Design and characterization of a cough simulator

The transmission of infectious respiratory diseases has always been a topic of public health, drawing significant attention from various disciplines for decades (Morawska 2006, Zhu et al 2013, Filipiak et al 2015). One of the most common paths of transmission is through inhaling pathogen-carrying airborne droplets from sneezing and coughing, such as in the common cold and influenza Moser et al (1979), in severe acute respiratory syndrome (Olsen et al 2003), in avian flu and, probably, Ebola (Funk and Kumar 2015, Osterholm et al 2015, WHO 2015). Due to the limited knowledge of Ebola transmission, even with modern technology, an unprecedented number of medical personnel in the front line are still unable to evade infection (WHO 2014). More importantly, though it is unrealistic if not impossible to predict new contagions in the future, it is reasonable to believe that the spread of most respiratory diseases follows a certain pattern. There is an urgent need to reveal the routes of transmission of those infectious diseases. To study the transmission, one of the most important approaches is simulating the spread of disease initiated by coughing.

Coughing has been studied as a challenging subject (Shaker and Hunt 2008, Taylor 2012, Dryahina et al 2014), especially for determining its transport characteristics. Some flow characteristics of a cough, such as flow rate, flow direction and mouth opening area, were preliminarily measured (e.g., Gupta et al 2009). The most important factor contributing to cough-induced disease transmission is the characteristics of droplets, which comes no surprise as the droplets are believed to be the pathogen-carrying vectors. Among all characteristics, the droplet size is a crucial factor for the following reasons. First, droplets with different sizes are generated from different depths of the respiratory tract as high-speed exhaling airflow skims over the mucus (Geng 1985, Zhang et al 2015), and atomization from lungs into upper airways (Ross 1955) as well as vocal chords vibration during coughing (Morawska et al 2008). Although it is difficult to determine the exact origin of expiratory droplets, the assumption that the larger droplets are generated from the upper tract is still widely accepted since the smaller droplets have a smaller Stokes number and tend to follow the gas flow more easily. This also indicates that different sized droplets may carry various types and densities of pathogens from the mucus surface of different respiratory tract depths. Second, the spread of droplets can be significantly affected by droplet size or associated inertia. The coughing-induced disease transmission could be classified as droplet transmission for large droplets and airborne transmission for comparatively smaller droplets, respectively (Gralton et al 2011). The former usually settle quickly to the surrounding surfaces (such as ground or garment) within 1 m from the site of generation (Garner 1996), and the latter may remain airborne for a longer period of time and therefore has a much broader spreading area (Duguid 1946, Wang et al 2005, Xie et al 2007). Third, the droplet size dominates its penetration depth and deposition inside the respiratory tract of the receptors. Studies show that the smaller droplets are more likely to penetrate deeper into the respiratory tract while the larger droplets are more likely to deposit onto the upper airway surface of the receptors. The critical diameter, also known as the cut-off diameter, is around 5–10 μm (WHO 2007, Gralton et al 2011). The penetration distance matters because the possibility and mortality of infection grow as the pathogens penetrate deeper (Thomas 2013). Due to their importance, the determination of droplet size and its distribution has been a focus of recent studies (Sze To et al 2009, Chao et al 2010). Many studies have agreed that the sizes of cough droplets have at least two main ranges (Wan et al 2007, Yang et al 2007): one group of fine droplets typically from sub-micron to about 10 μm, and the other of coarse droplets from several tens of microns to sub-millimeters. A cough simulator generating such size characteristics of droplets is defined here as a bimodal cough simulator.

The desired characteristics of a bimodal cough simulator should include the following features: coughing duration about 1 s (Wan et al 2007, Sze To et al 2009), volumetric air capacity per cough around 0.4 l (Wan et al 2007, Sze To et al 2009), total mass of cough droplets per cough from 6.7 to 75 mg (Sze To et al 2009, VanSciver et al 2011), area of mouth opening in coughing from about 2 to 5 cm² (Gupta et al 2009, Chao et al 2010, VanSciver et al 2011), coarse droplet size from 100 to 300 μm (Wan et al 2007), fine droplet size around 5 μm (Wan et al 2007), the maximum coughing velocity at mouth opening from 1.5 to 30 m s⁻¹ (Zhu et al 2006, VanSciver et al 2011) and droplet size distributions in probability density function (PDF) over a range from 2 μm to 2 mm (Sze To et al 2009, Chao et al 2010, Gralton et al 2011). There have been few reported prototypes of bimodal cough simulators, nevertheless, none meeting all the desired features mentioned above. Lindsley et al (2013) suggested to use an airbrush to generate coarse droplets and a nebulizer for fine droplets. However, the system was not operated by using both an airbrush and a nebulizer simultaneously. It also failed to give the coughing velocity, which is an important factor for pathogen spreading. In summary, many studies on fine droplet size and distribution have been reported, however few have been reported on size distributions of coarse droplets by coughing. Hence, it is imperative to design and build a two size ranges or called bimodal cough simulator, as the one delineated here.

The entire experiment system has three sub-systems, namely, the bimodal cough simulator with a control unit, a velocity measurement system, and a droplet sizing system, as shown in figure 1.

Zoom In Zoom Out Reset image size

The cough simulator consists of a pressurized gas tank, a nebulizer and an ejector, connected in series. The cough air flow is supplied by the pressurized gas tank. The fine droplets are generated inside a cylinder container by a nebulizer (Mistmakerworld^®, MDC01). The coarse droplets are generated by an ejector.

The core of our design is a coarse droplet generator, which is an ejector composed of a venturi nozzle and reservoir. As shown in figure 2, the coughing air with fine aerosol droplets flows through the venturi nozzle. The solution is sucked up from the reservoir by the negative gauge pressure at the venturi throat, and consequently broken into liquid segments or coarse droplets by the passing air steam. The hydrophobic material of our ejector minimizes the liquid attachment of the solution sucked from the ejector reservoir. For the air humidity of flow, due to the intermittent but repeated generation of fine droplets and breaking up of coarse droplets, the relative humidity of the jet flow is close to saturation. To match with the human body temperature (37 °C), the temperature of the jet flow is controlled between around 35 °C–40 °C by the heating of the nebulizer. Some operation conditions and nozzle design dimensions are critical to the droplet generation, including but not limited to: the gas pressure from the air cylinder (around 10 kPa), the solution level in the ejector reservoir (7–10 mm), as well as the diameters of nozzle outlet (25 mm), nozzle throat (12 mm) and the capillary (0.75 mm) at throat, as shown in figure 2.

Zoom In Zoom Out Reset image size

The coughing frequency and duration are controlled by programmed solenoid shut-off valves. The process logic operations are shown in the figure 3. Based on the published data (Wan et al 2007, Sze To et al 2009), each coughing duration is set to be 1 s with an interval of 3 s between coughs. The periodic repeated coughing is achieved by automatically switching on/off of the solenoid valve (Grainger, P251SS-120-A), where the whole control system is consisted of an electrical counter, a repeat cycle timer, a time delay relay and a solenoid valve. The dimension of this cough simulator is about 800 × 300 × 300 mm, which is much smaller than the one from Lindsley et al (2013), which is about 1500 × 700 mm (2D projection). The cough simulator supporting structure is made of steel, the nebulizer container is made of acrylic, and the nozzle is produced by 3D printing using acrylonitrile butadiene styrene. Other design dimensions can be referred in the appendix, such as in figure A1. The saliva is made of distilled water, glycerin and sodium chloride with a mass ratio of 1000:76:12, referring to the mucus content reported in Nicas et al (2005).

Zoom In Zoom Out Reset image size

A few limitations of the current cough simulator were identified to be further improved in the future. First, though the peak velocity, volumetric gas can be properly designed, the detailed velocity profile of coughing is not a controllable character yet. A more realistically designed control system to regulate coughing flow shall be implemented. Secondly, the combined effect of surface hydrophilicity/composition of saliva droplets under vibrating conditions shall be evaluated, which may require further modeling and experiment development.

The characterization of this cough simulator includes the determination of coughing velocity, coarse and fine droplet size and size distributions, and air volume per cough. The volumetric air released per cough can be readily determined thermodynamically over the gas released from compressed gas tank. The averaged air volume per cough is calculated based on the total pressure difference of the air cylinder before and after the experiment, the pressure regulated for air release, and the total number of coughs. The determination of coughing velocity and droplet size are explained in detail in the following.

3.1. Coughing velocity

The velocity of coughing is difficult to be determined. This is because the transient and non-uniform release of air per cough makes the common 'steady-state' types of velocity measurement techniques, such as laser Doppler velocimetry and pitot-tube method, inapplicable to our case. Some common 'instant-velocity' measurement techniques, such as hot-wire anemometry (HWA) and particle-image velocimetry (PIV), also fail to be applied. The HWA, whose measurement principle is based on the empirical correlation of convective heat transfer coefficient to Reynolds number over a cylinder, may not be accurate in the presence of fine and coarse droplets from the coughing. The traditional PIV, requiring the timely tracking of fine droplets in a two-dimensional plane (typically illuminated by a laser sheet), may have difficulties in field and time-domain resolutions, in addition to the required synchronization with the coughing. It is realized that some advanced PIV systems can reach kHz time resolution and the volumetric velocimetry (3D PIV) can also provide certain capability of capturing a 3D flow field. However, considering the overall constrains (with respect to the wide ranged droplet size and transient velocity) in size of view field of measurement, spatial resolution, time resolution, synchronization, and other limitations in real-time digital image process, in addition to the facility cost, 3D PIV may not be an effective or realistic choice for the measurements in this particular application.

The dual-beam cross-correlation method (Zhu et al 1991), which is also economically viable, can be used to characterize the coughing velocity. The basic principle is to best correlate the time-shifting of light scattered by non-uniformly-distributed tracers (droplets in our case) over a given distance along their trajectory path. As shown in figure 4, a laser beam is split by a 50–50 reflection and penetration mirror, and one of two split beams goes through a full-reflection mirror. Thus, a single laser beam becomes two parallel beams. The two laser beams are set at a given separation distance (3 cm in this case) along the spray path close to the nozzle opening (1 cm away in this case). The laser beams are received by two photodiodes, respectively, with light intensity converted into electric signals in terms of current or voltage. During a cough simulation, the transient flow with a mist of fine droplets generated by nebulizer is flushed out of the nozzle. The change in intensity is then affected by the light scattering by aerosols (mostly droplets) passing through the beams. The received signal from photodiodes is analyzed by cross-correlation method to determine the distance-averaged transient coughing velocity.

Zoom In Zoom Out Reset image size

The measured coughing velocity most likely represents that of fine droplets, which is also assumed to be the same as gas velocity. This non-slip in velocity between fine droplets and air can be justified since the size of the fine droplets is very small, which is validated by the laser diffraction measurement, and the corresponding Stokes number (Stk = 0.154) is much less than unity. The coarse droplets, once passing through the laser beam, also generate light scattering. However, their contribution to the light scattering is believed to be low due to the scarcity in total number of coarse droplet generated per cough, the low fraction of surface area among all droplets, and the wide dispersion in space by their inertia-driven trajectories.

Based on the large size of coarse droplets (a few hundred microns), the direct measurement for velocity of coarse droplets may be achievable by image-streaking technique in an illuminated plane (typically by laser-sheet). However, the number of coarse droplet generated per cough is too scarce, and the wide dispersion in space makes the image capture and streaking track in a 2D plane to be very difficult if not impossible. Hence in this study, the velocity of coarse droplets is indirectly estimated by the acceleration of coarse droplets from their formation site (location of sucking channel from reservoir inside ejector) to the nozzle opening, driven by the coughing air.

3.2. Droplet size distribution

Our cough simulator has two distinctively different mechanisms for droplet generation: fine droplets generated by ultrasonic membrane vibration in a nebulizer, and coarse droplets generated by flow instability and shearing in the nozzle ejector. Based on our measurements, there is little overlap in size range between fine droplets and coarse droplets. The fine droplet size and size distributions are measured by the principle of laser diffraction using a commercial instrument (Sympatec^®, HELOS/KR) at the outlet of nozzle, whereas the coarse droplet size and its size distribution are determined by a fibrous collector and image analysis system developed in this study. The HELOS/KR has a measuring range of 0.1–8790 μm, and the absolute accuracy is typically within ±1%. The measurement setup is illustrated in figure 1. The following is focused on the design and measurement principles of our method for the size determination of coarse droplets by coughing.

Figure 5 shows the schematic diagram of a collector of coarse droplets and the subsequent image of wetted spots for droplet size analysis. This collector of coarse droplets is a water-absorbing paper made of hydrophilic fibers (in our case, we use a reed-based calligraphic paper, DoubleDeer^®, Xuan). To enhance the visibility for image analysis, some dyes (fountain pen ink) are added into the cough liquid. The fibrous collector is located in front of the nozzle opening at a short distance L, which is designed to allow the fine droplets to be averted from direct collision while trap the coarse droplets by inertia impaction without overlapping of collected coarse droplets. This is because the fine droplets follow the gas phase closely (Stk ≪ 1) and hence diffuse out before hitting on the fibrous collector, while the coarse droplets with much larger momentum penetrate the distance of L can collide on the collector. Besides, the fine droplets are very likely fully evaporated within a distance L as the computational fluid dynamics (CFD) results predict in the later section. The size of collector is also large enough to capture all coarse droplets. The captured droplets are soaked into the collector by the capillary effect but kept locally by the surface tension. The image of wetted spots is then digitally analyzed to yield droplet size and size distributions.

Zoom In Zoom Out Reset image size

The image analysis method is based on the mass conservation of individual droplets and the total mass conservation. As schematically shown in figure 6, once a droplet is collected and soaked into the fibrous collector, the spherical shape of droplet becomes a round disc, with a typically slightly larger wetted image (A_a) on the collision-front surface and a slightly smaller area (A_b) in the back surface. For the ith droplet, we have

$$\begin{eqnarray}&&{m}_{i}={\rho }_{{d}}{\alpha }_{i}{\delta }_{i}{A}_{i}\equiv {K}_{i}{A}_{i},\end{eqnarray} \tag{ 1 }$$

where m, A and ρ represent mass, droplet soaking area and density of the droplet, respectively, while δ and α represent thickness and voidage of the fibrous collector. The area A_i can be calculated based on the soaked area from both sides A and B of the fibrous collector as in the subscript from equation (2):

$$\begin{eqnarray}&&{A}_{i}=\left(\displaystyle \frac{{A}_{iA}+\sqrt{{A}_{iA}{A}_{iB}}+{A}_{iB}}{3}\right).\end{eqnarray} \tag{ 2 }$$

Consequently, the total mass of collected droplets can be determined by:

$$\begin{eqnarray}&&{\rm{\Delta }}{m}_{{T}}={m}_{{T}1}-{m}_{{T}0}=\displaystyle \sum _{i}{m}_{i}=\displaystyle \sum _{i}{\rho }_{{d}}{A}_{i}{\delta }_{i}{\alpha }_{i},\end{eqnarray} \tag{ 3 }$$

where the subscript T, 0, 1 stands for total, before and after the collection, respectively. Assuming thickness δ and voidage α of the fibrous collector are constant, and the density of droplet is known (which can be measured easily), the equation (3) becomes:

$$\begin{eqnarray}&&{\rm{\Delta }}{m}_{T}={\rho }_{d}{\bar{\delta }}_{i}{\bar{\alpha }}_{i}\displaystyle \sum _{i}{A}_{i}=K{A}_{T},\end{eqnarray} \tag{ 4 }$$

where the parameter $K\equiv {\rho }_{d}{\bar{\delta }}_{i}{\bar{\alpha }}_{i},$ which represents the true soaked mass of individual droplet inside the fibrous collector. K can be easily determined by the measured total mass and wet area:

$$\begin{eqnarray}&&K=\displaystyle \frac{{\rm{\Delta }}{m}_{T}}{\displaystyle \sum {A}_{i}}.\end{eqnarray} \tag{ 5 }$$

For a single droplet, its mass can be determined by individual wet area and parameter K:

$$\begin{eqnarray}&&{m}_{i}=\displaystyle \frac{\pi }{6}{d}_{i}^{3}{\rho }_{d}=K{A}_{i},\end{eqnarray} \tag{ 6 }$$

where d represents the diameter of the droplet. Thus, combining (5) and (6), the size of a single droplet can be correlated by the total mass, individual wet area, and liquid density by:

$$\begin{eqnarray}&&{d}_{i}=\sqrt[3]{\displaystyle \frac{6{\rm{\Delta }}{m}_{T}}{\pi {\rho }_{d}}\left(\displaystyle \frac{{A}_{i}}{\displaystyle \sum {A}_{i}}\right)}.\end{eqnarray} \tag{ 7 }$$

Consequently, the size distribution can be obtained by including all droplet sizes. Each and every wet areas A_i can be analyzed by a self-compiled image analysis program.

Zoom In Zoom Out Reset image size

The developed method above has an important assumption, which is no droplet break-up or rebounding during the droplets impingement on the fibrous collector. To verify this, a calibration using a dripping-tube has been conducted. The uniformity in K value of the fibrous collector and the direct comparison between true droplet size and the size calibrated by the proposed method have also been studied. And results show that the effect of bounce-back or break-up of droplets is considered to be insignificant under our current experiment conditions. The individual K_i value is reasonably uniform, with a discrepancy within 7% to the averaged K value. The comparison between the droplet mass calculated by (6) and its true mass shows a good match with very minor discrepancy (<2.5%). The details can be referred in the appendix.

While the fine and coarse droplets are very much independently generated by nebulizer and ejector, respectively, there could be some coupling effects such as coagulation of fine droplets into larger droplets or break-up of coarse droplets. However, the coagulation of fine droplets can be ignored here. This is because, to form a coarse droplet of 50 μm (lower bound in coarse size distribution) from the coagulation of 5 μm (medium-size) of fine droplets, it would require a continuous collision of about 10³ times, which is nearly impossible with our low volumetric concentration of fine droplets and very low relative velocity among the fine droplets to make such collisions. The break-up of coarse droplet, if any, would likely to be at the same order of magnitude of the original mother droplet, i.e., the daughter droplets are unlikely to be in the same size range of fine droplets, nor would affect the mass conservation of coarse droplets if some break-up are indeed in the size range of fine droplets. Hence, the coupling effects on mass or size redistribution of fine droplets and coarse droplets are neglected in both modeling analysis and experimental data analysis.

In our study, each collector is used for only one cough sample, hence it is unlikely to have the sever overlapping of droplets, especially with a sampling distance far enough from the coughing nozzle origin (L/D > 10). Another concern is the effect of droplet evaporation during the droplet travel between the nozzle exit and the fibrous collector. The evaporation effect, which turns out to be insignificant in our study, will be detailed in the following section.

Above section introduces a method to determine the size and size distribution of droplets collected at the distance of L from the nozzle. The question is: how does the droplet evaporation affect the droplet size at the nozzle exit by the droplet size measurement at a downstream distance? Since a direct measurement of this evaporation effect is very difficult, we adopt a modeling approach. The modeling effort also helps to determine the required size of droplet collector to be used in the experiment.

The d²-law model of droplet vaporization (Goldsmith 1955, Sirignano 1999) is among the simplest and the most popular, which considers the evaporation of a saturated spherical droplet by pure diffusion in a quiescent gas environment. The classical d²-law can be directly applied to estimate the evaporation for moving droplets with very small relative Re number (around or less than 1), such as the case of fine droplets (1–10 μm) in this study. However, the aforementioned assumptions of quiescent gas environment and pure diffusion are not adequate for the higher Re number scenario, such as the case of the coarse droplets (100–700 μm with corresponding Re number around 10–50) in this study. Hence, a modified d²-law has been used to include the convection effect (Sirignano 1999):

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}m}{{\rm{d}}t}=4\pi D\rho {R}_{D}\left(1+\displaystyle \frac{{Re}Sc}{2}\right)\mathrm{ln}\left(1+\displaystyle \frac{{Y}_{{R}_{D}}-{Y}_{\infty }}{1-{Y}_{{R}_{D}}}\right),\end{eqnarray} \tag{ 8 }$$

where D, Y, R_D, Re and Sc represent the mass diffusivity, mass fraction of water vapor in the gas mixture, droplet radius, droplet Reynolds number, and Schmidt number, respectively. It should be noted that ${Y}_{{R}_{D}}$ depends on the droplet temperature and Re depends on both droplet velocity and ambient gas flow velocity. In this study, the ambient gas flow is acquired by a CFD simulation with the coupling effect of droplet vaporization. The droplet transport is modeled by a Lagrangian approach, with coupled momentum and energy equations of droplet.

The droplet momentum equation can be expressed by:

$$\begin{eqnarray}m\displaystyle \frac{{\rm{d}}{\vec{u}}_{d}}{{\rm{d}}t} & = & \displaystyle \frac{\pi }{8}{C}_{D}{d}^{2}\rho ({\vec{u}}_{g}-{\vec{u}}_{d})| {\vec{u}}_{g}-{\vec{u}}_{d}| \\ & & +\mathop{g}\limits^{\longrightarrow}-{\vec{u}}_{d}\displaystyle \frac{{\rm{d}}m}{{\rm{d}}t},\end{eqnarray} \tag{ 9 }$$

where g is the gravitational force and C_D is a function of Re (Mandø and Rosendahl 2010):

$$\begin{eqnarray}{C}_{D}\,=\left\{\begin{array}{ll}\displaystyle \frac{16}{{Re}}\left\{1+\left[\displaystyle \frac{8}{{Re}}+\displaystyle \frac{1}{2}\left(1+\displaystyle \frac{3.315}{\sqrt{{Re}}}\right)\right]\right\} & 100\gt {Re}\gt 1,\\ \displaystyle \frac{48}{{Re}}\left(1-\displaystyle \frac{2.21}{\sqrt{{Re}}}\right) & 100\lt {Re}.\end{array}\right.\end{eqnarray} \tag{ 9a }$$

The energy equation of droplet, with the effects of heat convection and droplet evaporation, can be expressed by:

$$\begin{eqnarray}&&{c}_{p}m\displaystyle \frac{{\rm{d}}{T}_{d}}{{\rm{d}}t}=\pi {d}^{2}h({T}_{g}-{T}_{d})+L\displaystyle \frac{{\rm{d}}m}{{\rm{d}}t},\end{eqnarray} \tag{ 10 }$$

where the convective heat transfer coefficient h is estimated using a simple heat transfer correlation of a non-evaporative sphere (Incropera and DeWitt 2011):

$$\begin{eqnarray}&&h=\displaystyle \frac{k}{d}\left(2\,+\,0.6\,* \,{{Re}}^{\displaystyle \frac{1}{2}}{{\Pr }}^{\displaystyle \frac{1}{3}}\right).\end{eqnarray} \tag{ 10a }$$

Hence, the droplet mass, velocity and temperature can be solved by the coupled equations (8)–(10), with the ambient gas velocity and humidity calculated via the CFD simulation.

Due to the transient nature of coughing (Zhu et al 2006, Chao et al 2010, VanSciver et al 2011), the ambient gas velocity and humidity of each individual coarse droplet have to be obtained through a CFD simulation. This transient, multi-component and turbulent flow is simulated between the mouth and the droplet collector, using ANSYS Fluent 14.5. Specifically, the species transport model is set active for the gas mixture of air and water vapor. The turbulence model is selected as realizable k–ε model, for its accurate prediction for spreading rate of round jets (ANSYS Inc. 2009). For simplification, the coughing is regarded as axisymmetric. In addition, the contribution of fine droplets evaporation to the ambient humidity is ignored. This assumption is not only for simplifying our simulation but also for allowing the maximum evaporation of coarse droplet in a drier environment without the humidification from fine droplet evaporation. The impact of ignoring fine droplet evaporation to the ambient gas velocity is considered small due to the insignificant amount of fine droplets in mass (<1%) comparing to that of gas per cough.

The simulation domain includes a mouth opening and a trap surface to represent the fibrous collector, with all geometric dimensions identical to those in the experiment. The total cell number of mesh is 4900. A grid-independency study of 14 200 cell shows almost identical result of flow field and the biggest difference of cough flow entrainment area is less than 3%. Thus the cell number is chosen as 4900 in this study. A summary of model selection, boundary condition, initial condition and governing equations are listed as table A1 in the appendix. Such combined approach of CFD and experiment measurements may also be an effective method to obtain a reliable flow field in determining dispersion characteristics, especially for a transient flow such as coughing.

5.1. Calibration in coughing characteristics

5.2. Impact of droplet evaporation on size measurement

Figure 7 shows the mass residual of droplet along x (up to L) for droplets of different initial sizes. The results suggest that, for coarse droplets (e.g., 500 μm), the evaporation can be ignored since the total evaporation is small (e.g., Δm_e < 1%). Hence, no modifications of evaporation effect on coarse droplet measurements are needed in this study. Figure 7 also suggests that, for fine droplets (e.g., 5 μm), the evaporation is relatively significant (>16% for Δm_e). However, since our size measurement of fine droplets is performed very close to the nozzle, there is also no need to take the evaporation effect into account on the fine droplet measurements. Another observation is that, for fine droplets, the evaporation rate is faster at the beginning, which might be due to the relatively large cooling from the initial droplet temperature of human body temperature that is higher to the ambient temperature. In addition, for very fine droplets (say < 5 μm), they may evaporate quite fast near the nozzle exit region, hence the evaporation rate may affect their hydrodynamic transport and humidification performance in that region, as suggested by Chen and Zhao (2010) and Gupta et al (2011). However, since this paper is mainly focused on the coarse droplet generation and size measurement, the evaporation effect of very fine droplets has little impact on our results.

Zoom In Zoom Out Reset image size

5.3. Size distribution of droplets

Table 2 gives an example of the bin size distribution of coarse droplets. It can be seen that, after reaching the peak between 137 and 197 μm, the number of droplets decreases as its size increases.

Table 2.  Bin data of coarse droplet measurement.

Droplet diameter (μm)	<77	(77,137]	(137,197]	(197,257]	(257,317]	(317,377]	(377,437]	(437,497]	(497,557]	(557,617]	(617,677]	(677,737]
Total number	177	542	756	406	322	184	90	42	23	12	6	3

Directly from the measured number of droplets, the residual probability of droplet size could be expressed as a number-based Rosin–Rammler distribution as expressed in equation (11), which is different from a mass-based distribution. These two kinds of distribution can be interchangeable:

$$\begin{eqnarray}&&P(X\gt x)=\exp \left[-{\left(\displaystyle \frac{x}{{\alpha }_{C}}\right)}^{{\beta }_{C}}\right],\end{eqnarray} \tag{ 11 }$$

where the x is in unit of μm. The corresponding PDF can be yielded as:

$$\begin{eqnarray}&&{f}_{N,C}(x)=\displaystyle \frac{{\rm{d}}}{{\rm{d}}x}(1-P)=\displaystyle \frac{{\beta }_{C}}{{\alpha }_{C}}{\left(\displaystyle \frac{x}{{\alpha }_{C}}\right)}^{{\beta }_{C}-1}\\ &&\,\,\,\,\times \exp \left[-{\left(\displaystyle \frac{x}{{\alpha }_{C}}\right)}^{{\beta }_{C}}\right],\end{eqnarray} \tag{ 12 }$$

where the constants α_C and β_C can be determined by the least-square method, and the subscripts N and C represent number-based and coarse, respectively. For the data set in table 2, these constants are calculated as:

$$\begin{eqnarray*}{\alpha }_{C} & \approx & 224.6,\\ {\beta }_{C} & \approx & \mathrm{2.092.}\end{eqnarray*}$$

Figure 8 shows a good agreement in the accumulative distribution between the fitted Rosin–Rammler curve and experimental data.

Zoom In Zoom Out Reset image size

Aforementioned number-based droplet size distribution can be converted into a mass-based function by:

$$\begin{eqnarray}&&{f}_{M}(x)=\displaystyle \frac{\pi {\rho }_{d}}{6}\displaystyle \frac{{N}_{T}}{{m}_{T}}{x}^{3}{f}_{N}(x),\end{eqnarray} \tag{ 13 }$$

where m_T, N_T, ρ_d represent total mass, total number and droplet density. And f_N(x) is expressed by equation (12), hence the mass-based PDF for coarse droplets can be derived as:

$$\begin{eqnarray}&&{f}_{M,C}(x)=\displaystyle \frac{\pi {\rho }_{d}{N}_{T}{\beta }_{C}}{6{m}_{T}{\alpha }_{C}}{x}^{3}{\left(\displaystyle \frac{x}{{\alpha }_{C}}\right)}^{{\beta }_{C}-1}\exp \left[-{\left(\displaystyle \frac{x}{{\alpha }_{C}}\right)}^{{\beta }_{C}}\right].\end{eqnarray} \tag{ 14 }$$

The droplet size range of fine droplets, generated by nebulizers, varies depending upon the different temperature and solution concentration conditions (Steckel and Eskandar 2003). In this study, the droplet size and size distribution of fine droplets are measured by laser diffraction using Sympatec^® HELOS/KR.

Figure 9 shows a mass-based distribution with a mean of 4.69 μm and a range between 0.8 and 16 μm. This result matches the specification of similar nebulizers under the operation of the same conditions (Justnebulizers 2015), which shows the droplet size of a mean diameter around 5 μm and a size range around 0.1–15 μm. The distribution in figure 9 is also in a good agreement with several similar studies reported (Shen et al 2008, Kuhli et al 2009, Beck-Broichsitter et al 2014). The measurement of this fine droplet PDF has been fitted by a log-normal distribution function:

$$\begin{eqnarray}&&{f}_{M,F}(x)=\displaystyle \frac{1}{\sqrt{2\pi }{\alpha }_{F}x}\exp \left[-\displaystyle \frac{1}{2}{\left(\displaystyle \frac{\mathrm{ln}x-{\beta }_{F}}{{\alpha }_{F}}\right)}^{2}\right],\end{eqnarray} \tag{ 15 }$$

where the constants α_F and β_F can be determined by the least-square method. For the data set in figure 9, these constants are calculated as:

$$\begin{eqnarray*}{\alpha }_{F} & \approx & 0.4643,\\ {\beta }_{F} & \approx & 1.481.\end{eqnarray*}$$

Since there is no overlap of mass-based PDF between coarse droplets and fine droplets, a combined mass-based PDF can be expressed by the linear relation of the two fractions (equations (14) and (15)) with different weights for fine and coarse droplets:

$$\begin{eqnarray}&&{f}_{M}(x)=\displaystyle \frac{{m}_{F}}{{m}_{T}}{f}_{M,F}(x)+\displaystyle \frac{{m}_{C}}{{m}_{T}}{f}_{M,C}(x),\end{eqnarray} \tag{ 16 }$$

where the m_F, m_C and m_T represent mass of fine droplets, mass of coarse droplets and total mass, respectively. In this study, the mass percentages of fine and coarse droplets are: 53.8% and 46.2%. Hence a combined mass-based PDF, with a partition of total mass fraction for each individual PDF of fine and coarse droplets, can then be plotted based on equation (16), as shown in figure 10. It shows two clear peaks of droplet mass PDF, one for fine droplets, and another for coarse droplets. This correctly represents the two major mechanisms of droplets generation from human coughing. In summary, the bi-modal cough simulator has the most complete hydrodynamics parameters and, more importantly, two main size ranges of droplets.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

A bimodal cough simulator has been successfully designed and constructed, which meets all basic features desired for human coughing characteristics. The cough velocity has been measured by a dual-beam cross-correlation method. The coarse droplet size distribution has been determined by an innovative method combining the first principles of mass-balance and image analysis, while the fine droplet size distribution has been measured by a commercial laser diffraction sensor. This cough simulator can generate droplet size from sub-micron to about 800 μm. The resulted PDF is reasonably fit to a mass-based combined function consisting of a traditional Rosin–Rammler distribution for coarse droplets and a log-normal distribution for fine droplets with no overlap in between. To evaluate the evaporation effect on the spray transport, a parametric model is established with its ambient conditions predetermined from a CFD simulation. The results show that, the evaporation effect is negligibly small for large droplets with relatively short distance of trajectory. However, for the micron size droplets, the evaporation could be significant and needs to be taken into consideration.

We deeply appreciate the kindly help from Sympatec Inc. for the measurement of fine droplet size distribution.

As shown in figure A2. The height of dripping should exceed the distance required to generate the equivalent impact velocity by gravity, which is a convenient approach under the condition that the impact velocity is less than the terminal velocity of the droplet. For the size range of droplets in this study (up to 700 μm from our measurements), the terminal velocity is up to 3 m s⁻¹, which is comparable to the measured impact velocity of coarse droplets on the fibrous collector. The corresponding height of dripping is less than 2 m. Hence, we performed the test with a height range between 0.5 and 2 m.

Table A1.  Summary of detailed CFD model setup.

CFD model selection
Steady/unsteady	Unsteady
Geometry	2D axisymmetric
Turbulence model	Realizable k–ε
Multi-component model	Species transport
Boundary conditions
Nozzle opening	Velocity inlet (10 m s−1)
Trapping paper	Wall (standard wall function)
Default condition	Pressure outlet (Pgauge = 0)
Initial conditions
Temperature	298 K
Multi-component model	50% relative humidity
Major governing equations
Continuity	$\displaystyle \frac{\partial }{\partial t}(\rho {Y}_{i})+{\rm{\nabla }}\cdot (\rho {\bf{U}}{Y}_{i})=-{\rm{\nabla }}\cdot {{\bf{J}}}_{{\bf{i}}}$
Momentum	$\displaystyle \frac{\partial }{\partial t}(\rho {\bf{U}})+{\rm{\nabla }}\cdot (\rho {\bf{U}}{\bf{U}})=-{\rm{\nabla }}p+{\rm{\nabla }}\cdot {\boldsymbol{\tau }}$
Energy	$\displaystyle \frac{\partial }{\partial t}(\rho E)+{\rm{\nabla }}\cdot [{\bf{U}}(\rho E+p)]={\rm{\nabla }}\cdot \left[k{\rm{\nabla }}T-\displaystyle {\sum }_{i}{h}_{i}{{\bf{J}}}_{{\bf{i}}}+({\boldsymbol{\tau }}\cdot {\bf{U}})\right]$
Species conservation	$\displaystyle \sum {Y}_{i}=1$
Diffusion flux	${{\bf{J}}}_{{\bf{i}}}=-\rho {D}_{i,m}{\rm{\nabla }}{Y}_{i}$

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The results of this experiment show undetectable droplet breaking and splashing, even with the dripping of 1–2 mm droplets. Hence, the effect of bounce-back or break-up of droplets is considered to be extremely insignificant under our current experiment conditions. The comparison between the droplet mass calculated by (6) and its true mass shows a good match with very minor discrepancy (<2.5%). The uniformity in K value of the fibrous collector and the direct comparison between true droplet size and the size calibrated by the proposed method have been presented in the section of results and discussion.

The calibration of coarse droplet measurement is performed using a dripping tube experiment in which the mass of individual droplet and the corresponding wetted areas on fibrous collector can be independently measured. This experiment is designed to check the possibility in droplet bounce or break-up during the droplet collection, the uniformity in K, and the calibration of droplet size.

It is noted that the accuracy of our proposed method is closely related to the uniformity in K value, where K is defined as the product of droplet density, thickness and porosity of fibrous collector. To check this, a number of droplet collection locations on the fibrous collector have been randomly chosen so that the individual K_i (from equations (1) or (6)) of the ith droplet location can be calibrated in terms of ratio of K_i over the averaged K value (from equation (5)), as illustrated in figure A3. This sample calibration shows that the individual K_i value is reasonably uniform, with a discrepancy within 7% to the averaged K value.

Zoom In Zoom Out Reset image size

The true droplet size of an individual droplet is obtained, based on equation (6), from the measurement of mass of that droplet. The collected droplet size of the same individual droplet is calculated, based on equation (7), from the measurement of overall mass of all droplets, wetted area of the individual droplet in concern, and wetted areas of all droplets collected. A direct comparison between the true droplet sizes and those calculated is illustrated in figure A4, which shows a very good agreement with a discrepancy less than 2.5%.

Zoom In Zoom Out Reset image size