach presented here, albeit with a semiempirical scaling factor. This intriguing possibility is currently under investigation. REFERENCES Armbruster, L. and Brauer, H. (1982) Ann. occup. Hyg. 26, 21. Bird, R. B., Steward, W. E. and Lightfood, E. N. (1960) Transport Phenomena. Wiley, New York. Erdal, S. and Esmen, N. A. (1990) J. Aerosol Sci. 21, 431. Fuchs, N. A. (1964) The Mechanics of Aerosols. Pergamon Press, New York. Gerald, C. F. and Wheatley, P. O. (1989) Applied Numerical Analysis, 4th Edition. Addison-Wesley, New York. Haldane, J. B. S. (1945) Biometrika 33, 222. IMSL (1989) IMSL Math~Library User's Manual, 1.1 Edition, U.S.A. Lamb, H. (1945) Hydrodynamics, 6th Edition. Dover, New York. Ogden, T. L. and Birkett, J. L. (1977) In Inhaled Particles IV(Edited by Walton, W. H.). Pergamon Press, Oxford. Ogden, T. L. and Birkett, J. L. (1978) Ann. occup. Hyg. 21, 41. Vincent, J. H. (1987) J. Aerosol Sci. 15, 487. Vincent, J. H. and Armbruster, L. (1981) Ann. occup. Hyg. 24, 245. Vincent, J. H., Hutson, D. and Mark, D. (1982) Atmos. Envir. 16, 1243. Vincent, J. H. and Mark, D. (1981) Ann. occup. Hyg. 24, 375. Vincent, J. H. and Mark, D. (1982) Ann. occup. Hyg. 26, 3. Vincent, J. H. and Mark, D. (1984) Ann. occup. Hyg. 28, 117. Vincent, J. H., Mark, D., Miller, B. G., Armbruster, L. and Ogden, T. L. (1990) J. Aerosol Sci. 21, 577.