An Instrument for the Classification of Aerosols by Particle Relaxation Time: Theoretical Models of the Aerodynamic Aerosol Classifier

Abstract

A new aerosol particle classifier, the aerodynamic aerosol classifier (AAC), is presented and its classifying characteristics are determined theoretically. The AAC consists of two rotating coaxial cylinders rotating at the same angular velocity. The aerosol to be classified enters through a gap in the inner cylinder and is carried axially by particle-free sheath flow. The centrifugal force causes the particles between the rotating cylinders to move in the radial direction and particles of a narrow range of particle relaxation times exit the classifier through a gap in the outer cylinder with the sample flow. Particles with larger relaxation times impact and adhere to the outer cylinder and particles with smaller relaxation times exit the classifier with the exhaust flow. Thus, the aerosol is classified by particle relaxation time from which the aerodynamic equivalent diameter can easily be found. Four theoretical models of the instrument transfer function are developed. Analytical particle streamline models (with and without the effects of particle diffusion), like those often used for mobility classifiers, are developed for the case when the centrifugal acceleration field is assumed to be uniform in the radial direction. More accurate models are developed when this assumption is not made. These models are the analytical limiting trajectory model which neglects the effects of diffusion and a numerical convective diffusion model that does not. It is shown that these models agree quite well when the gap between the cylinders is small compared to the radii of the cylinders. The models show that, theoretically, the AAC has a relatively wide classification range and high resolution.

Copyright 2013 American Association for Aerosol Research

1. INTRODUCTION

Aerosol classifiers are used to produce a monodisperse aerosol, that is, they select a narrow range of particles from a larger distribution of particles. These devices are used for many applications including: nanoparticle generation, measuring particle count distributions, measuring the deposition of particles in filters and other devices, and measuring particle properties.

Currently, the most commonly used classifier is the differential mobility analyzer (DMA; Knutson and Whitby 1975). The DMA classifies particles based on their electrical mobility, that is, the motion of a charged particle in an electrostatic field. By controlling the electrostatic field and rates of aerosol and sheath flow, the particles are classified by their electrical mobility, which is related to the number of electric charges on the particle and the drag experienced by the particle, which is a function of the particle's size and shape. To classify particles with the DMA, an electric charge must be placed on these particles using charging methods such as radioactive charge neutralizers, X-ray sources, or corona discharge. However, when the particles are charged they do not receive just a single elementary charge but rather they can obtain an integer number of elementary charges or no charge at all, depending on the charging mechanism. Therefore, a smaller particle with one charge may have the same electrical mobility as a larger particle with two charges. Thus, the aerosol sample that is classified by the DMA will not be truly monodisperse in terms of particle size, but rather, it will have a mix of sizes corresponding to an integer number of charged particles. For some applications, like measuring size distributions, the error introduced by the charge distribution can be corrected using inversion techniques, but its uncertainty cannot be eliminated. In other applications and experiments, the larger particles with more than one elementary charge can degrade performance or skew results.

Techniques that have been used to classify particles by their mass-to-charge ratio are the aerosol particle mass analyzer (APM; Ehara 1995; Ehara et al. 1996) or the centrifugal particle mass analyzer (CPMA; Olfert and Collings 2005). With these instruments charged particles are classified between two concentric rotating cylinders by balancing electrostatic and centrifugal forces. As with the DMA, a charging mechanism is used to charge the particles. Therefore, particles of the same mass-to-charge ratio will be classified. For example, a particle with one charge will be classified in the same way as a particle with twice the mass and twice the number of charges. Therefore, under many circumstances the CPMA or APM do not produce a truly monodisperse aerosol.

Many aerosol instruments can be used to measure the relaxation time of a particle from which the aerodynamic equivalent diameter can be determined. The particle relaxation time is defined as the particle's mobility (B) multiplied by its mass (m) (Hinds 1999). The aerodynamic equivalent diameter is defined as the diameter of a spherical particle with a density of 1000 kg/m³ that has the same terminal settling velocity as the actual particle. Except for very irregular particles in a shear flow, particles with the same aerodynamic diameter follow the same path in a flow. Therefore, if the aerodynamic diameter of a particle is known, the motion of the particle can be described independent of particle morphology and density, when the diffusion effect is neglected and in the absence of phoretic forces. The aerodynamic diameter can be related to the particle relaxation time, τ, by assuming that the particle is spherical with standard density (i.e., the assumptions made in defining the aerodynamic equivalent diameter),

where μ is the viscosity of the gas, C _c is the Cunningham slip correction factor, and d _ae is the aerodynamic diameter of the particle.

Instruments that classify particles by particle relaxation time (or aerodynamic diameter) include various kinds of impactors (Marple et al. 1991; Keskinen et al. 1992) and virtual impactors (Conner 1966). However, these methods only provide a means of collecting a size-selected aerosol sample, where particles larger than the cut-off point are classified in one direction (i.e., impacted onto the impaction plate) and particles smaller than the cut-off point continue with the flow. Often, several of these stages are stacked together to provide classification into several large bins. The relaxation time of individual particles can be measured using time-of-flight methods, like the aerodynamic particle sizer (Wilson and Liu 1980), when the flow is in the Stokes regime. The vacuum aerodynamic diameter (the aerodynamic diameter of the particle when it is in the free-molecular regime; DeCarlo et al. 2004) can be measured using time-of-flight with aerodynamic lens systems (Liu et al. 1995a,b), such as those found on several aerosol mass spectrometers (Jimenez et al. 2003). Another technique is to employ an external oscillatory field and Laser Doppler velocimetry to measure the relaxation time of the particles (Mazumder and Kirsch 1977). Also, different centrifuges have been used to deposit particles, sorted by relaxation time, on a foil for size measurement (Sawyer and Walton 1950; Goetz 1957; Stöber and Flachsbart 1969).

Although there are several methods to measure the relaxation time of a particle, there has been little success making a practical instrument that produces a monodisperse aerosol classified by relaxation time. A combination of a virtual impactor and a standard impactor has been used to produce relatively broad aerodynamically classified distributions (Chein and Lundgren 1993). Also, it has been suggested that a monodisperse aerosol could be generated using an opposed migration aerosol classifier (OMAC) using a centrifugal force rather than an electrostatic force (Flagan 2004) or by using an aerodynamic lens with a sheath flow (Kiesler and Kruis 2012). This paper reports theoretical models of a new instrument, called the aerodynamic aerosol classifier (AAC), which produces a monodisperse aerosol classified by particle relaxation time. The AAC does not rely on particle charging so it produces a true monodisperse aerosol without classifying multiply-charged particles like the DMA, CPMA, or APM. Another interesting feature of the AAC is that it can be combined in series with a DMA or CPMA in order to measure other important particle properties including: mobility diameter, particle mass, effective density, mass-mobility exponent, and dynamic shape factor. Two diffusion models and two nondiffusion models have been used to predict the transfer function of the AAC. The limiting trajectory and particle streamline models are analytical methods and do not include particle diffusion. To demonstrate the diffusion effect, a convective diffusion model has been developed by solving the convective-diffusion equation for the AAC using the numerical Crank–Nicolson method. Particle diffusion was also modeled using the diffusing particle streamline model, which is an analytical model which models particle diffusion as a Gaussian cross-stream profile about the corresponding nondiffusing particle streamline. The theoretical models show the instrument has good classification properties for most aerosol applications including a relatively wide classification range, high resolution, and high penetration efficiency.

FIG. 1 Schematic of the AAC.

2. WORKING PRINCIPLE

As shown in Figure 1 the AAC consists of two concentric cylinders rotating in the same direction and at the same rotational speed ¹ . The particles, carried along by the aerosol flow, Q _a, enter the gap between the two cylinders through a slit in the inner cylinder wall. A sheath flow of particle-free air, Q _sh, is also introduced between the two cylinders. In the absence of rotation, the particles will travel between the inner cylinder wall and the aerosol streamline, as shown in Figure 2. However, when the cylinders are rotated, the particles experience a centrifugal force and drag force in the radial direction. The particles are also carried in the axial direction by the sheath and aerosol flows. The radial forces cause the particles to move toward the outer cylinder and particles of a narrow range of particle relaxation times exit with the sample flow. Particles with larger relaxation times impact and adhere to the outer cylinder and particles with smaller relaxation times exit the classifier with the exhaust flow. Models describing the classification of these particles by particle relaxation time are described below.

FIG. 2 Details of the particle trajectory and flows between the cylinders.

3. NONDIFFUSION MODELS

3.1. Limiting Trajectory Model

The velocity of the particles in the classification region of the instrument (Figure 2) can be derived by Newton's second law using a noninertial reference frame. In the radial direction, the friction force, which is given by Stokes law ( v _r /B, where v _r is the velocity of particle with respect to the fluid and B is mechanical mobility), will be equal to the centrifugal force. ² Therefore, the particle's velocity in the radial direction, v_r , will be,

where r is the radial position of the particle with respect to the axis of rotation, ω is the rotational speed of the cylinders, m is the mass of the particle, and τ is the particle relaxation time. The particle velocity in the axial direction, v _z, will simply be equal to the velocity of the carrier gas in the axial direction, u,

The velocity profile is assumed to be uniform (i.e., u is constant). As will be shown in the diffusion model, this assumption has no effect on the calculated instrument transfer function compared to a varying fluid velocity (as was also shown in the derivation of the transfer function of the DMA by Knutson and Whitby [1975], Hoppel [1978]). Fluid flow in this instrument is also assumed to be laminar. Meseguer and Marques (2002) studied flow instability between two rotating coaxial cylinders with an imposed axial pressure gradient (spiral Poiseuille flow). They have shown that for rigid-body rotation the flow is stable for all rotational speeds if the axial Reynolds number is lower than a critical value. Typical axial flows in the AAC will be relatively low and are far below the critical value given by Meseguer and Marques (2002).

Using the chain rule and differentiating, the radial position of the particle can be found as a function of the axial position

where r _in is the initial position of the particle when it enters the classifier, which will be between r ₁ and r _a.

The instrument transfer function is defined as the probability that an aerosol particle that enters the aerosol inlet with relaxation time, τ, will exit with the sampling flow. The transfer function can be found by determining the limiting trajectory of the particles in a similar method as was done by Wang and Flagan (1990) for the DMA.

The largest particle (i.e., the largest τ) that will pass through the classifier, exiting the classifier in the sample flow, Q _s, will start at r _in=r ₁ and will reach r ₂ at the end of the classifier (z=L). Therefore, from Equation (4),

The smallest particle that will be classified, τ_min, will enter the classifier at r _in=r_a and will reach r_s at the end of the classifier. The radii r_a and r_s can be related to the radii r ₁ and r ₂, realizing that for uniform flow

Therefore,

Particles with τ>τ_max will intercept the outer cylinder wall before reaching the exit slit and will adhere to the cylinder surface, while particles with τ<τ_min will flow past the exit slit and be carried out of the instrument with the exhaust flow.

The transfer function is determined by calculating the probability that an aerosol particle that enters the aerosol inlet will exit with the sample flow. To be classified, a particle must migrate into the sample flow, defined by the sample streamline (r_s ⩽r<r ₂), by the time the particle has reached the end of the classifier (z = L). For particles with relaxation times greater than τ _max or less than τ _min, the probability they will exit with the sample flow is zero. For particles with relaxation times between τ _max and τ _min, only a fraction will be classified. For particles with relaxation times greater than τ_min, only particles with an initial radial position between the aerosol streamline, r _a, and a critical initial position, r _c, will be classified. The trajectory of a particle with initial position r _c is defined as the limiting trajectory: particles with initial position r _c < r _in < r _a will be classified and particles with initial position r _in < r _c will not. It is assumed that particles entering the classifier will be uniformly distributed between the inner radius and the aerosol streamline (r ₁ < r _in < r _a). Therefore, the probability that the aerosol will be classified is simply the fraction (f) of the aerosol flow that enters between the critical radius and the aerosol streamline

The limiting trajectory for τ>τ_min will be the particle that starts at r _c and reaches r_s . Substituting this condition (r _in=r _c and r(L)=r _s) into Equation (4) and solving for the aerosol fraction that is classified, f ₁, reveals

Likewise, for particles with τ<τ_max, the particles starting at the critical radius, r _c, must reach r ₂ by the end of the classifier. The fraction f ₂ is the aerosol fraction with τ<τ_max that enters between r _c and r ₁ and exits the classifier in the sample flow,

Substituting r _in=r _c and r(L)=r ₂ into Equation (4) and solving for the aerosol fraction, f ₂, gives

Furthermore, if the sample flow rate is smaller than the aerosol flow rate, then the transfer function cannot be larger than, f ₃=Q _s/Q _a. The transfer function, Ω, will be the minimum of these three fractions or one. Therefore, the transfer function can be expressed as, Ω=max [0, min(f ₁, f ₂, f ₃, 1)].

3.2. Particle Streamline Model

Knutson and Whitby (1975) proposed a mathematical model using a stream function, ψ, and defining an electric flux function, φ, to derive the DMA transfer function. When the flow is axisymmetric, laminar, and incompressible (∇·u=0), the stream function can be defined as:

Similarly, the DMA electric flux function, φ, can be defined because ∇·E=0, where E is the electrostatic field. In the AAC, a centrifugal acceleration field, a, is used to classify the particles where the acceleration in the radial direction is a_r = ω² r and there are no acceleration in the axial and azimuthal directions (a_z = a_θ = 0). This acceleration field is not a solenoidal vector field (∇·a≠0). However, if the gap between the two cylinders, h, is small compared to the mean radius of the cylinders, , in other words, when is small, then the acceleration field can be assumed to be constant in the gap, and the acceleration field will approximately be solenoidal. Under these approximations φ can be defined as:

Integrating Equation (13), the change in φ from the aerosol inlet to the outlet is:

Following Knutson and Whitby (1975) and Stolzenburg (1988) the nondiffusing transfer function Ω_nd is obtained and the nondimensional form can be expressed as:

where β and δ are dimensionless flow parameters expressed as:

and the dimensionless particle relaxation time is defined as:

The value τ* is the particle relaxation time at the maximum of the transfer function and is defined as, τ*=(τ_max+τ_min)/2. In the particle streamline model, this value (τ*_PS) can be found from

FIG. 3 AAC transfer function for two non-diffusion models with balanced flows (δ = 0) and aerosol to sheath flow ratio of 0.1 (β = 0.1). The solid line shows the particle streamline model, and dashed and dashed-dot lines represent the limiting trajectory model.

3.3. Discussion of Nondiffusion Models

Figure 3 shows the normalized nondiffusion transfer function with balanced flows (Q _sh = Q _exh and Q _a = Q _s; or δ=0) and aerosol to sheath flow ratio of 0.1 (β = 0.1) for the particle streamline model and the limiting trajectory model for two values of . One of the values of represents the proposed dimensions of the AAC shown in Table 1. For this case, is relatively small (the gap is relatively small with respect to the cylinders’ radii) with a value of 0.0455. The other case shows an example when the gap is ten times larger than the proposed dimensions and all other dimensions are the same ( = 0.455). In this figure τ is normalized with respect to τ* for each respective model (i.e., or ).

TABLE 1 Proposed dimensions and operating conditions

Property	Value
r 2	45 mm
r 1	43 mm
L	210 mm
Q sh	3 L/min
Q a	0.3 L/min
Q s	0.3 L/min
Temperature	20°C
Pressure	101.325 kPa

Recall that the particle streamline model assumes the centrifugal force between the cylinders is constant and it neglects the divergence of the centrifugal acceleration field. This model is a simplification of the actual classification in the AAC, which is better represented with the limiting trajectory model. As shown in the figure, the particle streamline model closely approximates the limiting trajectory model when is small, but it is a poor approximation when is large. For small values of the width of the transfer function (τ _max and τ _min) between the two models are very similar and the particle streamline model slightly overestimates the amplitude of the transfer function (Ω_LT(τ*_LT)=0.96Ω_PS(τ*_PS)). For large values of the difference between the widths of the transfer functions become apparent and the particle streamline model greatly overestimates the amplitude of the transfer function (Ω_LT(τ*_LT)=0.72Ω_PS(τ*_PS)). It should also be noted (because it cannot be seen in the figure) that there is also a very small difference in the location of the peak of the transfer functions (τ*) between the two models. For the difference between τ*_PS and τ*_LT is 0.007% and for the difference between τ*_PS and τ*_LT is 0.7%. This small difference suggests that it is reasonable to approximate the centrifugal force as a constant using the value of the force at .

From Figure 3, it is apparent that the largest difference between the models is the amplitude of the transfer functions. The maximum amplitude of the particle streamline model will always be 1 when using balanced flows because this model assumes that the centrifugal acceleration field does not diverge (it is analogous to the work of Stolzenberg (1988) who showed that the electric flux function in the DMA does not diverge, ∇·E=0). However, in reality the centrifugal force used in the AAC causes the particles to diverge and the radial distance between the particles increases as the particles move in the axial direction. This causes particles with relaxation time τ* that enter the classification region near r = r _a to impact the outer cylinder and those that enter near r = r ₁ will exit the classifier in the exhaust flow, resulting in a transfer function less than 1. However, as becomes small this effect becomes negligible and the maximum amplitude of the transfer function approaches 1.

FIG. 4 Transfer function of the AAC using the limiting trajectory model (thick lines) and particle streamline model (thin lines) for (a) unbalanced flows (δ≠0) when β=0.1 and (b) balanced flows (δ=0) at different aerosol to sheath flow ratios.

Like the DMA, it is expected that the AAC would normally be operated with balanced flows and at an aerosol to sheath flow ratio of 0.1 as shown in Figure 3. Figure 4 shows examples of the transfer function where the flows are not balanced (Figure 4a) or where the aerosol to sheath flow ratio is varied (Figure 4b) for the proposed geometry in Table 1. Like the DMA, using unbalanced flows results in trapezoidal transfer functions and using lower aerosol to sheath flow ratios (smaller β) results in higher resolution. As before the two models in these cases agree very well.

4. DIFFUSION MODELS

The previous models of the AAC did not include particle diffusion. Therefore, two diffusion models have been developed to show the effect of diffusion on the transfer function of the AAC.

4.1. Convective Diffusion Model

Here, an Eulerian diffusion model has been used to model the diffusion effect in the AAC. The convective diffusion of particles can be modeled using the convective diffusion equation as given by Friedlander (2000). Olfert and Collings (2005) have used a similar model to show the diffusion effect in the APM and Couette CPMA. The convective diffusion equation can be written as:

where n is the particle concentration, u is the gas velocity, c is the particle migration velocity, and D is the diffusion coefficient, which can be calculated using the Stokes-Einstein equation (D = kTB, where k is Boltzmann's constant and T is the temperature). The particle migration velocity is the particle velocity, v, relative to the fluid velocity, u, so that . For the AAC the particle migration velocity will be equal to the particle velocity in the radial direction c _r=τω² r (Equation (2)).

The convection diffusion equation is rewritten for a two-dimensional fully developed flow. To simplify the analysis, diffusion in the z-direction is neglected, since it will be small compared to the diffusion term in the r-direction and the convection term in the z-direction. Therefore, the nondimensional equation can be written for the two-dimensional space as:

where and are nondimensional lengths. is the nondimensionalized particle concentration, where n ₀ is the initial particle concentration in the aerosol flow. Flow velocity, u(r), in the z-direction is considered parabolic. The dimensionless constant, η, is defined as the ratio of radial diffusion length to radial convection length

where, h is the gap between two cylinders, r ₂−r ₁. Another dimensionless constant, ζ, is defined as the ratio of the characteristic time for a particle to travel the length of the classifier to the characteristic time for it to travel the gap between the cylinders

Aerosol particles adhere to a wall surface due to the existence of London–van der Waals forces between particles and the surface (Friedlander 2000). Hence, in aerosol science the concentration of particles at a wall is conventionally considered to be zero, which is usually an extremely good approximation for common aerosol particles and geometries ³ . Therefore, the initial and boundary conditions for this system will be:

Equation (21) has been solved numerically using the Crank-Nicolson method (Smith 1978). The Crank–Nicolson method is a finite difference method which is unconditionally stable for diffusion equations. This method converges faster than other implicit methods and has second-order accuracy in all dimensions.

The value of the transfer function for a given relaxation time can be found by calculating the ratio of the flux of particles exiting with the sample flow to the flux of particles entering the classifier

The radial position of the sample streamline, r _s, can be calculated by solving the following equation:

Equation (21) is solved repeatedly for different τ at a given τ* to obtain the transfer function shape.

4.2. Diffusing Particle Streamline Model

Assuming that the nondiffusing particle streamline model is a good approximation of the AAC transfer function (as discussed in Sections 3.2 and 3.3), a theoretical diffusion model based on the streamline model can be developed. Stolzenburg (1988) developed a particle streamline diffusion model for the DMA assuming that diffusion spreads particles in a Gaussian cross-stream profile about the corresponding nondiffusing particle streamline. The same method can be used here for the AAC and following the work of Stolzenburg (1988) and the derivation of the nondiffusion streamline model shown in Section 3.2, the diffusion transfer function, Ω_d can be given

where and erf(x) is the error function, and the standard derivation, σ, is given by

G _AAC is a nondimensional geometry factor (see the Appendix), it can be calculated from Stolzenburg (1988) and its value for the proposed geometry in Table 1 is 61.7. As discussed in Sections 3.2 and 3.3 this method is only a good approximation when the gap between cylinders is considerably smaller than the mean of cylinders’ radii.

4.3. Discussion of Diffusion Models

In the convective diffusion model Equation (21) shows that the solution only depends on two variables; η and ζ. In the convective diffusion model, η characterizes the effect of diffusion in the classifier and ζ determines the location of the transfer function. By assuming Q _a and Q _s≪Q _sh, τ* can be approximated using Equations (5) and (7), which is results in,

Adding the assumption and using a first-order Taylor series expansion, τ* is approximated by,

Thus,

Also, using Equation (19) and assuming gives . Thus,

which shows how the convective diffusion model and non-diffusion models are linked.

In Equations (22) and (29), η and σ characterizes the effect of diffusion for the convective diffusion model and diffusing particle streamline model; respectively. For balanced flows, when , from Equations (22) and (29) we have

and for the proposed geometry σ=1.23η.

Figure 5 shows the transfer function for the AAC for the non-diffusion limiting trajectory model, the convective diffusion model for η*=0 and η*=0.05, and its corresponding value (σ*=0.0616) in the diffusing particle streamline model. The equivalent aerodynamic diameter for the diffusion models is 50 nm for the dimensions shown in Table 1. Here, η* and σ* is the η and σ value, where τ=τ*. As η* and σ* increases the diffusion effect increases which leads to a broader transfer function and greater diffusional losses. The effect of diffusion on resolution is more clearly seen in Figure 6, which shows the classifier resolution as a function of the dimensionless numbers η* and σ* from the convective diffusion model and diffusing particle streamline respectively; when δ = 0 and β = 0.1. The resolution, R, is defined as R = τ*/FWHM, where FWHM is the full-width half maximum of the transfer function. If diffusional effects are neglected, is small, and the flows are balanced, then R = 1/β. As shown above, the resolution decreases for smaller particle sizes (large η* and σ*) because of the diffusion effect; and the resolution for larger particles (small η* and σ*) asymptotically approaches the value of 1/β. Also, Figure 6 shows that the convective diffusion and the diffusing particle streamline models agree very well.

FIG. 5 AAC transfer function models with δ = 0 and β = 0.1. The models include the nondiffusion limiting trajectory model (solid line), the convective diffusion model when the diffusion term is zero (dotted line), the diffusing particle streamline model (dashed-dot line), and the convective diffusion model (dashed line).

FIG. 6 AAC resolution as a function of η* and σ* from the convective diffusion model and diffusing particle streamline model, respectively, for δ = 0 and β = 0.1.

Figure 7 shows examples of the convective diffusion transfer function for particles with (a) 30 nm, (b) 100 nm, and (c) 300 nm aerodynamic diameters for the operating conditions shown in Table 1. It can be seen that the transfer function with the convective diffusion model is not symmetric. The transfer function on the left side is wider, where the particle size is slightly smaller. The reason for this is that smaller particles have a higher diffusitivity than larger particles.

FIG. 7 Transfer functions calculated with the convective diffusion model (δ = 0 and β = 0.1) of spherical particles with an aerodynamic diameter of (a) 30 nm, (b) 100 nm, (c) 300 nm with particle densities of 500, 1000, and 2000 kg/m³ for the instrument specifications in Table 1.

It should also be noted that the diffusion coefficient (and thus the diffusive transfer function) is a function of the mobility of the particle. The relationship between aerodynamic diameter and mobility diameter for spherical particles can be shown to be

where, d _mo is the mobility equivalent diameter and ρ_p is the particle density including internal voids (DeCarlo et al. 2004) ⁴ . As the particle density increases, for constant aerodynamic diameter, the mobility equivalent diameter decreases, and the diffusion coefficient increases. Therefore, particles with higher particle densities (at a given aerodynamic diameter) will have broader transfer functions and higher diffusional losses as shown in Figure 7 for spherical particles where the diffusion coefficient is calculated from the Stokes–Einstein equation (D = kTB) using the mobility-equivalent diameter calculated in Equation (35). This effect is also seen in particle mass classifiers (CPMA and APM), which classify particles by mass-to-charge ratio yet their diffusive transfer functions are dependent on particle mobility. One application for the AAC would be to measure the aerodynamic-equivalent size distribution of an aerosol by stepping or scanning the rotational speed and measuring the downstream aerosol concentration with a condensation particle counter. However, to invert the AAC measurements using the diffusive transfer function, the mobility of the particle would have to be known or assumed since the diffusive transfer function is dependent on particle mobility.

5. CONCLUSION AND SUMMARY

The AAC is an aerosol classifier that classifies particles by their particle relaxation time, from which the aerodynamic equivalent diameter of the particles can be found. Two diffusion models and two nondiffusion models have been used to predict the transfer function of the AAC. An analytical limiting trajectory model and a particle streamline model were used to predict the nondiffusion transfer function. A numerical convective diffusion model and a diffusing particle streamline model were used to model the effects on particle diffusion on the transfer function. The particle streamline models (diffusion and nondiffusion) neglect the divergence of the centrifugal acceleration field in the classifier, but are able to closely approximate the actual classification in the instrument when the gap between the cylinders is small with respect to the radii of the cylinders. In most practical embodiments of the instrument this is expected to be true.

Although the limiting trajectory model and the convective diffusion model are more accurate representations of the classification in the instrument, it is expected that the particle streamline models (diffusion and nondiffusion) will be more widely used for the following reasons: (1) the limiting trajectory non-diffusion particle streamline models are both analytical, but the streamline model has a simpler form and it is simple to see the relationship between the flow rates and the instrument resolution through the parameter β. (2) The convective diffusion model is numerical and modern computers require several minutes to calculate the transfer function. The diffusive streamline model is analytical and the transfer function is quickly calculated. (3) The streamline models for the AAC have very similar form to the streamline models of the DMA, which are widely used and are familiar to many.

Like the DMA is it expected that the AAC will often be used with balanced flows (Q _a = Q _s and Q _sh = Q _exh; δ = 0) and with an aerosol to sheath flow ratio of 0.1 (β = 0.1), since this is simple to implement in practice and gives good resolution. Using the proposed dimensions (Table 1), the AAC would be able to classify particles over an extremely wide range of 50 nm to 10 μm using rotational speeds ranging from 6800 to 75 rpm and smaller particle sizes could be classified by using higher rotational speeds. However, the range of an actual instrument will be dependent on its design. The lower size limit will be sensitive to particle diffusion and the maximum rotational speed obtainable and the upper limit will depend on particle impaction in the inlet and outlet of the classifier. Future work will demonstrate a prototype of the AAC.

APPENDIX

The geometric factor G _AAC is the nondimensionalized integral of v ² r ² dt in an annular cross section, where t is time (Stolzenburg and McMurry 2008). G _DMA (Appendices B and C of Stolzenburg, 1988) can be rearranged for AAC as:

where

and

α_a and α_s can be calculated by

where F _γ is

NOMENCLATURE

v	=	particle velocity
u	=	gas velocity
a	=	centrifugal acceleration
u	=	gas velocity in z direction
m	=	particle mass
f	=	fraction of particles that pass through the classifier
n	=	particle concentration
	=	nondimensional particle concentration
C c	=	slip correction factor
d ae	=	aerodynamic equivalent diameter
d*ae	=	d ae at the center of transfer function (τ=τ*)
d ve	=	volume equivalent diameter including internal voids
d mo	=	mobility equivalent diameter
z	=	distance in the axial direction
r	=	distance from axis of rotation
r 1	=	inner cylinder radius
r 2	=	outer cylinder radius
r in	=	initial position of the particle at the classifier entrance
	=	=(r 1+r 2)/2
r c	=	critical radius
r a	=	aerosol flow streamline radius
r s	=	sample flow streamline radius
	=	nondimensional radius (r/h)
	=	nondimensional length (z/L)
Q sh	=	sheath flow rate
Q a	=	aerosol flow rate
Q exh	=	exhaust flow rate
Q s	=	sample flow rate
B	=	mechanical mobility
D	=	diffusion coefficient
L	=	classifier length
T	=	temperature
r	=	classifier resolution
h	=	radial distance between two cylinders (r 2−r 1)
	=
η	=	nondimensional number defining diffusion effect in convective diffusion model
η*	=	η at the center of transfer function (τ=τ*)
ζ	=	nondimensional number defining transfer function location
k	=	Boltzmann's constant
τ	=	relaxation time
τmax	=	largest τ that passes through the classifier
τmin	=	smallest τ that passes through the classifier
Δτ	=	=(τmax−τmin)/2
τ*	=	τ at the maximum of the transfer function
	=	dimensionless particle relaxation time ()
ω	=	rotational speed
μ	=	gas dynamic viscosity
ρp	=	particle density including internal voids
ρ0	=	standard density (1000 kg/m3)
Ω	=	transfer function
χ	=	dynamic shape factor
G AAC	=	nondimensional geometry factor of AAC
δ	=	=Q s−Q a/Q s+Q a
β	=	=Q s+Q a/Q sh+Q exh
σ	=	nondimensional number defining diffusion effect in streamline model
σ*	=	σ at the center of transfer function (τ=τ*)
ϵ(x)	=	and erf(x) is the error function
FWHM	=	full-width half maximum of transfer function
LT	=	limiting trajectory (model)
CD	=	convective diffusion (model)
PS	=	particle streamline (model)
DPS	=	diffusing particle streamline (model)

Acknowledgments

Funding for this project was provided by Cambustion Ltd. The authors would like to thank Dr. Peter McMurry and an anonymous peer reviewer for their discussions on this article.

Notes

A separate embodiment could employ cylinders rotating at different rotational speeds, but that is not investigated here.

In an inertial reference frame, it would be said that the friction force will be equal to the mass of the particle multiplied by the acceleration of the particle in the radial direction, where the acceleration is ω² r.

There are some studies that suggest the particle concentration is proportional to the particle flux to the wall (Gallis et al. 2008); however, Tavakoli et al. (2011) showed that this boundary condition does not have any significant effect on aerosol transport in channels.

For nonspherical particles the relationship between aerodynamic and mobility diameter is , where d _ve is volume equivalent diameter including internal void spaces.