Inversion of electrical mobility measurements using bipolar or unipolar chargers for the arbitrary distribution of channels
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摘要
The inversion of the particle size distribution from electrical mobility measurements is analyzed.Three different methods are adapted for a dot-matrix approach to the problem,especially for non-square or singular matrices,and applied to electrical mobility measurements from fixed or scanning voltages.Multiply charged particles,diffusion losses,arbitrary voltage steps and noise were considered,which results in non-adjoining and overlapping transfer functions.The individual contribution of the transfer functions in each size interval was geometrically estimated,which requires only its characteristic mobilities.The methodology is applied to mobility measurements from particles charged with unipolar and bipolar chargers.However,the method can be extrapolated to any charging method with a defined charge distribution,and retrieval of the singly charged particle distribution and mean charge from a tandem differential mobility analysis configuration was successfully demonstrated.
The inversion of the particle size distribution from electrical mobility measurements is analyzed.Three different methods are adapted for a dot-matrix approach to the problem,especially for non-square or singular matrices,and applied to electrical mobility measurements from fixed or scanning voltages.Multiply charged particles,diffusion losses,arbitrary voltage steps and noise were considered,which results in non-adjoining and overlapping transfer functions.The individual contribution of the transfer functions in each size interval was geometrically estimated,which requires only its characteristic mobilities.The methodology is applied to mobility measurements from particles charged with unipolar and bipolar chargers.However,the method can be extrapolated to any charging method with a defined charge distribution,and retrieval of the singly charged particle distribution and mean charge from a tandem differential mobility analysis configuration was successfully demonstrated.
The inversion of the particle size distribution from electrical mobility measurements is analyzed.Three different methods are adapted for a dot-matrix approach to the problem,especially for non-square or singular matrices,and applied to electrical mobility measurements from fixed or scanning voltages.Multiply charged particles,diffusion losses,arbitrary voltage steps and noise were considered,which results in non-adjoining and overlapping transfer functions.The individual contribution of the transfer functions in each size interval was geometrically estimated,which requires only its characteristic mobilities.The methodology is applied to mobility measurements from particles charged with unipolar and bipolar chargers.However,the method can be extrapolated to any charging method with a defined charge distribution,and retrieval of the singly charged particle distribution and mean charge from a tandem differential mobility analysis configuration was successfully demonstrated.
引文
Adachi,M.,Kousaka,Y.,&Okuyama,K.(1985).Unipolar and bipolar diffusion charging of ultrafine aerosol particles.Journal of Aerosol Science,16(2),109-123.
Alofs,D.J.,&Balakumar,P.(1982).Inversion to obtain aerosol size distributions from measurements with a differential mobility analyzer Journal of Aerosol Science,J3(6),513-527.
Alonso,M.,&Kousaka,Y.(1996).Mobility shift in the differential mobility analyzer due to Brownian diffusion and space-charge effects.Journal of Aerosol Science,27(8),1201-1225.
Bazan,F.S.V.,&Francisco,J.B.(2009).An improved fixed-point algorithm for determining a Tikhonov regularization parameter.Inverse Problems,25(4),045007.
Biskos,G.,Reavell,K.,&Ceilings,N.(2005).Unipolar diffusion charging of aerosol particles in the transition regime.Journal of Aerosol Science,36(2),247-265.
Boisdron,K.,&Brock,J.R.(1970).On the stochastic nature of the acquisition of electrical charge and radioactivity by aerosol particles.Atmospheric Environment,4(1),35-50.
Chien,C.L,Tsai,C J.,Chen,H.L,Lin,G.Y.,&Wu,J.S.(2011).Modeling and validation of nanoparticle charging efficiency of a single-wire corona unipolar charger.Aerosol Science and Technology,45,1468-1479.
Collins,D.R.,Flagan,R.C,&Seinfeld,j.H.(2002).Improved inversion of scanning DMA data.Aerosol Science and Technology,36(1),1-9.
Crump,J.,&Seinfeld,J.(1981).A new algorithm for inversion of aerosol size distribution data.Aerosol Science and Technology,1(1),15-34.
Domat,M.,Kruis,F.,&Fernandez-Diaz,J.(2014a).Determination of the relevant charging parameters for the modelling of unipolar chargers.Journal of Aerosol Science,71,16-28.
Domat,M.,Kruis,F.,&Fernandez-Diaz,J.(2014b).Investigations of the effect of electrode gap on the performance of a corona charger having separated corona and charging zones.Journal of Aerosol Science,68,1-13.
Dubey,P.,&Dhaniyala,S.(2008).Analysis of scanning DMA transfer functions.Aerosol Science and Technology,42(7),544-555.
Dubey,P.,&Dhaniyala,S.(2013).Improved inversion of scanning electrical mobility spectrometer data using a new multiscale expectation maximization algorithm.Aerosol Science and Technology,47(1),69-80.
Fiebig,M.,Stein,C,Schroder,F.,Feldpausch.P.,&Petzold,A.(2005).Inversion of data containing information on the aerosol particle size distribution using multiple instruments.Journal of Aerosol Science,36(11),1353-1372.
Fuchs,N.(1964).The mechanics of aerosols.New York:Pergamon Press.
Gysel,M.,McFiggans,G.B.,&Coe,H.(2009).Inversion of tandem differential mobility analyser(TDMA)measurements.Journal of Aerosol Science,40(2),134-151.
Hagen,D.E.,&Alofs,D.J.(1983).Linear inversion method to obtain aerosol size distributions from measurements with a differential mobility analyzer.Aerosol Science and Technology,2(4),465-475.
Hansen,P.C,&O'Leary,D.P.(1993).The use of the L-curve in the regularization of discrete ill-posed problems.SIAM Journal on Scientific Computing,14,1487-1503.
He,M.,&Dhaniyala,S.(2013).A multiple charging correction algorithm for scanning electrical mobility spectrometer data.Journal of Aerosol Science,61,13-26.
Hontanon,E.,&Kruis,F.E.(2008).Single charging of nanoparticles by UV photoionization at high flow rates.Aerosol Science and Technology,42,310-323.
Hoppel,W.A.(1978).Determination of the aerosol size distribution from the mobility distribution of the charged fraction of aerosols.Journal ofAerosolScience,9(1),41-54.
Intra,P.(2012).Corona discharge in a cylindrical tri ode charger for unipolar diffusion aerosol charging.Journal of Electrostatics,70(1),136-143.
Johnston,P.R.,&Gulrajani,R.M.(2000).Selecting the corner in the L-curve approach to Tikhonov regularization.IEEE Transactions on Biomedical Engineering,47,1293-1296.
Kaminski,H.,Kuhlbusch,T.A.J.,Fissan,H.,Ravi,L.,Horn,H.-G.,Han,H.-S.,et al.(2012).Mathematical description of experimentally determined charge distributions of a unipolar diffusion charger.Aerosol Science and Technology,46(6),708-716.
Karlsson,M.N.,&Martinsson,B.G.(2003).Methods to measure and predict the transfer function size dependence of individual DMAs.Journal of Aerosol Science,34(5),603-625.
Kim,D.S.,Kim,Y.M.,Kwon,Y.T.,&Park,K.(2011).Evaluation of a softX-ray unipolar charger for charging nanoparticles.Joumal ofNanoparticle Research,13,579-585.
Knutson,E.O.,&Whitby,K.T.(1975a).Accurate measurement of aerosol electric mobility moments Journal of Aerosol Science,6(6),453-460.
Knutson,E.O.,&Whitby,K.T.(1975b).Aerosol classification by electric mobility:apparatus,theory,and applications.Journal of Aerosol Science,6(6),443-451.
Kwon,S.,Fujimoto,T.,Kuga,Y.,Sakurai,H.,&Seto,T.(2005).Characteristics of aerosol charge distribution by surface-discharge microplasma aerosol charger(SMAC).Aerosol Sdence&Technology,39(10),987-1001.
Kwon,S.,Sakurai,R,&Seto,T.(2007).Unipolar charging of nanoparticles by the surface-discharge microplasma aerosol charger(SMAC).Journal ofNanoparticle Research,9(4),621-630.
Lawson,C.L,&Hanson,R.J.(1974).Solving least squares problems.Englewood Cliffs,NJ:SIAM(revised republication of the first published in Prentice-Hall).
Li,L,&Chen,D.R.(2011).Performance study of a DC-corona-based particle charger for charge conditioning.Journal of Aerosol Science,42(2),87-99.
Li,W.,Li,L,&Chen,D.-R.(2006).Technical note:A new deconvolution scheme for the retrieval of true DMA transfer function from tandem DMA data.Aerosol Science and Technology,40(12),1052-1057.
Liu,B.Y.R,Pui,D.Y.R,Whitby,K.T.,Kittelson,D.B.,Kousaka,Y.,&McKenzie,R.L.(1978).The aerosol mobility chromatograph:A new detector for sulfuric acid aerosols.Atmospheric Environment,12(1),99-104.
Lloyd,J.J.,Taylor,C.J.,Lawson,R.S.,&Shields,R.A.(1997).The use of the L-curve method in the inversion of diffusion battery data.Journal of Aerosol Science,28,1251-1264.
Pfeifer,S.,Birmili,W.,Schladitz,A.,Muller,T.,Nowak,A.,&Wiedensohler,A.(2013).A novel inversion algorithm for mobility particle size spectrometers considering non-sphericity and additional aerodynamic/optical number size distributions.Atmospheric Measurement Techniques and Discussions,6(3),4735-4767.
Qi,C,Chen,D.R.,&Pui,D.Y.(2007).Experimental study of a new corona based unipolar aerosol charger Journal of Aerosol Science,38(7),775-792.
Rader,D.J.,&McMurry,P.H.(1986).Application of the tandem differential mobility analyzer to studies of droplet growth or evaporation.Journal of Aerosol Science,17(5),771-787.
Reist,P.C.(1993).Aerosol science and technology.New York:McGraw-Hill,Inc.
Rojas,M.,&Steihaug,T.(2002).An interior-point trust-region-based method for large-scale non-negative regularization.Inverse Problems,18,1291-1307.
Stolzenburg,M.R.(1988).An ultrafine aerosol size distribution measuring system(Doctoral dissertation).USA:University of Minnesota.
Stolzenburg,M.R.,&McMurry,P.H.(2008).Equations governing single and tandem DMA configurations and a new lognormal approximation to the transfer function.Aerosol Science and Technology,42(6),421-432.
Stratmann,F.,Kauffeldt,T.,Hummes,D.,&Fissan,H.(1997).Differential electrical mobility analysis:A theoretical study.Aerosol Science and Technology,26(4),368-383.
Talukdar,S.S.,&Swihart,M.T.(2003).An improved data inversion program for obtaining aerosol size distributions from scanning differential mobility analyzer data.Aerosol Science and Technology,37(2),145-161.
Tsai,C.J.,Lin,G.Y.,Chen,H.L,Huang,C.H.,&Alonso,M.(2010).Enhancement of extrinsic charging efficiency of a nanoparticle charger with multiple discharging wires.Aerosol Science and Technology,44(10),807-816.
TSI Inc.(2007).Operation and service manual for condensation particle counter(CPC),model 3775,revision E.
Twomey,S.(1975).Comparison of constrained linear inversion and an iterative nonlinear algorithm applied to the indirect estimation of particle size distributions.Journal of Computational Physics,18(2),188-200.
Vivas,M.,Hontanon,E.,&Schmidt-Ott,A.(2008).Reducing multiple charging of submicron aerosols in a corona diffusion charger.Aerosol Science and Technology,42(2),97-109.
Voutilainen,A.(2001).Statistical inversion methods for the reconstruction of aerosol size distributions(Doctoral dissertation).Finland:University of Kuopio.
Wiedensohler,A.(1988).An approximation of the bipolar charge distribution for particles in the submicron size range.Journal of Aerosol Science,19(3),387-389.
Wiedensohler,A.,Birmili,W.,Nowak,A.,Sonntag,A.,Weinhold,K.,Merkel,M.,et al.(2012).Mobility particle size spectrometers:harmonization of technical standards and data structure to facilitate high quality long-term observations of atmospheric particle number size distributions.Atmospheric Measurement Techniques,5(3),657-685.
Willoughby,R.A.(1979).Book reviews:Solutions of ill-posed problems(by A.N.Tikhonov&V.Y.Arsenin,New York:John Wiley).SIAM Review,21,266-267.
Abbreviations:CN,condition number;CPC,condensation particle counter;DMA,differential mobility analyzer;DMPS,differential mobility particle sizer;FWHM,full width at half maximum;GSD,geometric standard deviation;GCV,generalized cross-validation;LC,L-Curve;LS,least squares;NNLS,non-negative least squares;NRMSE,normalized root mean square error;PSD,particle size distribution;SMPS,scanning mobility particle sizer;SVD,singular value decomposition;TDMA,tandem differential mobility analysis;TF,transfer function.
Alofs,D.J.,&Balakumar,P.(1982).Inversion to obtain aerosol size distributions from measurements with a differential mobility analyzer Journal of Aerosol Science,J3(6),513-527.
Alonso,M.,&Kousaka,Y.(1996).Mobility shift in the differential mobility analyzer due to Brownian diffusion and space-charge effects.Journal of Aerosol Science,27(8),1201-1225.
Bazan,F.S.V.,&Francisco,J.B.(2009).An improved fixed-point algorithm for determining a Tikhonov regularization parameter.Inverse Problems,25(4),045007.
Biskos,G.,Reavell,K.,&Ceilings,N.(2005).Unipolar diffusion charging of aerosol particles in the transition regime.Journal of Aerosol Science,36(2),247-265.
Boisdron,K.,&Brock,J.R.(1970).On the stochastic nature of the acquisition of electrical charge and radioactivity by aerosol particles.Atmospheric Environment,4(1),35-50.
Chien,C.L,Tsai,C J.,Chen,H.L,Lin,G.Y.,&Wu,J.S.(2011).Modeling and validation of nanoparticle charging efficiency of a single-wire corona unipolar charger.Aerosol Science and Technology,45,1468-1479.
Collins,D.R.,Flagan,R.C,&Seinfeld,j.H.(2002).Improved inversion of scanning DMA data.Aerosol Science and Technology,36(1),1-9.
Crump,J.,&Seinfeld,J.(1981).A new algorithm for inversion of aerosol size distribution data.Aerosol Science and Technology,1(1),15-34.
Domat,M.,Kruis,F.,&Fernandez-Diaz,J.(2014a).Determination of the relevant charging parameters for the modelling of unipolar chargers.Journal of Aerosol Science,71,16-28.
Domat,M.,Kruis,F.,&Fernandez-Diaz,J.(2014b).Investigations of the effect of electrode gap on the performance of a corona charger having separated corona and charging zones.Journal of Aerosol Science,68,1-13.
Dubey,P.,&Dhaniyala,S.(2008).Analysis of scanning DMA transfer functions.Aerosol Science and Technology,42(7),544-555.
Dubey,P.,&Dhaniyala,S.(2013).Improved inversion of scanning electrical mobility spectrometer data using a new multiscale expectation maximization algorithm.Aerosol Science and Technology,47(1),69-80.
Fiebig,M.,Stein,C,Schroder,F.,Feldpausch.P.,&Petzold,A.(2005).Inversion of data containing information on the aerosol particle size distribution using multiple instruments.Journal of Aerosol Science,36(11),1353-1372.
Fuchs,N.(1964).The mechanics of aerosols.New York:Pergamon Press.
Gysel,M.,McFiggans,G.B.,&Coe,H.(2009).Inversion of tandem differential mobility analyser(TDMA)measurements.Journal of Aerosol Science,40(2),134-151.
Hagen,D.E.,&Alofs,D.J.(1983).Linear inversion method to obtain aerosol size distributions from measurements with a differential mobility analyzer.Aerosol Science and Technology,2(4),465-475.
Hansen,P.C,&O'Leary,D.P.(1993).The use of the L-curve in the regularization of discrete ill-posed problems.SIAM Journal on Scientific Computing,14,1487-1503.
He,M.,&Dhaniyala,S.(2013).A multiple charging correction algorithm for scanning electrical mobility spectrometer data.Journal of Aerosol Science,61,13-26.
Hontanon,E.,&Kruis,F.E.(2008).Single charging of nanoparticles by UV photoionization at high flow rates.Aerosol Science and Technology,42,310-323.
Hoppel,W.A.(1978).Determination of the aerosol size distribution from the mobility distribution of the charged fraction of aerosols.Journal ofAerosolScience,9(1),41-54.
Intra,P.(2012).Corona discharge in a cylindrical tri ode charger for unipolar diffusion aerosol charging.Journal of Electrostatics,70(1),136-143.
Johnston,P.R.,&Gulrajani,R.M.(2000).Selecting the corner in the L-curve approach to Tikhonov regularization.IEEE Transactions on Biomedical Engineering,47,1293-1296.
Kaminski,H.,Kuhlbusch,T.A.J.,Fissan,H.,Ravi,L.,Horn,H.-G.,Han,H.-S.,et al.(2012).Mathematical description of experimentally determined charge distributions of a unipolar diffusion charger.Aerosol Science and Technology,46(6),708-716.
Karlsson,M.N.,&Martinsson,B.G.(2003).Methods to measure and predict the transfer function size dependence of individual DMAs.Journal of Aerosol Science,34(5),603-625.
Kim,D.S.,Kim,Y.M.,Kwon,Y.T.,&Park,K.(2011).Evaluation of a softX-ray unipolar charger for charging nanoparticles.Joumal ofNanoparticle Research,13,579-585.
Knutson,E.O.,&Whitby,K.T.(1975a).Accurate measurement of aerosol electric mobility moments Journal of Aerosol Science,6(6),453-460.
Knutson,E.O.,&Whitby,K.T.(1975b).Aerosol classification by electric mobility:apparatus,theory,and applications.Journal of Aerosol Science,6(6),443-451.
Kwon,S.,Fujimoto,T.,Kuga,Y.,Sakurai,H.,&Seto,T.(2005).Characteristics of aerosol charge distribution by surface-discharge microplasma aerosol charger(SMAC).Aerosol Sdence&Technology,39(10),987-1001.
Kwon,S.,Sakurai,R,&Seto,T.(2007).Unipolar charging of nanoparticles by the surface-discharge microplasma aerosol charger(SMAC).Journal ofNanoparticle Research,9(4),621-630.
Lawson,C.L,&Hanson,R.J.(1974).Solving least squares problems.Englewood Cliffs,NJ:SIAM(revised republication of the first published in Prentice-Hall).
Li,L,&Chen,D.R.(2011).Performance study of a DC-corona-based particle charger for charge conditioning.Journal of Aerosol Science,42(2),87-99.
Li,W.,Li,L,&Chen,D.-R.(2006).Technical note:A new deconvolution scheme for the retrieval of true DMA transfer function from tandem DMA data.Aerosol Science and Technology,40(12),1052-1057.
Liu,B.Y.R,Pui,D.Y.R,Whitby,K.T.,Kittelson,D.B.,Kousaka,Y.,&McKenzie,R.L.(1978).The aerosol mobility chromatograph:A new detector for sulfuric acid aerosols.Atmospheric Environment,12(1),99-104.
Lloyd,J.J.,Taylor,C.J.,Lawson,R.S.,&Shields,R.A.(1997).The use of the L-curve method in the inversion of diffusion battery data.Journal of Aerosol Science,28,1251-1264.
Pfeifer,S.,Birmili,W.,Schladitz,A.,Muller,T.,Nowak,A.,&Wiedensohler,A.(2013).A novel inversion algorithm for mobility particle size spectrometers considering non-sphericity and additional aerodynamic/optical number size distributions.Atmospheric Measurement Techniques and Discussions,6(3),4735-4767.
Qi,C,Chen,D.R.,&Pui,D.Y.(2007).Experimental study of a new corona based unipolar aerosol charger Journal of Aerosol Science,38(7),775-792.
Rader,D.J.,&McMurry,P.H.(1986).Application of the tandem differential mobility analyzer to studies of droplet growth or evaporation.Journal of Aerosol Science,17(5),771-787.
Reist,P.C.(1993).Aerosol science and technology.New York:McGraw-Hill,Inc.
Rojas,M.,&Steihaug,T.(2002).An interior-point trust-region-based method for large-scale non-negative regularization.Inverse Problems,18,1291-1307.
Stolzenburg,M.R.(1988).An ultrafine aerosol size distribution measuring system(Doctoral dissertation).USA:University of Minnesota.
Stolzenburg,M.R.,&McMurry,P.H.(2008).Equations governing single and tandem DMA configurations and a new lognormal approximation to the transfer function.Aerosol Science and Technology,42(6),421-432.
Stratmann,F.,Kauffeldt,T.,Hummes,D.,&Fissan,H.(1997).Differential electrical mobility analysis:A theoretical study.Aerosol Science and Technology,26(4),368-383.
Talukdar,S.S.,&Swihart,M.T.(2003).An improved data inversion program for obtaining aerosol size distributions from scanning differential mobility analyzer data.Aerosol Science and Technology,37(2),145-161.
Tsai,C.J.,Lin,G.Y.,Chen,H.L,Huang,C.H.,&Alonso,M.(2010).Enhancement of extrinsic charging efficiency of a nanoparticle charger with multiple discharging wires.Aerosol Science and Technology,44(10),807-816.
TSI Inc.(2007).Operation and service manual for condensation particle counter(CPC),model 3775,revision E.
Twomey,S.(1975).Comparison of constrained linear inversion and an iterative nonlinear algorithm applied to the indirect estimation of particle size distributions.Journal of Computational Physics,18(2),188-200.
Vivas,M.,Hontanon,E.,&Schmidt-Ott,A.(2008).Reducing multiple charging of submicron aerosols in a corona diffusion charger.Aerosol Science and Technology,42(2),97-109.
Voutilainen,A.(2001).Statistical inversion methods for the reconstruction of aerosol size distributions(Doctoral dissertation).Finland:University of Kuopio.
Wiedensohler,A.(1988).An approximation of the bipolar charge distribution for particles in the submicron size range.Journal of Aerosol Science,19(3),387-389.
Wiedensohler,A.,Birmili,W.,Nowak,A.,Sonntag,A.,Weinhold,K.,Merkel,M.,et al.(2012).Mobility particle size spectrometers:harmonization of technical standards and data structure to facilitate high quality long-term observations of atmospheric particle number size distributions.Atmospheric Measurement Techniques,5(3),657-685.
Willoughby,R.A.(1979).Book reviews:Solutions of ill-posed problems(by A.N.Tikhonov&V.Y.Arsenin,New York:John Wiley).SIAM Review,21,266-267.
Abbreviations:CN,condition number;CPC,condensation particle counter;DMA,differential mobility analyzer;DMPS,differential mobility particle sizer;FWHM,full width at half maximum;GSD,geometric standard deviation;GCV,generalized cross-validation;LC,L-Curve;LS,least squares;NNLS,non-negative least squares;NRMSE,normalized root mean square error;PSD,particle size distribution;SMPS,scanning mobility particle sizer;SVD,singular value decomposition;TDMA,tandem differential mobility analysis;TF,transfer function.