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How to Compute the Mean Particle Diameter from a LISST Volume Distribution

[Sequoia, March 26, 2010]

In the table below, the computation of the mean particle size from a LISST volume distribution is outlined.

Column A is the size class number, from 1-32.
Column B is the MID POINT of the size bin in µm. In this example we are using the size classes associated with a LISST type B randomly shaped inversion (LISST Bin Sizes).
Column C is the Volume Distribution in µl/l.

Each row in column D (dSum) is computed as the size class # for that row multiplied with the volume concentration in the size class. For example, for size class 8, dSum is equal to 8 * 3.035 = 24.28.

A	B	C	D
Size class	Size class mid point (µm)	Volume Distribution (µl l^-1)	dSum (Size class # * VC in size class)
1	1.09	0.190	0.190
2	1.28	0.256	0.512
3	1.51	0.360	1.080
4	1.79	0.616	2.464
5	2.11	0.853	4.265
6	2.49	1.041	6.246
7	2.93	1.429	10.003
8	3.46	3.035	24.280
9	4.09	4.071	36.639
10	4.82	4.419	44.190
11	5.69	4.818	52.998
12	6.71	5.714	68.568
13	7.92	6.218	80.834
14	9.35	6.252	87.528
15	11.03	6.151	92.265
16	13.02	6.360	101.76
17	15.36	6.288	106.896
18	18.13	6.311	113.598
19	21.39	6.732	127.908
20	25.25	7.726	154.52
21	29.79	8.252	173.292
22	35.16	9.573	210.606
23	41.49	10.666	245.318
24	48.96	11.581	277.944
25	57.77	11.843	296.075
26	68.18	12.214	317.564
27	80.45	11.177	301.779
28	94.94	9.434	264.152
29	112.04	6.993	202.797
30	132.21	4.750	142.500
31	156.02	3.128	96.968
32	184.11	2.300	73.600
		?VD = VC	?dSum
		180.751	3719.339

By definition, the mean particle size in terms of size class number is now ?dSum/?VD = ?dSum/VC, in this example 3719.339 / 180.751 = 20.57714.

We can now see that the mean particle size, in terms of size class numbers is somewhere between size class 20 and 21, i.e. somewhere between 25.25 and 29.79 µm.

Now round down ?dSum/VC to the nearest integer, in this case it would be 20.

Compute the remainder; in this case it would be 20.57714 – 20 = 0.57714.

The mean particle size is now computed as the midpoint of size class 20 (from rounding down ?dSum/VC) multiplied by the ratio of the bin midpoints (200^(1/32) = 1.1801) raised to the power of the remainder: 25.25 * 1.1801^0.57714 = 27.78µm.

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