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# Equivalent diameter

Many properties of particles which are used as measures of dispersity can be expressed in terms of equivalent diameters (Table 2.1). These are the dimensions of a particle of defined geometrical shape which has the same property as the property of interest of the particle under investigation. The equivalent diameters are then introduced as measures of dispersity; for instance, the equivalent settling rate or drag diameter is the diameter of a sphere with the same terminal velocity and density as the particle when it

Table 2.1 The most important equivalent diameters

 Geometric equivalent diameters Diameter of the sphere with the same volume Jtv Diameter of the sphere with the same surface area Diameter of the sphere with the same specific surface area X sv Diameter of the circumscribing sphere X en From the projection of the particle Particle orientation Stable Average Diameter of the circle with the same area X p,s X p,m Diameter of the circle with the same perimeter pe,s pe,m Diameter of the inscribed circle •*-in,s X in,m Diameter of the circumscribed circle ci,s -*• ci,m Hydrodynamic equivalent diameters Diameter of the sphere with the same resistance (drag) XD Diameter of the sphere with the same terminal velocity Xu Diameter of the sphere with the same terminal velocity in the Stokes law range (Stokes diameter) Xst Other equivalent diameters Diameter of the sphere scattering light at the same intensity X sea Diameter of the sphere causing the same change in electrical resistance (Coulter counter) *el

is falling in the same fluid. If the velocity of the particle is measured in the Stokes law region, then the drag diameter is called the Stokes diameter. If the projected area of a particle is compared with a circle of the same area, then the equivalent diameter is that of the circle corresponding to the particle lying in either its stable or its average position. The introduction of an equivalent diameter proves to be very appropriate in many cases, particularly when, for instance, a linear measurement must be converted into an area and a volume.

If we are interested in the motion of a particle in a fluid then the appropriate measure of dispersity is the settling or terminal velocity. It is usually measured in the Stokes law region. The Stokes diameter thus obtained is truly characteristic of the behaviour of the particle, provided also that the particle is actually moving in the Stokes law range. If the motion occurs at higher Reynolds numbers (see Chapter 3, section 3.1.1(b)), we generally again use the Stokes diameter to specify it, together with the laws for the drag force in the intermediate and Newtonian ranges. However, this is not exact. Strictly speaking that equivalent diameter which corresponds to the effective particle resistance should be chosen. It will probably deviate from the Stokes diameter because at higher Reynolds numbers drag plays a more important role than in the Stokes law range. We could also consider using the equivalent diameter of the circle with the same projected area; however, it is not certain that this would reduce the error. Using the carefully measured Stokes diameter is often the most suitable approximation.

Frequently an equivalent diameter is used which has been obtained by sifting.

If this is done even larger errors can arise.

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