`
`UNITED THERAPEUTICS, EX. 2007
`WATSON LABORATORIES V. UNITED THERAPEUTICS. |PR2D17—D1621
`Page 1 of 7
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`

`

`The Mechanics of Inhaled
`
`Pharmaceutical Aerosols
`
`An Introduction
`
`Warren H. Finlay
`
`L-‘m'rw'xr'Ill' UfAz’bm'Hr
`Edmomon, (‘mmu’a T66 368
`
`ACADEMIC PRESS
`
`
`A Harcourt Smence and Technology Company
`
`San Diego
`
`San Francisco NewYcrk Boston
`London
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`
`UNITED THERAPEUTICS, EX. 2007
`WATSON LABORATORIES V. UNITED THERAPEUTICS, IPR2017-D1621
`Page 2 of 7
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`

`

`This book is printed on acid-free paper.
`
`Copyright rig: 200] by ACADEMIC PRESS
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`All Rights Reserved.
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`ISBN 0-!2-256971-7
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`A catalogue record for this book is available from the British Libraryr
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`Typeset by Paston PrePress Ltd. Beccles. Suffolk. UK
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`UNITED THERAPEUTICS, EX. 2007
`WATSON LABORATORIES V. UNITED THERAPEUTICS, |PR2017—D1621
`Page 3 of 7
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`3. Motion of a Single Aerosol Particle in a Fluid
`
`19
`
`Here. v is the kinematic viscosity of the fluid surrounding the particle and is given by
`
`\' = it"pfimd
`
`{3-8)
`
`where it and paw are the dynamic viscosity and mass density. respectively. of the fluid
`surrounding the particle. Various empirical equations for CdtRe} based on experimental
`data are normally used {Crowe er al. l998). one such correlation being
`
`Cd : 24(1 — 0.15 Reo'fimlyRe
`
`(3.9)
`
`However. most inhaled pharmaceutical aerosol particles have very small diameters dand
`low velocities vm. so that Re is small. If Re << 1. the drag coefficient of a sphere is given
`by
`
`Cd = 243m
`
`which for Re < 0.1. gives a value of Cd that is accurate to within 1%.
`Combining Eqs (3.4}—{3.10). for Re << I we can write
`
`Fdrag = ‘33di-‘(V _ "fluid)
`
`(3.10)
`
`(3-1 1)
`
`Equation (3.11) is often referred to as Stokes lawl. 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 d 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 of a 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. Le.
`
`”1g : Fdrug
`
`(3-12}
`
`where dee is the magnitude ofthe drag force. Assuming the Reynolds numbers Re << 1.
`we can use Eq. (3.] l} for FMS. in which the air velocity is zero (vflmd = 0}. so that Eq.
`(3.1 1) reduces to
`
`Fdrag = 3fidpvscttltng
`
`(313)
`
`Also. the gravity force is
`
`(3.14}
`”lg _—_ 91:31-1thch
`_______—_————————
`
`'11 is named after George Stokes. who first determined the flow field due to a rigid sphere in translational
`motion through a fluid for very low Reynolds number flowiStokes 185]).
`
`UNITED THERAPEUTICS, EX. 2007
`WATSON LABORATORIES V. UNITED THERAPEUTICS, IPRZD1Y—D1621
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`20
`
`The Mechanics of Inhaled Pharmaceutical Aerosols
`
`where V = mix-'6 is the volume of the spherical particle and g is the acceleration of
`gravity. Equation {3.14) can thus be written
`
`Substituting Eqns (3.13}and [3.15] into Eq. {3.12). we have
`
`mg = paractnd-‘tig
`
`01'
`
`3mipt'smlmg = pmn.dc(nd3.-'6_lg
`
`waning = ppautttgdfl’lstz
`
`{3.15)
`
`(3.16]
`
`(3.17)
`
`Equation (3.1?) gives the settling velocity for a spherical particle settling under the
`action ofgravity under the condition that Rt” << | and diameter >> mean free path. Most
`inhaled pharmaceutical aerosols readily satisfy the condition diameter 5-) mean free
`path. and many inhaled pharmaceutical aerosols also satisfy the condition that Re << I.
`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 10).
`
`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 = [.26] settling in room
`temperature air?
`
`Solution
`
`We have
`
`pparticle = l-lfi X density of water = 1260 kg 111" 3
`
`viscosity ol'airit = 1.8 ><l0"‘ltg1'ti'ls_1
`
`d= 10 x lD’ém
`
`which gives
`
`t‘scltting = (1260 kg whom 111 s-liuo x10‘h inf/(is x 1.3 x m-5 kgm."1 s-')
`= 3.8 x 10-3 m s.—1
`
`= 3.8 mm s"
`
`This gives us a Reynolds number of
`
`Re = Urcldfu
`
`: (3.8 x [0" m 5-11 x (10 x1043 mmls x104 m2 s“)
`
`where we have used Eq. (3.8) for the kinematic viscosity of air with the density of air
`being ,9 = [.2 kg 111-3. Calculating the numbers. we have
`Re = 00025
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`UNITED THERAPEUTICS, EX. 2007
`WATSON LABORATORIES V. UNITED THERAPEUTICS, |PR2017—D1621
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`3. Motion of a Single AerOsol Particle in a Fluid
`
`21
`
`This is very much lower than 1 and so we are quite Justified in using Eq. {3.11) for the
`drag force, and Eq. (3.17} that results from Eq. {3.11).
`
`3.2.1 Settling velocities for droplets
`
`The above discussion and Eqs (3.9). [3.10). {3.11) and (3.1?) all assume solid spherical
`particles. If the particle is not solid. but is instead a liquid droplet. then it is possible for the
`relative motion ofthe air flowing past the droplet to induce fluid flow [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 1974 for
`some discussion on this]. Even if surface impurities did not prevent internal circulation.
`the magnitude of the drag force including such circulation can be shown to be given by
`
`Fd = inpdr .1
`1'38
`rt
`
`ML]
`
`1 + pair/“drop
`
`om
`
`where am, is the viscosity ot‘the air surrounding the drop and ,ude is the viscosity ofthe
`liquid in the drop (this result was derived independently by both Hadamard (1911] and
`Rybczynski ( 191 l )1. This equation dificrs from Stokes law by the factor in curly brackets.
`For water droplets in air. as well as HFA 134a propellant droplets in air at their wet bulb
`temperature (211 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. (3.17)) because the particles travel in each other‘s
`wakes, rather than in an undisturbed fluid. This effect is often referred to as ‘hindered
`settling‘.
`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 [e.g. Di Felice 1994, Crowe er of. 1998).
`These results suggest that for aerosols with particle Reynolds numbers Re << 1. hindered
`settling alters the Stokes drag formula by a factor [’13 7. i.e.
`
`Fdrug = —3fld}l(\' —' ‘JfiuitilI-xw
`
`(3.19}
`
`where 1 is the volume fraction of the continuous phase t_i.e. air]. and is always < 1.
`Specifically.
`
`a: = volume of airstvolume of 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. 5:. 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. Le. the aerosol needs to occupy more than 2.5% of the volume. Thus. in a cubic
`meter of aerosol, 0.025 m3 would need to be occupied by aerosol. At a particle density of
`1000 kg m”. this implies that 25 kg of particles must be present per m3. which is
`
`UNITED THERAPEUTICS, EX. 2007
`WATSON LABORATORIES V. UNITED THERAPEUTICS, |PR2017—D1621
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`22
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`The Mechanics of Inhaled Pharmaceutical Aerosols
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`25 gl". 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
`
`is derived from the Navier—Stokes
`(3.11)]
`As mentioned earlier. Stokes law (Eq.
`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 molecuies surrounding the particle. For air at room temperature and 1
`atmosphere pressure.
`the mean free path is 0.06? pm. For inhaled pharmaceutical
`aerosols. particles ofinterest 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. (3.1 l) is required for these small particles. This
`correction was first suggested by Cunningham in 1910. 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
`1
`C = — —
`d
`C. x Re
`
`where C: is the Cunningham slip correction factor. This is an empirically determined
`factor. The drag force is then
`
`3d v— -
`Frag—w (Re<< 1)
`
`(3.21)
`
`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 particie as
`we did before to obtain the terminal settling velocity ofa spherical particle, we obtain
`
`A simple. approximate formula for CE when d > 0.] pm is
`
`I‘settling = Ccppttrlicll! gdli'll 81“
`
`Q = 1 +2.52 an:
`
`(d b 0.1 ttm]
`
`(3122)
`
`(3.23}
`
`where i. is the mean free path of moiecules in the fluid. For air. the mean free path at
`room temperature and 1 atm pressure is 0.0671 pm. At other temperatures and pressures
`it is different. eg. at body temperature [37:0 2‘. = 0.072 pm. More general and complex
`formuiae for Cc and also for i. 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. Le. noncontinuum effects 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.? pm does the Cunningham
`slip factor result in a correction to the drag coefficient that is larger than 10%.
`
`Example 3.2
`
`Calcutatethe settling velocity in air of a 0.5 pm diameter spherical droplet of nebulized
`Ventolin“ respiratory solution (2.5 mg ml ‘1 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—D1621
`Page 7 of 7
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