Mass Balancing (Three-Product Formula)

Description

This article describes the classical two-product formula for estimating the separation efficiency of a process.

Model theory

The three-product formula estimates the mass flow split of solids and the recovery of two assayed components from a process step with one feed and three product streams.

The formula is derived from a consideration of the mass balance of solids flow and the assay components around the process. That is:

$F.f_{A}=C.c_{A}+M.m_{A}+T.t_{A}$ for component A

$F.f_{B}=C.c_{B}+M.m_{B}+T.t_{B}$ for component B

where:

• $F$, $C$, $M$ and $T$ are the mass flow rates of the feed, concentrate, middlings and tailings streams, respectively.
• $f$, $c$, $m$ and $t$ are the assay values of a single component in the feed, concentrate, middlings and tailings streams, respectively.
• $A$ and $B$ indicate each of the two components assayed.

By rearranging the three-product mass balance equations, the solids split to concentrate and middlings is:

$S_{C}={\dfrac {C}{F}}={\dfrac {(f_{A}-t_{A})(m_{B}-t_{B})-(f_{B}-t_{B})(m_{A}-t_{A})}{(c_{A}-t_{A})(m_{B}-t_{B})-(c_{B}-t_{B})(m_{A}-t_{A})}}$

$S_{M}={\dfrac {M}{F}}={\dfrac {(f_{A}-t_{A})(c_{B}-t_{B})-(f_{B}-t_{B})(c_{A}-t_{A})}{(m_{A}-t_{A})(c_{B}-t_{B})-(m_{B}-t_{B})(c_{A}-t_{A})}}$

The recovery of the two assayed components to concentrate and middlings is:

$(R_{A})_{C}={\dfrac {C}{F}}.{\dfrac {c_{A}}{f_{A}}}$

$(R_{B})_{C}={\dfrac {C}{F}}.{\dfrac {c_{B}}{f_{B}}}$

$(R_{A})_{M}={\dfrac {M}{F}}.{\dfrac {m_{A}}{f_{A}}}$

$(R_{B})_{M}={\dfrac {M}{F}}.{\dfrac {m_{B}}{f_{B}}}$

The assays must show a reasonable degree of degree of separation across the process for the three-product formula to return sensible results.

Mineral compositions, particle size distributions and water fractions can be substituted in place of metal assays in the formula.

Excel

The three-product formula may be invoked from the Excel formula bar with the following function call:

=mdMassBal_ThreeroductFormula(Assay as Range, Optional AbsSD as Range)

Invoking the function with no arguments will print Help text associated with the model, including a link to this page.

The input parameters and calculation results are defined below in matrix notation, along with an example image showing the selection of the same cells and arrays in the Excel interface:

${\begin{aligned}{\mathit {Assay}}&={\begin{bmatrix}f\\c\\m\\t\\\end{bmatrix}}\\\\\end{aligned}}$ ${\begin{aligned}{\mathit {mdMassBal\_ThreeProductFormula}}&={\begin{bmatrix}S_{C}{\text{ (frac)}}&(R_{A})_{C}{\text{ (frac)}}&(R_{B})_{C}{\text{ (frac)}}\\S_{M}{\text{ (frac)}}&(R_{B})_{M}{\text{ (frac)}}&(R_{B})_{M}{\text{ (frac)}}\\\end{bmatrix}}\end{aligned}}$

Figure 1. Example showing the selection of the Assay (blue frame) and Results (light blue frame) arrays in Excel.

See also

• Mass Balancing (Two-Product Formula)
• Mass Balancing (n-Product Formula)