Mixing of Granular Media-Calculation of
Stress at the Blade Front
Štefan Gužela*, Marián Peciar and František Dzianik
Institute of Process Engineering, Faculty of Mechanical Engineering, Slovak University of Technology in Bratislava, Slovak Republic
Submission: June 26, 2021; Published: July 20, 2021
*Corresponding Author: Štefan Gužela, Institute of Process Engineering, Faculty of Mechanical Engineering, Slovak University of Technology in Bratislava, Námestie slobody 17,812 31 Bratislava, Slovak Republic
How to cite this article: Štefan G, Marián P, František D. Mixing of Granular Media-Calculation of Stress at the Blade Front. Civil Eng Res J. 2021; 12(1):
555826. DOI 10.19080/CERJ.2021.12.555826
Particulate materials are processed in various machines and equipment where it needs to ensure a motion of machines parts in a form of the flat blade through a layer of the grainy material. Conveyers, feeders, driers and especially mixers of granular materials can be mentioned as examples. Markedly, mixing elements of various sizes and shapes are moved through a bed of the granular material that resists this movement. Knowledge of interaction between the granular material and element moving through a bed of the granular media is necessary for the regular design and calculation of such machines. The paper deals with exact determination of force effects of the mixed material on moving blade submerged in a grain material. The solution is based on a yield equilibrium tensions of the particulate material described by theory of Mohr’s circle. A box in form of rectangular prism will be used in experiments which enables uniform rectilinear motion of the blending elements of a simple design.
Keywords: Mohr’s circle; Rectilinear motion of blade; Homogenizer; Particulate material; Yield equilibrium; Failure zone
Mixing is wide-spread process in the various branches of industry. For example, in the chemical, food, and pharmacy industry, in the agriculture, or energetics. The special case of blending is mixing of the systems consisting of rigid particles. Mixing elements, various of sizes and shapes, are transferred through a bed of the grainy material. Knowledge of interaction between the grainy material and moving object is necessary for a construction of such machines. However, a theoretical knowledge of this problem is only little analyzed. Target of the paper is to give a theoretical and experimental basis for study of the grainy material motion in such cases. Designed experimental equipment enables to measure a force vector if grainy material acts against displacement of the blade. It makes possible to observe formation of failure zone of the particulate material in front of the moving blade. Steady state motion of the grainy material over a wide flat blade was studied in work . Blade of the rectangular shape moves with constant velocity in horizontal direction through layer of the particulate material. Target of research is to explain the action of force of the granular medium on the flat blade moving through the granular medium layer. Normal of the blade can be deviationed from direction of its motion.
Some authors attempted to describe the energetical aspect of mixing process according to relations well-known in hydrodynamics. The authors looked for analogy between flow off a solid body by liquid and the movement of the mixing element through the layer of particulate material [2-4] or mixing of the particulate materials and mixing of liquids [5,6]. However, these efforts were mostly marked by big errors due to inadequate simplification of mechanics laws of the particulate materials. It was confirmed in these works that it is not possible to continue this way. For estimation of energetical aspect of the particulate material mixing it is necessary to start with tension state of the particulate material. It is possible to find several research works in literature dealing with these problems. Their authors base the solution of the energetical aspect of the mixing process on the tension state of the particulate materials [7-15]. They used a simple geometric models in their experiments which made it possible to solve action of force between the grainy material and the mixing element in two-dimensional problem. Derived
functions, containing an influence of internal and external angles
of friction of the particulate material, are complicated also for
design of the mixing elements of simply form. Their determination
assumes that the material is in the yield equilibrium. This state
may be described by the theory of Mohr’s circle. Based on the
works mentioned above, it is possible to draw several conclusions.
Region of the failure zone forms in front of the moving blade
through a layer of the granular media (Figure 1). A course of
the slip lines is possible to determine by the theory of Mohr’s
circle in this region. Then, a force acting on the blade can be
estimated by form of the slip lines. This force is very important
for calculation of the blender input power. The force acting on
the blade depends on the form and the size of the plastic zone.
The form and the size of the failure zone depends especially on
mechanical-physical properties of the grainy material (namely on
and external (ϕ) angles of friction of the particulate
material), geometric dimensions of the blade and depth of the
blade immersion (z) in a batch. In these works, it was investigated
that bulk density(ρs)
, internal (ϕi)
and external (ϕ) angles of
the friction of the particulate material belong among the most
important parameters characterizing grainy material. Theoretical
studies and experimental results proved that the internal (ϕi)
external (ϕ) angles of the friction are at movement of the granular
media markedly assert, and hence it is not possible to ignore
them, which is declared by some authors.
Theoretical determination of the stress acting on blade was
performed for a following case. The blade of rectangular shape is
situated so that its shorter edge(B) is vertical and perpendicular
to a horizontal plane (Figure 1). The blade width(L) is considerably
bigger than its height (B) . Problem is to define a relationship for
calculation of the normal stress component(sn) acting on the blade
front (Figure 2) (Figure 2b). The stress distribution between
the mixing element and cohesion particulate material is simply
plotted by Mohr’s circle (Figure 2a). In this case the circle is
Mohr’s circle of stresses, the straight-line OS is an axis of normal
stresses σ and the straight line, perpendicular to the line OS and
passing through point O, is the axis of tangential stresses τ.
These relationships were derived for the cohesion particulate
material and for the adhesive forces between the particulate
material and the surface of the blade which is not possible to
neglect. Let’s to consider a case where 0 τ = 0and 0 0 s τ = (Figure 3). It
is cohesionless particulate material where between the particulate
material and the surface of the blade are only the friction bonds.
Then the relationships (6) and (13) will be simplified into the form
of the relations (14) and (15). The relationship for calculation of
the angle β will be:
The relationship (13) was derived for the rectilinear motion of
the blade. Shorter edge of the blade was perpendicular to direction
of its motion and a plastic zone was formed in front of the blade (Figure 5). The shape and size of plastic zone significantly affects
the force acting on moving blade through layer of particulate
material. Shear zone in front of the blade (Figure 6) rises with
angles of internal friction (ϕi)
and external friction(ϕ) . The
influence of the angle of internal friction (ϕi)
is possible to follow
in (Figures. 6 (a, b, c)) at constant value of the angle of external
friction(ϕ) . The angle of internal friction(ϕi
influences the angle
at which the slip lines are intersecting. A region of the shear zone
rises with the angle of internal friction (ϕi)
. The angle of external
friction(ϕ) influences the Rankins zone shape and also additional
regions. The region of shear zone rises again with the angle of
external friction(ϕ) (Figures. 6 (d, e, f)).
The influence of angle of the blade inclination(α) towards the
horizontal plane (Figure 7) on forming of the shear zone is also
significant. Inclination of the blade normal towards a vector of its
motion is changed with variation of the angle of blade inclination.
As well as the angle of external friction(ϕ) could be the significant
parameter affecting mutual action of the blade and the granular
material. All regions of the shear zone are changing (Figure 8)
with gradual rising value of the blade inclination(α) . It follows
that the force needed for move of the blade through a layer of the
granular material must also change. We will try to explain this fact.
The value of the normal stress component sn acting on front
of the blade can be calculated by relationship (13). On the basis
of experimental measurements, it was obtained that magnitude
of the horizontal component of the force(H ) decreases with
increasing value of angle of the blade inclination(α) at the
constant height of the blade in projection(B) (Figure 9). This
means that when the blade is inclined under the angleα < 90o
, then the value of the horizontal force (H(α>90))
is bigger than the
value of the force(H(α=90)) . On the contrary, when the blade
is inclined under the angleα > 90o, then the value of the horizontal
force (H ( α > 90 ))
is smaller than the value of the force(H) for α = 90o .
Generally, it is possible to suppose that total force (F) is diverted
from the normal of the blade by external angle of the friction(ϕ)
. For various angles of the blade inclination(α) it is necessary to
consider an orientation of the total force(F ) acting on the blade.
Orientation of the resistance force(F), as is plotted in (Figure
10), was determined by experimental measurements . The
horizontal force(H ) , or (H( α <90 )), or (H ( α > 90 )), necessary to the movement
of the blade through the granular material, was estimated by the
resistance force(F) (Figure 10).
Value of the stress sn is calculated by the relationship (13).
The horizontal component of the force (H) acting on the blade
plate with length L and height B for the case of α = 90o is
calculated by the value of the stress sn :
From the balances of the forces (Figure 11) in direction of the
blade normal, the following is valid:
The introduction of article deals with available results of the
investigation, from those some conclusions were deduced. It was
confirmed that for calculation of the energetical aspect of the
grainy material mixing it is needed to start from tension state of
the particulate material. The bulk density, internal and external
angles of the material friction, geometric dimensions of the
blade, and depth of the blade immersion belong among the most
important parameters characterizing the grainy matters which
have an influence on the magnitude of the horizontal component
of the force acting on the blade. On the basis of these information’s,
in this paper, the relationship including the influence of these
above-mentioned parameters, was designed. It is written in the
part of analytical calculation of the stress acting on the blade. Next
part of the article considers the influence of the blade inclination
angle on the form and largeness of the shear zone in front of
the moving blade, which significantly affects the magnitude of
the force acting on the moving blade through the layer of the
particular matters. The given relationship specifies theoretical
value of the horizontal component of the force. If this relationship
would give a total value of the force horizontal component, so
then a ratio of the measured and theoretically calculated values
would be close to one. If this ratio differs significantly from the
value 1, then a computing model will be dependent on another, so
far the unknown parameters. These ideal results will be the basis
for formation of the dimensionless arguments by dimensional
analysis of the homogenization process. Design of the criterion
dependency for the calculation of the dimensionless horizontal
component of the force acting on the blending element of the
various form and for different types of the grainy material will be
main goal of the next research. Knowledge of real force interaction
between homogenized material and mixing element facilitates
qualified design of the homogenizer drive.
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