Grinding power : step by step calculation

Comminution (particle size reduction) is one of the industrial activities that is requiring the highest amount of energy worldwide, particularly through mining activities. Estimating the power required by a grinding mill is then particularly interesting for industrial looking to invest in a new grinding process in order to define the CAPEX and OPEX costs.

The calculations below mainly apply to ore mills.

1. Work-index

What is the work-index of a mill ?

The work-index is a parameter used in ore milling processes. It is defined as the energy needed to reduce ore from infinite size to the state where 80% of the milled ore will pass through a 100 mesh screen.

What is a mesh, how does it convert to micron ?

It is very common in milling application to refer to the size as per the "mesh" measurement. The correspondance mesh to microns is given below :

US Mesh*	Microns	Inches	Millimeters
35	500	0.0197	0.5
40	400	0.0165	0.4
45	354	0.0138	0.354
50	297	0.0117	0.297
60	250	0.0098	0.25
70	210	0.0083	0.21
80	177	0.007	0.177
100	149	0.0059	0.149
120	125	0.0049	0.125
140	105	0.0041	0.105
170	88	0.0035	0.088
200	74	0.0029	0.074
230	63	0.0025	0.063
270	53	0.0021	0.053
325	44	0.0017	0.044
400	37	0.0015	0.037
450	32	0.0013	0.032
500	25	0.0010	0.025
635	20	0.0008	0.020

2. Calculation of the work-index

2.1 STEP 1 : Gather the data

In order to calculate the work index for a particular material in a mill, it is necessary to carry out a grindability test :

• Get a sample of the material to grind
• Select the mesh P₁ at which the material is to be grinded during the test : for example 200 mesh (= 75 microns)
• Measure the 80% passing size F (microns) for the feed
• Run the test and measure the Bond grindability G in g/rev of the mill
• Measure the 80% passing size P (microns) for the product

2.2 STEP 2 : Calculate the work index

Wet milling [Chopey]

W_i = 44.5/[P₁^0.23*G^0.82*(10/P^0.5-10/F^0.5)]

Dry milling [Chopey]

W_i = 44.5/[P₁^0.23*G^0.82*(10/P^0.5-10/F^0.5)] *4/3

With :

W_i = work index (kWh/t)
P1 = size of the mesh of grind of the grindability test (microns)
F = 80% passing size of the feed to the grindability test (microns)
P = 80% passing size of the result of the grindability test (microns)

3. Estimation of the power consumption of a grinding mill

3.1 STEP 3 : Estimate the power consumption of a grinding mill for a particular application

The Bond 3rd theory of comminution is estimating the power required to mill a particular ore thanks to the following formula [Chopey] :

W = 10*W_i*(P^-0.5-F^-0.5)

With :

W = Power consumption of the mill (kWh/t)
W_i = work index (kWh/t)
F = 80% passing size of the feed to the mill (microns)
P = Required 80% passing size of the milled product (microns)

F and P are different than on those of the grindability test

4. Example : Step by step estimation of the power required for a grinding mill

A mining company is investing in a new line of production of an ore. The material must be sold as fine powder, with 80% of the material passing through a 80 microns mesh. The feed material is however much bigger since 80% of the feed 8passes through a 2000 microns mesh.

What is the estimated power required to run the mill ? The milling is done Dry.

4.1 STEP 1 : Gather data

In order to gather the necessary data to size the mill, the company is running 1st a grindability test. The results are the following :

• The material is grinded at a mesh P₁ of 200 mesh (= 75 microns)
• Measure the 80% passing size F (microns) for the feed = 2000 microns
• Run the test and measure the Bond grindability G in g/rev of the mill = 2.5 g/rev
• Measure the 80% passing size P (microns) for the product = 42 microns

4.2 STEP 2 : Calculate the Work-Index

The work index is calculated thanks to the following formula :

W_i = 44.5/[P₁^0.23*G^0.82*(10/P^0.5-10/F^0.5)] *4/3= 44.5/[75^0.23*2.5^0.82*(10/42^0.5-10/2000^0.5)]*4/3 = 7.86 kWh/t

4.3 STEP 3 : Estimate the power required for the particular application

W = 10*W_i*(P^-0.5-F^-0.5) = 10*7.86*(80^-0.5-2000^-0.5) = 7.03 kWh/t

5. Free Excel calculation tool for estimating grinding mill power consumption

The required power consumption of a grinding mill based on the calculations shown above can be estimated thanks to this free Excel calculator : Calculation Tool - Grinding Mill Power Calculation

Warning : this calculator is provided to illustrate the concepts mentioned in this webpage, it is not intended for detail design. Please consult a reputable designer for all detail design you may need.

Source

[Chopey] Handbook of Chemical Engineering Calculations, 13.5, Chopey, 2004, MacGraw Hill