A preliminary simulation model for fine grinding in high speed hammer mills

doi:10.1016/j.powtec.2004.04.017

Powder Technology

Volumes 143–144, 25 June 2004, Pages 240-252

https://doi.org/10.1016/j.powtec.2004.04.017 Get rights and content

Abstract

High speed hammer mills break particles suspended in gas by high velocity impact against a hammer surface. Assuming that the action is comparable to a stream stagnating against a surface, it is shown that the net mill power can be calculated and it is predicted that the power varies approximately proportional to the hammer speed cubed. Literature data on the distribution of particle strength of tested particle sizes were generalized to enable the distribution to be measured for any size. Considering the mill section containing the hammers to be fully mixed, the algorithms are developed to predict the product size distribution under known hammer velocity and exit classification conditions. Particles that are too strong to break under the impact forces, as indexed by specific impact energy, will leave the mill unbroken as “hard” (strong) particles. However, applying the concepts of damage mechanics, the algorithms are modified to allow for weakening of hard material by repeated impacts. Examples show that the predicted change in product size distribution is very significant for a damage accumulation constant appropriate for limestone, and it is concluded that a correct prediction cannot be obtained without allowing for damage.

Introduction

Recent studies [1], [2], [3], [4], [5] of the breakage of particles under impact have made it possible to develop an analysis of the kinetics of breakage in hammer mills, thus leading to a simulation model applicable to this widely used type of mill. Such a simulation model allows the prediction of the product size distribution of a mill for any size distribution of feed and any mill operating condition, providing that the required fracture properties of the material being milled are determined by impact breakage studies or estimated by back-calculation from test data from the mill.

The model has seven main components:

(1)
The net mill power is derived from the number, area and geometry of the hammers and the tip velocity, based on the impact forces of the hammers against the air–solid suspension.
(2)
The mass striking the hammers per unit time and the part of the net mill power due to impact with solid give the specific energy (J/kg) of impact of the solid particles.
(3)
The mass fraction of impacting particles that break at a given specific impact energy is assumed to have the same form of variation with particle sieve size and specific impact energy as found from laboratory studies of breakage (3) of various sizes at various specific energies, representing the distribution of strengths to one impact of the particles with respect to size and specific impact energy.
(4)
The repetitive breakage algorithm developed by Austin [6] is used to develop the size distribution of the fragments produced from any feed particle size during a single impact, assuming a true primary breakage distribution function. Those fragments formed by fracture in a single impact are assumed to have the same strength distribution as the original feed particles, whereas those particles that do not fracture are the “hard” (strong) fractions of the feed.
(5)
The weak product fragments and the strong unbroken particles mix into the air–solid suspension and are re-impacted, and the size–mass–rate balance can be computed assuming the mill is a fully mixed system at steady state.
(6)
Allowance is made for the weakening of the hard particles due to repeated impacts that leave internal damage, using the methods of damage mechanics (4).
(7)
Since larger particles can be size classified (preferentially retained) by the centrifugal action of the rotating hammers, it is necessary to incorporate an exit classification equation. The final product size distribution leaving the mill (or fully mixed section of a mill) thus depends on the feed size distribution, the impact energy, the distribution of strengths, the true primary breakage distribution, weakening of the particles and the exit classification.

Section snippets

Mill power

The system studied is taken to be a smooth cylindrical case with sets of hammers rotating in the case around the central axis, which can be vertical or horizontal. Fine grinding in this type of mill is produced principally by impact of particles against the hammers, as distinct from hammer crushers where a substantial amount of breakage can be produced by nipping of lumps between the hammers and breaker bars or exit grates located in the case. In a hammer mill designed for fine grinding [7], it

Fracture by impact

When a particle of a given size is impacted at a relative velocity v, the total specific kinetic energy (J/kg) at the instant of impact is E=v²/2. As the particle is compressed and slows down (with respect to the hammer), kinetic energy is converted to strain energy. If the particle is too strong to break at this impact energy, all the kinetic energy will be converted to strain energy and the fully compressed particle will decompress and reach a relative velocity acting away (a rebound) from

Fully mixed hammer milling

Consider a hammer mill (or a section of a hammer mill) that is fully mixed by the rotating action, and is receiving a known feed size distribution f_i at a known mass flow rate F (kg/s). It is expected that the rotating action will cause size classification so that larger particles will be preferentially held in the mill as compared to finer particles (exit size classification). The system is illustrated in Fig. 2. Let $ws_{i}$ be the mass fraction of the mill contents that can be rebroken by

Damage accumulation

It has been shown [4] that the effective Young's modulus of a mineral particle decreases as the particle is compressed by impaction. This is ascribed to irreversible internal damage (microcracking) that is insufficient to cause disintegrative fracture. It means that an impact at a given specific impact energy will produce a lower maximum impact stress but a higher fractional deformation than if the material behaved strictly elastically. A damage accumulation factor D is defined by $D=1− effective$

Illustrative example

The parameters used in the example (see Table 1) are those for the limestone tested by Datta and Rajamani [3], as interpreted by Cho and Austin [9] and Austin [6], plus a value of 8 for the damage accumulation coefficient, that is appropriate for a medium hard material [4]. The feed size distribution as fraction by mass less than size is given in Table 2, with a top size of 19.2 mm (3/4 in.).

The mass of solid striking the hammer faces per second is $M ̇ =ρ_{a} sA(1−Γ) V ̄$ where V̄ is the arithmetic mean

Discussion and conclusions

It must be understood that the theoretical model presented here is preliminary because a number of simplifications have been made to avoid computational complexity. The function of this report is to show the physical concepts and computational factors involved, and further work will be necessary to remove those simplifications so that the effects of the assumptions can be estimated. In addition, the parameters chosen for the example are in some cases somewhat arbitrary since the required data

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A preliminary simulation model for fine grinding in high speed hammer mills

Abstract

Introduction

Section snippets

Mill power

Fracture by impact

Fully mixed hammer milling

Damage accumulation

Illustrative example

Discussion and conclusions

Single particle fracture under impact loading

Int. J. Miner. Process.

Modeling of particle fracture by repeated impacts using continuum damage mechanics

Powder Technol.

A direct approach of modeling batch grinding in ball mills using population balance principles and impact energy distribution

Int. J. Miner. Process.

Int. J. Miner. Process.

A treatment of impact breakage

Powder Technol.