Impactors and Particle Size Distribution (2)

1 1 Impactors and Particle size Distribution (2)Ju-Hyeong Park, , , Institute for Occupational Safety and HealthDivision of Respiratory Disease StudiesField Studies BranchParticle size statistics2 Aerosol measurement and size distributionDiameterConcentration1. Concentration Particle mass, surface area or number per unit volume2. size Distribution Concentation versus Particle sizeIncreasing ComplexityAerosol Particle particlesAssign to size binBinsDiameter / m < d < mParticles are assigned to bins according to Particle diameter3 Conversion of a discrete Particle size Distribution to a continuous Particle Diameter / Particle Diameter / Discrete DistributionNumber concentation is proportional to the height of each barParticle size binsNumber 234 56 Number Concentation / Bin Particle Diameter / Particle Diameter / Continuous DistributionNumber concentation is proportional to the area of

2 Each 234 56 Number Concentation / Bin Particle Diameter / Particle Diameter / Continuous DistributionThe smooth continuous Distribution is obtained by joining the bin mid-pointsdn/d(d)Continuous graph Y-axis Differential Particle concentration (or number) Particle number normalized by range of Particle diameter of the interval (or bin) dn/dd X-axis Particle diameter range of the interval dd Integrated area under the curve=total # of particledndd dd=n5 Properties of Particle size Distribution Asymmetrical (or skewed) Distribution Long tail to the right Large number (fraction) of small particles Small number (fraction) of large particles Large range of Particle size Several orders of magnitude in Particle diameter No negative Particle size Mode < Median < Mean Geometric mean (dgor GM)Log dg=( nilogdi)/N dg=exp{( nilogdi)/N} Arithmetic mean and GM Arithmetic mean da=( nidi)/N = (summation of all areas of the bars) / (total number of particles ) Geometric mean Log dg=( nilogdi)/N dg=exp{( nilogdi)}

3 /N} 234 56 Number Concentation / Bin Particle Diameter / Particle Diameter / Continuous DistributionThe smooth continuous Distribution is obtained by joining the bin mid-pointsdn/d(d)Asymmetric (skewed) 23 45dn/d(d)Diameter / mLognormalormal size DistributionNormalized Distribution (n=1)Area inder curve = 1 CMDM odal DiameterNote that when plotte d as dn/ d(d), the modal and count median diameters of a lognormal Distribution are diff erentDistribution skewed to smaller Particle diametersModeGeometric mean (Median)Arithmetic 23 45dn/d(d)Diameter / m+ - d p68% of all particles are between Normal size DistributionNormalized Distribution (n=1)Area inder curve = 1 Distribution characterized by n, and d pNormal distributionMean = Mode = MedianNormal Distribution dp: arithmetic mean Particle diameter : standard deviation ddp: Particle diameter interval df.

4 Frequency of occurrence of particles of diameter dpdf=n 2 e dp d p()22 2ddp8 Y-axis Differential Particle concentration (or number) Particle number normalized by range of log-transformed Particle diameter of the interval (or bin) dn/dlog(d) X-axis Range of log-transformed Particle diameter of the interval dlog(d) Integrated area under the curve=total # of Particle {dn/dlog(d)}dlog(d)Log transformation of continuous graphLognormal Distribution with arithmetic 23 45dn/d(d)Diameter / mLognormalormal size DistributionNormalized Distribution (n=1)Area inder curve = 1 CMDM odal DiameterNote that when plotte d as dn/ d(d), the modal and count median diameters of a lognormal Distribution are diff erentDistribution skewed to smaller Particle diametersCount median diameter (CMD)=GM9 Mathematical function of lognormal Distribution df=n2 Log( g)e Log dp() Log CMD()()22 Log g()2dLog dp()df=n 2 e dp d p()

5 22 2ddpTwo ways of log-transformed graph Transform the original Particle size data using logarithm, and then plot them on normal arithmetic scale of the graph To calculate all statistics mathematically Exponentiate log-transformed statistics Transform x-axis scale of the graph, and then plot the original Particle size data on it Do not transform the data Only change the scale of the 23 45dn/d(d)Diameter / m+ - d p68% of all particles are between Normal size DistributionNormalized Distribution (n=1)Area inder curve = 1 Distribution characterized by n, and d pLog normal Distribution (first approach)Mean = Mode = MedianLognormal Distribution with log (d)Diamet er / mCMD gCMD/ g68% of all particles are between CMD / g and CMD gLognormal size DistributionNormalized Distribution (n=1)Area i nder curve = 1 Distribution characterized by n, CMD and gCount M edian Diameter (CMD)

6 16% of all particles are less than CMD/ g84% of all particles are less than CMD* g50% of all particles are less than CMD11 Cumulative size of particles smaller than dDiameter / mCMD gCMD/ gLognormal size DistributionNormalized Distribution (n=1)Area inder curve = 1 CMD84%16%50%Probability scale of lognormal Distribution Percent of particles less than a given Particle diameter <CMD/(2 g)- 5% <CMD/ g- 16 % <CMD (median)- 50% <CMD* g- 84% <CMD*(2 g)- 95% Any pattern?? Symmetry of probability12 Lognormal Distribution with log og(d)Diamet er / mCMD gCMD/ g68% of all particles are between CMD / g and CMD gLognormal size DistributionNormalized Distribution (n=1)Area i nder curve = 1 Distribution characterized by n, CMD and gCount M edian Diameter (CMD)16% of all particles are less than CMD/ g84% of all particles are less than CMD* g50% of all particles are less than CMDC umulative plotto log-probability of particles smaller than dDiameter / mCMD gCMD/ gLognormal size DistributionNormalized Distribution (n=1)

7 Area inder curve = 1 CMD84%16%50%Switch and change to probability scale13 Log-probability paperProbability scale (probit)12 51020304050607080909598 Log scale ( Particle size : m)234567892030405060708090110100 Log-probability plot of lognormal / m% Distribution less than diameter d (probit)16%50%84%CMD gCMD/ gCMDL ognormal size Distribution1231/2 = g= 2/314 Count median diameter and g CMD (count median diameter) 50% of all particles are less than CMD Geometric standard deviation CMD* g/CMD or CMD/(CMD/ g) CMD, SMD, and weighted (X=n)Surface-area weighted (X=s)Mass weighted (X=m)dX/dLog(d)

8 Diameter / mNormalized distributionsCMDSMD MMD15 Log-probability plots for count, mass, and surface area lognormal weightedSurface-area weightedMass weightedDiameter / m% Distribution less than diameter d (probit)16%50%84%Plots are parallel - g (given by gradient) is the same for each weightingCascade impactor data ( m) > Range( m) Fraction (%)MassFraction(%)Net Mass (mg)Final Mass (mg)Initial Mass (mg)Stage #16An example using count dataDataWilliam Hinds, Aerosol Technology, 2nd(1999)17 Plotting upper size range vs. fraction/ m Particle size ( m)0 102030405060 Fraction/ scale of Particle diameter (lognormal Distribution ) Particle size ( m)246820406080110100 Fraction/ lognormal plot (cum vs.)

9 size ) Particle size ( m)110100 Cumulative fraction (%)020406080100 Cumulative fraction in probability scale/ m1 251020 30 405060 70 8090 9598 Particle size in log scale ( m)234567892030405060708090110100 Log-probability plot of count data84%50%16%19 Take-home practiceStage cut-point (diameter d) Mass concentration Cumulative mass concentration greater than diameter d % Cumulative mass concentration greater than diameter % Cumulative mass concentration less than diameter ( m) (mg/m3) (mg/m3) 10 1 Sum (mg/m3) out the blank out MMD (Mass Median Diameter)

10 And Geometric standard deviation (GSD).