Consider a planar OMAC of length L and electrode spacing b, as shown in Fig. 2.2.
Variations across the width, which is of extent W, are ignored here in this minimal plug flow (uniform velocity) model. The flows are balanced, so the aerosol flow Qa is equal to the exiting classified sample flow Qs. The entering and exiting crossflows are both equal to Qc. A useful metric of the performance of this and many other differential electrical mobility classifiers is the resolving power, R = Z∗/∆ZF W HM, where Z∗ is the mobility of maximal transmittance and ∆ZF W HM is that full width at half maximum of the transfer function. As was the case for DMAs, the nondif- fusive resolving power is the ratio of the separation gas flow to that of the aerosol, Rnd =Qc/Qa. The behavior of the diffusive planar plug flow OMAC is governed by the convective-diffusion equation. At steady-state, aerosol is advected through the channel in the x-direction by the primary (aerosol/sample) flow. In the y-direction, displacement by the crossflow is counteracted by that of the electric field, where the target electrical mobility is defined by Z∗ =Qcb/(V W L), the value where the elec- trical displacement exactly counteracts the crossflow displacement, where V is the applied potential difference. For large aspect ratio devices where (L/b)2 1, which is stipulated here, diffusion is negligible in the x-direction, but merits consideration in the thin y-direction that is aligned with the field. For plug flow, the convective- diffusion equation is then written
Qa W b
∂C
∂x + Qc
W L − ZV b
∂C
∂y =D∂2C
∂y2, (2.1)
with boundary conditions
C[0, y] =C0(H[y]−H[y−b]) and C[x,0] =C[x, b] = 0. (2.2) Here,Zis the electrical mobility of the particle of interest,Dis its diffusion coefficient, C0 is its concentration at the aerosol inlet,V is the applied voltage,His the Heaviside step function, and the porous walls are taken to be perfect particle sinks.
The governing equation and its boundary conditions may be rendered dimension- less by defining the variables
Cˆ =C/C0, xˆ=x/L, and ˆy=y/b, (2.3)
substituting them in Eqs. (2.1) and (2.2), and rearranging to obtain
∂Cˆ
∂xˆ −ζ∂Cˆ
∂yˆ = σ2 2
Z Z∗
∂2Cˆ
∂yˆ2, (2.4)
with
Cˆ[0,y] =ˆ H[ˆy]−H[ˆy−1] and ˆC[ˆx,0] = ˆC[ˆx,1] = 0, (2.5) where ζ = Rnd(Z/Z∗−1) and the square of the dimensionless diffusion parameter is σ2 = 2Rnd/Pe. The migration P´eclet number Pe is the ratio of the characteristic time for diffusion to that for the field to displace a particle the distance of the gap in the absence of a crossflow. Here, Pe =eV /kT, where e is the elementary charge, k is the Boltzmann constant, and T is the absolute temperature, as only singly charged particles are considered.
The transmission probability may then be found by solving for the concentration by separation of variables, and then integrating the concentration over the outlet to obtain the probability that a particle of mobilityZ will be transmitted when targeting particles of mobility Z∗. This probability, which is known as the instrument transfer function Ω, can be found by separation of variables to be
Ω = X∞ n=1
n2π2σ4 λ4n
Z Z∗
2
exp
−λ2n 1−(−1)ncosh
ζ σ2(Z/Z∗)
, (2.6)
where the eigenvalues are
λ2n= σ2 2
Z Z∗
n2π2+
ζ σ2(Z/Z∗)
2!
. (2.7)
As is commonly the case with transport phenomena, the most interesting part of
modeling the OMAC is in the scaling analysis. The Schmidt number, Sc = ν/D∗, where ν is the kinematic viscosity and D∗ is the diffusion coefficient of the target particle, is a function only of σ2 and a dimensionless flow parameter κ = Reb/L, where Re is the Reynolds number, and the relationship is
Sc = 2
σ2κ. (2.8)
Since ν is only a function of the gas composition, temperature, and pressure, and the Stokes-Einstein-Sutherland relationship relatesD∗ to the particle diameterDp, a general operating diagram that has Dp as a function of σ2 and κ may be made once the gas properties are defined [29]. Beyond enabling the construction of a general operating diagram for the OMAC, the dimensionless groups σ2 and κ govern the performance of many differential electrical mobility classifiers. It has been shown theoretically that both scale favorably for OMACs (and IGMAs) relative to DMAs as a function of the the nondiffusive resolving power Rnd, with an O(Rnd) edge in σ2 and an O(R2nd) advantage in κ[29]. The testing of the present and future OMAC instruments enables the experimental validation of this theoretical result.
The theoretical value of σ2 represents a lower bound, ideal value of the square of the dimensionless diffusion coefficient for real instruments. Nonidealities, such as imperfections in the fabricated components and field nonuniformities near edges, give rise to greater dispersion than that predicted by theory. The practice in the field has been to account for all nonidealities with an empirical multiplicative factor fσ ([21]; see also [30]) or an additive distortion factor σdistor2 , i.e., σobs = fσσ or σobs2 = σ2 +σ2distor, where σobs2 is the experimentally observed (fitted) square of the dimensionless diffusion parameter.
The utility of a mobility classifier is also influenced by the efficiency with which particles that enter the instrument are counted. Losses within the DMA occur pri- marily in the entrance and exit regions. Because particles migrate across a clean sheath flow, diffusion to the walls of the classification region is minimal except at the lowest classification voltages of the DMA.
In contrast, the porous walls of the OMAC act as a particle sink along the entire classification channel. The transmission efficiency can be defined as the ratio of the integral over the diffusive transfer function to that over the nondiffusive one, i.e.,
ηtrans = R∞
−∞Ω [Z|Z∗]dZ R∞
−∞ΩND[Z|Z∗]dZ. (2.9)
The transmission efficiency is O(1) for σ2 <(8 ln 2)−1, but drops precipitously above this value.
The minimal model on which Eq. (2.6) was based neglected the effects of viscous dissipation at the porous electrode surfaces. However, it was shown by Brownian dynamics simulation that the performance, as measured by the resolving power and efficiency, does not change appreciably from the plug flow model when the full velocity profile is considered [29]. The favorable theoretical results for the OMAC and a desire for experimental validation of its performance motivated the design and fabrication of a prototype.