AEROSOL STATISTICS LOGNORMAL DISTRIBUTIONS AND dn/dlogd p

Transcription

1 Concentration (particle/cm3) [e5] Concentration (particle/cm3) [e5] AEROSOL STATISTICS LOGNORMAL DISTRIBUTIONS AND dn/dlod p APPLICATION NOTE PR-001 Lonormal Distributions Standard statistics based on normal distributions are frequently not suitable for most airborne particle (aerosol) size distributions. Generally, lonormal distributions tend to be the best fit for sinle source aerosols. There is no real theoretical reason as to why aerosol size distributions are lo normal, it s merely empirically the best fit. When statistical analysis is applied to aerosol size distributions, it is routinely based on lonormal distributions. Lonormal statistics are not widely used, and so this slihtly skewed point of view from standard statistics can be quite confusin to those who have not been exposed to it. Lonormal distributions are most useful where the data rane (the difference between the hihest and lowest values) of the x-axis is reater than about 10. If the data rane is narrow, the lonormal distribution approximates a normal distribution. Example Aerosol Size Distribution Lonormal Distribution Example Aerosol Distribution Linear Scale Lo Scale - Diameter (nm) Linear Scale - Diameter (nm) Fiure 1: a) Aerosol data displayed usin a lo scale for particle diameter and b) usin a linear scale for particle diameter. TSI, TSI loo, Scannin Mobility Particle Sizer, SMPS, Enine Exhaust Particle Sizer, EEPS, Fast Mobility Particle Sizer, FMPS, Aerodynamic Particle Sizer, APS, Ultraviolet Aerodynamic Particle Sizer, and UV-APS are trademarks of TSI Incorporated.

2 Lonormal Statistics Count Median Diameter (CMD) When statistically describin lonormal distributions, the eometric mean diameter (D ) of normal distributions is replaced by the count median diameter (CMD). In lonormal distributions, the lo of the particle size distribution is symmetrical, so the mean and the median of the lonormal distribution are equal. The median of the lonormal distribution and normal distribution are equal, since the order of the values does not chane when convertin to a lonormal distribution. Therefore, for a lonormal distribution, D = CMD. D n n n nn 1/ N 1 1 D D3 DN ) CMD ( D where: D = eometric mean diameter D i = midpoint particle size n i = number of particles in roup i havin a midpoint size D i N = n i, the total number of particles, summed over all intervals Geometric Standard Deviation ( or GSD) In aerosol lonormal distributions, the lo of the particle diameter is normally distributed. The normal, linear based standard deviation () with which most are familiar with is replaced with the standard deviation of the loarithms, called the eometric standard deviation ( or GSD). The eometric standard deviation is always reater than 1 ( 1). lo n i (lo Di lo D N 1 ) 1/ where: = eometric standard deviation (GSD) D i = midpoint particle diameter of the i th bin n i = number of particles in roup i havin a midpoint size D i N = n i, the total number of particles, summed over all intervals What Do the Lonormal Statistics Mean? The eometric standard deviation describes how spread out the values are in the distribution. In a normal distribution, 95% of the particle diameters fall within D p. In a lonormal distribution, 95% of the particle diameters fall within a size rane expressed as: 10 [locmdlo ] or CMD < 95% of all particle diameters < CMD 10 =.5nm and i.e., if =.0 and CMD = 10nm; 95% of the particle diameters lie between 10 4 = 4 40nm. Monodisperse aerosols have narrow size distributions and lower eometric standard deviation values ( ). Polydisperse aerosols have wider size distributions and hiher eometric standard deviation values ( ). The eometric standard deviation value at which an aerosol can be considered enerally monodisperse is quite subjective, but a ood rule of thumb is: Monodisperse: 1.5 Polydisperse: >

3 Many scientists prefer a more strinent definition of monodispersity with 1.1. A 1.05 is considered hihly monodisperse. Normalized Concentrations: dn/dlod p Limitations of Concentration (dn) vs. Particle Diameter Data Displays As established above, aerosol distributions are predominantly lonormal in character, so data is typically plotted on a lonormal X-axis. In the simplest technique, particle data is plotted as a function of the concentration (dn) for each particle size bin. The mode concentration of the size distribution is often estimated by the concentration in the peak bin. In aerosol sizin instruments, the number of size bins is finite. This simple concentration vs. lo particle diameter approach works well only if you are exclusively usin one type of instrument, or if you are comparin instruments with identical resolution. However, if you try to compare aerosol size distributions taken of the same aerosol usin instruments of different resolutions, plots of concentration vs. particle diameter can be confusin. In the example below you will see two different plots of data taken with our Scannin Mobility Particle Sizer (SMPS ) Spectrometer. On the left the distribution is plotted usin 64 channel resolution. On the riht is the exact same data plotted with 3-channel resolution. At first lance, it looks like the aerosol represented by the data on the riht is rouhly double the concentration. However, this is just an illusion from the difference in resolution. The concentration on the riht is double because the width of the bin is twice as lare as the channel on the left. The Resolution Problem with Concentration vs. Particle Diameter Data Display Fiure : a) SMPS concentration vs. particle diameter data taken with 64 channel resolution and b) 3 channel resolution. The solution: dn/dlod p The method statisticians use to avoid this problem is to plot the data usin normalized concentration (dn/dlod p ). dn (or N) is the number of particles in the rane (total concentration) and dlod p (or lod p ) is the difference in the lo of the channel width. dlod p is calculated by subtractin the lo of the lower bin boundary from the lo of the upper boundary for each channel (normalizin for bin width). The concentration is divided by the bin width, ivin a normalized concentration value that is independent of the bin width. dn / d lo D p dn d lo D p dn lod p, ulo D p, l where: dn = particle concentration D p = midpoint particle diameter D p,u = upper channel diameter D p,l = lower channel diameter -3-

4 With normalized concentrations the concentration values at the mode will be similar even on instruments with very different resolution. Below is the same SMPS spectrometer data plotted usin normalized concentration. You can see that the peak concentration values are very similar for both raphs. Note, that normalized concentrations are several orders of manitude larer than the concentrations not normalized. This is due to the hih resolution and small bin width of TSI particle instruments. The width of the bins (which is the divisor in normalized concentration) is much less than one. Fiure 3: a) SMPS normalized concentration vs. particle diameter data taken with 64-channel resolution and b) 3-channel resolution. The small difference in mode concentration is due to sliht particle bin chanes in the reions near the bin boundaries. TSI Instrumentation and dn/dlod p The simple way to use dn/dlod p with TSI particle sizers is to understand that for most every instrument we have equally spaced size channels (on a lo scale). The differences between the boundaries for any one channel are the same as any other channel. The Scannin Mobility Particle Sizer (SMPS ) Spectrometer is typically used with 64 channels per decade resolution dn/dlod p = 1/64. Therefore, to calculate the normalized concentration, the concentration value for each bin is divided by 1/64 (i.e., multiplied by 64). The SMPS data can be displayed in 64, 3, 16, 8, or 4 channel resolution. The Aerodynamic Particle Sizer, the Ultraviolet Aerodynamic Particle Sizer, and the Sinle-Box Scannin Mobility Particle Sizer (SMPS ) Spectrometer have 3 channels per decade resolution. Therefore, to calculate the normalized concentration, the concentration value for each bin is divided by 1/3 (i.e., multiplied by 3). The Enine Exhaust Particle Sizer and Fast Mobility Particle Sizer have 16 channels per decade resolution. The displayed resolution in the Laser Aerosol Spectrometer is determined by the channel boundaries set by the user. Description Model Resolution Multiplier Scannin Mobility Particle Sizer (SMPS ) Spectrometer 3936xxx Adjustable 64, 3, 16, 8, or 4 Aerodynamic Particle Sizer (APS ) Spectrometer 331 Fixed 3 Ultraviolet Aerodynamic Particle Sizer (UV-APS ) Spectrometer 3314 Fixed 3 Sinle-Box Scannin Mobility Particle Sizer (SMPS ) Spectrometer 3034 Fixed 3 Enine Exhaust Particle Sizer (EEPS ) Spectrometer 3090 Fixed 16 Fast Mobility Particle Sizer (FMPS ) Spectrometer 3091 Fixed 16 Laser Aerosol Spectrometer 3340 Adjustable User determined -4-

5 Common loarithm vs. Natural loarithm Loarithms can be conceptualized as the inverse of exponents. The loarithm (with defined base) of a number is the power to which the base must be raised in order to produce the number. Loarithms: x = b y y = lo b (x) Common loarithm lo(x) = lo 10 (x) Natural loarithm ln(x) = lo e (x)* *The irrational, mathematical constant e is ~ The natural loarithm (ln) has many convenient computational properties (i.e. derivative = 1/x) which make it attractive to mathematicians. Common lo (lo) tends to be attractive to enineers since numbers are typically base 10 and mental math is easier. Fundamentally, loarithms (reardless of the base) are similar to each other. You can easily chane from one base to the other. Recall the followin loarithm definitions and properties: 1. lo( x) lo10( x). ln( x) loe ( x) y b 3. lo ( x ) y lo ( x) 4. So x b lo ( x) b b lo ( x) lo10( x) ln( x) ln(10 10 ) loe (10 ) lo10( x)loe (10) lo( x).305 ln(x) =.305 lo(x) The only real difference between these two functions is a scale factor (.305). Obviously, if you multiply both sides of the equation by the same number the relative values of the constants remains the same on both sides. In TSI aerosol data analysis, dlod p is used as the divisor for normalized concentration values. If dlnd p was used instead, the absolute values of the normalized concentration numbers would chane by a scale factor, but the shape of the distribution would not be affected. -5-

6 The calculated Fiure 4: SMPS Aerosol Instrument Manaer (AIM) software statistics box. Final Note The primary reference for this application note was the book Aerosol Technoloy: Properties, Behavior, and Measurement of Airborne Particles, by William C. Hinds. This text is a old standard for aerosol technoloy fundamentals and is hihly recommended to anyone who will be venturin into the field of aerosol science. It is easy to read and understand and covers a thorouh workin knowlede of modern aerosol theory and applications. TSI Incorporated Visit our website for more information. USA Tel: UK Tel: France Tel: Germany Tel: India Tel: China Tel: Sinapore Tel: PR-001 Rev. B 01 TSI Incorporated Printed in U.S.A.