Sauter diameter

The Sauter diameter is a parameter of a particle size distribution . It is defined as follows: if the entire volume of the particles in a bed were to be converted into spheres of equal size, the volume to surface ratio of which is to be reproduced, these spheres would have the Sauter diameter.

meaning

The Sauter diameter is mainly used to describe the particle size distributions of solids (e.g. sand) and comminuted liquids (e.g. drops in sprays or emulsions ). The correct designation for the Sauter diameter according to DIN ISO 9276-2 is:

${\displaystyle {\bar {x}}_{1,2}}$

The following terms are also widely used in the literature (d is equivalent to x, with and without a comma, often without a macron due to software limitations):

${\displaystyle {\bar {x}}_{1,2}=d_{32}=SMD=D_{S}}$

The Sauter diameter is formulated mathematically as follows:

${\displaystyle {\bar {x}}_{1,2}={\frac {6}{S_{V}}}}$

being the specific surface area

${\displaystyle S_{V}={\frac {S_{ges}}{V_{ges}}}}$

is the quotient of the total surface and the total volume of the particles. Similar to the equivalent diameter , the specific surface can be described by form factors .

Another application is the description of porous solids. Here, more emphasis is placed on porosity in the mathematical formulation :

${\displaystyle {\bar {x}}_{1,2}={\frac {6\cdot (1-\epsilon )}{\sigma }}}$

where denotes the porosity and the volume-related inner surface. ${\displaystyle \epsilon }$ ${\displaystyle \sigma }$

The porous solid body can consist, for example, of densely packed particles adhering or sintered together at the points of contact, between which spatially crosslinked through pores are free.

${\displaystyle \epsilon }$is the free volume (pore volume) in relation to the total volume (total particle volume + pore volume) and the total inner surface in relation to the total volume. ${\displaystyle \sigma }$

If the porous solid consists of a packing of spheres of equal size with a radius in volume , the following applies: ${\displaystyle n}$ ${\displaystyle R}$ ${\displaystyle V}$

${\displaystyle \epsilon =1-{\frac {n\cdot 4\cdot \pi \cdot R^{3}}{3\cdot V}}}$

${\displaystyle \sigma ={\frac {n\cdot 4\cdot \pi \cdot R^{2}}{V}}}$

For the Sauter diameter it follows:

${\displaystyle {\bar {x}}_{1,2}=2\cdot R}$

so just the ball diameter.

The Sauter diameter is an important parameter for the statistical description of liquid or gas flows through porous solids. There is a linear relationship between the locally averaged flow velocity over several pore diameters and the locally averaged pressure gradient : ${\displaystyle \mathbf {v} }$ ${\displaystyle \nabla p}$

${\displaystyle \nabla p={\frac {\eta \cdot \mathbf {v} }{B}}}$

where the permeability of the porous solid and the viscosity of the liquid or gas denotes. With a geometrically similar structure, i.e. also the same porosity, the permeability is proportional to the square of the Sauter diameter: ${\displaystyle B}$ ${\displaystyle \eta }$

${\displaystyle B\sim {\bar {x}}_{1,2}^{2}}$

Individual evidence

1. ↑ Presentation of the results of particle size analyzes - Part 2: Calculation of mean particle sizes / diameters and moments from particle size distributions (DIN ISO 9276-2), 2009.