Comminution Circuit Specific Energy (Bond)

Description

This article describes the Bond method for estimating the specific energy of rod and ball mill comminution circuits.

Model theory

The Bond method estimates the specific energy required for comminution based on ore grindability and the extent of size reduction. It is derived from Bond's third theory of comminution, which states that energy consumption is proportional to the reduction in the square root of particle size. The Bond method applies to both rod mills and ball mills.

The specific energy consumption, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W} (kWh/t), is given by the Bond equation:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W = 10 \cdot {\rm WI} \left( \frac{1}{\sqrt{P_{80}}} - \frac{1}{\sqrt{F_{80}}} \right)}

where:

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\rm WI}} is the Bond Rod Work Index or Bond Ball Work Index, depending on the application (kWh/t)
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_{80}} is the 80% passing size of the feed (μm)
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_{80}} is the 80% passing size of the product (μm)

To correct for variations from the standard test conditions (wet, closed-circuit grinding in a 2.44 m diameter mill), the model includes eight efficiency factors:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_{\rm actual} = W \cdot {\rm EF}1 \cdot {\rm EF}2 \cdot {\rm EF}3 \cdot {\rm EF}4 \cdot {\rm EF}5 \cdot {\rm EF}6 \cdot {\rm EF}7 \cdot {\rm EF}8}

The correction factors are described below. A conversion factor of 1 tonne = 1.102 short tonnes is applied to work index terms.

The dry grinding factor (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\rm EF1}} ) increases the specific energy required for dry grinding compared to wet grinding:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\rm EF1} = \begin{cases} 1.3, & \text{for dry grinding}\\ 1.0, & \text{for wet grinding}\\ \end{cases} }

The open circuit ball mill factor (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\rm EF2}} ) increases the specific energy required for open circuit grinding compared to closed circuit grinding. The factor accounts for the reduced efficiency when no classifier or closing screen is used to control the maximum product size.

The control size is the hypothetical closing screen size that would be used in an equivalent closed circuit. If the mill is in open circuit, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\rm EF2}} is interpolated from the table below based on the percentage of the product passing the control size. For closed-circuit grinding (including all rod mill configurations), Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\rm EF2} = 1.0} .

Table 1. Open-circuit inefficiency factors based on % passing control size (after Rowland, 2006).

% Passing Control Size	Inefficiency Multiplier, EF2
50%	1.035
60%	1.050
70%	1.100
80%	1.200
90%	1.400
92%	1.460
95%	1.570
98%	1.700

The diameter efficiency factor (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\rm EF3}} ) scales specific energy for mill diameter, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D} (m):

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\rm EF3} = \begin{cases} \left( \dfrac{2.44}{D} \right)^{0.2}, & D < 3.81 \text{ m}\\ 0.914, & D \geq 3.81 \text{ m}\\ \end{cases} }

The oversized feed factor (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\rm EF4}} ) applies when the mill is to grind a feed that is coarser than an optimum 80% passing feed size, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_{\rm o}} (μm):

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\rm EF4} = \begin{cases} \dfrac{R_{\rm r} + \left ( \dfrac{\rm WI}{1.102} - 7 \right ) \cdot \dfrac{F_{80} - F_{\rm o}}{F_{\rm o}}}{R_{\rm r}}, & F_{80} > F_{\rm o}\\ 1.0, & F_{80} \leq F_{\rm o} \end{cases} }

where:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_{\rm o} = \begin{cases} 16,000 \sqrt \left ( \dfrac{13}{\rm WI} \cdot 1.102 \right ), & \text{ for rod milling}\\ 4,000 \sqrt \left ( \dfrac{13}{\rm WI} \cdot 1.102 \right ), & \text{ for ball milling}\\ \end{cases} }

and:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {R_{\rm r}} = \dfrac{F_{80}}{P_{80}}}

The fineness of grind factor (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\rm EF5}} ) applies when the $P_{80}$ is finer than 75 μm:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\rm EF5} = \begin{cases} \dfrac{P_{80} + 10.3}{1.145 \, P_{80}}, & P_{80} < 75 \text{ }\mu\text{m}\\ 1.0, & P_{80} \geq 75 \text{ }\mu\text{m}\\ \end{cases} }

The high or low ratio of reduction rod milling factor (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\rm EF6}} ) accounts for reduced efficiency when the reduction ratio in rod milling is too high or too low:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{cases} {\rm EF6} = 1 + \dfrac{(R_{\rm r} - R_{\rm ro})^2}{150}, & \text{if rod milling and } \left |R_{\rm r} - R_{\rm ro} \right | \geq 2 \text{ or } {\rm WI} \geq 7 \cdot 1.102\;\text{ (i.e., 7.714 kWh/t)}\\ 1.0, & \text{otherwise} \end{cases} }

where:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_{\rm ro} = 8 + \dfrac{5L}{D}}

and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L} is rod length (m).

The low ratio of reduction ball milling factor (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\rm EF7}} ) applies to ball milling when the ratio of reduction (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_{\rm r}} ) is low:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\rm EF7} = \begin{cases} \dfrac{2 \, (R_{\rm r} - 1.35) + 0.26}{2 \, (R_{\rm r} - 1.35)}, & \text{if ball milling and } R_{\rm r} < 6\\ 1.0, & \text{otherwise}\\ \end{cases} }

The rod milling factor (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\rm EF8}} ) accounts for rod milling inefficiencies:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\rm EF8} = \begin{cases} 1.4, & \text{rod mill only, open-circuit crushing}\\ 1.2, & \text{rod mill only, closed-circuit crushing}\\ 1.2, & \text{rod–ball mill circuit, open-circuit crushing}\\ 1.0, & \text{all other configurations} \end{cases} }

Excel

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

=mdMillPower_BondWI(Parameters as Range)

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

The Parameters array and model results are defined below in matrix notation, along with an example image showing the selection of the same arrays in the Excel interface:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Parameters= \begin{bmatrix} {\rm WI}\text{ (kWh/t)}\\ F_{80}\text{ (}\rm{\mu}\text{m)}\\ P_{80}\text{ (}\rm{\mu}\text{m)}\\ \text{Dry grinding (True/False)}\\ \text{Open circuit (True/False)}\\ \text{Control fraction passing (frac)}\\ D\text{ (m)}\\ L\text{ (m)}\\ \text{Rod mill (True/False)}\\ \text{Rod mill only circuit (True/False)}\\ \text{Closed circuit crushing (True/False)}\\ \end{bmatrix},\;\;\;\;\;\; \mathit{mdMillPower\_BondWI} = \begin{bmatrix} W\text{ (kWh/t)}\\ {\rm EF1}\\ {\rm EF2}\\ {\rm EF3}\\ {\rm EF4}\\ {\rm EF5}\\ {\rm EF6}\\ {\rm EF7}\\ {\rm EF8}\\ \end{bmatrix}\;\;\;\;\;\;\;\;\;\;\;\; } where: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\rm WI}\text{ (kWh/t)}} is the Bond Work Index of the ore being processed (kWh/t) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_{80}\text{ (}\rm{\mu}\text{m)}} is the 80% fraction passing size of the feed (µm) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_{80}\text{ (}\rm{\mu}\text{m)}} is the 80% fraction passing size of the product (µm) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{Dry grinding}} indicates if the mill is dry grinding Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{Open circuit}} indicates if the mill is operating in open circuit Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{Control fraction passing}} is the product size control reference fraction passing value (frac) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D\text{ (m)}} is the mill diameter (m) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L\text{ (m)}} is the rod length (m) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{Rod mill}} indicates if the comminution device is a rod mill Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{Rod mill only circuit}} indicates if the circuit includes a rod mill only Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{Closed circuit crushing}} indicates if the feed is prepared with closed circuit crushing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W} is the operating work index (kWh/t) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\rm EF}1-{\rm EF}8} are efficiency factors applied to the calculation of operating work index.

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

See also

• Comminution Circuit Specific Energy (Morrell)