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Optimal length of the plain Loscertales mobility analyzer
J. Aemol Sci. Vol. 29. Suppl. 1. pp. S63S64. 1998 8 1998 Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain 0021-85W98 519.00 + 0.00
OPTIMAL LENGTH OF THE PLAIN LOSCERTALES MOBILITY ANALYZER H. TAMMET Department of Environmental Physics, University of Tat-m, 18 Ulikooli Str., Tartu, EE2400 Estonia
KEYWORDS
DMA; Mobility resolution; Diffusion broadening; Particle sizing
Loscertales (1998) proposed a new method of improving the diffusion-limited mobility resolution by means of the longitudinal electric field in an aspiration mobility analyzer. The plain Loscertales analyzer is explained in Figure 1. The plates are not equipotential and the electric field is not perpendicular to the Particle air flow as expected in traditional inlet mobility analyzers. The deviation from a traditional analyzer is characterized by the ratio of longitudinal and transversal components of the electric field Lo+
(1) ‘Y
called the Loscertales number below. When discussing the difI%Outlet sive broadening of the transfer function, the aerosol flow rate is Figure 1. The plain Loscertales mobility analyzer. assumed negligible when compared with the sheath gas flow rate. The electric field between the plates is inclined but uniform, and the voltage I’between any pair of facing points in Figure 1 is the same along the plates. Due to the Brownian diffusion, the cloud of simultaneously entering particles of mobility Z expands during the passage as illustrated in Figure 1. The center of the cloud is deposited in distance
I=(+Q)h.
(2)
The mobility of outlet particles Z, corresponds with the length of the analyzer I,. The mobility resolution S = az Z, is to be estimated according to the Brownian standard deviation of the sedimentation length a/ and the derivative aI dl uh2 _---_dZ Z;v I 1 oz
(3)
In a traditional analyzer Id/ dZI = /.,Z and S = q /I, When the analyzer is too short, the denominator C suppresses the resolution with decrease in the length. In a Loscertales analyzer S63
S64
Abstracts of the 5th International Aerosol Conference 1998
the derivative Idi/dZj remains finite even if the length of the analyzer is approaching zero. Thus a further increase in resolution is possible. The one-dimensional standard deviation of particles from the center of the cloud is cr=&%=,/m=h,/m,
where
D
is the diffusion
coefficient,
k
is the
Boltzmann constant, T is temperature, and q is particle charge (see Fuchs, 1964). The standard deviation of the sedimentation length is increased by the ratio of the hypotenuse and the left leg of the small gray triangle drawn in Figure 1 near the outlet slit: n,=f$hdl+oZ. The expression factors:
of 6 derived
(4)
from Equations
(2-4)
can be spliced into three dimensionless
6 = SC-‘/2 Re-‘/2 F where Schmidt number SC = qv’(kTZ,), viscosity), and the third factor F
Reynolds
=
> (9 number Re = uh/v (v is the gas kinematic
21+VoW2 Lo+l,lh
(6)
The controls of diffusion-limited resolution are the numbers SC, Re, Lo, and the relative length LJh. When SC, Re, and Lo are fixed, the factor F reaches a minimum at the optimal relative length of the analyzer Iopt = (&Xl
0
l_oslcertales2number3Lo =
5 Ex4/E,
Figure 2. Optimal relative length 1,/h, optimal resolution factor, and resolution factor for a zero-length analyzer depending on E,/E,
- Lo)
(7)
The variation of optimal relative length and the value of the factor F with the Loscertales number is shown in Figure 2. The factor F is shown as for an analyzer of optimal length and for a zero-length analyzer. The optimal relative length of a traditional plain mobility analyzer is 1 (RosellLlompart et al., 1996) and the corresponding value of factor F is 2. Lo = 0.5 is enough to keep the resolution of the zerolength analyzer on the same level as in a traditional analyzer of optimal length. The zero length is formally never optimal but the resolution of a zero-length analyzer approaches the resolution of optimal-length analyzer when Lo > 2.
REFERENCES Fuchs, N. A. (1964) The Mechanics of Aerosols. Pergamon Press, Oxford. Loscertales, I. G. (1998) Drift differential mobility analyzer. J. Aerosol Sci. 29 (in press). Rosell-Llompart, J., Loscertales, I. G., Bingham, D., Fernandez de la Mora, J. (1996) Sizing nanoparticles and ions with a short differential mobility analyzer. J. AerosoZ Sci. 27, 695 719.
OPTIMAL LENGTH OF THE PLAIN LOSCERTALES MOBILITY ANALYZER H. TAMMET Department of Environmental Physics, University of Tat-m, 18 Ulikooli Str., Tartu, EE2400 Estonia
KEYWORDS
DMA; Mobility resolution; Diffusion broadening; Particle sizing
Loscertales (1998) proposed a new method of improving the diffusion-limited mobility resolution by means of the longitudinal electric field in an aspiration mobility analyzer. The plain Loscertales analyzer is explained in Figure 1. The plates are not equipotential and the electric field is not perpendicular to the Particle air flow as expected in traditional inlet mobility analyzers. The deviation from a traditional analyzer is characterized by the ratio of longitudinal and transversal components of the electric field Lo+
(1) ‘Y
called the Loscertales number below. When discussing the difI%Outlet sive broadening of the transfer function, the aerosol flow rate is Figure 1. The plain Loscertales mobility analyzer. assumed negligible when compared with the sheath gas flow rate. The electric field between the plates is inclined but uniform, and the voltage I’between any pair of facing points in Figure 1 is the same along the plates. Due to the Brownian diffusion, the cloud of simultaneously entering particles of mobility Z expands during the passage as illustrated in Figure 1. The center of the cloud is deposited in distance
I=(+Q)h.
(2)
The mobility of outlet particles Z, corresponds with the length of the analyzer I,. The mobility resolution S = az Z, is to be estimated according to the Brownian standard deviation of the sedimentation length a/ and the derivative aI dl uh2 _---_dZ Z;v I 1 oz
(3)
In a traditional analyzer Id/ dZI = /.,Z and S = q /I, When the analyzer is too short, the denominator C suppresses the resolution with decrease in the length. In a Loscertales analyzer S63
S64
Abstracts of the 5th International Aerosol Conference 1998
the derivative Idi/dZj remains finite even if the length of the analyzer is approaching zero. Thus a further increase in resolution is possible. The one-dimensional standard deviation of particles from the center of the cloud is cr=&%=,/m=h,/m,
where
D
is the diffusion
coefficient,
k
is the
Boltzmann constant, T is temperature, and q is particle charge (see Fuchs, 1964). The standard deviation of the sedimentation length is increased by the ratio of the hypotenuse and the left leg of the small gray triangle drawn in Figure 1 near the outlet slit: n,=f$hdl+oZ. The expression factors:
of 6 derived
(4)
from Equations
(2-4)
can be spliced into three dimensionless
6 = SC-‘/2 Re-‘/2 F where Schmidt number SC = qv’(kTZ,), viscosity), and the third factor F
Reynolds
=
> (9 number Re = uh/v (v is the gas kinematic
21+VoW2 Lo+l,lh
(6)
The controls of diffusion-limited resolution are the numbers SC, Re, Lo, and the relative length LJh. When SC, Re, and Lo are fixed, the factor F reaches a minimum at the optimal relative length of the analyzer Iopt = (&Xl
0
l_oslcertales2number3Lo =
5 Ex4/E,
Figure 2. Optimal relative length 1,/h, optimal resolution factor, and resolution factor for a zero-length analyzer depending on E,/E,
- Lo)
(7)
The variation of optimal relative length and the value of the factor F with the Loscertales number is shown in Figure 2. The factor F is shown as for an analyzer of optimal length and for a zero-length analyzer. The optimal relative length of a traditional plain mobility analyzer is 1 (RosellLlompart et al., 1996) and the corresponding value of factor F is 2. Lo = 0.5 is enough to keep the resolution of the zerolength analyzer on the same level as in a traditional analyzer of optimal length. The zero length is formally never optimal but the resolution of a zero-length analyzer approaches the resolution of optimal-length analyzer when Lo > 2.
REFERENCES Fuchs, N. A. (1964) The Mechanics of Aerosols. Pergamon Press, Oxford. Loscertales, I. G. (1998) Drift differential mobility analyzer. J. Aerosol Sci. 29 (in press). Rosell-Llompart, J., Loscertales, I. G., Bingham, D., Fernandez de la Mora, J. (1996) Sizing nanoparticles and ions with a short differential mobility analyzer. J. AerosoZ Sci. 27, 695 719.