I am trying to understand the measurements of a particle size analyzer that could for example look like this:

Particle size distribution

For this image I have three values for each error bar (min, max and mean).

The y axis is here defined as $\frac{dN}{dlog(Dp)}$. Why is the particle number for the distribution defined like this and not just plainly as $N$?

As I want to use these measurements as input to a simulation I need to lump the particle numbers to a certain amount of particle diameters. E.g. I could use 10 particle sizes as input in the simulation software. How could I calculate the corresponding particle numbers to these 10 particle sizes from the particle size distribution?


In the manual of the particle size analyzer is the following page on this topic:

Page in manual concerning the particle number calculation

  • $\begingroup$ You need to tell us what is the quantity measured, then we may be able to tell why it is plotted like this. at the moment, I have no idea what is $N$ or $Dp$. $\endgroup$ – Adam Latosiński May 7 '19 at 14:41
  • $\begingroup$ The measured quantity is the number of particles per cubic centimeter of air that is flowing through a device. $\endgroup$ – Axel May 7 '19 at 14:42
  • $\begingroup$ And on the x axis the particle size. However I don't understand why the y axis is defined as it is instead of just the particle number $N$. $\endgroup$ – Axel May 7 '19 at 14:54

I think it's because the number of particles of exactly the diamiater you specify will always be 0. What is more useful, is the number of particles with diameter in some range. Of course the wider the range, the more particles you get. So what they plot $\frac{dN}{d\log D_p}$ is an extrapolated limit $$ \frac{dN}{d\log D_p} = \lim_{\Delta D_p \rightarrow 0}\frac{\Delta N}{\Delta (\log D_p)}$$ where $\Delta N$ is the number of particles with the diamter in the range from $D_p$ to $D_p +\Delta D_p$. Of course, they don't actually measure the limit, they are limited by the sensitivity of their apparatus.

They use $\Delta (\log D_p)$ in the denominator instead of just $\Delta D_p$ because they also use logarithmic scale on the $x$ axis. They do so so that the integral under the curve will still be equal to the total number of particles in a given range.

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  • $\begingroup$ In the manual of the particle size analyzer it is also stated that total particle number can be calculated by $N=\frac{1}{16}\sum_{D_{p1}}^{D_{p2}}\frac{dN}{dlog(D_p)}$. How can I use this formula to calculate the corresponding numbers for e.g. 10 particle size classes? $\endgroup$ – Axel May 7 '19 at 15:15
  • $\begingroup$ Sorry, I don't know what re the 10 particle size classes, and I don't undertand from where this factor $\frac{1}{16}$ comes from. This whole topic isn't actually my area of expertise. $\endgroup$ – Adam Latosiński May 7 '19 at 21:49
  • $\begingroup$ The $\frac{1}{16}$ comes according to the manual from the reason they used 16 bins or ranges per decade. I added an image of the page from the manual to the question. If you have time maybe you can have a short look at it? Thank you already for the help you offered. $\endgroup$ – Axel May 8 '19 at 7:06
  • $\begingroup$ The manual says it all: you just need to sum the values of $\frac{d N}{d\log D_p}$ corresonding to the 10 particle size classes that you want to count. Every point with error bars on the graph gives you one value (notice that you have 16 points between particle size $10^1$ and $10^2$, another 16 between $10^2$ and $10^3$ - they clearly correspond to these szie classes they are speaking of. I don't know what labels 'DPF in' and \DPF out\ refer to, so I can't tell you which of the two data point for each size class you should use. $\endgroup$ – Adam Latosiński May 8 '19 at 7:18
  • $\begingroup$ So as input for my simulation I could for example sum the first 5 values in my graph and then lump then to the mean particle size of these 5 measurements? Would that be a reasonable thing to do? Would I need to divide this by 5 then instead of 16? $\endgroup$ – Axel May 8 '19 at 7:23

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