Skip to main content

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

Image
By on

 


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

Labels:

Comments

Post a Comment

Feel Free To Leave Your Comment Below!

Popular posts from this blog

Dust clogged up a baghouse

By on
  We were called in by a customer on one cold morning. Their dust collection system did not work when they turned on power as usual. First, we checked the ductwork to see if there was any dust accumulated inside from the sucking points, to the fan, then to the baghouse. It seemed the ductwork was not the problem. After being in the baghouse, the filtered air went back to the workshop. Then we opened the access door of the baghouse hopper, and what we saw showed us the problem: it was packed with wood dust (Figure 1).  Figure 1 baghouse hopper packed with wood dust  Later, the access door on the clean side of the baghouse was opened, and one bag was taken out, it was very clear that the dust even accumulated between bags (Figure 2). Figure 2 dust clogged up between bags After the dust was removed from the baghouse, the system was turned on. In the first 20 minutes, the baghouse was warmed up without dust loading.  When this happened, the daytime temperature outdoors w...
Post a Comment
Read more

Baghouse dye test for leak detection - 2

By on
  In a periodic test of a baghouse dust collection system, dye test for leak detection was done to the baghouse by following the test procedure. The system collects wood dust, and the bags were installed 4 years ago. The air after the baghouse is recirculated back to the workshop to save energy, especially in winter. The dye test easily found glowing spots on some bags adjacent to the tubesheet with the help of a black light wand (Figure 1&2) . These spots were thought to be caused by the bleed-through of the green dye powder; they were not indicators of broken bags.  We put the dye test results in the comprehensive report for this periodic system test and explained our concerns to the customer.  Figure 1 Glowing spots under black light on bags, image 1 Figure 2 Glowing spots under black light, image 2 Leak Test, Air Balancing, Trouble-shooting, and System Evaluation services by Airvate Dye test for bag/cartridge leak  Dust collection system and industrial ventil...
Post a Comment
Read more

Troubleshooting: Air Header got no air for the pulse jet cleaning system

By on
  One customer told us that their dust collection system had a very weak suction on the grinding process line, and we went to their workshop on-schedule to do troubleshooting.  Introduction of the dust collection system The dusty air from the plastic grinder first ran through a cyclone and the separated dust from the cyclone directly went back to the process; the airflow with fine particles from the cyclone went to a cartridge dust collector, and then the filtered air was released to outside of the building. Since the plastic dust is combustible, an explosion panel and a vent duct were installed on the dust collector, and a non-return valve on the ductwork just in front of the dust collector inlet. The cartridge dust collector was equipped with a pulse jet cleaning system controlled by a Photohelic pressure gauge. The Photohelic Gauge measures the Differential Pressure and provides the capability for “on-demand” cleaning in conjunction with a Timer Board. Troubleshooting proce...
Post a Comment
Read more