Mass Balancing (n-Product Formula)

Description

This article describes a methodology for estimating the solids mass split and recovery of multiple assayed components from a process step with any number of product streams.

Model theory

The method of mass flow errors is applied to estimate the solids split and recovery of multiple assayed components to an arbitrary number of product streams.

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

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

Unweighted estimates

The solids mass flow split to each product stream is found by minimising the unweighted sum of squares of the differences between the mass flows of each assayed component in the feed and product streams, i.e.

$\mathrm {argmin} \left\{\sum \limits _{i=1}^{m}{\Delta _{i}}^{2}\right\}$

where:

$\Delta _{i}={f_{i}-\sum \limits _{j=1}^{n}{\beta _{j}p_{ij}}}\quad :\quad 0\leq \beta _{j}\leq 1,\quad \sum \beta _{j}=1,\quad m\geq n-1$

and:

• $m$ is the number of component assays
• $n$ is the number of product streams
• $f_{i}$ is the assay of component $i$ in the feed stream
• $p_{ij}$ is the assay of component $i$ in product stream $j$
• $\beta _{j}$ is solids mass split to product stream $j$ (frac)

The feed stream assays may be reconstituted by summing the product stream assays at the estimated solids mass splits:

${\hat {f}}_{i}=\sum \limits _{j=1}^{n}\beta _{j}p_{ij}$

where ${\hat {f}}_{i}$ is the reconstituted assay of component $i$ in the feed stream.

The recovery of each assayed component $i$ to each product stream $j$, $R_{ij}$ (frac), is then:

$R_{ij}={\dfrac {\beta _{j}p_{ij}}{{\hat {f}}_{i}}}$

Weighted estimates

A weighted sum of squares for minimisation can be generated which takes into account measurement uncertainty.

The propagation of variance approximation states that the variance of a function is the sum of the product of the variance of each input and the square of its partial derivative, and so:

${\sigma _{\Delta i}}^{2}={\sigma _{{\rm {f}}i}}^{2}+\sum \limits _{j=1}^{n}{\beta _{j}}^{2}{\sigma _{ij}}^{2}$

where:

• $\sigma _{{\rm {f}}i}$ is the absolute standard deviation of the assay of component $i$ in the feed stream
• $\sigma _{ij}$ is the absolute standard deviation of the assay of component $i$ in product stream $j$

The weighted solids splits is therefore found from:

$\mathrm {argmin} \left\{\sum \limits _{i=1}^{m}{\dfrac {{\Delta _{i}}^{2}}{{\sigma _{\Delta i}}^{2}}}\right\}$

and the reconstituted feed assays and component recoveries are computed as for the unweighted case.

Additional notes

Absolute and relative standard deviations

The assay standard deviations required for the weighted estimates, ($\sigma _{\rm {f}},\sigma _{ij}$), are absolute standard deviations, and posses the same units as the assay measurement.

However, many other literature sources quote relative standard deviations, usually as percentages, which must be converted to absolute values. For example:

$\sigma _{\rm {fi}}=\mathrm {RSD} _{\rm {fi}}.f_{i}$

for the feed stream assays, where $\mathrm {RSD}$ is the Relative Standard Deviation (frac).

Care must be taken to differentiate between absolute and relative values when applying error models, e.g. the Whiten approach for particle size distributions.

Component assay magnitude

Component assays with larger values will tend to dominate the results of the method of mass flow errors. For example, component assays with values of thousands of parts per million will affect the solids mass split more than components with value of a fraction of a weight percent.

This can be addressed as follows:

• For the unweighted estimates case, the measured assay values can be scaled to more appropriate units, e.g. 100g/t rather than g/t.
• For the weighted estimates case, the absolute standard deviations can be scaled to produce the same effect.

The scaling factors applied in either case are arbitrary, and should be selected to reflect the relative "importance" or "confidence" of the component measurement.

Excel

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

=mdMassBal_nProductFormula(Assays as Range, Optional AbsSD as Range, Optional Transposed as Boolean = False)

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 example images showing the selection of the same cells and arrays in the Excel interface.

Default

The following default arrangement of inputs and results ranges apply when the formula argument Transposed is false or omitted:

${\begin{aligned}{\mathit {Assay}}&={\begin{bmatrix}f_{1}&\cdots &f_{m}\\p_{11}&\cdots &p_{m1}\\\vdots &\ddots &\vdots \\p_{1n}&\cdots &p_{mn}\\\end{bmatrix}}\\\\{\mathit {AbsSD}}&={\begin{bmatrix}\sigma _{\rm {f1}}&\cdots &\sigma _{{\rm {f}}m}\\\sigma _{11}&\cdots &\sigma _{m1}\\\vdots &\ddots &\vdots \\\sigma _{1n}&\cdots &\sigma _{mn}\\\end{bmatrix}}\\\end{aligned}}$ ${\begin{aligned}{\mathit {mdMassBal\_nProductFormula}}&={\begin{bmatrix}\beta _{\rm {f}}{\text{ (frac)}}&{\hat {f}}_{1}{\text{ (frac)}}&\cdots &{\hat {f}}_{m}{\text{ (frac)}}\\\beta _{1}{\text{ (frac)}}&R_{11}{\text{ (frac)}}&\cdots &R_{m1}{\text{ (frac)}}\\\vdots &\vdots &\ddots &\vdots \\\beta _{n}{\text{ (frac)}}&R_{1n}{\text{ (frac)}}&\cdots &R_{mn}{\text{ (frac)}}\\\end{bmatrix}}\end{aligned}}$ where: $\beta _{\rm {f}}$ is total mass fraction of solids in the feed (frac), i.e. $\beta _{\rm {f}}=\sum \nolimits _{i}^{n}{\beta _{i}}=1$ the unweighted sum of squares approach is used if $AbsSD$ is omitted.

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

Transposed

The following arrangement of inputs and results ranges apply when the formula argument Transposed is true:

${\begin{aligned}{\mathit {Assay}}&={\begin{bmatrix}f_{1}&p_{11}&\cdots &p_{1n}\\\vdots &\vdots &\ddots \\f_{m}&p_{1m}&\cdots &p_{mn}\\\end{bmatrix}}\\\\{\mathit {AbsSD}}&={\begin{bmatrix}\sigma _{{\rm {f}}1}&\sigma _{11}&\cdots &\sigma _{1n}\\\vdots &\vdots &\ddots \\\sigma _{{\rm {f}}m}&\sigma _{1m}&\cdots &\sigma _{mn}\\\end{bmatrix}}\\\end{aligned}}$ ${\begin{aligned}{\mathit {mdMassBal\_nProductFormula}}&={\begin{bmatrix}\beta _{\rm {f}}{\text{ (frac)}}&\beta _{1}{\text{ (frac)}}&\cdots &\beta _{n}{\text{ (frac)}}\\{\hat {f}}_{1}{\text{ (frac)}}&R_{11}{\text{ (frac)}}&\cdots &R_{1n}{\text{ (frac)}}\\\vdots &\vdots &\ddots &\vdots \\{\hat {f}}_{m}{\text{ (frac)}}&R_{m1}{\text{ (frac)}}&\cdots &R_{mn}{\text{ (frac)}}\\\end{bmatrix}}\end{aligned}}$

Figure 2. Example showing the selection of the Assay (blue frame), AbsSD (red frame), and Results (light blue frame) arrays in Excel when ${\mathit {Transposed}}=\mathrm {True}$.

See also

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