# Equivalent diameter

The equivalent diameter (from Latin : aequus = equal + valere = to be worth) is a measure of the size of an irregularly shaped particle such as a grain of sand. It is calculated by comparing a property of the irregular particle with a property of a regularly shaped particle. Depending on the selection of the property used for comparison, a distinction is made between different equivalent diameters. So is z. B. a division into geometric and physical equivalent diameter is possible. The equivalent diameter is an important parameter in mechanical process engineering .

If information about the particle shape is to be taken into account in addition to the size of a particle, so-called shape factors can be defined using several equivalent diameters .

## Geometric equivalent diameter

A geometric equivalent diameter is obtained by determining the diameter of a sphere or a circle with the same geometric property ( surface , volume or projection area) as the irregularly shaped particle.

### Volume-equivalent sphere diameter

The volume-equivalent sphere diameter (symbol ) indicates the diameter of a sphere with the same volume as the particle under consideration. For simple geometric bodies it can easily be calculated: ${\ displaystyle d_ {v}}$${\ displaystyle d_ {v}}$

• Cube : The volume of a cube with edge length is . By equating to the volume of a sphere with the same volume and diameter , the equivalent diameter is obtained${\ displaystyle a}$${\ displaystyle V = a ^ {3}}$${\ displaystyle V = {\ frac {\ pi} {6}} d_ {v} ^ {3}}$${\ displaystyle d_ {v}}$
${\ displaystyle d_ {v} = {\ sqrt [{3}] {\ frac {6} {\ pi}}} \ cdot a \ approx 1 {,} 241 \ cdot a}$
• Octahedron : An octahedron with edge length has the volume , resulting in an equivalent diameter of${\ displaystyle a}$${\ displaystyle V = {\ frac {a ^ {3} {\ sqrt {2}}} {3}}}$
${\ displaystyle d_ {v} = {\ sqrt [{3}] {\ frac {2 {\ sqrt {2}}} {\ pi}}} \ cdot a \ approx 0 {,} 9656 \ cdot a}$
• Tetrahedron : For the tetrahedron with the result is analogous${\ displaystyle V = {\ frac {a ^ {3} {\ sqrt {2}}} {12}}}$
${\ displaystyle d_ {v} = {\ sqrt [{3}] {\ frac {\ sqrt {2}} {2 \ pi}}} \ cdot a \ approx 0 {,} 6083 \ cdot a}$

### Surface equivalent ball diameter

Analogous to the volume-equivalent sphere diameter, the surface-equivalent sphere diameter (symbol ) is defined as the diameter of a sphere that has the same surface area as the examined particle. Here, too, an equivalent diameter can be calculated for simple geometric bodies with the aid of the formula for the spherical surface : ${\ displaystyle d_ {s}}$${\ displaystyle S = \ pi d_ {s} ^ {2}}$

• Cube : With you get${\ displaystyle S = 6a ^ {2}}$
${\ displaystyle d_ {s} = {\ sqrt {\ frac {6} {\ pi}}} \ cdot a \ approx 1 {,} 382 \ cdot a}$
• Octahedron : Over the surface results${\ displaystyle S = 2 {\ sqrt {3}} a ^ {2}}$
${\ displaystyle d_ {s} = {\ sqrt {\ frac {2 {\ sqrt {3}}} {\ pi}}} \ cdot a \ approx 1 {,} 050 \ cdot a}$
• Tetrahedron : The surface of the tetrahedron is so that becomes${\ displaystyle S = {\ sqrt {3}} a ^ {2}}$
${\ displaystyle d_ {s} = {\ sqrt {\ frac {\ sqrt {3}} {\ pi}}} \ cdot a \ approx 0 {,} 7425 \ cdot a}$

### Circle equal to the projection area

The following applies to the area A of a circle: with : diameter of the circle with the same projection area. So it follows: ${\ displaystyle A = {\ frac {\ pi} {4}} d_ {p} ^ {2}}$${\ displaystyle d_ {p}}$

${\ displaystyle d_ {p} = {\ sqrt {\ frac {4A} {\ pi}}}.}$

In extinction particle counters , for. B. the signal generated by a circle equal to the projection area is used for calibration and measurement.

## Physical equivalent diameter

If one compares physical properties of the particle such as the sinking speed in a liquid, the resistance in an electric field or the scattered light intensity, one speaks of physical equivalent diameters.

### Aerodynamic diameter

The aerodynamic diameter of a particle corresponds to the diameter of a sphere with a density of 1 g / cm 3 , which has the same rate of descent in air as the particle.

### Equivalent diameter in the fluid

The sedimentation speed of a ball in a fluid at rest depends on its diameter and the Reynolds number . If one considers non-spherical particles, an equivalent diameter can also be given here. For different flow ranges (Stokes, transition and Newton range), the different Reynolds number results in a different formula for this. So z. B. for the Stokes range (Reynolds number Re <approx. 0.25 depending on literature):

${\ displaystyle d_ {St} = {\ sqrt {{\ frac {18 \ eta} {(\ rho _ {p} - \ rho _ {f}) g}} v_ {St}}}}$

with : equivalent diameter in the Stokes range,: density of the particle ,: density of the liquid, g: gravitational acceleration,: dynamic viscosity ,: sedimentation speed . ${\ displaystyle d_ {St}}$${\ displaystyle \ rho _ {p}}$${\ displaystyle \ rho _ {f}}$${\ displaystyle \ eta}$${\ displaystyle v_ {St}}$

## literature

• Walter Müller: Basic mechanical operations and their laws. Oldenbourg Wissenschaftsverlag GmbH, Munich et al. 2008, ISBN 978-3-486-57842-3 .

## Individual evidence

1. a b Matthias Stieß: Mechanical process engineering. Volume 1: Particle Technology. 3rd, completely revised edition. Springer, Berlin et al. 2009 (published 2008), ISBN 978-3-540-32551-2 .