size distributions many processes and properties depend on particle size –fall velocity...
TRANSCRIPT
Size Distributions• Many processes and properties depend on particle size
– Fall velocity– Brownian diffusion rate– CCN activity– Light scattering and absorption– Others
• There are a number of quantitative ways to represent the size distribution– Histogram– Number Distribution– Number Distribution function– Volume, area, mass distributions– Cumulative distributions– Statistics of size distributions: Median, mode, averages, moments…– Others
• We will review a number of these next
• Our primary goal is to explain the Number Distribution Function, which is the most widely used
Size DistributionsThe Histogram
NB = Number of size binsDi = Lower-bound particle diameter for bin i Di+1 = Upper-bound particle diameter for bin i Ni = concentration of particles in bin i (cm-3)
NiDi –Di+1
BNi . . . 2, 1,
Simplest form of distribution – Very instrument-based
Lots of structure at small sizesFew particles at largest sizes
Size DistributionsCumulative Properties from Histogram
NiDi –Di+1
BN
iit NN
1
Total Concentration
Total Surface Area
Total Volume
Total Mass
BN
iiiit NDDS
11
BN
iiiit NDDV
1
2/3
16
BN
iiiiit NDDM
1
2/3
16
i = density of aerosol substance in bin i
Note: We don’t have an “average” diameter for the bin – only the bin boundaries. Above I use the geometric mean.
Sometimes it makes sense to estimate where the particles are within the bin based on the concentrations of neighboring bins, and then calculate the effective mean diameter.
N
ii
NN
iig
AN
AA
1
/1
1
ln1
exp
Size DistributionsCumulative Distributions
NiDi –Di+1
i
jji NDN
11)(
1
11)(
i
jjjji NDDDS
1
1
2/3
16)(
i
jjjji NDDDV
1
1
2/3
16)(
i
jjjjji NDDDM
A Cumulative distribution gives the concentration (or some other property) of particles smaller than diameter Di
• Cumulative values are properties at bin boundaries, not bin centers!• They are monotonically increasing in size
• N(DNB+1) = Nt
•
• Different instruments should report the same function, just sampled differently
1
1
)(i
jji NDNCum. Concentration
Cum. Surface Area
Cum. Volume
Cum. Mass
)()( 1 iii DNDNN
Size DistributionsThe Number Distribution
ni = aerosol number distribution for bin iDi = Di+1–Di is the bin widthNi = niDi
ni = has units of (cm-3 m-1)
NiDi –Di+1 in
i
jji NDN
01)(
i
jjji DnDN
01)(
Area under the curve = total aerosol concentration, N
More uniform way to present instrument data
Instruments with different Di would produce very different histograms, but similar number distributions
Histogram Number distribution
Size DistributionsThe Number Distribution
Small bin width at small sizes leads to amplification of concentrations here relative to histogram
Size DistributionsThe Log Number Distribution
Aerosol distributions span orders of magnitude in size, and are often best shown as a function of log-diameter. Now, the area under curve is NOT equal to total concentration.
To remedy this, we can create a log number distribution (not shown above)
in
ii
i
i
ii DD
N
D
Nn
/loglog 1
0
Size DistributionsThe Number Distribution Function
Distributions are often represented in models or analytically, as continuous functions of diameter. This is as if we had an number distribution with perfectly precise resolution
in
dD
DdND
Dd
DdNDn p
)(303.2
log
)()(0
ii
i
DDp DD
NDn
ii
11
lim)(
This looks a lot like the definition of the derivative.If we use the cumulative distribution, we get…
ii
ii
DD DD
DNDN
dD
DdNDn
ii
1
1
1
lim)(
)(
We think of the number distribution function as the derivative with diameter of the cumulative distribution
When n(D) is plotted vs. D (NOT logD), then the area under the curve = total concentration
The log-diameter distribution is the derivative of the cumulative distribution with log of diameter
Size DistributionsOther Distribution Functions
Aerosol distributions span orders of magnitude in size, and are often best shown as a function of log-diameter. We must use the identity
This lowers the power of Dn in the functions above.
Note the “shifting of the peaks” from number area volume
dD
dNDn )(Number Distribution
Surface Distribution
Volume Distribution
Mass Distribution
)()( 2 DnDdD
dSDnS
)(6
)( 3 DnDdD
dVDnV
)()(6
)( 3 DnDDdD
dMDnM
)(303.2)(log0 DDnDn NN
Statistics of Size distributions
The “moments” will come in when you do area, volume distributions
2/1
11
1
BN
iiii
t
DDNN
DMean Diameter
Standard Deviation
Geometric Mean
nth moment
Histogram Discrete distribution Continuous dist.
2/1
11
1
BN
iiiii
t
DDDnN
D dDDDnN
Dt
)(1
BN
iiii
tD DDDN
N 1
22/11
1 dDDnDDN t
D )()(1 2
BN
iiii
tg DDN
ND
11ln
2
11exp
dDDnD
ND
tg )()ln(
1exp
2/
11
1nN
iiii
tn
B
DDNN
D
dDDnDN
D n
tn )(
1
More Statistics
In-class…
• Power-law distributions
• Log-normal distributions
• Properties of each