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Aqueous aerosol delivery devices For drugs that are soluble in water, perhaps the simplest approach to lung delivery is via inhalation of an aqueous aerosol. This can be achieved by taking a bulk aqueous solution of the drug and creating aerosol droplets, a process referred to as atomization. The literature on atomization is vast (see, e.g., Ashgriz, 2011), although most of it is unrelated to inhaled pharmaceutical aerosols. Indeed, because of the unique requirements of respiratory drug delivery devices and particularly the need for droplet diameters of a few micrometers, only a handful of the many approaches to atomization have been successfully commercialized and accepted in clinical use with inhaled pharmaceutical aerosols. In this chapter, we explore the fundamental mechanics of several such atomization methods.

8.1 Actively vibrating mesh nebulizers Extruding liquid droplets from small holes in a solid surface is a well-known method of atomization that is used in many applications, including inkjet printing. For inhaled pharmaceutical aerosols, the most widely adopted approach in this regard is to vibrate a piezoelectric element that directly forces oscillation of a perforated diaphragm containing many small orifices, as shown in Fig. 8.1. Aqueous aerosol drug delivery devices that use this approach are usually called “vibrating mesh nebulizers” or “vibrating membrane nebulizers.” The adjective “active” is sometimes added to highlight the fact that the diaphragm is actively vibrated and to contrast them with “passive vibrating mesh nebulizers” in which the diaphragm is not actively forced, but instead merely receives acoustic pressure waves that extrude droplets from a passive (unforced) mesh. For active vibrating mesh nebulizers, the motion of the diaphragm is caused by the motion of the piezoelectric actuator in direct contact with a portion of the diaphragm. The motion of a vibrating diaphragm is a problem in classical mechanics that has a closed form solution for diaphragms of uniform thickness and whose displacement has small slopes (see, e.g., Leissa, 1969). For a circular diaphragm, the displacement of the diaphragm can be written as a linear The Mechanics of Inhaled Pharmaceutical Aerosols. https://doi.org/10.1016/B978-0-08-102749-3.00008-7 © 2019 Elsevier Ltd. All rights reserved.

183

184 The Mechanics of Inhaled Pharmaceutical Aerosols

Piezoelectric actuator

Aqueous solution containing drug Aerosol droplets Perforations In diaphragm

Vibrating diaphragm FIG. 8.1 A diaphragm that contains many small holes and that is made to vibrate by the motion of a piezoelectric actuator can produce aerosol droplets from an aqueous drug formulation as shown schematically here.

superposition of an infinite sequence of modes of vibration each having displacement of the form w ¼ ½An Jn ðknm r Þ + Bn In ðknm r Þ cos nθ cos ωnm t

(8.1)

Here, w gives the displacement of the diaphragm (in the z-direction) at location r,θ due to the given mode; n indicates the azimuthal wave number of the mode (n ¼ 0, 1, 2, …); ωnm is the given mode’s natural frequency of vibration (ω ¼ 2πf where f is the frequency in Hz); Jn is the nth-order Bessel function of the first kind; In is the nth-order modified Bessel function of the first kind; and An and Bn are coefficients determined by the initial and boundary conditions. The parameter knm is related to the mode’s natural frequency ωnm by 4 ¼ knm

ρhω4nm De

(8.2)

where ρ is the density of the material making up the diaphragm, h is the diaphragm’s thickness, and De is the flexural rigidity parameter defined by De ¼

Eh3 12ð1 ν2 Þ

(8.3)

where E and ν are Young’s modulus and Poisson’s ratio of the material making up the diaphragm. For each value of the azimuthal wave number n, there are an infinite number of vibration modes, each having a different value of the circular frequency ωnm whose value is determined from an eigenvalue problem based on the boundary conditions. The subscript m allows the identification of the different modes for each n and also corresponds to the number of nodal circles of that mode (i.e., the number of concentric circles that have zero displacement at all times for that mode). To clarify the above discussion, Fig. 8.2 shows the displacement w at time t ¼ 0 specified by Eq. (8.1) for n ¼ 0 and m ¼ 2, which is one of the most important vibration modes as we will soon discuss. This mode will cause the diaphragm to oscillate up and down with the shape shown in Fig. 8.2 but with the amplitude of the displacement modulated by the cos(ωnmt) term in Eq. (8.1).

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w

y x

FIG. 8.2 The diaphragm displacement w from Eq. (8.1) is shown at time t ¼ 0 for a mode with n ¼ 0 and m ¼ 2.

8.1.1

Natural frequencies of a diaphragm vibrating in a vacuum

Before turning to the more complicated case of a diaphragm vibrating with a liquid on one side and air on the other, let us first consider the simpler case of a diaphragm vibrating in a vacuum. The values of the natural frequencies of vibration ωnm of the different modes are known to depend on whether there is tension T in the diaphragm (Hong et al., 2006). This should be evident when one recalls, for example, that the pitch of a kettledrum becomes higher when its tension rods are tightened. The response of the diaphragm then depends on the magnitude of the tension T compared with the parameter De/R2, where De is given in Eq. (8.3) and R is the radius of the diaphragm. If T is large compared with De/R2, the vibration of the diaphragm results from its internal tension (rather than its internal stiffness), and the vibration frequencies are obtained from the eigenvalues for a membrane, given as (Hong et al., 2006) sﬃﬃﬃﬃﬃ αnm T (8.4) fnm ¼ 2πR ρh where ωnm ¼ 2πfnm

(8.5)

and αnm is a numerical constant depending on n and m. Values of αnm for a few n and m are given in Table 8.1 for vibration in a vacuum. If instead T is zero or small compared with De/R2, the vibration of the diaphragm results from its internal stiffness (not its tension), and the values of ωnm are obtained from the eigenvalues for a plate clamped at its outer edge and given in Leissa (1969). In this case, the natural frequencies of vibration are given by sﬃﬃﬃﬃﬃﬃ αnm De (8.6) fnm ¼ 2πR2 ρh

186 The Mechanics of Inhaled Pharmaceutical Aerosols

TABLE 8.1 Values of αnm in Eq. (8.4) are given for a diaphragm in a vacuum and behaving as a membrane n 50

n 51

n 52

m ¼0

2.405

3.832

5.134

m ¼1

5.520

7.016

8.417

m ¼2

8.654

10.174

11.620

TABLE 8.2 Values of αnm in Eq. (8.6) are given for a diaphragm in a vacuum and behaving as a plate n 50

n 51

n 52

m ¼0

10.22

21.26

34.88

m ¼1

39.77

60.82

84.58

m ¼2

89.104

120.08

153.81

where the αnm now take on the values given for a few n and m in Table 8.2 (see Leissa, 1969 for values at other n and m). Eqs. (8.4), (8.6) apply to a diaphragm of uniform thickness, whereas we are interested in the vibration of a diaphragm that is pierced by many holes. The latter case does not have a closed form solution, but instead requires numerical solution of the governing equations. However, Olszewski et al. (2016) studied vibrating diaphragms atomizing droplets of the size appropriate for pharmaceutical inhalation. They found that frequencies predicted for the n ¼ 0 and m ¼ 2 mode by Eq. (8.6) were within 5% of the values obtained by numerical simulations that included the presence of the extrusion holes, indicating that the extrusion holes had little effect on the diaphragm’s resonant frequency.

8.1.2 Resonant frequencies of a diaphragm vibrating with liquid on one side In order to produce aerosol droplets from a vibrating diaphragm, one side of the diaphragm must be in contact with the liquid that is to be aerosolized, while the other side is exposed to air. The presence of a fluid medium next to a diaphragm can affect the frequencies of the vibration modes if the inertia of the fluid is not small compared with the inertia of the diaphragm, since in this case the fluid’s mass interferes with the motion of the diaphragm, reducing the vibration

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frequency. The relevant parameter that must be considered is the dimensionless frequency Ω0 (Chapman and Sorokin, 2005) given by hω ρ 2 (8.7) Ω0 ¼ c B ρ0 Here, cB is the bending-wave characteristic speed, which for a plate is given by sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ E cB ¼ (8.8) 12ρð1 ν2 Þ If Ω0 is large, the diaphragm is said to have “light loading,” and the fluid has little effect on the vibration frequency, so that, for example, Eq. (8.6) is a reasonable approximation for the mode frequencies of a vibrating plate. On the other hand, if Ω0 is not large, the diaphragm undergoes significant (Ω0 1) or heavy (Ω0 ≪ 1) loading, and its frequency of vibration is reduced. Kwak (1991) gives an analysis that allows calculation of the change in vibration frequency due to fluid loading, with the vibration frequency now given by fv fw ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1+β

(8.9)

Here, fw is the natural frequency of vibration of a given mode when in contact with liquid on one side, fv is the natural frequency when air is present on both sides, and β is the ratio of vibrational kinetic energy of the liquid vs that of the plate and is given by β ¼ Cnm

ρ0 R ρh

(8.10)

Here, Cnm is a numerical constant whose value depends on the vibration mode (i.e., n and m from Eq. 8.1) and the boundary conditions. Values of Cnm are given for various modes by Kwak (1991), and Table 8.3 gives a few of these values for a circular plate clamped at its outside edge.

TABLE 8.3 Values of Cnm in Eq. (8.10) are given for a circular plate (with Poisson’s ratio ν 5 0.3) clamped at its outer edge, with water on one side n 50

n 51

n 52

m ¼0

0.4667

0.2731

0.1986

m ¼1

0.2033

0.1523

0.1234

m ¼2

0.1265

0.1045

0.08996

From Kwak, M.K., 1991. Vibration of circular plates in contact with water. J. Appl. Mech. 58, 480–483.

188 The Mechanics of Inhaled Pharmaceutical Aerosols

Liquid Diaphragm

Diaphragm

Air

d

Diaphragm

Diaphragm

Liquid

(A)

Air

(B)

FIG. 8.3 A close-up view of one hole in the diaphragm of Fig. 8.1 is shown. The hole has a diameter d in the neighborhood of the outlet. The hole is intentionally tapered (see text). Liquid is above, while air is below the diaphragm. In (A), the diaphragm is moving into the liquid, whereas in (B), the diaphragm is moving away from the liquid.

Experiments on vibrating diaphragms (Olszewski et al., 2016) show that the largest displacement of a diaphragm occurs when the diaphragm is forced at a resonant frequency corresponding to the natural frequency of the n ¼ 0 and m ¼ 2 mode shown in Fig. 8.2. Thus, design of vibrating diaphragms for inhaled pharmaceutical aerosol production normally aims for piezoelectric excitation at the resonant frequency of this mode. An aerosol delivery device that uses a vibrating diaphragm operating at its resonant frequency extrudes liquid as the holes are forced into the aqueous formulation, as shown in Fig. 8.3A. The piezoelectric actuator motion is normally tuned so that on each vibration period, a droplet with a diameter approximately equal to the diameter d of the hole is formed, giving an aerosol volume production rate that is proportional to the product of the number of holes, the vibration period, and the volume of a single droplet. Note that the presence of solid particulates in the liquid is typically undesirable with vibrating mesh nebulizers, since such particulates can clog the extrusion holes over time and impair aerosol production (Rottier et al., 2009). Example 8.1 Determine the resonant frequency of the n ¼ 0 and m ¼ 2 mode shown in Fig. 8.2 for a circular plate with zero tension (T ¼ 0) (a) vibrating in air, (b) vibrating in contact with an aqueous formulation with a density of 998 kg/m3 on one side. The plate is made from a nickel chromium alloy with density ρ ¼ 8000 kg/m3, Young’s modulus E ¼ 200 GPa, Poisson’s ratio ν ¼ 0.3, thickness h ¼ 125 μm, and diameter 1 cm.

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Solution Since there is no tension in the diaphragm, we may use Eq. (8.6) with a value of αnm ¼89.104 taken from Table 8.2 for n ¼ 0 and m ¼ 2 to find the frequency of this mode when vibrating in a vacuum. Substituting in the numbers, we obtain f02 ¼ 107 kHz for vibration in a vacuum. To determine whether we expect the vibration frequency to be different in air or water compared with a vacuum, we must determine the value of Ω0 in Eq. (8.7). For part (a), assuming an air density of ρ0 ¼ 1.2 kg/ m3, we find Ω0 ¼ 2.4 106, which is much larger than 1, and so, the diaphragm operates in the light loading regime. Thus, the presence of air is not expected to have much effect on the vibration frequency, and the resonant frequency will be close to the value of 107 kHz that we determined for a vacuum. For part (b), if we instead calculate Ω0 with a density ρ0 ¼ 998 kg/m3, we find Ω0 ¼3.6. This corresponds to significant/heavy loading, and the diaphragm will now vibrate at a lower frequency than given by Eq. (8.6). To find this frequency, we use Eqs. (8.9), (8.10) where we read from Table 8.3 a value of C02 ¼ 0.1265. Eq. (8.10) then gives β ¼ 0.631, and we find from Eq. (8.9) that fw ¼ 0.782 fa. Thus, the n ¼ 0 and m ¼ 2 mode has a natural frequency of 0.782 107 kHz ¼ 84 kHz when in contact with water on one side. A piezoelectric element should therefore vibrate at 84 kHz in order to produce resonance and thereby achieve the largest possible displacement of this diaphragm, associated with optimal aerosol production of the given aqueous formulation.

8.1.3

Interference by cavitation bubbles

Because of surface tension, the pressure in the liquid near the extrusion outlets of a perforated vibrating diaphragm is considerably different from that in the ambient air into which the droplets exit. For example, when a droplet forms at the extrusion hole as in Fig. 8.3A, the curvature and surface tension of the air-liquid surface of the forming droplet result in the pressure in the liquid being higher than in the surrounding air by an amount given by the Laplace pressure Δp ¼

2σ d

(8.11)

where σ is the surface tension of the air-liquid interface (σ ¼ 0.073 N/m for airwater at room temperature). This pressure can be quite high, for example, for water and d ¼ 3 μm, Δp ¼ 97.3 kPa. Of more concern however is the fact that during the other half of a vibration period when the diaphragm moves away from the liquid, as shown in Fig. 8.3B, the pressure on the liquid side of the air-liquid interface is now below ambient pressure by a similar amount. If the pressure in a liquid drops to values below its vapor pressure, gas microbubbles (which are nearly always dispersed throughout liquids) can grow explosively in a process called cavitation (see Chapter 10). This can result in the creation of bubbles in the extrusion region that can interfere with droplet creation (Maehara et al., 1986). Vapor pressure varies rapidly with temperature (see Chapter 4), but water at 25°C has a vapor pressure of 3.1 kPa so that a

190 The Mechanics of Inhaled Pharmaceutical Aerosols

Laplace pressure, for example, of 97.3 kPa (for a 3 μm diameter extrusion hole noted above) may create cavitation bubbles when ambient air pressures fall below 100.4 kPa, which is likely to be a common occurrence. To avoid this issue, it is usual to use tapered extrusion holes so that the radius of curvature of the air-liquid interface occurring when the diaphragm moves away from the liquid is larger, as shown in Fig. 8.3.

8.1.4 Effects of liquid viscosity and electrical conductivity An important aspect of aerosol production with a perforated vibrating diaphragm is the volume that is extruded on each period of the vibration, since this determines the diameter of the resulting droplets. As we have previously noted, the piezoelectric actuator normally operates at the resonant frequency of the n ¼ 0 and m ¼ 2 mode with a displacement that results in the production of a single droplet on each vibration cycle, with each droplet having a diameter approximately equal to the hole diameter. However, droplet size is known to vary with the viscosity and electric conductivity of the liquid being aerosolized (Beck-Broichsitter et al., 2015). The effect of viscosity on droplet sizes can be understood by realizing that the higher the viscosity, the greater is the tendency of the fluid in the extrusion region to be dragged along by the walls due to viscous shear stresses. For example, as the diaphragm switches direction at the bottom of a cycle, the fluid continues to move downward through the hole because of its inertia and will extrude through the hole as the diaphragm begins its upward motion, as in Fig. 8.3A. However, viscous shear stresses at the wall now act upward as the diaphragm moves upward and will tend to drag the fluid upward. If these viscous stresses are large, they can inhibit fluid from extruding though the hole and thereby reduce the volume that is extruded, resulting in smaller droplet sizes for higher liquid viscosities, with droplet production ceasing at high enough viscosities, as is indeed observed (Ghazanfari et al., 2007). An interesting dependency on liquid electric conductivity is also found with vibrating mesh nebulizers. This is thought to be due to an electrokinetic phenomenon related to the presence of an electric double layer at the walls of the extrusion region (Beck-Broichsitter, 2017; Zhang et al., 2007). To understand this effect, consider an aqueous formulation in which ions of the same charge (e.g., positive ions) preferentially bind to the walls of the diaphragm, either due to chemical affinity or due to the diaphragm being electrically charged to a nonzero potential (i.e., not grounded). The presence of these bound ions will attract an excess of counterions in the immediate vicinity of the wall, leading to a so-called electric double layer. As a droplet extrudes from a hole, the walls of the nozzle move upward (as in Fig. 8.3A) and carry counterions in the electric double layer away from the forming droplet. This sets up an imbalance of charge within the liquid in this region, leading to a “streaming potential” and associated “streaming current” whereby counterions in the bulk of the liquid (i.e., not in the immediate vicinity of the walls) undergo motion opposite to

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that of the diaphragm motion. If the liquid has high electric conductivity, this streaming current has little effect on the motion of the liquid through the extrusion region. However, when the electric conductivity is low enough, the counterions in the streaming current collide frequently enough with other molecules in the liquid as they attempt to move opposite to the counterions in the electric double layer that they exert considerable drag on the bulk liquid. This results in the liquid in the extrusion region not moving along with the walls as easily as it would without this electrokinetic effect and leads to the formation of larger droplets at low conductivities, as is indeed observed (Beck-Broichsitter and Oesterheld, 2017). The effect of a streaming potential on the motion of liquids through microchannels is well understood for simple, steady cases such as steady pressure driven flow through a channel (Li, 2004). However, the present case is more complex because the diaphragm moves at radio-wave frequencies, complicating the electromagnetic analysis, so that a predictive understanding of this effect has yet to be developed for vibrating mesh nebulizers.

8.2 Capillary instability of a liquid thread One of the principal droplet production mechanisms employed by several aqueous pharmaceutical aerosol delivery devices involves first producing liquid threads or ligaments, which subsequently break up into droplets. While different methods can be used to produce such threads, as will be discussed shortly, let us first examine what causes such a thread to break into droplets. Referring to Fig. 8.4, consider what happens when the surface of the thread is perturbed by an axisymmetric disturbance with wavelength λ. The pressure inside the liquid will differ from that in the air outside the thread due to surface tension σ at the curved air-liquid interface, with the pressure inside the concave side being higher than outside by an amount given by the Laplace pressure (White, 2016) 1 1 (8.12) + Δp ¼ σ R1 R 2 For now, let us assume a long wavelength disturbance so that λ ≫ dt and R2 ≫ R1. In that case, it is clear that regions where R1 is smaller will have higher pressure than regions with larger R1. This will lead to liquid moving away from regions where the thread is narrower and toward regions that are already bulbous. This results in growth of the surface disturbance, pinching off the narrow regions, and breakup of the initially cylindrical thread into droplets. Because this instability is caused by surface tension effects, it is referred to as a capillary instability. It was first studied in the 1800s by Plateau and subsequently by Rayleigh, so that it is also commonly referred to as the Rayleigh-Plateau instability. Note that if we relax our assumption of long wavelengths, then we must also consider the effect of the second radius of curvature R2. From Fig. 8.4, we see that in pinched regions where R1 is lower, the second radius of curvature is convex so that R2 is negative. From Eq. (8.12), this can reverse the pressure gradient and lead to fluid moving toward the pinched region, thereby reversing the

192 The Mechanics of Inhaled Pharmaceutical Aerosols

dt

R2

d

l

d R1

(A)

(B)

(C)

FIG. 8.4 Schematic of a cylindrical liquid thread of initial diameter dt in (A) is perturbed by a disturbance so that the surface of the thread varies along its length with wavelength λ as in (B), resulting in the thread breaking into droplets of diameter d shown in (C). The surface in (B) is characterized by two different radii of curvature R1 and R2 whose value varies along the length of the thread.

surface disturbance and stabilizing the surface. Thus, only wavelengths longer than some critical value will result in instability. To be more specific, a linear stability analysis can be performed. This is done by introducing a small disturbance to the smooth interface at r ¼ a (where a ¼ dt/2), so that the perturbed interface is at r ¼ a + η, where in order for this to be a small disturbance, we must have η ≪ a. The goal of a linear stability analysis is to find out whether a given disturbance will grow and, if so, the rate at which it grows. To achieve this goal, one must solve the equations governing the liquid and gas phases. However, because a small amplitude disturbance is assumed, all the equations governing the disturbance can be linearized, and so Fourier transforms in z and θ, and Laplace transforms in time can be used. As a result, one ends up analyzing what happens to disturbances that are of the form, for example, η ¼ η0 f ðr Þ exp ðikzÞ exp ðstÞ

(8.13)

where η0 is the initial amplitude of the disturbance to the interface. Similar forms are assumed for the perturbations to the base velocity and pressure. Here, k ¼ 2π/λ is a wave number in the z-direction. The parameter s is a growth rate that governs how fast the disturbance grows in time. If we neglect viscous effects and aerodynamic effects of the surrounding gas, the analysis is straightforward (see, e.g., Ashgriz and Yarin, 2011), and we find that only disturbances

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with ka < 1 are stable, while the most unstable wavelength is λmax ¼ 4.51 dt. Assuming one droplet forms per wavelength, then equating the volume in a thread of diameter dt and length λmax to the volume of a droplet with diameter d, we find that breakup of the thread via the most unstable wavelength results in droplets of diameter d ¼ 1:9 dt

(8.14)

This result is surprisingly close to typical droplet sizes observed experimentally (Rutland and Jameson, 1970). While some research and development have gone into inhalation devices that rely directly on Rayleigh-Plateau breakup of a cylindrical thread by extruding a pressurized liquid formulation through an array of small-diameter nozzles (Schuster et al., 1997; de Boer et al., 2008), this approach is not currently used in any marketed inhalation aerosol product. Instead, our interest here is in realizing that the above noted capillary instability causes ligaments to be unstable and to break up into droplets, a process relied upon by the remaining atomization methods we discuss in this chapter.

8.3 Jet nebulizers While vibrating mesh nebulizers are currently one of the most commonly used methods of producing aqueous aerosols, other methods are preferred for some formulations, for example, for high-viscosity liquids or colloidal suspensions. In these cases, jet nebulizers offer an alternative method in which aqueous aerosol is instead created by a jet of air at high speed that disrupts a small stream of the liquid formulation. The kinetic energy of the air supplies the energy needed to break up the liquid stream into large primary droplets. To produce droplets with small enough diameter for delivery to the lungs, in jet nebulizers these primary droplets are typically impacted on a baffle, yielding secondary droplet breakup. This two-step approach to droplet production has been used in nebulizers since the mid-1800s (Nikander and Sanders, 2010). Jet nebulizers are a subset of the more general twin-fluid atomizers and belong to the air-assist/air-blast atomizer classes of atomizers discussed by Omer and Ashgriz (2011). However, unlike traditional air-assist/air-blast atomizers, a pressurized liquid feed line is not normally used with nebulizers due to cost, safety, and portability issues. Instead, the fluid is drawn passively into the droplet production region by the low pressure of the air jet in this region.

8.3.1

Basic operation

Although there are a variety of different designs of jet nebulizers, the basic geometry of a typical “unvented” jet nebulizer is shown in Fig. 8.5. The basic operating principle of an “unvented” nebulizer is as follows. A pressurized air source (either from a pump/compressor or from a wall source)

194 The Mechanics of Inhaled Pharmaceutical Aerosols

Mouthpiece Aerosol to patient

Additional inhaled air Secondary baffle Primary baffle Liquid feed tube

Primary droplet production region

Liquid reservoir containing drug

Pressurized air supply FIG. 8.5 Schematic of a typical “unvented” jet nebulizer design.

supplies high-pressure air that flows through a nozzle (or orifice or venturi, depending on the design) where the air accelerates to high speed. The pressure near the nozzle is lower than in the liquid reservoir, and this draws liquid up the liquid feed tube. The nozzle region is designed so that the high-speed air here meets a liquid stream supplied by the liquid feed tube, shattering it into ligaments that form droplets. This is the primary droplet production region. The droplets produced in this region then splash off of primary baffles, producing smaller droplets, which then flow with the air, possibly passing through secondary baffles (which impact out the larger particles) and into a mouthpiece (or mask) for patient inhalation of the aerosol. For most patients, the air supply from the pressurized source does not supply enough air to make up a typical inhalation flow rate, so additional ambient air is inhaled through the mouthpiece. If the additional air that is brought in to make up the patient’s full inhalation flow rate comes through the primary droplet production region, then the nebulizer is referred to as a “vented” or “breath-enhanced” nebulizer (also called an “active venturi” nebulizer, since the low pressure in the nozzle or “venturi” region may actively draw air into the nebulizer even without a patient present). By placing a one-way valve on the vent, additional air is entrained into the nebulizer only during inhalation, which lowers the flow rate of air through the primary production region during exhalation and reduces the amount of exhaled aerosol somewhat. A schematic of a basic vented nebulizer, with such a valve in place, is shown in Fig. 8.6.

Aqueous aerosol delivery devices Chapter

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195

Additional inhaled air Valve (closed on exhalation)

Secondary baffle

Primary baffle Liquid feed tube

Primary droplet production region Liquid reservoir containing drug

Pressurized air supply FIG. 8.6 Schematic drawing of a valved “vented” jet nebulizer design.

It should be noted that for all inhalation aerosol devices requiring multiple breaths, there is a small “connection” volume between the entrance to the respiratory tract (either the mouth or the nose) and the aerosol containing volume of the device. After the first breath from the device, this connection volume will be filled on exhalation with exhaled air that does not contain significant amounts of aerosol. This exhaled air will then be rebreathed immediately on the next tidal breath through the device, causing the amount of aerosol inhaled to be smaller than would be expected if the connection volume was absent. For small tidal volumes (such as with toddlers and infants), this can cause a significant reduction in the amount of aerosol being inhaled from a nebulizer and is a reason for using as small a connection volume as is possible for such patients. The constant supply of air through the pressurized air supply line with jet nebulizers also introduces an age-dependence to the dose delivered to very young subjects (DiBlasi, 2015). In particular, for young subjects, inhalation flow rates may be below the air flow rate supplied by the nebulizer (the excess air exits out the exhalation route of the nebulizer), so that even during inhalation, there is aerosol exiting the nebulizer. Because this wastage does not occur until inhalation is below a certain flow rate (the value depends on the nebulizer but is typically 4–8 L/min), patients with inhalation flow rates above this flow rate will inhale the full dose, but patients inhaling below this flow rate will receive only a portion of this dose (with the inhaled dose depending on their flow rate). Weight/age correction of doses should bear this phenomenon in mind.

196 The Mechanics of Inhaled Pharmaceutical Aerosols

8.3.2 Primary droplet diameters Various correlations are available to predict the diameters of droplets produced by air-assist/air-blast atomizers with specific nozzle geometries (Omer and Ashgriz, 2011). However, the geometry through which liquid and air enter and mix in the primary droplet production region varies considerably between different jet nebulizer designs. As a result, it is difficult to develop a generalpurpose, predictive correlation that can be expected to be accurate for all types of jet nebulizers. However, the following simplified analysis, after Lefebvre (1992), is useful for providing a general understanding of the parameters affecting primary droplet production. For simplicity, let us consider a jet nebulizer design in which the liquid stream to be atomized is supplied to the primary droplet production region via a small, circular duct of diameter D. At the exit of this duct, the liquid initially exits with a cylindrical shape to the air-liquid interface. If the total mass of liquid that flows through this duct is mld, then the total area of newly created airliquid interface here (prior to breakup into droplets) is the area of a cylinder of length L, that is, Al 1 ¼ π D L

(8.15)

The volume Vl of this cylinder is mld/ρl where ρl is the density of the liquid being nebulized, where Vl ¼ π D2 L=4

(8.16)

Eliminating L from Eq. (8.15) by using Eq. (8.16), we find Al1 ¼

4mld ρl D

(8.17)

Upon exiting the duct, the liquid mass mld is broken into a total of n droplets. To simplify the analysis, let us assume these droplets all have the same diameter d, so that we also have mld ¼ nρl πd 3 =6

(8.18)

Then, the surface area of these n primary droplets is simply. Al2 ¼ n πd 2

(8.19)

Combining Eqs. (8.18) and (8.19), the primary droplets thus have a total surface area of Al2 ¼

6mld ρl d

(8.20)

The spherical droplets have a larger surface area than the initial cylindrical area exiting the liquid supply duct, and the creation of this new area requires an energy given by

Aqueous aerosol delivery devices Chapter

E l ¼ σ ð A l 2 Al 1 Þ

8

197

(8.21)

where σ is the surface tension of the air-liquid interface. Substituting in Eqs. (8.17), (8.20) and simplifying, we find 2σmld 3 2 (8.22) El ¼ ρl d D Assuming negligible kinetic energy of the liquid stream, then the energy given by Eq. (8.22) is supplied by the kinetic energy of the air jet mj vj2/2, where mj is the total mass of air that flows through the air jet nozzle and vj is the air jet’s velocity. Of course, only a fraction F of the air jet’s kinetic energy is used to produce droplets. We may thus write mj v2j 2σmld 3 2 (8.23) ¼F ρl 2 d D Solving for primary droplet diameter d, we find d¼

3 2 2 Fρl vj mj + 4σ mld D

(8.24)

The ratio mj/mld here may be replaced by the ratio of air vs liquid mass flow rates m_ j =m__ ld , yielding d¼

3 2 2 Fρl vj m_ j + 4σ m_ ld D

(8.25)

Eq. (8.25) is instructive from a qualitative perspective, since it suggests smaller primary droplets can be achieved by increasing the air jet mass flow rate or by decreasing (a) surface tension, (b) the liquid mass flow rate, or (c) the liquid stream duct diameter D. However, it should be noted that in a jet nebulizer, the liquid mass flow rate m_ ld can be affected by the jet velocity vj, since the latter affects the pressure in the droplet production region and which is responsible for drawing liquid into this region, so that vj (and thus m_ j Þ and m_ ld are not independent parameters. Note that the liquid stream duct diameter D and the liquid mass flow rate m_ ld are also not independent, since for a given pressure drop in a duct, the liquid mass flow rate decreases with D. While we have not explicitly included viscous effects in the above analysis, increases in liquid viscosity result in decreased liquid mass flow rate m_ ld in the liquid supply duct (for the low Reynolds number flows expected here), so that increases in liquid viscosity can be expected to give smaller primary droplets. However, this will come at the expense of a reduced nebulizer output rate, since m_ ld is the mass production rate of primary droplets, which itself directly relates to the rate of secondary droplet production.

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Example 8.2 Eq. (8.25) lacks utility since the factor F is unknown and because the form of this equation will differ for different liquid supply duct geometries. However, consider a jet nebulizer that has a value of F ¼ 0.007 (typical of certain twin-fluid atomizers—see Lefebvre, 1992), a jet velocity vj of 120 m/s, an air volume flow rate of 4 L/min (typical of jet nebulizers), and a liquid duct volume flow rate of 17.5 mL/min (again typical for a jet nebulizer—see Carrigy et al., 2017) nebulizing a formulation with liquid density 1000 kg/m3 and surface tension 0.072 N/m. If the liquid is supplied to the primary droplet production via a 1 mm circular duct, use Eq. (8.25) to estimate the diameter of the primary droplets produced by this jet nebulizer. Solution The given volume flow rate of air in SI units is Q_ j ¼4 L/min (103 L/m3)/(60 s/ min) ¼ 6.67 105 m3/s. Assuming an air density of 1.2 kg/m3, we have m_ j ¼ 1.2 Q_ j ¼8 105 kg/s. Similarly, we convert 17.5 mL/min to find m_ ld ¼2.9 104 kg/s. The air-to-liquid mass flow rate ratio is thus m_ j /m_ ld ¼0.27. Inputting the given numbers into Eq. (8.25), we find a primary droplet diameter of d ¼ 30.6 μm. As expected, this droplet diameter is too large for an inhalation aerosol, which is of course why jet nebulizers impact their primary droplets on baffles to yield secondary droplet breakup, a topic to which we now turn.

8.3.3 Secondary droplet diameters As noted above, droplets in the primary production region of jet nebulizers are usually too large in diameter to be inhaled for delivery to the lungs. To achieve desired diameters of a few micrometers, the primary droplets in a jet nebulizer are normally impacted on a baffle separated by a short distance from the primary droplet production region, as shown in Figs. 8.5 and 8.6, yielding secondary droplet breakup. Droplet breakup due to wall impact occurs in diverse applications and has been studied by many authors (see, e.g., Moreira et al., 2010 for a review). For walls with relatively thin layers of liquid on them (thin compared with the droplet diameter), splashing results because the impacting drop forms a circular crown-like sheet coming out of the wall, which is unstable and results in droplets forming at the free rim of this sheet, as shown schematically in Fig. 8.7. For smooth surfaces, it is found that splashing can occur by this process only when 2=5 ≫1 (8.26) Wedl 2 Redl

FIG. 8.7 Typical sequence of events occurring during droplet impaction and splash on a dry or thinly wetted wall.

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where Wedl is a droplet Weber number based on droplet diameter db before impact Wedl ¼ ρl db U0 2 =σ

(8.27)

Redl ¼ ρl db U0 =μl

(8.28)

and the Reynolds number is

where U0 is the droplet velocity just prior to impaction. Various correlations in the literature have been developed to predict the diameter of the secondary droplets produced by splashing (Moreira et al., 2010), with one convenient such correlation given by Han et al. (2000): rﬃﬃﬃﬃﬃ da 3 ρl (8.29) ¼ 0:5 0:25 db Wedl Redl ρa Here, da is secondary droplet diameter (i.e., droplet diameter after impact), and db is the primary droplet diameter (i.e., before impact), while ρa is the surrounding gas (i.e., air) density. The dependence of secondary droplet diameter on the various dimensional parameters can be seen by substituting Eqs. (8.27), (8.28) into Eq. (8.29) and simplifying to yield da ¼

3db0:25 μ0:25 σ 0:5 l 0:25 1:25 ρ0:5 U0 a ρl

(8.30)

Here, we see that secondary droplet diameter da decreases with reductions in surface tension σ or increases in velocity U0, which we found was also true for primary droplet diameters given by Eq. (8.25). Given that Eq. (8.30) indicates reductions in primary droplet diameter db cause reductions in secondary droplet diameter, surface tension and impact velocity will therefore doubly affect secondary droplet diameter. Earlier we noted that Eq. (8.25) predicts that primary droplet diameters will decrease with increases in viscosity, which opposes the already weak dependence on viscosity seen in Eq. (8.30), so that viscosity effects are not expected to be strong. Secondary droplet production in jet nebulizers is relatively inefficient, since much of the splashed spray hits the inner walls of the nebulizer and drips back into the liquid reservoir, to then be again drawn up into the primary production region and reatomized. Indeed, Carrigy et al. (2017) estimate that a typical parcel of liquid cycles through the primary droplet production region on average 100 times before finally exiting the nebulizer in aerosol form. Such extensive repetitive atomization of the liquid formulation can degrade large molecules (Lentz et al., 2005) or biologics such as bacteriophages (Carrigy et al., 2017) due to the large local strain rates and fluid accelerations associated with droplet formation during ligament breakup and wall impact. The breakup of ligaments (which in jet nebulizers occurs both in the case of primary droplet formation by the air jet and in secondary droplet formation by

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wall impact) is thought to occur via the Rayleigh-Plateau instability discussed in Section 8.2. While combined use of Eqs. (8.25) and (8.30) does allow estimation of droplet sizes, in jet nebulizers, the resulting droplet size distribution is quite polydisperse due to variation in the ligament breakup dynamics during droplet formation (Villermaux, 2007). Eqs. (8.25) and (8.30) also do not include any size selection by secondary baffles or walls as the droplet exits the nebulizer, although these effects will depend on the detailed internal geometry of the nebulizer’s aerosol flow paths. Example 8.3 Use Eq. (8.29) to estimate secondary droplet diameters produced after the primary droplets in Example 8.2 impact on a baffle. Recall that the air jet velocity was 120 m/s, liquid density ρl ¼ 1000 kg/m3, σ ¼ 0.072 N/m, and we found a primary droplet diameter of 30.6 μm. Assume the liquid formulation has a viscosity of 0.001 kg/m/s and an air density of 1.2 kg/m3. Solution Substituting in the given numbers, we find the Weber number We dl ¼ ρl db U0 2 =σ ¼ 1000 30:6 106 ð120Þ2 =0:072 ¼ 6120 and the Reynolds number Redl ¼ ρldb U0/μl ¼ 1000 (30.6 106) 120/ 0.001 ¼ 3672. Eq. (8.29) then gives a secondary droplet diameter of 4.4 μm, which, unlike the primary droplet diameter, is now in the diameter range suitable for inhalation into the lungs and is typical of that produced by commercially available jet nebulizers.

8.3.4 Cooling and concentration of jet nebulizer solutions It has long been known that jet nebulizers become cooler than their surroundings and that the concentration of drug in solution increases during operation (Mercer et al., 1968). Cooling occurs because the droplets inside the nebulizer evaporate to come into equilibrium with the air entering the nebulizer (which is generally not saturated). This causes the droplets to cool, as we saw in Chapter 4. Most of these droplets impact on baffles and walls in the nebulizer and return to the liquid reservoir in the nebulizer, cooling the nebulizer and its contents. Because the droplets lose water to the air as they evaporate and humidify this air, water ends up leaving the nebulizer as water vapor, while drug is left behind with the droplets that impact before leaving. As a result, an amount of water leaves the nebulizer as water vapor, resulting in water leaving the nebulizer at a faster rate than the drug. This results in concentration of the drug in the nebulizer. We can be more specific about cooling of the nebulizer with the following analysis. In particular, if we consider a control volume V with surface S that surrounds a nebulizer, the energy equation for this volume is given by ð ð ð d ρ^ udV (8.31) qdS ρhV n^dS ¼ dt S

S

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where we have neglected differences in kinetic and gravitational energy between the inlets and outlets compared with differences in enthalpy. Here, q is the rate of heat transfer through the nebulizer walls over the surface S surrounding the nebulizer, h is the specific enthalpy of the gas (air + water vapor) and droplets entering or exiting the nebulizer, and the right hand side is the rate of change of internal (“thermal”) energy of the nebulizer and its contents (including plastic walls, liquid in reservoir, droplets, and air). This can be written more simply as X X dT hm_ out ¼ mc (8.32) Q_ + hm_ in dt where Q_ is the rate of heat transfer to the nebulizer from the ambient room, and the two summation terms give the rate at which enthalpy is convected in and out of the nebulizer where the air flows in or out of the nebulizer. The sums are to be done over the different components of the material entering or exiting the nebulizer, that is, they give a term for air, for water vapor, and for the droplets, where m_ in or m_ out are the mass flow rates of each of these components. On the right hand side, we have replaced the right hand side of Eq. (8.31) using a mass-averaged specific heat, c, for the nebulizer and its contents that we assume are temperature-independent for the range of temperatures we expect, where T is a mass-averaged temperature of the nebulizer and its contents and m is its mass. Note that the enthalpy of the air, the water vapor, and the droplets entering or exiting the nebulizer can all be written as functions of temperature as h ¼ cp T + arbitrary constant Also, we must account for the fact that the mass flow rates of water vapor and droplets are different at the entrance and exit (since no droplets enter the nebulizer but some do leave and since the droplets will humidify the air to some extent, so the water vapor concentration of the air is different on leaving than entering the nebulizer), while for air, the mass flow rates going in and out of the nebulizer are the same. Including these considerations, we can write Eq. (8.32) as dT m_ air cl Tout m_ l ¼ mc (8.33) Q_ + cpa ðTin Tout Þm_ air + cpw ðcsin Tin csout Tout Þ ρair dt where cpa is the specific heat of air; cpw is the specific heat of water vapor; cl is the specific heat of liquid water; Tin is the temperature of the air and water vapor entering the nebulizer; Tout is the temperature of the air, water vapor, and droplets exiting the nebulizer; m_ air is the mass flow rate of air through the nebulizer; and m_ l is the mass flow rate of liquid droplets leaving the nebulizer. The water vapor concentration in the air at the entrance is csin , while that at the exit is csout (where both of these are a function of temperature via a Clausius-Clapeyron

202 The Mechanics of Inhaled Pharmaceutical Aerosols

equation like that given in Chapter 4 and note that they are to be evaluated using the same ambient pressure that ρair is evaluated at since strictly speaking it is the mass fraction cs/ρair at the inlet or outlet that should appear in this equation). In Eq. (8.33), the inlet temperature Tin can be considered known and equal to the ambient temperature T0. In addition, the inlet water vapor concentration csin can also be assumed known (based on the relative humidity and temperature in the air supply line). The air mass flow rate through the nebulizer m_ air can also be considered known. Eq. (8.33) can be simplified by making two additional assumptions. In particular, if we assume the rate of heat transfer to the nebulizer, Q_ can be obtained from a thermal resistance hres (see, e.g., Incropera and DeWitt, 2011), that is, Q_ ¼ hres ðT0 T Þ

(8.34)

and if we also assume the outlet temperature Tout is equal to the temperature of the nebulizer T, then Eq. (8.33) can be written as d m_ air ðmcT Þ ¼ T0 hres + cpa m_ air + cpw csin ρair dt m_ air (8.35) T k + cpa m_ air + cpw csout + cl m_ l ρair Defining

m_ air a ¼ T0 hres + cpa m_ air + cpw csin ρair

(8.36)

m_ air + cl m_ l ρair

(8.37)

and b ¼ hres + cpa m_ air + cpw csout then this equation can be written in the form d ðmc T Þ=dt ¼ a bT

(8.38)

If we assume m, c, a, and b are independent of T or time t, Eq. (8.38) can be solved to show that the temperature obeys T ¼ a=b + ðT0 a=bÞebt=mc

(8.39)

In this case, the temperature will follow an exponential decay from initial temperature T0 to an equilibrium temperature given by Tss ¼ a/b, where Tss < T0 (since Tss/T0 ¼ a/(bT0) < 1 from the definition of a and b), as shown in Fig. 8.8. The actual time dependence of the nebulizer temperature will not be given exactly by Eq. (8.39), since the mass of the nebulizer and its contents, m, and the specific heat c are not constant (e.g., m decreases with time). In addition, the value of b in Eq. (8.38) will depend on T because of the dependence of the water

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Temperature T

T0

Tss = a/b

Time t FIG. 8.8 Jet nebulizer temperature vs time.

vapor concentration csout on T (recall from Chapter 4 that water vapor concentration cs varies as e1/T), so that solution of Eq. (8.38) is not so straightforward. These factors could be included, and a more general solution to Eq. (8.38) could be sought, but the value of the thermal resistance, hres, in Eq. (8.34) is not usually known since it will be affected by conductive heat transfer from the patient’s hand holding the nebulizer and convective motion of air next to the nebulizer and cannot be readily predicted. Despite these difficulties, it is clear from this analysis that nebulizer temperature can be expected to decay in an approximately exponential manner over time to a constant value, as has indeed been observed by many researchers. A straightforward consideration of mass conservation can be used to detail how the concentration of solute (which includes drug and usually NaCl) in a jet nebulizer increases with time. In particular, the mass of solute in the nebulizer is related to the volume of liquid in the nebulizer by ms ¼ C Vl

(8.40)

where C is the concentration (e.g., in kg/m3) of the solute in the liquid. The mass of solute in the nebulizer can only change due to solute being carried out of the nebulizer by liquid droplets exiting the nebulizer. However, we have said that the rate at which mass leaves the nebulizer as droplets is m_ l . The volume flow rate of these drops is m_ l /ρl, and their concentration is C (since the concentration of the drops is nearly the same as the concentration of the liquid in the reservoir (Stapleton and Finlay, 1995)), so the rate at which solute leaves the nebulizer will be dms m_ l ¼ C dt ρl

(8.41)

Substituting Eq. (8.40) into Eq. (8.41), we obtain an equation for the rate of change of solute concentration: C

dV l dC m_ l + Vl ¼ C dt Vl dt

(8.42)

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Notice that to solve this equation for C(t), we must know the volume of liquid Vl in the nebulizer. However, we can develop an equation for Vl from mass conservation. In particular, the volume of liquid in the nebulizer will change because the air picks up water vapor as it travels through the nebulizer (since the drops in the nebulizer try to bring the water vapor concentration in the air up to the same level that is present at their surfaces, which is usually nearly saturated for the isotonic solutions that are usually used in nebulizers). In addition, liquid is lost as droplets exit the nebulizer. More specifically, we can write ρl

dV l m_ air ¼ ðcs csout Þ m_ l dt ρair in

(8.43)

where ρair and cs are to be evaluated at ambient pressure, and we have neglected any density changes in the liquid due to increases in solute concentration. To determine how the concentration of solute C changes with time, we must solve Eq. (8.43) and put our solution for Vl(t) into Eq. (8.42). We can then solve Eq. (8.42) to obtain C(t). However, the right hand side of Eq. (8.43) will depend on the temperature of the nebulizer T, since csout is a function of the temperature. Thus, we must solve the equations we wrote down earlier for the nebulizer temperature T before we can solve Eq. (8.43). However, we saw above that this cannot be done easily, so instead, to obtain an idea of how the concentration changes with time, let us assume that the right hand side of Eq. (8.43) does not vary with time. This means we are assuming that the rate of water vapor transport and liquid droplet transport out of the nebulizer are constant in time. With this assumption, we can solve Eq. (8.42) to obtain 1 m_ air ðc s c s Þ (8.44) m_ l + Vl ¼ Vl0 λt where λ ¼ ρair out in ρl and Vl0 is the initial volume of liquid in the nebulizer. This equation clearly cannot be valid for all times, since the liquid volume Vl decreases linearly with time according to this equation and will become negative after some time. Thus, it is clear that our assumption of a constant right hand side to Eq. (8.43) is incorrect. However, if we substitute this equation into Eq. (8.42), we obtain a linear equation for C that can be solved to give CðtÞ ¼ C0

Vl0 Vl0 λt

1 m_ l

λρl

(8.45)

where C0 is the initial concentration of solute in the nebulizer. This equation predicts an algebraic increase of concentration with time. The actual concentration variation with time will not obey this equation exactly, since the rate of change of volume in the nebulizer is not constant. However, Eq. (8.45) can

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be used to fit experimental data (Mercer et al., 1968), where indeed concentration increases with time, as noted by many authors.

8.4 Other atomization methods While vibrating mesh and jet nebulizers make up the majority of aqueous aerosol delivery devices used clinically for inhaled pharmaceutical aerosol delivery, several other atomization methods have been the focus of varying levels of research and development work. Let us consider two of these as follows.

8.4.1

Impinging liquid jet atomization

The production of droplets by the collision of two liquid jets, referred to as impinging jet atomization, has seen diverse industrial use (Ashgriz, 2011). Although many researchers have examined its use in bipropellant rocket engines, it is in fact the droplet production mechanism used in the Respimat® device for inhaled pharmaceutical aerosol delivery (Dalby et al., 2011). In the Respimat®, the essential aspects of droplet production involve the use of high pressures (e.g., 25 MPa, Hochrainer et al., 2005) to extrude a liquid formulation at high speed through a pair of small nozzles that are a few micrometers in diameter. The nozzles are positioned so that the liquid jets emanating from them collide approximately 25 μm downstream, forming an unstable sheet that subsequently breaks up into droplets, as shown schematically in Fig. 8.9. With this approach, a small volume (15 μL) of formulation can be delivered as an aerosol in a single breath. Liquid sheet breakup and subsequent droplet formation with impinging jet atomizers are known to depend on the jet Weber number We ¼ ρl D U 2 =σ

(8.46)

Re ¼ ρl D U=μl

(8.46a)

the jet Reynolds number

and the angle 2θ subtended by the colliding jets. Here, D is the nozzle/jet diameter, while ρl is the density of the liquid formulation, and μl is its dynamic viscosity. For the Respimat®, the angle subtended by the jets is 2θ ¼ 90°. Wachtel (2016) gives values of We ¼ 1840 and Re ¼ 738 and also notes that the Respimat® operates in the “impact wave” regime of sheet breakup whereby unstable waves form on the surface of the sheet, leading to ligaments breaking up via capillary instability (Chen et al., 2013). Correlations for predicting droplet diameters with impinging jets in the impact wave regime remain a topic of current research. Rodrigues et al. (2015) find that droplet diameters depend on the velocity profile in the jets. For a uniform jet velocity profile, a linear stability analysis applied to a liquid sheet (Dombrowski and Jones, 1963), with an assumption that the most unstable

206 The Mechanics of Inhaled Pharmaceutical Aerosols

High-pressure liquid vessel

U Liquid jet

2q

U Liquid jet

Liquid sheet (seen side-on) Sheet breakup

Droplets

FIG. 8.9 Schematic of the colliding impinging jet geometry viewed along the plane of the resulting liquid sheet that is formed where the jets collide.

wavelength is responsible for producing droplets, leads to the following correlation (Rodrigues et al., 2015; Ryan et al., 1995) for droplet diameter d: d 1:14 ¼ " #1=3 D ρa 1=6 ð1 cos θÞ2 We ρl sin 3 θ

(8.47)

Here, D is the nozzle/jet diameter, and ρa is the ambient gas density (i.e., air density in our case). While Ryan et al. (1995) find Eq. (8.47) disagrees with their experimental data, Rodrigues et al. (2015) suggest this is due to a nonuniform velocity profile in the experiment of Ryan et al. (1995). Rodrigues et al. (2015) extend Eq. (8.47) to consider other jet profiles, but they only examine the accuracy of this correlation at higher We than occurs in the Respimat® so that its validity for our purposes remains unknown.

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Example 8.4 Estimate the droplet diameters produced by a modified Respimat® in which the jets meet at an angle of 2θ ¼ 60° instead of the usual 90°. Assume a liquid formulation density of 998 kg/m3, air density 1.2 kg/m3, and values of We ¼ 1840 and nozzle diameter D ¼ 6.6 μm given by Wachtel (2016) for the Respimat®. Assume a uniform velocity profile within the jets and use Eq. (8.47). Compare with the droplet diameter seen for 2θ ¼ 900. Solution This is a straightforward matter of plugging the following numbers into Eq. (8.47): ρa ¼1.2 kg/m3, ρl ¼ 998 kg/m3, We ¼ 1840, and θ ¼ 30°, D ¼ 6.6 106 m. In doing so, we find droplet diameter d ¼ 3.6 μm. Using instead the marketed Respimat’s actual jet angle 2θ ¼ 90°, Eq. (8.47) gives d ¼ 3.0 μm. Wachtel (2016) measured a value of MMAD ¼ 4 μm experimentally for 2θ ¼ 90°. Given the polydispersity of the aerosol seen with impinging jet aerosols, the prediction of Eq. (8.47) is not unreasonable considering that the complex nonlinear breakup dynamics that occur in the impact wave regime are not considered by this equation.

8.4.2

Ultrasonic wave nebulization

Ripples on an air-liquid interface in a container are readily generated by forced vibratory vertical motion of the bottom wall of container. For forcing at ultrasonic frequencies f0, the surface waves that result have high wave numbers (i.e., short wavelengths) and are called “capillary waves” since surface tension supplies the restoring force that propagates these waves. Assuming an incompressible, inviscid, irrotational flow, a plane wave analysis can be used to show that these capillary waves obey the relation (Kundu and Cohen, 2008): ω2 ¼

σk3 tanh kH ρl

(8.48)

where H is the depth of liquid in the container and ω ¼ 2πf is the circular frequency, while f ¼ 1/T is the wave frequency, and T is the wave period. For high wave numbers, kH ≫ 1, and tanh kH is well approximated by 1, giving the following relation between wavelength and frequency of the wave: 2πσ 1=3 (8.49) λ¼ ρl f 2 Waves for which an integer number of half wavelengths fit inside the container form standing waves due to reflection of the sides of the container, and these standing waves are unstable to a Faraday instability (Faraday, 1831) when the forcing frequency f0 is above a certain value for a given amplitude of forcing. Nonlinear growth of unstable Faraday waves leads to the wave tips forming

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ligaments that break up due to a Rayleigh-Plateau mechanism (see Section 8.2), thereby producing droplets. To decide if viscous effects need to be considered when examining the Faraday instability, a viscous timescale may be defined (Goodridge et al., 1997) by the ratio Tv ¼ λ2 =νl

(8.50)

where νl is the kinematic viscosity of the liquid formulation. Viscous effects will be negligible if the vibration period T ≫ Tv. Realizing that T ¼ 1/f and using Eq. (8.49) for λ, we find that viscous effects can be neglected (Donnelly et al., 2004) if 2 3 σ (8.51) f0 ≪ νl ρl For water, the right hand side of Eq. (8.51) is 5.2 GHz, while typical ultrasonic nebulizers operate at frequencies below a few MHz; thus, we may neglect viscous effects for formulations with viscosity, surface tension, and density near that of water. An inviscid linear stability analysis (see, e.g., Kumar and Tuckerman, 1994) then finds that the most unstable modes have a frequency half that of the forcing frequency f0, that is, f ¼ f0/2. Substituting f ¼ f0/2 into Eq. (8.49) and making the assumption that the diameter of the droplets forming from the Faraday waves has a diameter that is an unknown fraction Cu of the wavelength of the most unstable Faraday waves, we find the following wellknown equation for droplet diameters produced by an ultrasonic nebulizer (Lang, 1962): 8πσ 1=3 (8.52) d ¼ Cu ρl f02 Experiments find a value of Cu ﬃ 0.35 (Donnelly et al., 2004), so we may write 8πσ 1=3 (8.53) d ¼ 0:35 ρl f02 Although there is considerable discussion in the literature as to whether droplet formation with ultrasonic nebulizers instead occurs due to pressure waves leading to cavitation bubble rupture at the surface, recent work supports the above Faraday wave mechanism in the parameter range at which medical ultrasonic nebulizers operate (Deepu et al., 2018; Donnelly et al., 2004). For water at room temperature, Eq. (8.53) indicates that a forcing frequency of f0 ¼ 1.9 MHz is needed to produce droplets with a diameter of 3 μm. Ultrasonic nebulizers normally use a piezoelectric transducer placed in contact with the formulation reservoir to achieve MHz vibration frequencies and thereby achieve atomization by the above discussed wave dynamics.

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Note that for high-viscosity formulations, Eq. (8.51) may no longer be obeyed, which moves the capillary waves into the viscous regime, rather than the low-viscosity regime considered above. High viscosity can result in cessation of droplet production, which is most readily seen by realizing that for a given forcing frequency, it is found that droplet production requires a minimum threshold acceleration a of the vibrations. When viscous effects are negligible, Goodridge et al. (1997) find the threshold acceleration is given by 1=3 σ ð2πf0 Þ4=3 (8.54) as 0:261 ρl but in the viscosity dominated region, they find the threshold acceleration is instead given by av 1:306ðνl Þ1=3 ð2πf0 Þ3=2

(8.55)

For a given forcing frequency f0, the ratio of threshold acceleration for the lowand high-viscosity regimes is av 5:0νl 1=3 ð2πf0 Þ1=6 ¼ 1=3 as σ

(8.56)

ρl where νl is the kinematic viscosity of the high-viscosity formulation while σ and ρl are the surface tension and density of the low-viscosity formulation. Considering the low-viscosity solution to have σ ¼ 0.073 N/m and ρl ¼ 998 kg/m3 and a high-viscosity formulation with νl ¼ 30 106 m2/s, we find av/as ¼ 56, that is, the minimum threshold acceleration needed to achieve droplet production is 56 times higher for the viscous formulation in this case. For an ultrasonic nebulizer with a piezoelectric transducer operating at a given frequency and amplitude, the acceleration is fixed, and while this acceleration may be sufficient to produce droplets from capillary waves with a dilute aqueous formulation, this same piezoelectric vibration may give insufficient acceleration to nebulize a highviscosity formulation. This may explain the usual recommendation that ultrasonic nebulizers not be used with high-viscosity formulations (Carvalho and McConville, 2016), although this is an oversimplified recommendation since increasing the acceleration of the piezoelectric transducer (by increasing either its amplitude or frequency) would allow successful atomization of highviscosity formulations. Note however that droplet diameters would then no longer obey Eq. (8.53), since that equation applies only when viscous effects are 2 3 σ ), diameters instead negligible. In the viscous regime (when f0 ≫ νl ρl vary as (Donnelly et al., 2004) 1=2

d∝νl 1=2 f0

(8.57)

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so that droplet diameters increase with increases in kinematic viscosity ν and are unaffected by surface tension. Eqs. (8.56) and (8.57) are valid at opposite 2 3 σ extremes of the dimensionless frequency νl , but of course in between ρl these two asymptotic regimes, neither equation alone is valid, which has led to some confusion regarding the effects of the relevant parameters on droplet sizes (Yeo et al., 2010). It should be noted that suspension formulations in which the suspended particles have diameters that are of the same order as the droplets may result in preferential nebulization of the liquid, leaving the suspended particulates behind (Nikander et al., 1999). This is presumably because such particles are not as readily entrained in the ligaments that form at the capillary wave crests prior to droplet ejection. For such formulations, other atomization methods (e.g., jet nebulizers) may be preferred. Note also that viscous dissipation of wave energy in ultrasonic nebulizers can result in the temperature of the formulation rising considerably over time, which may be a concern for biologics and thermally labile drug molecules such as proteins (Hertel et al., 2015).

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