MATHEMATIC
SIMULATION OF CUTTING UNLOADING FROM THE BUNKER
Serhii Yermakov
State Agrarian and Engineering University in Podilya, Ukraine
E-mail: ermkov@gmail.com
Taras Hutsol
State Agrarian and Engineering University in Podilya, Ukraine
E-mail: tarashutsol@gmail.com
Oleh Ovcharuk
State Agrarian and Engineering University in Podilya, Ukraine
E-mail: ovcharuk.oleh@gmail.com
Iryna Kolosiuk
State Agrarian and Engineering University in
Podilya, Ukraine
E-mail: ikolosyk@ukr.net
Submission: 14/12/2018
Revision: 08/02/2018
Accept: 27/02/2019
ABSTRACT
The peculiarities of cutting movement at unloading
them from the hopper are described. The analysis of the scientific researches
on bulk materials movement and bridging is given. To develop the mathematical
model of cutting unloading the layer should be described as a pseudoliquid,
that consists of discrete components (cuttings) and gaseous medium (air). The
Navier-Stokes equation can be applied to the process of cutting unloading and
velocity field. The equation of pseudoliquid motion is a nonlinear integral and
differential equation. The initial and boundary conditions for speed of cutting
movement are identified. As a result
of research has been theoretically obtained a formula, that evaluates the rate
of planting material unloading, the adequacy of which has already been
partially tested in experimental experiments carried out by the authors on the
way to creating an automatic planting machine.
Keywords: cutting; movement; mathematical model; pseudoliquid; modeling of motion; velocity field
1. INTRODUCTION
Many
machines that are used in production lines, deal with such workflows, as
loading, unloading, handling, selection, transportation and etc. of different
bulk materials. Efficiency and high-quality operation of such machines depends,
primarily, on the speed of the unloading, the parameters and operation modes of
handling devices (ADAMCHUK; BARANOV; BARANOVSKYI, 2001; VOITYUK; YACUN;
DOVZHYK, 2008). The issue is complicated by the need to ensure a
uniform and continuous unloading of material one size (length) of which is
significantly higher than the other two sizes.
An
example of such a material is cutting.
This issue is an up-to-date one due to increased popularity of fuel on the
basis of bioenergy crops and, consequently, the need in fast and efficient
machines to create so called energy plantations. One of the most common crops
is energy osier. The osier is planted by vegetative way with the help of cutting 20-25 cm long and 8-20 mm in diameter (graph
1).
Graph 1: Energy willow planting material
Planting
is carried out by machines in which the planting material is fed manually
(graph 2). This significantly limits the possibility of improving the
efficiency of the units.
Graph 2: Traditional energy willow planting with manual feed
cuttings. Source: Willowpedia; Probstdorfer
To
create a machine for planting such material the cutting should be transported fast and accurate. The study is an
attempt to find strategies for justifying the cutting movement during their unloading from the accumulative
capacity (YERMAKOV, 2017; YERMAKOV; BORYS, 2015; YERMAKOV; TULEJ;
SHEVCHUK, 2018).
In
general, the tasks associated with the loading and unloading of bulk and lump
material operations, that are aimed at reduction of manual labor, at increase
of performance and transport load factor, are of great importance. A lot of
scientists and researchers focused on the issue of process stability in
unloading material from the storage hopper.
To
achieving positive results of their scientific research would not have been
possible without the in-depth study on the patterns of granular material
characteristics. The phenomenon of bridging takes place in the technological
process of hoppers and vehicles with bodies of hopper construction. As a
result, the time of complete cleaning of storage hopper and bulk cargo
transportation is increased. This leads to violations of safety requirements in
carrying out the activities and sufficiently large financial losses.
2. METHODS AND MATERIALS
It is
obvious that this issue is connected not only with the raising of the level of
technical and technological reliability of hopper devices, but also with the
ensuring of the requirements of the occupational safety, health and the
economic factor as well. The theoretical basis of the study is the researches
of Ukrainian and foreign scientists on scientific methods of bulk material
unloading from containers, issues of bridging problems and continuous flow of
material.
To
model the cutting movement in the middle of the hopper methods of hydrodynamics
of multiphase systems is used. According to this approach, the combination of
cutting is considered to be pseudoliquid that consists of two phases: a
discrete phase formed by cuttings and continuous phase formed by gaseous medium
(air). Each of these phases is regarded to be medium that allowed us to
consider uploading as the movement of viscous incompressible pseudoliquid. The
velocity speed of this pseudoliquid must satisfy Navier-Stokes equation.
3. RESULTS AND DISCUSSIONS
3.1.
The
process of granular materials floe in modern science and methodology
Numerous
researches on bridging process made it possible to identify the dependencies
that explain the essence of the process. The degree of influence of a huge
number of various interrelated factors on bridging is difficult to assess and
predict theoretically. It is about the geometry and hopper outlet and physical
and mechanical properties of materials, and loading conditions, storage and
release. It is due to the complexity of ensuring uniform continuous motion that
excludes bridging, there is no universal feeding device, effectively working
with any loose material.
At
the same time the variety of material requires further contribution to the
studies of motion of material. It is also difficult to overstate the scientific
and practical value of the research on the mechanism of granular materials
movement under its own weight because physical and mechanical properties of
materials and their expiry date patterns have a decisive impact on the design
of hoppers, as well as exhaust devices and mechanisms that enable expiration.
The
researches of native scientists are devoted to physical and mechanical
properties of bulk materials and patterns of their movement. Among the most
significant the works of Bagnold (1954), Schulz (1967)
Sokolovskiy (1954), Klein (1956), Sokolovskyi (1954),
Zenkov (1964) should be considered. The works of Borshhev (2005), Dolgunin
(2005), Gjachev (1992), Jenkins (1979), Protodjakonov
(1981), Savage (1999), Yermakov (2018) and
others deepened the previous researches. Their
contribution to the theory of scientific basics of calculation theory involved
the development of various unloading constructions.
Research
on dynamics of bulk material flow from tanks, antibridging strategies and
development of bridging equipment was done by the following scientists Bogomjagkih
(1985, 2000), Geniev (1958), Gorjushinskyi (2003), Gyachev (1968, 1992), Jansen
(1895), Keneman (1960), Kunakov (2000), Pepchuk (1985),
Semenov (1980), Varlamov (2011), Zenkov (1964) et
al. The basic characteristics, physical and mechanical properties of bulk
materials that influence the bridging, research on the smooth functioning of
hopper devices and development of equipment for bulk cargoes with a wide range
of physical and mechanical properties are given in their work.
It
should be noted that there is no unified theory in bulk material flow and processes
of bridging in the hopper. For example, Zenkov (1964) note a significant
influence of height of material layer on the velocity of bulk material flow.
Keneman
(1960) determine the absence of such influence. In the process of storing chips
in the hopper, they seal, which later after long storage leads to increasing
adhesion between particles that in its turn decreases their mobility and
promotes the growth of shear forces.
The
experience of the hopper operation showed that Jansen formula underestimated
the values of pressures on the bottom and walls of the hopper. This is due to
the fact that the formula does not take into account the change in the density
of the raw materials during storage in the hopper.
The
theory by Jansen (1895) includes the case where the area of pressure is the
whole bottom of the hopper, and lateral friction occurs between dissimilar
bodies: loose body and the material of the hopper walls.
According
to Jenike (1968) body is the union of homogeneous absolutely solid flat discs
that are stacked in the correct rows.
The
theory by Schulz (1967) considers cargo to be a bending beam. He pointed out
that if homogeneous layers have little traction, each layer above goafing will
bend itself under the influence of its own weight. Just like the loaded beam
bents.
Protodjakonov
(1981) supposed that load on vault was considered vertical and distributed
across its surface, and therefore a set of curve was parabolic. The issue of
sets development in hoppers when the particle size of loose body is not too
large in comparison with the size of the possible sets, is highlighted in the
works by Vasilev (), but his research did not take into account the factors
that influence the process of bridging.
The
issue of adhesion between particles has been analyzed by Zenkov (1964). On the
basis of stress state, he came to the conclusion that under certain conditions
over the vent formed a set of matching paths from the greatest stress. The
disadvantage of this theory is that it does not take into account the effect of
overlying layers of loose material on the elementary volume selected over vent
as well as the influence of the particle size.
Zenkov
research on pressure distribution in bulk material was later developed in the
monograph of Gjachev (1968). The scientist developed the differential equations
of motion of elementary and ending amounts of granular material in hoppers of
various shapes. The solution of these equations allowed to set the pressure
distribution on the bottom and walls of the hoppers both in motion and calm
state of loose materials. The theoretical dependency coincides with the
experimental data for the dry crops, mineral fertilizers.
Adequacy
by Gjachev model is explained, in particular, by the fact that parameters
characterizing the bulk material: external and internal friction angles, angle
stacking grains, grain size, etc. are included. This conclusion made it
possible to study the effect of each option separately for granular material on
pressure distribution laws. The proposed model allows scientists to explore
some extreme cases.
According
to Gjachev (1968) model, the infinitesimal amount of loose body grains
turns into "liquid", which provides Coulomb friction between grains.
In case of zero external and internal friction angles, this "fluid"
turns into a so-called "perfect" liquid. It should be noted that the
properties of such fluid differ from the usual properties of ideal fluid.
The
study by Bogomjakih, Kunakov and Voronoj (2000) determines all phenomena that
take place in the hopper in terms of static and dynamic state of granular
materials on the basis of equivalent dynamic bridging. Analysis of the
theoretical material shows that the bridging model of a loose body corresponds
better to practical research.
Varlamov
(2011) notes that the theory by Bogomjagkih (2000) describes the processes in
bulk body when it is unloaded from the hoppers quite accurately, it reveals the
theoretical basis for the relationship between the parameters of a loose body
and accumulating parameters of the hopper, it allows providing the calculations
of hopper systems. According to this theory dynamic and static bridging has a
significant impact on the bulk movement of the body that is limited by the
walls of the hopper. Dynamic bridging slows down the process of unloading and
static stops it.
Analyzing
the research papers by Alferov (1966), Zenkov (1966), Gjachev (1968),
Bogomjagkih (1974), Semenov (1980) it may be noted that the main parameters
that influence the flow of bulk materials from hoppers are: the size of hopper
hole, physical and mechanical properties of bulk materials. The values of
certain coefficients and constants are determined according to flowing material
from the hopper and relations of constructive plan in terms of machines with
big time performance are identified in a number of analytical dependencies.
Therefore, the use of these dependencies in the hopper with the feeder in
riddled installation with significantly reduced performance is not appropriate.
3.2.
The
preconditions for the implementation and the choice of technique process for
unloading of cuttings from the hopper
There
are disadvantages existing of granular materials. First of all, it is
significant damage of the assembled product and the formation of the statically
stable pickups. Second of all, increased speed of unload material with
different physical and mechanical properties, etc. is the unsolved part of the
research on machine productivity. One of the main obstacles in solving of this
problem is the complexity of bridging.
Having
analyzed the existing theories that reflect the essence of bridging, we may
conclude that the majority of them describes the behavior of the material
itself, but offers no solutions to the problems. In addition, since material
properties vary greatly, it is obvious that there are no common approaches to
solving the problems of bridging.
In
research papers of Gorjushinskyi (2003) the bridging is defined as the process
of bridging development in containers in terms of loose cargo release.
The
scientists distinguish two main areas to ensure smooth unloading of bulk cargo
from tanks:
1. To prevent bridging that can be achieved by
proper choice of parameters of capacity;
2. To
destroy the formed arches with the help of different bridging devices.
Both
directions are up-to-date ones. But the most progressive is the first one. It
is better to prevent bridging, than to deal with it. The simulation of particle
motion of loose material being unloaded, as well as the choice of means for the
destruction of the bridging formed in the tank depend on the physical and
mechanical properties of the material and the tank capacity.
This
paper deals with the way the cuttings act in the process of unloading according
to the influence of gravitational forces.
Let
us assume that the layer of cuttings consists of circular cylinders with length,
density and
radius. On the basis
of research, we can conclude that during the pouring out of such bodies the
issues connected with the position of cuttings in the longitudinal and
transverse planes arise. However, the selection of parameters of an unloading
device guaranties even and continuous motion of the material. To explore the
process and build a mathematical model of movement of material it is important
to determine the physical essence of a set of cuttings and determine the
corresponding theory for describing her movements.
The
process of unloading from the tank could be designed on the basis of methods
for hydrodynamics of multiphase systems (SOUS, 1971; NYHMATULIN, 1978).
According to this approach the total number cuttings that is influenced by the
gravitational field and seismic fluctuations is modeled by a two-phase
structure. This structure consists of discrete components (a set of cuttings)
and continuous components (gaseous medium). These components in terms of
mechanism of multiphase systems are treated as solid mediums.
These
mediums are characterized by two effective coefficients of the viscosity caused
by the interaction between cuttings and the interaction of cuttings with
gaseous medium (air). We suppose that the bulk concentration of (discrete
component) is bigger than similar values for continuous components. In this
case the viscosity that is associated with the interaction with gas medium can
be neglected. Therefore, the movement of discrete components can be designed as
a movement of viscous incompressible pseudoliquid.
The field rate of this pseudoliquid
must satisfy Navier-Stokes equation.
The
solution of this equation in the linear approximation is a mathematical process
model of pouring out the cuttings from the hopper.
3.3.
The
model of the tank and two-phase pseudoliquid that models a set of cuttings
As a
model for cutting unloading hopper we consider two half-planes that are located
at and
angles according to
the horizontal plane. The width of the unloaded window can be identified as
. Let us introduce the Cartesian coordinate system
with the axis
that coincides with
the line passing through the center of the unloaded window.
Graph 3: Design scheme
of the hopper for cuttings
Graph 3 shows the cross section of the hopper at the
platitude of . Let us suppose that the movement of cuttings in the hopper
does not depend on the coordinate
, i.e., we will focus on two-dimensional model of cutting unloading process. This
limitation implies the existence of parallel plane walls that limit the
movement of cutting along an
axis. The combination of cutting
in a hopper is considered to be influenced by the gravitational field and the
vibrations. It is supposed that on one of the walls of the hopper is affected
by harmonics with
amplitude and
frequency. The direction of these vibrations is perpendicular
to the wall of the hopper.
To
simulate the movements of a set of cuttings
the methods of hydrodynamic multiphase systems are used (SOUS,
1971; NYHMATULIN, 1978). According to this approach, we believe that the
combination of cuttings is
regarded as a pseudoliquid that
consists of two phases: a discrete phase formed by cuttings and continuous phase ‑ gaseous medium (air). Each of
these phases is regarded as solid medium. Let us introduce the density of these
solid mediums. is considered
to be the density averaged across all cuttings,
is the density
of the gas (air) medium. Then the discrete phase density is determined by the
formula (SOUS, 1971)
(1)
and the density of continuous phase
(2)
Here ‑ is the volumetric
concentration of cuttings in
the hopper. In addition to
and
parameters and pseudoliquid is characterized by
efficient dynamic coefficients vibrio viscosity:
is the coefficient of
discrete phase viscosity that is characterized by the interaction between cuttings;
‑ is dynamic viscosity
coefficient of gaseous medium. Let's assume that the volumetric concentration
of cuttings is significantly more similar to values of continuous phase. In
this case, the effective coefficient of vibrio viscosity that is determined by
the interaction between cuttings and gaseous medium can be neglected. Thus, the
movement of a set of cuttings will simulate a two-phase movement pseudoliquid.
3.4.
Equation
of motion of a two-phase pseudoliquid
Let
us introduce the velocity of field: is the field speed of
discrete phase and
is the velocity field
of the continuous phase. According to the works of Sous
(1971), the equations of motion of a two-phase pseudoliquid can be represented in the following form
(3)
(4)
(5)
(6)
are unit vectors of
Cartesian coordinate system,
– is discrete and
continuous pressure phases,
is acceleration of
free fall,
is the power that influences on mass unit of pseudoliquid and, according to the works
of Sous (1971) has the form of:
(7)
where
(8)
is the radius of the
circle that matches with an average area of cross-sections in cuttings,
is the coefficient of
kinematic viscosity of continuous phase. Equations (3) to (6) are nonlinear
integro-differential equations. The solution of these equations is possible
only by numerical computer methods (KRYLOV; BOBKOV; MONASTYRSKYI, 1976). It
follows from (3) in the process of hydrodynamic simulation of the cuttings
movement, we should use such important characteristic as an effective
coefficient of viscosity
.
3.5.
Initial
and boundary conditions for speed movement of pseudoliquid
Mathematically
correct statement about the flow of cuttings in equations (3) to (6) must be
supplemented by the relevant initial and boundary conditions. Since the field
velocity of a two-phase pseudoliquid ,
and depend on a
temporary variable, the initial conditions should be formulated. As a starting
point of time, with no loss of generality, we choose time
. Let's assume that the initial time
and speed
, pressure
and
are equal to zero
(9)
(10)
Since
the movement of pseudoliquid
occurs in a limited volume, we should put the appropriate boundary conditions
on boundary surfaces of the hopper and free surface of cuttings that border
with the atmosphere. In order to formulate these conditions, let us introduce
stress tensors of discrete and continuous phases of pseudoliquid. According to the works of Sous (1971) and
Nigmatulin (1978), we have
(11)
(12)
Here and
dynamic factors of viscosity in discrete and continuous
phases of pseudoliquid.
are unit vectors of
Cartesian coordinate system (see graph 3). According to the established model
of the hopper (see description above) equations describing its borders, have a
look
(13)
(14)
At
these borders, taking into account the slippage of cuttings that form the
discrete phase, equal tangents of discrete phase must be performed according to
resistance force of discrete phase per
unit area of hopper boarders.
According
to relatively free surface let’s assume the process of cuttings motion remain
flat. Then the equation can be represented as
(15)
where function depends only on the temporary variable and
its value coincide with the distance from the free surface to plane unloaded
window.
At free surface, you must
meet the following boundary conditions. We will neglect the influence of the
atmosphere on the dynamics of pseudoliquid.
Then on the free surface of the discrete voltage phases must apply to zero
(16)
(17)
Here and
are normal unit vector
and tangent unit vector to the free surface.
In
addition to the conditions (16), (17) so called kinematic boundary conditions
must be carried out. On free surface the velocity field of discrete phases must
satisfy the condition
(18)
where
Let’s
transform conditions (16), (17) and (18). Vectors and
have the appearance of
(19)
Substituting
(19), (16) and (17), we have
(20)
(21)
From
(20), (21) with the help of (11), (12) we get
(22)
(23)
These
ratios must be executed when . Next (18) have.
(24)
where the dot indicates operation of differentiation
Thus,
on free surface of
velocity field the discrete phases of pseudoliquid must satisfy conditions (22) to (24).
According
to velocity of continuous
phase let’s assume that the same conditions are met. Let us now consider
boundary conditions on
and
walls of a hopper. Let's believe that
boundary is exposed to
harmonic oscillation with
amplitude and
frequency. The direction of these fluctuations coincides with
the direction of the unit vector to
(24).
Then
the normal components of velocity of the
discrete phase on
border must satisfy
condition.
(25)
where unit normal vector is
condition
(25) can be represented as
(26)
This
condition must be met when
Parallel
with the condition (26) velocity must meet the
condition of equality of tangential stresses to force resistance of
surface movement of discrete phase, transported to middle
square
(27)
(28)
Here
(29)
(30)
where are vectors that are
perpendicular and tangent to
surface.
Let’s
substitute the expression (11) for stress tensor in (27), (28) and will perform
the necessary calculations.
Will
have:
(31)
(32)
and
resistance forces
according to the works of Tyshchenko (2011), and if we assume that
and
borders and are
smooth, coincide with the pendant dry sliding
(33)
where is the coefficient of friction,
is the normal
pressure.
Normal
pressure of discrete phases on and
surfaces can be
calculated by using formulas
(34)
Let’s
plug (34) in (31) and (32) and finally get the boundary conditions at and
borders in the hopper
(35)
(36)
Here is the kinetic
coefficient vibro viscosity. Conditions (35) and (36) must be carried out when
и
where is the maximum
thickness of the cutting layer at
zero time.
Consequently,
the velocity of two-phase of pseudoliquid that stimulates a set of cuttings
must satisfy the initial conditions (9), (10) and (2) to (4), (26), (35), (36)
boundary conditions.
4. CONCLUSION AND RECOMMENDATIONS
The review of information sources
showed that the research on the regularities of the mechanics of granular body
flow deals with a number of theoretical and experimental research. The analyzed
scientific issues are quite complex and are covered with the help of classical
mechanics, theory of plasticity, soil mechanics and rheology on the basis of
mathematical modeling. Moreover, the majority of papers explore of small-sized
materials, and process of unloading of such materials as cuttings hasn’t been
studied yet.
To simulate the movements of a set
of cuttings the hopper model was presented as two half-planes located at and
angles to the horizontal plane. The width of the unloaded
window will be marked as
. For mathematical processing methods of hydrodynamics of
multiphase systems in which the totality of cuttings is regarded as a
pseudoliquid and consists of two phases: discrete phase formed by cuttings and
continuous phase-gaseous medium (air) were used.
The patterns of movements that are
nonlinear integra differentiate equations, which can be processed on the basis
of software, are presented for this pseudoliquid. Two-phase velocity field of
pseudoliquid model the combination of cuttings, must meet the primary and
regional conditions, which are presented in the form of equations. As the
result, the preconditions for the establishment of mathematical model for
unloading the layer of cuttings from the hopper are created in the study.
Mathematical simulation of energy
willow cutting unloading from the
hopper allows theoretically substantiating the possibility of increasing
the planting process efficiency, up to its full automation. As a result of research
has been theoretically obtained a formula, that evaluates the rate of planting
material outflow, the adequacy of which has already been partially tested in
experimental experiments carried out by the authors on the way to creating an
automatic planting machine.
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