Determining solids fraction

To determine the percent solids (or solids fraction) of a slurry from the density of the slurry, solids and liquid^[7]

${\displaystyle \phi _{sl}={\frac {\rho _{s}(\rho _{sl}-\rho _{l})}{\rho _{sl}(\rho _{s}-\rho _{l})}}}$

where

${\displaystyle \phi _{sl}}$ is the solids fraction of the slurry (state by mass)

${\displaystyle \rho _{s}}$ is the solids density

${\displaystyle \rho _{sl}}$ is the slurry density

${\displaystyle \rho _{l}}$ is the liquid density

In aqueous slurries, as is common in mineral processing, the specific gravity of the species is typically used, and since specific gravity of water is taken to be 1, this relation is typically written:

${\displaystyle \phi _{sl}={\frac {\rho _{s}(\rho _{sl}-1)}{\rho _{sl}(\rho _{s}-1)}}}$

even though specific gravity with units tonnes/m³ (t/m³) is used instead of the SI density unit, kg/m³.

Liquid mass from mass fraction of solids

To determine the mass of liquid in a sample given the mass of solids and the mass fraction: By definition

${\displaystyle \phi _{sl}={\frac {M_{s}}{M_{sl}}}}$

therefore

${\displaystyle M_{sl}={\frac {M_{s}}{\phi _{sl}}}}$

and

${\displaystyle M_{s}+M_{l}={\frac {M_{s}}{\phi _{sl}}}}$

then

${\displaystyle M_{l}={\frac {M_{s}}{\phi _{sl}}}-M_{s}}$

and therefore

${\displaystyle M_{l}={\frac {1-\phi _{sl}}{\phi _{sl}}}M_{s}}$

where

${\displaystyle \phi _{sl}}$ is the solids fraction of the slurry

${\displaystyle M_{s}}$ is the mass or mass flow of solids in the sample or stream

${\displaystyle M_{sl}}$ is the mass or mass flow of slurry in the sample or stream

${\displaystyle M_{l}}$ is the mass or mass flow of liquid in the sample or stream

Volumetric fraction from mass fraction

${\displaystyle \phi _{sl,m}={\frac {M_{s}}{M_{sl}}}}$

Equivalently

${\displaystyle \phi _{sl,v}={\frac {V_{s}}{V_{sl}}}}$

and in a minerals processing context where the specific gravity of the liquid (water) is taken to be one:

${\displaystyle \phi _{sl,v}={\frac {\frac {M_{s}}{SG_{s}}}{{\frac {M_{s}}{SG_{s}}}+{\frac {M_{l}}{1}}}}}$

So

${\displaystyle \phi _{sl,v}={\frac {M_{s}}{M_{s}+M_{l}SG_{s}}}}$

and

${\displaystyle \phi _{sl,v}={\frac {1}{1+{\frac {M_{l}SG_{s}}{M_{s}}}}}}$

Then combining with the first equation:

${\displaystyle \phi _{sl,v}={\frac {1}{1+{\frac {M_{l}SG_{s}}{\phi _{sl,m}M_{s}}}{\frac {M_{s}}{M_{s}+M_{l}}}}}}$

So

${\displaystyle \phi _{sl,v}={\frac {1}{1+{\frac {SG_{s}}{\phi _{sl,m}}}{\frac {M_{l}}{M_{s}+M_{l}}}}}}$

Then since

${\displaystyle \phi _{sl,m}={\frac {M_{s}}{M_{s}+M_{l}}}=1-{\frac {M_{l}}{M_{s}+M_{l}}}}$

we conclude that

${\displaystyle \phi _{sl,v}={\frac {1}{1+SG_{s}({\frac {1}{\phi _{sl,m}}}-1)}}}$

where

${\displaystyle \phi _{sl,v}}$ is the solids fraction of the slurry on a volumetric basis

${\displaystyle \phi _{sl,m}}$ is the solids fraction of the slurry on a mass basis

${\displaystyle M_{s}}$ is the mass or mass flow of solids in the sample or stream

${\displaystyle M_{sl}}$ is the mass or mass flow of slurry in the sample or stream

${\displaystyle M_{l}}$ is the mass or mass flow of liquid in the sample or stream

${\displaystyle SG_{s}}$ is the bulk specific gravity of the solids