Slurry

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 ${\displaystyle SG_{water}}$ 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/m^3 (t/m^3) is used instead of the SI density unit, kg/m^3.

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

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

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