Comminution Circuit Specific Energy (Morrell)

Description

This article describes the Morrell method for estimating the specific energy of comminution circuits (Morrell et al., 2016).

Model theory

This section is currently under construction. Please check back later for updates and revisions.

Excel

Coarse particle tumbling mill specific energy

The coarse particle tumbling mill specific energy calculation may be invoked from the Excel formula bar with the following function call:

=mdSMC_Wa(K1 as Double, Mia as Double, x1 as Double, Optional returnCalcs 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 an example image showing the selection of the same cells and arrays in the Excel interface:

${\begin{aligned}K1&={\big [}K_{1}{\big ]}\\{\mathit {Mia}}&={\big [}M_{\rm {ia}}{\text{ (kWh/t)}}{\big ]}\\x1&={\big [}x_{1}{\text{ (}}\mu {\text{m)}}{\big ]}\\{\mathit {returnCalcs}}&={\big [}({\text{True / False)}}{\big ]}\end{aligned}}$ ${\mathit {mdSMC\_Wa}}={\begin{bmatrix}W_{\rm {a}}{\text{ (kWh/t)}}\\f(x_{1})^{*}\\f(x_{2})^{*}\\\end{bmatrix}}$ where: $K_{1}=1$ for all circuits without a recycle pebble crusher, and $K_{1}=0.95$ where circuits have a pebble crusher $M_{\rm {ia}}$ is the coarse ore (> 750 μm) work index in tumbling mill circuits (kWh/t) $x_{1}$ is the 80% passing size of the feed (μm) ${\mathit {returnCalcs}}$ indicates whether to return the optional results marked with the $^{*}$ superscript (default is to omit) $f(x_{1})$ and $f(x_{2})$ are exponents of the F80 and P80 equation terms, respectively

Figure 1. Example showing the selection of the input parameters (shaded blue, red and purple frames), and Results (green frame) array in Excel. The parameter ${\mathit {returnCalcs}}={\text{False}}$ in this example.

Fine particle tumbling mill specific energy

The fine particle tumbling mill specific energy calculation may be invoked from the Excel formula bar with the following function call:

=mdSMC_Wb(Mib as Double, x3 as Double, Optional returnCalcs 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 an example image showing the selection of the same cells and arrays in the Excel interface:

${\begin{aligned}{\mathit {Mib}}&={\big [}M_{\rm {ib}}{\text{ (kWh/t)}}{\big ]}\\x3&={\big [}x_{3}{\text{ (}}\mu {\text{m)}}{\big ]}\\{\mathit {returnCalcs}}&={\big [}({\text{True / False)}}{\big ]}\end{aligned}}$ ${\mathit {mdSMC\_Wb}}={\begin{bmatrix}W_{\rm {b}}{\text{ (kWh/t)}}\\f(x_{1})^{*}\\f(x_{2})^{*}\\\end{bmatrix}}$ where: $M_{\rm {ib}}$ is the fine ore (< 750 μm) work index in tumbling mill circuits (kWh/t) $x_{3}$ is the 80% passing size of the final grind product (μm) ${\mathit {returnCalcs}}$ indicates whether to return the optional results marked with the $^{*}$ superscript (default is to omit) $f(x_{1})$ and $f(x_{2})$ are exponents of the F80 and P80 equation terms, respectively

Figure 2. Example showing the selection of the input parameters (shaded blue and red frames), and Results (green frame) array in Excel. The parameter ${\mathit {returnCalcs}}={\text{False}}$ in this example.

Conventional crushing specific energy

The conventional crushing specific energy calculation may be invoked from the Excel formula bar with the following function call:

=mdSMC_Wc(K2 as Double, Mic as Double, x1 as Double, x2 as Double, Optional returnCalcs 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 an example image showing the selection of the same cells and arrays in the Excel interface:

${\begin{aligned}K2&={\big [}K_{2}{\big ]}\\{\mathit {Mic}}&={\big [}M_{\rm {ic}}{\text{ (kWh/t)}}{\big ]}\\x1&={\big [}x_{1}{\text{ (}}\mu {\text{m)}}{\big ]}\\x2&={\big [}x_{2}{\text{ (}}\mu {\text{m)}}{\big ]}\\{\mathit {returnCalcs}}&={\big [}({\text{True / False)}}{\big ]}\end{aligned}}$ ${\mathit {mdSMC\_Wc}}={\begin{bmatrix}W_{\rm {c}}{\text{ (kWh/t)}}\\f(x_{1})^{*}\\f(x_{2})^{*}\\{S_{\rm {c}}}^{*}\\\end{bmatrix}}$ where: $K_{2}=1$ for closed circuit crushing, and $K_{1}=1.19$ for open circuit crushing $M_{\rm {ic}}$ is the ore work index in crusher circuits (kWh/t) $x_{1}$ is the 80% passing size of the feed (μm) $x_{2}$ is the 80% passing size of the product (μm) ${\mathit {returnCalcs}}$ indicates whether to return the optional results marked with the $^{*}$ superscript (default is to omit) $f(x_{1})$ and $f(x_{2})$ are exponents of the F80 and P80 equation terms, respectively $S_{\rm {c}}$ is a parameter that accounts for the decrease in ore strength in coarse crushing applications

Figure 3. Example showing the selection of the input parameters (shaded blue, red, purple and green frames), and Results (green frame) array in Excel. The parameter ${\mathit {returnCalcs}}={\text{False}}$ in this example.

HPGR specific energy

The HPGR specific energy calculation may be invoked from the Excel formula bar with the following function call:

=mdSMC_Wh(K3 as Double, Mih as Double, x1 as Double, x2 as Double, Optional returnCalcs 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 an example image showing the selection of the same cells and arrays in the Excel interface:

${\begin{aligned}K3&={\big [}K_{3}{\big ]}\\{\mathit {Mih}}&={\big [}M_{\rm {ih}}{\text{ (kWh/t)}}{\big ]}\\x1&={\big [}x_{1}{\text{ (}}\mu {\text{m)}}{\big ]}\\x2&={\big [}x_{2}{\text{ (}}\mu {\text{m)}}{\big ]}\\{\mathit {returnCalcs}}&={\big [}({\text{True / False)}}{\big ]}\end{aligned}}$ ${\mathit {mdSMC\_Wh}}={\begin{bmatrix}W_{\rm {h}}{\text{ (kWh/t)}}\\f(x_{1})^{*}\\f(x_{2})^{*}\\{S_{\rm {h}}}^{*}\\\end{bmatrix}}$ where: $K_{3}=1$ for closed circuits with a screen, and $K_{3}=1.19$ in open circuits $M_{\rm {ih}}$ is the ore work index in HPGR circuits (kWh/t) $x_{1}$ is the 80% passing size of the feed (μm) $x_{2}$ is the 80% passing size of the product (μm) ${\mathit {returnCalcs}}$ indicates whether to return the optional results marked with the $^{*}$ superscript (default is to omit) $f(x_{1})$ and $f(x_{2})$ are exponents of the F80 and P80 equation terms, respectively $S_{\rm {h}}$ is a parameter that accounts for the decrease in ore strength in coarse crushing applications

Figure 4. Example showing the selection of the input parameters (shaded blue, red and purple frames), and Results (green frame) array in Excel. The parameter ${\mathit {returnCalcs}}={\text{False}}$ in this example.

Specific energy correction for size distribution

The specific energy correction for size distribution calculation may be invoked from the Excel formula bar with the following function call:

=mdSMC_Ws(Mia as Double, x1 as Double, x2 as Double, Optional returnCalcs 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 an example image showing the selection of the same cells and arrays in the Excel interface:

${\begin{aligned}{\mathit {Mia}}&={\big [}M_{\rm {ia}}{\text{ (kWh/t)}}{\big ]}\\x1&={\big [}x_{1}{\text{ (}}\mu {\text{m)}}{\big ]}\\x2&={\big [}x_{2}{\text{ (}}\mu {\text{m)}}{\big ]}\\{\mathit {returnCalcs}}&={\big [}({\text{True / False)}}{\big ]}\end{aligned}}$ ${\mathit {mdSMC\_Ws}}={\begin{bmatrix}W_{\rm {s}}{\text{ (kWh/t)}}\\f(x_{1})^{*}\\f(x_{2})^{*}\\\end{bmatrix}}$ where: $M_{\rm {ia}}$ is the coarse ore (> 750 μm) work index in tumbling mill circuits (kWh/t) $x_{1}$ is the 80% passing size of the feed (μm) $x_{2}$ is the 80% passing size of the product (μm) ${\mathit {returnCalcs}}$ indicates whether to return the optional results marked with the $^{*}$ superscript (default is to omit) $f(x_{1})$ and $f(x_{2})$ are exponents of the F80 and P80 equation terms, respectively

Figure 5. Example showing the selection of the input parameters (shaded blue, red and purple frames), and Results (green frame) array in Excel. The parameter ${\mathit {returnCalcs}}={\text{False}}$ in this example.

General comminution specific energy

The general comminution specific energy calculation may be invoked from the Excel formula bar with the following function call:

=mdSMC_W(K as Double, Mi as Double, x1 as Double, x2 as Double, Optional returnCalcs 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 an example image showing the selection of the same cells and arrays in the Excel interface:

${\begin{aligned}K&={\big [}K{\big ]}\\{\mathit {Mi}}&={\big [}M_{\rm {i}}{\text{ (kWh/t)}}{\big ]}\\x1&={\big [}x_{1}{\text{ (}}\mu {\text{m)}}{\big ]}\\x2&={\big [}x_{2}{\text{ (}}\mu {\text{m)}}{\big ]}\\{\mathit {returnCalcs}}&={\big [}({\text{True / False)}}{\big ]}\end{aligned}}$ ${\mathit {mdSMC\_Wa}}={\begin{bmatrix}W_{\rm {a}}{\text{ (kWh/t)}}\\f(x_{1})^{*}\\f(x_{2})^{*}\\\end{bmatrix}}$ where: $K$ is an adjustment factor $M_{\rm {i}}$ is the ore work index (kWh/t) $x_{1}$ is the 80% passing size of the feed (μm) $x_{2}$ is the 80% passing size of the product (μm) ${\mathit {returnCalcs}}$ indicates whether to return the optional results marked with the $^{*}$ superscript (default is to omit) $f(x_{1})$ and $f(x_{2})$ are exponents of the F80 and P80 equation terms, respectively

Figure 6. Example showing the selection of the input parameters (shaded blue, red, purple, green and pink frames), and Results (green frame) array in Excel. The parameter ${\mathit {returnCalcs}}={\text{True}}$ in this example.

Calculating P80

The P80 calculation may be invoked from the Excel formula bar with the following function call:

=mdSMC_P80(K as Double, Mi as Double, F80 as Double, W as Double, Optional returnCalcs 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 an example image showing the selection of the same cells and arrays in the Excel interface:

${\begin{aligned}K&={\big [}K{\big ]}\\{\mathit {Mi}}&={\big [}M_{\rm {i}}{\text{ (kWh/t)}}{\big ]}\\F80&={\big [}F_{80}{\text{ (}}\mu {\text{m)}}{\big ]}\\W&={\big [}W{\text{ (kWh/t)}}{\big ]}\\{\mathit {returnCalcs}}&={\big [}({\text{True / False)}}{\big ]}\end{aligned}}$ ${\mathit {mdSMC\_P80}}={\begin{bmatrix}P_{80}{\text{ (}}\mu {\text{m)}}\\f(x_{1})^{*}\\f(x_{2})^{*}\\\end{bmatrix}}$ where: $K$ is an adjustment factor $M_{\rm {i}}$ is the ore work index (kWh/t) $F_{80}$ is the 80% passing size of the feed (μm) $W$ is the specific energy available to the comminution stage (kWh/t) ${\mathit {returnCalcs}}$ indicates whether to return the optional results marked with the $^{*}$ superscript (default is to omit) $P_{80}$ is the 80% passing size of the product (μm)

Figure 7. Example showing the selection of the input parameters (shaded blue, red, purple and green frames), and Results (green frame) array in Excel. The parameter ${\mathit {returnCalcs}}={\text{False}}$ in this example.

Estimating breakage parameters

The breakage parameter estimation may be invoked from the Excel formula bar with the following function call:

=mdSMC_AbToSMC(A as Double, b as Double, SG as Double, BWI as Double, Optional conversionMethod as Integer = 0)

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:

${\begin{aligned}A&={\big [}A{\text{ (}}\%{\text{)}}{\big ]}\\b&={\big [}b{\big ]}\\{\mathit {SG}}&={\big [}\rho _{\rm {S}}{\text{ (t/m}}^{3}{\text{)}}{\big ]}\\{\mathit {BWI}}&={\big [}{\mathit {Wi}}_{\rm {BM}}{\text{ (kWh/t}}{\big ]}\\{\mathit {conversionMethod}}&={\big [}{\text{0 (Doll) or 1 (Chitalov)}}{\big ]}\end{aligned}}$ ${\mathit {mdSMC\_AbToSMC}}={\begin{bmatrix}{\rm {DW}}_{\rm {i}}{\text{ (kWh/m}}^{3}{\text{)}}\\M_{\rm {ia}}{\text{ (kWh/t)}}\\M_{\rm {ib}}{\text{ (kWh/t)}}\\M_{\rm {ic}}{\text{ (kWh/t)}}\\M_{\rm {ih}}{\text{ (kWh/t)}}\\t_{\rm {a}}{\text{ (-)}}\\{\rm {SCSE}}{\text{ (kWh/t)}}\\\end{bmatrix}}$ where: $A$ is a JK Drop Weight Test impact breakage parameter (%) $b$ is a JK Drop Weight Test impact ore breakage parameter $\rho _{\rm {S}}$ is the density of solids (t/m3) ${\mathit {Wi}}_{\rm {BM}}$ is the Bond Ball Work Index of the ore (kWh/t) ${\mathit {conversionMethod}}$ indicates the correlation set to utilise (0 = Doll 2016, 1 = Chitalov 2020) ${\rm {DW}}_{\rm {i}}$ is the Drop Weight Index (kWh/m3) $M_{\rm {ia}}$ is the coarse ore (> 750 μm) work index in tumbling mill circuits (kWh/t) $M_{\rm {ib}}$ is the fine ore (< 750 μm) work index in tumbling mill circuits (kWh/t) $M_{\rm {ic}}$ is the ore work index in crusher circuits (kWh/t) $M_{\rm {ih}}$ is the ore work index in HPGR circuits (kWh/t) $t_{\rm {a}}$ is the JK Drop Weight Test abrasion breakage parameter ${\rm {SCSE}}$ is the SAG Circuit Specific Energy parameter (kWh/t)

Figure 8. Example showing the selection of the input parameters (shaded blue, red, purple and green frames), and Results (green frame) array in Excel. The parameter ${\mathit {returnCalcs}}={\text{False}}$ in this example.

See also

• Comminution Circuit Specific Energy (Bond)