Discrete Element Method - Mathematical Basis
Discrete element simulation of granular flows involves following the
trajectories and spins of all the particles and objects in the system and
predicting their interactions with other particles and their environment.
We can simulate the wide ranges of particle distributions found in
industrial applications (from several meters diameter down to 500 microns)
and their interactions with the complex shaped objects that form their
environment.
The motion of boundary objects such as mixing vessels or excavator
buckets can either be specified as controls or predicted. The method
essentially involves detecting particle collisions, predicting the
collisional forces and then moving the particles and objects according to
these forces. The heart of the method is the spring and dashpot contact
force model (shown in the figure below). The normal force has a spring to
provide the repulsive force and a dashpot to provide the inelasticity in
the collision.
The tangential force has an incrementing spring that models tangential
elastic deformation of the contacting surfaces and a dashpot to model
plastic deformation. The tangential force is limited by the Coulomb
friction. This is the limiting friction that can be withstood by the
contact before sliding of one particle over the other commences. The
friction is the prime generator of particle spin, which is an important
but often unconsidered part of these flows.
The collisional forces can be schematically represented as follows:
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where
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Dx is the particle overlap
k is the spring constant
vn and vt are the normal and tangential
velocities
C is the dashpot damping coefficient
m is the friction coefficient
Fn = - k Dx + C vn
Ft = min{ m Fn ,  k
vt dt + C vt }
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Once simulated the granular flows can be analysed with a wide array of
visualisation and data processing tools. Many manufacturing processes and
pieces of equipment involving granular materials are closed and dust
filled. Visualisation of these flows can provide enormous insights into
the nature of the flows and can be crucial in understanding the
requirements for process improvement and optimisation. They allow you to
look below the surface into the flows and understand them.
Depending on the concerns relating to the nature of each application,
we have many tools with which to analyse these flows and with which to
predict useful quantitative information. For processes where mixing or
segregation is important, we can analyse the degree and rates of mixing of
components or the rate of de-mixing by flow or vibration. For milling
operations, where mill efficiency, downtime and maintenance are concerns,
we can predict torques, power consumption, liner wear rates and
distributions. Sensitivity of milling performance to particle size and
variation of wear patterns with liner geometry can be evaluated. Flow
rates from hoppers and bins can be predicted and time variations of
particle size distributions within regions of flowing material can be
determined. Many other forms of output can be obtained depending on the
needs of the customer.
Such simulation capability can be used for optimisation and improvement
of existing processes and equipment and the development of new ones. This
approach can be an important part of any project to improve process
efficiency, lower costs, increase output and when used at the design stage
offers an opportunity to reduce capital expenditure.
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