The relationship between the arithmetic mean and quadratic mean for particle size distribution

 

Introduction

Particle size distribution data of a material to be collected is very important for the selection and sizing of a dust collector.

If a population of particles is represented by a single number (or mean), there can be many different measures of mean sizes (or central tendency): arithmetic, geometric, quadratic, cubic, biquadratic, and harmonic to name just a few, each appropriate to specific uses.

The mean particle size is rarely quoted in isolation: it is usually related to some application and used as a single number to represent the full size distribution. It represents the distribution by some property which is vital to the application or process under study; if two size distributions have the same mean, the two materials are likely to behave in the process in the same way.

The arithmetic mean

The arithmetic mean is the measure of central tendency most widely used in general statistics, and is essential to a few procedures (such as defining a normal probability distribution). In the precipitation of fine particles due to turbulence, the most relevant mean size is the arithmetic mean of the mass distribution.

The quadratic mean

The relaxation time and terminal velocity of a particle are both proportional to squared particle size. Most procedures in dynamic separation do not specifically require the use of the arithmetic mean. For example, in prediction of the total efficiency of settling chambers, cyclone separator, or other dynamic separators, quadratic means of the mass distribution is most appropriate.

The relationship between them

Usage of the phrase "average particle diameter" has often been very loose, and unwary readers often take it to mean the arithmetic mean, when in fact the value given is the quadratic mean. It is therefore good practice to be specific. The quadratic mean gives greater weight to bigger particles and is equal to or greater than the arithmetic mean by an amount that depends on the variance (s²) according to the relationship below:

(QMD)² = <x>² + s²

Where, QMD is Quadratic Mean Diameter,

<x> is arithmetic mean diameter and,

s is the spread of particle diameters distribution.

Normally, when <x> and s are presented in a particle size distribution report, the above expression can be used and obtain the quadratic mean easily and conveniently.

Two examples of particle size distribution

Here are two particle size distribution examples; the difference between the two means is clear.

Example 1 is milled corn; the arithmetic mean diameter is 839.27 micron, while the quadratic mean is 1041.38 micron. The spread of the distribution is 616.64. 

Particle diameter (micron)	Percentage of mass (%)
53	0.10%
73	0.91%
103	3.24%
150	3.44%
212	6.67%
297	8.09%
420	11.73%
594	15.17%
841	18.20%
1191	19.62%
1680	7.99%
2380	3.24%
3360	1.62%

Example 2 is the particle size distribution of sand; the arithmetic mean diameter is 102.83 micron, while the quadratic mean is 116.5 micron. The spread of the distribution is 54.75.

Particle diameter (micron)	Percentage of mass
10	5.50%
53	2.70%
63	5.60%
75	28.40%
106	23.30%
125	20.00%
150	12.00%
210	0.40%
250	0.50%
300	0.90%
420	0.40%
500	0.20%
850	0.10%

 Summary

The difference between the arithmetic mean and quadratic mean determines their different usages. However, if the variance of the particle size distribution is given, quadratic mean is conveniently available from their simple relationship.

In any case, users should be conscious of the difference between quadratic and arithmetic mean diameters (which usually is not large) and be specific in defining the value used.

In stands with small diameters and narrow range in diameters, the differences are slight between arithmetic mean and quadratic mean diameters. In stands with large diameters and a wide range of diameters present or with strongly skewed diameter distributions, the differences can be substantial.

Keyword

Particle size, mean particle size, Particle size distribution, arithmetic mean diameter (size), quadratic mean diameter (size), dynamic separator, settling chamber, cyclone separator