`
`UNITED THERAPEUTICS, EX. 2007
`WATSON LABORATORIES V. UNITED THERAPEUTICS. |PR2D17—D1622
`Page 1 of 7
`
`

`

`The Mechanics of Inhaled
`
`Pharmaceutical Aerosols
`
`An Introduction
`
`Warren H. Finlay
`
`['nh'ersf!_1‘ ofrlfberm
`Edmonton. ('c'nmdu T66 3G8
`
`ACADEMIC PRESS
`
`
`A Harcourt SCIencc 0nd [ethnology Company
`
`San Diego
`
`Sam Francisco New‘l'ork Boston
`London
`Sydney Tokyo
`
`UNITED THERAPEUTICS, EX. 2007
`WATSON LABORATORIES V. UNITED THERAPEUTICS, IPR2017-D1622
`Page 2 of 7
`
`

`

`This book is primed on acid-free paper.
`
`Copyright :1“; 200] by ACADEMIC PRESS
`
`All Rights Reserved.
`No part of this publication may be reproduced or transmitted in any form or by any
`means. electronic or mechanical. including photocopying recording. or any
`information storage and retrieval system. without permission in writing from the
`publisher.
`
`Academic Press
`A Harrow-r Science and Technology Compnnr
`Harcourt Place. 32 Jamestown Road. London NW1 TBY. UK
`http: .-'.-'www .ac ademicpresscom
`
`Academic Press
`A Harcourt-r Science and Homology ("ontpanyt'
`525 B Street. Suite I900. San Diego. California 930141495. USA
`http:.-":‘wwwacademicpresscOm
`
`ISBN 0-]2-2569'l'l-T
`
`Library of Congress Catalog Number: 2001090350
`
`A catalogue record for this book is available from the British Library
`
`Typeset by Paston PrePress Ltd. Beccles, Sulfolk. UK
`Printed and bound in Great Britain by Bookcral’t. Bath. Somerset
`
`0|02030405063C93?65432|
`
`UNITED THERAPEUTICS, EX. 2007
`WATSON LABORATORIES V. UNITED THERAPEUTICS, |PR2017—D1622
`Page 3 of 7
`
`

`

`3. Motion of a Single Aerosol Particle in a Fluid
`
`1?
`
`Here. v is the kinematic viscosity ofthe fluid surrounding the particle and is given by
`
`V = li’fiflutd
`
`(38}
`
`where ,u and pamd are the dynamic viscosity and mass density. respectively. of the fluid
`surrounding the particle. Various empirical equations for C‘dtRe) based on experimental
`data are normally ttsed (C rowe oi oi. 1998). one such correlation being
`
`Ccl = 240 + 0.15 wishing
`
`{3.9)
`
`However. most inhaled pharmaceutical aerosol particles have very small diameters d and
`low velocities rm. so that Re is small. If Re << 1. the drag coefficient of a sphere is given
`by
`
`Q=Mm
`
`which for Re c 0.1. gives a value of Cd that is accurate to within l%.
`Combining Eqs{3.4}—(3.10). for Re ((1 we can write
`
`Fang = —3Ttd;t[t‘ — Vfluidl
`
`om
`
`(3.“)
`
`Equation (3.! l} is often referred to as Stokes law'. it is derived from the continuum
`equations of fluid motion [since Eq.
`(3.10} comes by solving the Navier—Stokes
`equations}. and so is valid only for particle diameters that are much greater than the
`mean free molecular path (which in air at typical inhalation conditions is near 0.07 pm).
`Extension of Eq. (3.1 l) to particles with diameter 0’ near the mean free path is considered
`later in this chapter, while extension to larger Reynolds number is readily accomplished
`with correlations such as Eq. (3.9).
`
`3.2 Settling velocity
`
`A particle in stationary air will settle under the action of gravity. and reach a terminal
`velocity quite rapidly. The settling velocity (also referred to as the ‘sedimentation
`velocity") is defined as the terminal velocity ofa particle in still fluid.
`Because the particle‘s velocity does not change once it reaches the settling velocity. the
`acceleration on the particle is zero at this velocity, so that the net force on the particle
`must also be zero. Assuming the only forces on the particle are the aerodynamic drag
`and gravity. then for a solid. nonrotating. spherical particle only a vertical drag force will
`be present. which must balance gravity. i.e_
`
`Mg = deg
`
`(112)
`
`where deg is the magnitude of the drag force. Assuming the Reynolds numbers Re << 1.
`we can use Eq. (3.11) for For“, in which the air velocity is zero (mm = 0). so that Eq.
`(3. l l) reduces to
`
`Fdrug : 3ndyvselthng
`
`{3.13)
`
`Also. the gravity force is
`
`(3.14)
`mg = pparticleVg
`___—__—_.—._—————-——-—
`
`'lt is named after George Stokes. who first deten'ntnecl the flow field due to a rigid sphere in translational
`motion through a fluid for very low Reynolds number flow {Stokes 135]).
`
`UNITED THERAPEUTICS, EX. 2007
`WATSON LABORATORIES V. UNITED THERAPEUTICS, |PR2017—D1622
`Page 4 of 7
`
`

`

`20
`
`The Mechanics of Inhaled Pharmaceutical Aerosols
`
`where l' -— 71:33.6 is the volume of the spherical particle and g is the acceleration of
`gravity. Equation [314'] can thus be written
`
`Substituting Eqns (3.13) and (315) into Eq. (3.12). we have
`
`me = pm...mtnd’-6ig
`
`01‘
`
`minimum = ppmmicth.-'6)g
`
`I'ammg = Pmtiicuegdzslsiz
`
`{3.15)
`
`(3.16}
`
`(3.17)
`
`Equation 13.17} gives the settling velocity For a spherical particle settling under the
`action of gravity under the condition that Re << 1 and diameter >> mean free path. Most
`inhaled pharmaceutical aerosols readily satisfy the condition diameter 3) mean free
`path. and many inhaled pharmaceutical aerosols also satisfy the condition that Re << 1.
`as seen in the example below. Exceptions to the condition Re << 1 are uncommon with
`inhaled pharmaceutical aerosols. but do occur in the entrainment of large carrier
`particles that occur in dry powder particles {discussed in Chapter 9). and high-speed
`metered dose propellant droplets (discussed in Chapter I0].
`
`Example 3.1
`
`What is the Reynolds number of a 10 micron diameter spherical. budesonide powder
`particle (a drug used in treating asthma. specific gravity 2 1.26} settling in room
`temperature air‘?
`
`Solution
`
`We have
`
`pmlmm = 1.26 x density of water = 1260 kg m '3
`
`viscosity ofairti = [.8 x 10—5 kg [Tl—i s“
`d= to x lD‘am
`
`which gives
`
`rm...“ = (1200 ltg m_3l{9.3l m 5‘3'1t10 x 10-“ m'flus x 1.8 x 10-5 kg m '1 5'1)
`= 3.8 x 10"1 in s‘1
`
`: 3.8 mm s“
`
`This gives us a Reynolds number of
`
`Re = Unidfv
`
`= (3.3 x it)" m 5"] x (10 x104 [Til/(1.5 )<1Cl_firtt2 5-1)
`
`where we have used Eq. (3.8) for the kinematic viscosity of air with the density of air
`being ,0 = 1.2 kg tn‘ 3, Calculating the numbers. we have
`R0 = 0.0025
`
`UNITED THERAPEUTICS, EX. 2007
`WATSON LABORATORIES V. UNITED THERAPEUTICS, |PR2017—D1622
`Page 5 of 7
`
`

`

`3. Motion of a Single Aerosol Particle in a Fluid
`
`ll
`
`This is very much lower than i and so we are quite justified in using Eq. [3.11) 1'01' [he
`drag force. and Eq. (ll?) that results from Eq. (3.11).
`
`3.2.1 Settling velocities for droplets
`
`The above discussion and Eqs (3.9}. (3. I0]. {3. | l) and [3.1T] all assume solid spherical
`particles. ll‘the particle is not solid. but is instead a liquid droplet. then it is possible for the
`relative motion oftbe air flowing past the droplet to induce Fluid fiow t internal circulation)
`inside the droplet. This lowers the drag force and increases the settling velocity compared
`to a solid sphere of the same mass and diameter. However. surface impurities on the
`droplet surface appear to hinder internal circulation for small droplets [see Wallis [974 for
`some discussion on this}. Even if surface impurities did not prevent internal circulation.
`the magnitude ofthe drag force including such circulation can be ShOWn to be given by
`
`deg = Satiri’ilr...‘
`
`l ‘l‘ 2luriirfg’l'ldropl
`
`l + Jl'l'alrxpill'op
`
`[3.l3l
`
`where tin”. is the viscosity oftlte air Surrounding the drop and iidml, is the viscosity of the
`liquid in the drop (this result was derived independently by both Hadamard {19] l) and
`Rybczynski (l9l ll]. This equation differs front Stokes law by the I‘actor in curly brackets.
`For water droplets in air, as well as HFA 134a propellant droplets in air at their wet bulb
`temperature (Ell K). this factor is 0.994. and is thus negligible for such droplets.
`
`3.2.2 Particle—particle interactions in settling of particles
`
`For dense aerosols (Le. high number concentrations]. settling velocities are lower than
`predicted by the standard analysis {Eq. (3.1?” because the particles travel in each other's
`wakes. rather than in an undisturbed fluid. This effect is often referred to as “hindered
`settiing'.
`The drag on particles in dense clouds undergoing hindered settling has not been well
`studied. However. we can obtain an estimate as to when this effect becomes important by
`using empirical correlations in the literature [c.g. Di Felice I994. Crowe er al. 1998).
`These results suggest that for aerosols with particle Reynolds numbers Re (i I. hindered
`settling alters the Stokes drag Formula by a factor 1 or" 7. i.e.
`
`thtlg : _ :ll'mlllth' _ Vfluidl' 1-3 I
`
`(3.]9}
`
`where a is the volume fraction of the continuous phase (Le. air). and is always <1.
`Specifically.
`
`a __ volume of airstvolurne ol'air + volume of particles]
`
`{3.20)
`
`in a given total volume of aerosol.
`Notice that the drag force in Eq. {3.19} increases as the volume of particles per unit
`volume is increased he as the air volume fraction. 1. is decreased). which is of course
`why it is called hindered settling.
`For the drag force to be 10% more than that for a single particle. a: must be 0.975 or
`less. to the aerosol needs to occupy more than 2.5% of the volume. Thus. in a cubic
`meter of aerosol. 0.025 in3 would need to be occupied by aerosol. At a particle density of
`1000 kg m'i. this implies that 35 kg of particles must be present per ml. which is
`
`UNITED THERAPEUTICS, EX. 2007
`WATSON LABORATORIES V. UNITED THERAPEUTICS, |PR2017—D1622
`Page 6 of 7
`
`

`

`22
`
`The Mechanics of Inhaled Pharmaceutical Aerosols
`
`25 g I". This is much higher than is normally encountered in inhaled pharmaceutical
`aerosol applications. and so hindered settling is negligible for such aerosols.
`
`3.3 Drag force on very small particles
`
`As mentioned earlier. Stokes law (Eq. {3.1”} is derived from the Navier—Stokes
`equations. which assume that the fluid surrounding the particle is a continuum. This is
`valid only ifthe diameter of the particle is very much greater than the mean free path of
`the fluid molecules surrounding the particle. For air at room temperature and 1
`atmosphere pressure.
`the mean free path is 0.067 urn. For inhaled pharmaceutical
`aerosols. particles of interest have diameters down to 0.5 pm or so. which gives radii of
`0.25 pm. This is in the range where the particle radius is not very much greater than the
`mean free path. and so a correction to Eq. (3.] l ) is required for these small particles. This
`correction was first suggested by Cunningham in I910. and is thus referred to as the
`Cunningham slip correction factor. It is defined so that the drag coefficient for a sphere
`used to obtain Stokes law is replaced by
`
`24
`l
`C = —- x —
`d
`CC
`Re
`
`where Cc is the Cunningham slip correction factor. This is an empirically determined
`factor. The drag force is then
`
`3 d
`-—
`-
`deg = —'i[‘rc.ilfll
`c
`
`{R9 (<1)
`
`(331)
`
`Here the only restriction is that Re << I in order that we can use the Stokes flow solution
`for zero Re flow past spheres. Equating the drag force with the weight of the particle as
`we did before to obtain the terminal settling velocity ofa spherical particle. we obtain
`
`A simple. approximate formula for C} when d > 0.1 pm is
`
`rsellllng = C‘eppartlcle Lea-{21518;}
`
`Cc = 1*- 2.52 Afr!
`
`(d 3’ 0.] pm}
`
`(3.22]
`
`(3.23)
`
`where 21 is the mean free path of molecules in the fluid. For air. the mean free path at
`room temperature and I aim pressure is 0 067 pm. At other temperatures and pressures
`it is different. e.g. at body temperature [3T0 .i = 0.072 pm. More general and complex
`formulae for CD and also for a”. are given in the literature (Willeke and Baron 1993).
`Note that since C: > 1. the settling velocity obtained with the slip correction is larger
`than when this factor is neglected. i.e. noncontinuum cficcts result in larger settling
`velocities than predicted with a continuum assumption. For air at typical inhalation
`conditions. only for particles with diameter smaller than 1.7 urn does the Cunningham
`slip factor result in a correction to the drag coefficient that is larger than [0%.
`
`Example 3.2
`
`Calculatcthe settling velocity in air ofa 0.5 pm diameter spherical droplet of nebulized
`VenLOIinJ’ respiratory solution (2.5 mg ml*' salbutamol sulfate with 9 mg ml‘I NaCl
`in water) both with and without the Cunningham slip correction factor.
`
`UNITED THERAPEUTICS, EX. 2007
`WATSON LABORATORIES V. UNITED THERAPEUTICS, |PR2017—D1622
`Page 7 of 7
`
`