Collisions between particles in multiphase flows: focus
on contact mechanics and heat conduction
–
International Journal of Heat and Mass Transfer 70 (2014), pp. 674–687,
doi:10.1016/j.ijheatmasstransfer.2013.11.052
Pawel Kosinski*, Prachi Middha, Boris V. Balakin, Alex C. Hoffmann
*Corresponding author: [email protected]
The University of Bergen
Department of Physics and Technology, Bergen, Norway
Abstract
The topic of heat transfer between colliding particles or between particles
impinging on solid surfaces is, in spite of its practical significance, still not
widely studied in literature: efforts have, till now, led to only a handful of
models. At the same time this problem is crucial not only for many types
of multiphase flows but also for other systems involving colliding bodies.
Therefore the first objective of the research was to study heat conduction
between colliding particles for the case when the local deformation in the
contact region can be treated as consisting of two regions: one deformating
elastically and the other plastically. This requires investigation of collision
dynamics and therefore this paper also contains a short review of the existing
models for modelling particle-particle and particle-wall collisions. The second
important result of the work presented here is a new relation for the coefficient
of restitution that was developed by fitting a dimensionless relation involving
the salient physical parameters to numerical experiments based on collision
models.
Keywords:
fluid-particle flows, direct numerical simulation, collisions, heat transfer,
plastic deformation, contact mechanics
Preprint submitted to Int. J. Heat Mass Transfer
January 1, 2014
Nomenclature
a
b1 and b2
ci
e
E∗
Ei
f~p
F
ki
m∗
mi
p
pg
Q
Qe
Qp
qw
R∗
R̄∗
Ri
t
tcoll
Ti
~ug
vo
v1
Y
contact radius
constant parameters
heat capacity
coefficient of restitution
effective Young’s modulus
particle Young’s modulus
interphase force per unit volume
force acting on the particles during contact
heat conductivity
effective mass
particle mass
contact pressure
fluid pressure
total heat transfer during collision
total heat transfer during collision (only elastic deformation)
total heat transfer during collision (only fully-plastic deformation)
heat flux
effective radius
modified effective radius due to plastic deformation
particle radius
time
collision duration
particle temperature
fluid velocity vector
initial velocity along the normal direction
final velocity along the normal direction
material yield stress
Greek symbols
δ
indentation
∆T initial temperature difference
νi
Poisson ratio
νg
fluid kinematic viscosity
ρi
particle density
ρp
fluid density
2
Subscripts
e
f
i
m
max
p
r
J
S
y
elastic
the end of collision
particle number (i = 1, 2)
average
the end of compression
transition between the elastic-plastic and fully plastic ranges
recovery period
model by Johnson
model by Stronge
transition between the elastic and elastic-plastic ranges
1. Introduction
In many applications involving multiphase flow the effect of particleparticle and particle-wall collisions is important. This is especially true for
relatively dense flows. Studying collisions is of interest both for practical
aspects and for scientific curiosity. Particle-particle interactions have been
widely studied in literature and various models have been proposed ranging
from very fundamental ones, such as soft-sphere models, to more practical,
such as the relatively simple models used in Eulerian approaches, and finally
to purely empirical expressions.
Applications where heat transfer in fluid-solid flows is crucial include
those that involve combustion. In such applications the heat transfer may be
significantly affected by the presence of particles, for example in dense fluidized beds and even, some researchers claim, in liquid-nanoparticles systems
(nanofluids).
When two solid particles of different temperature approach each other
they begin to exchange heat due to convection, conduction and radiation.
During any contact heat conduction through the particle material plays the
dominating role and the total heat transfer depends on the contact duration
as well as their contact area. This mechanism for heat transfer in the system is more significant for systems involving particles made of materials of
relatively high thermal conductivity and for systems with frequent particleparticle collisions. Heat conduction during contact is the focus of this paper,
and to model it a description of impact dynamics is necessary.
Another thermal effect in systems involving colliding particles is the heat
generated due to dissipative processes, such as friction and plastic deforma3
tion, during collisions. This is especially significant for contact-dominated
flows, such as granular flows, and has already been researched for some
time (see e.g. Popov, 2010). Nevertheless, for multiphase flows that are not
contact-dominated, such heat generation can often be neglected.
1.1. Heat transfer between solid bodies during contact
The objective of this research is to focus on heat conduction during contact
between particles or between a particle and a solid surface, such as a wall.
A literature review relevant for this research is given in the paper by Sun
and Chen (1988) in which a model for heat conduction between colliding
particles was also derived. Their model assumed the collision to be purely
elastic and described by Hertz theory. More details are given in the following
paragraphs.
Sun and Chen also performed experiments to investigate heat transfer
effects associated with collisions (see Sun and Chen, 1995). This was done
by studying a stream of particles impinging on a solid surface, which made
it possible to assess the influence of a variety of parameters, such as particle
velocity and amount of particles, on the heat transfer.
The model by Sun and Chen (1988) has later been used by several other
researchers for various applications. For example, Li and Mason (2002) simulate a particulate flow with heat transfer effects using the discrete element
method. They do not, however, use the Hertzian contact force and thus
slightly modify the model by Sun and Chen in this respect. A similar strategy was used by the same group of researchers in subsequent works (see Li
et al., 2003a,b).
The model by Sun and Chen has also been applied for e.g. bubbling
fluidized beds with combustion (see Zhou et al., 2004). Mansoori et al. (2005)
used the model of Sun and Chen for simulating turbulent flows with particles.
The model has even been adopted for the Eulerian approach by Chang
et al. (2011). Recently it has been revisited by Li et al. (2012) who used
an analytic solution of the mathematical model for heat conduction for any
Fourier number.
Central assumptions in the model of Sun and Chen are that the contact surface between the particles is flat and that the heat transfer can be
considered one-dimensional. Zhou et al. (2008) improved the model by Sun
and Chen by relaxing these assumptions to create a more realistic boundary
between the particles. This was done by carrying out numerical simulations
using the finite element method. New expressions were found that were later
4
used also by Zhou and Yu (2009) and Zhou and Yu (2010). According to
Zhou et al the original model by Sun and Chen yields similar results as using the more complex numerical simulations for low values of the Fourier
number, as one might also suspect. When the Fourier number increases, the
model by Sun and Chen overestimates the amount of transferred heat.
Some attempts have been made to account for other types of collisions,
i.e. not only elastic. Ben-Ammar et al. (1992) suggested an analytical model
for plastic deformation for strain hardened metals. This model was validated
experimentally by investigating particles colliding against a solid surface in
vacuum so that the heat transfer between the bodies in contact was limited
to conduction. This paper forms a basis for the present research as shown
later in this paper. Also in earlier research by the present authors (Kosinski
et al., 2013) a similar strategy was used for studying heat conduction for
impacts with viscoelastic deformation.
Building on the work described above, the present paper describes a headon collision of two non-rotating particles, where it is assumed with Sun and
Chen that the collision process is relatively rapid so that the heat conduction
between bodies is one-dimensional. It is further assumed that the surfaces
are perfectly smooth so that there is no heat resistance at the boundary
between the colliding bodies.
These assumptions limit the applications of the derived models somewhat
but at the same time they make it possible to focus more on analytical or
simple numerical techniques and thus avoid complex computer simulations
during each collision. The results can be relevant for, among others, modelling of flows with many particles and frequent collisions.
1.2. Particle contact during collision
In order to investigate heat transfer during collisions the impact process
has to be precisely described. One of the earliest and well-known is Cundall
and Strack (1979) who considered the particle-particle contact to be modelled
by a spring-dashpot system. This strategy was later followed by numerous
researchers, e.g. Kuwabara and Kono (1987), Tsuji et al. (1992) and Brilliantov et al. (1996) (see also the review paper by Stevens and Hrenya, 2005)
who accounted for more realistic contact forces, or added other mechanisms
such as cohesion between colliding particles, see e.g. Brilliantov et al. (2007).
An important issue that is highly relevant for the present research is
plastic deformation of the colliding bodies. For some materials, such as
steel, permanent, i.e. plastic, deformation in the point of contact takes place
5
even for very low collision velocities. Therefore plastic deformation occurs
frequently in systems involving this type of particles.
In the following, a short summary of the different models relevant to this
work is presented.
Johnson (1985) in derives a model for particle-particle collisions where
only fully plastic deformation occurs in the region just around the contact
during impact (the deformation is elastic elsewhere). Johnson derived under
this assumption values for the coefficient of restitution (the ratio between
the rebound and the initial velocity) which he later related to experimental
observations. Thornton (1997) used a similar approach but made the issue
slightly more complex by considering also elastic deformation around the
contact region upon impact.
Stronge (2000) combined models from literature that involve different
regimes: elastic, elastic-plastic and fully plastic deformation just around the
contact region and presented a detailed model for particle-particle impact.
This combined model also forms the basis for the research described in this
paper.
The models mentioned above have later been used by many researchers,
who, in addition to theoretical analysis, have used computer simulations
(e.g. finite element analysis of the deforming bodies) and experimental research (Zhang and Vu-Quoc, 2002; Wu et al., 2003a,b, 2005; Weir and McGavin, 2008; Wu et al., 2009).
1.3. The objectives of the paper
The structure of the paper is as follows: we start with analysis of two
mathematical models (called later as models A and B) of particle-particle
interactions with a focus on plastic deformation. Even though many aspects
have already been addressed in literature, we also elucidate issues not investigated yet. Basing on the model analysis, we formulate a third model (called
as model C) that is a combination of models A and B. Later heat conduction
between particles is added into these models and the results are discussed.
Finally models for coefficient of restitution and the amount of heat transferred are built that base on dimensional analysis. Note that this type of
technique has not been widely used in literature yet in spite of its simplicity
and robustness.
The models are illustrated by running a computer simulation of a solidfluid flow.
6
2. Heat conduction between colliding particles: modelling strategy
Collisions between two particles (or a particle with a surface) lead to
strain and stress fields in the bodies that can be modelled using a variety
of theoretical, numerical and experimental techniques, see e.g. the classic
textbook by Johnson (1985) or the more recent one by Stronge (2000). The
dynamics of the extent of deformation or indentation, which can be expressed
as the extent of “overlap” of the non-deformed spheres, δ, can be solved from
the fundamental equation of motion:
X
X
m1 δ̈1 = m2 δ̈2 =
F
⇒
m∗ δ̈ =
F
(1)
where m∗ is the effective mass: m1 m2 /(m1 + m2 ) with m1 and m2 being the
masses of the particles and δ1 and δ2 summing to δ are the deformations of
the individual particles. In case one of the bodies is a plane
P wall on which
a particle, with mass m, collides, then m∗ reduces to m.
F is sum of all
forces that act on the particles in the normal direction as they deform.
Please note that only deformation along the normal to the plane of collisions is considered and not tangential forces or deformation.
The forces that act on the deforming particles may be of different origin,
this will be discussed in the following section. Generally speaking, solving
Eq. (1) results in the temporal history of the indentation δ and also the
contact radius a.
The model of Sun and Chen (1988) will be used for describing heat conduction between colliding bodies. It is assumed that the collision occurs
along the normal axis to the plane of collision. This axis is denoted as z
and the origin of this axis is in the point where the bodies meet. The initial
temperature difference between the colliding bodies is denoted by ∆T .
The model of Sun and Chen (1988) assumes, as mentioned, the heat
transfer to be one-dimensional, the contact surface to be plane and also that
the bodies are semi-infinite. The heat flux can then be calculated as (see Sun
and Chen, 1988):
qw =
(πt)0.5
∆T
,
(ρ1 c1 k1 )−0.5 + (ρ2 c2 k2 )−0.5
(2)
where index i = 1 or 2 indicates the colliding bodies, ρ, c and k are density,
heat capacity and heat conductivity of the particles, respectively, and t is
the contact time.
7
Equation (2) can be combined with the aforementioned collision dynamics
(represented by Eq. (1) resulting in a relation for the contact radius as a
function of time, a(t)), making it possible to find the total heat transferred
between the particles during the collision:
Z amax Z tcoll −2t
Q=
2πaqw dtda,
(3)
0
0
where tcoll is the total collision duration and amax is the maximum contact
radius.
Equation (3) can be used generally for many types collisions providing
that the history of the contact radius, a(t), is available.
The following section gives an analysis of two selected models describing
impact mechanics between two particles or between a particle and a solid
surface and thus allowing a(t) to be found. Even though descriptions of
these models are available in literature, the main steps are given here since
they form an input for modelling heat conduction between particles in the
subsequent sections.
3. Contact mechanics: synopsis
3.1. Compression
3.1.1. Full analysis as given in Stronge (2000)
In the analysis described in this section, the deformation in the vicinity
of the contact region, if large enough, goes through three stages: (i) elastic,
(ii) elastic-plastic and (iii) fully plastic. These three stages are now discussed
in turn.
Upon contact, when the indentation is still relatively small, the deformation is elastic. The equation of motion, Eq. (1), is:
p
m∗ δ̈ = −4/3 R∗ E∗ δ 3/2 ,
(4)
where the term on the right-hand-side is the elastic force that decelerates
the particles and δ is the indentation. This force is described using the wellknown Hertz theory (Hertz (1881), see also e.g. Johnson (1985)) where E∗
is the effective Young’s modulus given by [(1 − ν12 )E1−1 + (1 − ν22 )E2−1 ]−1 .
The material properties of the particles are thus described by their Young’s
moduli and Poisson ratios, E1 , ν1 and E2 , ν2 , respectively. The collision
process is illustrated in Fig. 1.
8
z
T1 , k1, c1, m1
1
δ
ΔT =|T1- T2|
R2
2
2
2
2
Figure 1: Collision between two particles
In addition it is possible to derive the relation between the contact radius,
a, between the colliding bodies and the indentation δ from Hertz theory:
δ=
a2
,
R∗
(5)
where R∗ is the effective radius given by R1 R2 /(R1 + R2 ), with R1 and R2
being the radii of the colliding bodies. As before this can also be used for a
particle of radius R colliding with a surface: then R∗ becomes equal to R.
According to, for example, Johnson (1985) the maximum value of normal
principal stresses in the colliding particles occurs slightly below the surface.
This issue becomes of importance when the deformation during impact is
large enough to initiate yield. As soon as that happens (i.e. the contact
force is large enough) the deformation becomes elastic-plastic: the plastic
deformation occurs under the surface while the deformation in the rest of
the colliding body is still in the elastic range.
According to empirical and numerical observation, supported by theory,
the transition to elastic-plastic deformation occurs when the average contact
pressure (the contact force divided by the contact area), denoted in this paper
as pm , exceeds b1 Y , where Y is the yield stress of the material. The parameter
b1 is widely assumed in literature to be 1.1 (see e.g. Johnson, 1985). In other
words, the elastic range terminates when pm becomes py = b1 Y = Fy /(πa2y ),
where the subscript y refers to the point where the transition occurs.
Equation (5) relates the contact radius and indentation at the boundary
between the elastic and the elastic-plastic ranges: a2y = δy R∗ . Using the
9
relations above:
δy =
3π
4
2 b1 Y
E∗
2
R∗ .
(6)
However, in the elastic-plastic range Hertz theory is not valid so different expressions are necessary. In this research, the model given by Stronge
(2000), which is based on on previous results collected by Johnson (1985), is
used.
The first relation in this model is an elastic-plastic equivalent to Eq. (5).
Stronge (2000) suggests:
1 a2
(7)
δ = δy ( 2 + 1).
2 ay
Equation (7) ensures continuity of the contact radius and indentation when
crossing the limit between the elastic and the elastic-plastic ranges.
In the elastic-plastic range the average pressure acting in the contact
surface between the particles is proposed to vary logarithmically as suggested
by Stronge (2000) based on a theoretical analysis by Johnson (1985). This
can then be used to calculate the force that acts on the colliding particles
in the elastic-plastic range. The equation of motion for this range becomes
then:
δ
1
δ
m∗ δ̈ = −Fy 2 − 1 1 +
ln 2 − 1 .
(8)
δy
3.3
δy
This equation is thus valid for indentations larger than δy , while Eq. (4)
is valid for 0 < δ < δy .
Further increase in the indentation leads to propagation of the plastic
yield until the plastic deformation reaches the surface. According to experimental observations this occurs when the average pressure, pm , reaches
values between 2.7Y and 3.0Y (in the following written in the general form
pm = b2 Y ). From this point the deformation close to the contact area becomes what is known as fully plastic.
All the parameters corresponding to the transition point between the
elastic-plastic and fully plastic deformation ranges are in this work denoted
with index p, i.e. ap , δp and Fp , in the same manner as the index y was used
above for the transition between the elastic and elastic-plastic ranges. These
parameters can be found using the relations above. Thus:
10
3
(b2 − b1 )
ap = ay exp
2
and:
δy
δp =
2
"
ap
ay
2
#
+1 .
(9)
(10)
In the fully plastic range, the pressure distribution on the particle surface
does not change significantly, i.e. it remains b2 Y . This fact was exploited
by Johnson (1985) who suggested a relatively simple model in which only
fully plastic deformation in the contact region was assumed to take place,
i.e. the elastic and elastic-plastic ranges were neglected (see the following
section for details).
Under this assumption the equation of motion becomes:
2δ
2
m∗ δ̈ = −πb2 Y ay
−1 ,
(11)
δy
which is valid for δ > δp .
The deformation process terminates if all the parameters: indentation,
contact radius, average contact pressure and contact force reach their maximum values. This may occur in the elastic, elastic-plastic or in fully plastic
range, depending on the momenta of the impacting particles and their physical properties. In a numerical algorithm that solves the model above, they
can be found by monitoring the speed δ̇ ≡ dδ/dt, which reaches zero at the
end of the deformation process. Alternatively, one can integrate the above
relations for contact force to find work and equate it to the initial kinetic
energy (see Appendix).
3.1.2. Simplified model of Johnson
The model presented in the previous section is based on the work by Stronge
(2000). In literature one can find other models, of which one of the most
well-known is that of Johnson (1985). This model is significantly simpler
and assumes that the collision process in the contact area is always in the
fully plastic range. The reasoning is that in many practical applications the
collisions are such that the fully plastic deformation dominates. An advantage is that the mathematical expressions, as shown above, are simpler in
the range of plastic deformation.
11
Thus the equation of motion for the model by Johnson is:
m∗ δ̈ = πa2 pm = πa2 b2 Y,
(12)
which is equivalent to this in the fully plastic range, Eq. (11). Also the
relation between the indentation δ and contact radius a is in Johnson’s model:
a2
.
(13)
2R∗
In this simplified model, however, the equation of motion is assumed
to be valid for the whole deformation process; for given particles the error
introduced decreases as the impact velocity increases.
The main advantage of this strategy is that Eq. (12) can be solved analytically. Doing so, using Eq. (13) to relate δ and a, Eq. (12) results in the
following relation for indentation history during compression:
2tmax vo
πt
δ(t) =
,
(14)
sin
π
2tmax
δ=
where vo ≡ δ̇(t = 0) is the initial relative speed between the colliding particles. Duration of the compression, the maximum indentation and the maximum contact radius are then given by:
s
s
2
m∗ vo
2m∗ vo2 R∗
πδmax
; δmax =
; a2max =
.
(15)
tmax =
2vo
2πb2 Y R∗
πb2 Y
3.2. Recovery
After compression, recovery begins at the point where the stored elastic
strain energy starts to be released. It is usually assumed that this process is
fully elastic so that Hertz theory can be used (see e.g. Tabor (1948) where this
was proved experimentally). If any plastic deformation took place during the
compression, a residual deformation is assumed to remain after the collision.
3.2.1. Model by Stronge (2000)
The initial indentation during recovery is δmax , while the final one (at
separation) is denoted as δf,S (where the index S refers to ”Stronge”). δf,S
is zero if the compression has been solely elastic, otherwise some residual
deformation remains at the end of the collision, as mentioned. The change of
indentation in the recovery period is thus: δr,S = δmax − δf,S . The radius of
12
curvature of the bodies is assumed to differ from R∗ due to this permanent
deformation. In the following the ”new” curvature is denoted R̄∗,S . In this
model these quantities are related to each other by assuming what Stronge
calls “geometric similarity”:
δr,S
δy
=
.
R∗
R̄∗,S
(16)
Since the recovery is elastic, Hertz theory can be used using these new parameters to find the contact force as a function of indentation, δ, which varies
from δmax to δf,S during the recovery:
4
1/2
Fr,S = E∗ R̄∗,S (δ − δf,S )3/2 .
3
(17)
The unknown parameters, δf,S and δr,S , can be evaluated in the following
way. Knowing the relation between the radius of the contact area and indentation for elastic deformation: δr,S = a2r,S /R̄∗,S (see Eq. 5) and assuming
that during unloading the contact radius, a, changes from amax to 0 such
that ar,S = amax − 0 (i.e. it is assumed that the plastic deformation, which
has taken place during compression, does not influence the contact radius)
one obtains:
δmax
2
amax = 2
− 1 a2y ,
(18)
δy
where Eq. (7) was used. Knowing that a2y = δy R∗ and making use of
Eq. (16) gives the following relation for δr,S :
δr,S = δy
δmax
−1
2
δy
1/2
.
(19)
The relation between contact radius, a, and the indentation, δ, in the
recovery period is not provided directly by Stronge (2000) and therefore the
following relation is used, which is consistent with elastic deformation:
a2 = (δ − δf,S )R̄∗,S .
(20)
This gives the necessary information about the force acting on the particles during recovery, Fr,S . The equation of motion during recovery is:
m∗ δ̈ = −Fr,S .
13
(21)
Assuming that the maximum indentation δmax is known (see Appendix for
more discussion), the equation of motion for the recovery range, i.e. Eq. (21)
can be solved analytically between the limits δ ∈ (δmax , δ) to obtain the
relative particle velocity as a function of δ. This is done by utilising dδ/dt =
v(δ(t)) together with the chain rule to rewrite the left-hand-side of Eq. (21):
dδ
m∗ dv
= m∗ dv
= m∗ dv
v, which makes (21) separable in v and δ. It is
dt
dδ dt
dδ
thus possible to find the relative velocity, v1,S , at the end of the recovery as
well as the duration of the recovery period by integrating between the limits
δ ∈ (δmax , δf,S ):
!0.5
0.5 5/2
16E∗ R̄∗,S
δr,S
v1,S = −
(22)
15m∗
having inserted that δr,S = δmax − δf,S .
Integrating (21) between the limits δ ∈ (δmax , δ) and substituting back
into v = dδ/dt, the resulting differential equation for δ(t) is separable, and
integration over the entire recovery period, t ∈ (0, tr,S ) (see Deresiewicz,
1968) gives an expression for the recovery time:
tr,S =
1.472δr,S
.
v1,S
(23)
While an analytical solution can thus be found for the duration of the
recovery period, analytical solutions for δ(t), and therefore a(t), are not available in general. The equation of motion for all the periods mentioned above
can, however, be solved numerically.
3.2.2. Model by Johnson
In the model of Johnson Hertz theory is adopted for the recovery period
in a similar way as in the model of Stronge. The main difference between the
model of Stronge and that of Johnson is determination of the new curvature
R̄∗,J and the new indentation δr,J during recovery. The index J refers to
“Johnson” analogous to the index S used in the previous section.
While Stronge used Eq. (16) to calculate the maximum contact radius
amax in the model of Johnson it is calculated in two ways:
1/2
amax = (2R∗ δmax )1/2 or amax = R̄∗,J δr,J
,
(24)
where δmax is the maximum indentation at the end of the plastic deformation,
while δr,J is the indentation in the beginning of the elastic recovery and is
14
equivalent to that used in the previous section. Since the process of recovery
is purely elastic δr,J can be determined from:
4
1/2 3/2
Fmax = πa2max b2 Y = E∗ R̄∗,J δr,J
3
(25)
(compare with Eqs. (12) and (17)), where the maximum contact force during
compression and recovery have been equated. Solving this equation for δr,J
leads, after some manipulation and using both expressions in Eq. (24), to:
1/2
δr,J =
3 πY b2 R∗ 1/2
δmax .
23/2
E∗
(26)
This parameter corresponds to δr,S in the previous section and: δf,J = δmax −
δr,J .
As before solving the equation of motion in the recovery period:
m∗ δ̈ = −Fr,J
(27)
we obtain the relative velocity at the end of the recovery period and the
duration of the recovery period (Johnson, 1985):
!0.5
0.5 5/2
16E∗ R̄∗,J
δr,J
v1,J = −
(28)
15m∗
and
tr,J =
1.472δr,J
.
v1,J
(29)
The relative speed at the end of the collision expressed in Eq. (28) can be
compared to that in Eq. (22) and the duration of the recovery period, tr , can
be compared to Eq. (23). The two models are different and do not lead to
the same results.
The contact radius is:
a2 = (δ − δf,J )R̄∗,J .
(30)
3.3. Other models
In literature there are other models that may be used for investigation of
collision dynamics. One of them is the model by Thornton (1997), where the
initial deformation is modelled using Hertz theory (i.e. in the same manner
15
as above). As the yield starts, it is assumed that the pressure distribution
in the contact area is still Hertzian except for a contact area with radius
ap (ap < a), where the pressure is uniform. Thornton shows that, under
these assumptions, the contact force becomes a linear function of indentation,
which corresponds to the models mentioned above for the fully plastic range.
The recovery process is modelled by equating: R∗ Fe = R̄∗ Fmax , where Fmax
is the maximum contact force during compression and Fe is the elastic force
that would lead to the same maximum indentation. R̄∗ is the equivalent of
R̄∗,S and R̄∗,J defined above. The model by Thornton makes it possible to find
analytical relations, in a similar way to the model by Johnson. Also Yigit
and Christoforou (1994) suggest a relation for the contact force that is a
simple linear function of δ.
In addition to plastic deformation, some researchers considered other aspects such as viscoelasticity and/or cohesion (see e.g. Stevens and Hrenya
(2005) or Brilliantov et al. (2007)), as also mentioned earlier in the paper.
These issues are not discussed here: use of these models for computing heat
transfer can, however, be achieved in the same manner as described below.
4. Heat conduction between colliding bodies: review of existing
models
4.1. Elastic collision: model by Sun and Chen
Sun and Chen (1988) assumed that the collision process was purely elastic.
Eqs. (4) with (3) were solved analytically and this led to the following relation
for the conductive heat transfer during impact:
Qe =
0.87πa2max ∆T t0.5
coll
.
(ρ1 c1 k1 )−0.5 + (ρ2 c2 k2 )−0.5
(31)
The collision duration, tcoll can be found analytically for elastic collisions
by solving Eq. (4) (see, for example, Stronge, 2000; Deresiewicz, 1968; Tsuji
et al., 1992):
1/5
m2∗
tcoll = 2.86
,
(32)
E∗2 R∗ vo
This relation is essentially valid for low velocities where the collision does not
involve plastic deformation. For some materials (e.g. steel) this assumption
limits the practical applications of this relation.
16
4.2. Fully plastic collision in the contact region: model by Ben-Ammar et al.
Ben-Ammar et al. (1992) extended this relation to collision of particles
where material strain-hardens and when the impact is solely plastic, i.e. no
elastic range occurs. This assumption is valid for high impact velocities, a
type of collision that occurs in various engineering applications. A similar
approach can be also used for collisions of perfectly plastic materials.
Assuming that the deformation process involves only fully plastic deformation in the contact region, the corresponding equation of motion is the
same as in Eq. (11), i.e. as in the model suggested by Johnson. This also
refers to other parameters developed in this model, such that the maximum
indentation, contact area and compression duration are found from Eq. (15).
These equations make it possible to find the history of the contact radius,
a(t), even though it is not possible to obtain an exact analytic solution. Nevertheless, Ben-Ammar et al. (1992) showed that the history of the contact
radius can be expressed using an approximate relation that involves trigonometric functions. This relation can then be substituted into Eq. (3) so that
the following relation for the heat transferred during collision is obtained:
Qp =
0.88πa2max ∆T (tmax + tr,J )0.5
.
(ρ1 c1 k1 )−0.5 + (ρ2 c2 k2 )−0.5
(33)
Here the collision duration is that of Johnson, tr,J . The expression by
Stronge, tr,S , can also be used in Eq. (33), and this is shown later in this
paper. As shown below, the two expressions for collision duration converge
to the same value for high-speed impacts where fully plastic deformation is
dominant.
4.3. Other aspects
In our previous work (Kosinski et al., 2013) we investigated heat conduction process for viscoelastic and cohesive particles. It was shown that
viscoelasticity and cohesion influences heat conduction between colliding particles due to their effects on the contact area and contact duration.
As mentioned earlier collisions between particles may also lead to heat
generation due to two factors: friction between the colliding bodies and viscoelasticity as they deform.
The former aspect is not relevant to the present research since the focus is
only on normal collisions where it is assumed that friction does not play any
role. As a matter of fact it is not always the case: an example is collisions of
17
particles made of different materials where friction may influence results also
for head-on collisions. Nevertheless, this aspect is assumed to play a minor
role in the context of the present research and not considered further.
The latter aspect requires modelling of dissipation, which can be done
by equating heat generation to the work done by the dissipation force. In
many applications of particle-particle collisions this issue can be neglected.
Nevertheless, this is not necessarily the case for contact-dominated flows.
5. Summary of the models used to model heat transfer in the
present paper
The strategies discussed in the previous section can be used for modelling
heat conduction between colliding particles. The main requirement is that
the history of contact radius, a(t), is available. This is also the objective of
the present paper.
The following contact models are used, while the transferred heat between
the colliding bodies are found from Eq. (3) and Eq. (2):
1. Model A. The compression is modelled using the following equations
of motion with corresponding relations between δ and a:
• for δ < δy : Eq. (4) with Eq. (5)
• for δy < δ < δp : Eq. (8) with Eq. (7)
• for δ > δp : Eq. (11) with Eq. (7)
while the recovery process is modelled using the relations:
• Eq. (21) between δmax and δf,S , where R̄∗,S and δr,S are found from
Eqs. (19) and (16), respectively. The contact radius is computed
from Eq. (20).
2. Model B. The compression is modelled using Eq. (11) with Eq. (13),
while the recovery processed is modelled using Eq. (27) between δmax
and δf,J , with Eqs. (26) and (24) to find δr,J and R̄∗,J , respectively.
The contact radius is calculated using Eq. (30).
3. Model C. The compression is computed in the same manner as in Model
A, while the recovery process is computed similarly to Model B. Below
the ramifications of modifying model B for recovery to take into account
18
the possibility that the compression may end in the elastic-plastic range
is briefly explored.
In the following few paragraphs we explore the consequences of applying
the Johnson recovery model to the case where the deformation terminates in
the elastic-plastic range. In Eq. (25) we drop the subscript J not to confuse
the recovery model in C with that in model B:
4
Fmax = E∗ R̄∗1/2 δr3/2 .
(34)
3
Recall that the left-hand-side represents the maximum force during compression, while the right-hand-side represents the recovery period, in other words,
it is assumed that during the transition from deformation to recovery, the
value of the maximum contact force does not change.
When the compression terminates in the elastic-plastic range, Fmax , i.e.
the left-hand-side of the equation, can be found from Eq. (8) (while it was
found from the right-hand-side of Eq. (11) if the compression ends in the
fully plastic range as in model B).
Similar to Eq. (24) we state that:
s
1/2
δmax
− 1 or amax = R̄∗ δr
,
(35)
amax = ay 2
δy
where the left-hand equation corresponds to the maximum contact area obtained in the compression period, see Eq. (18), while the right-hand-side
models the same contact area but at the beginning of the recovery.
Equations (34) and (35) lead to:
δr =
3Fmax
4E∗ amax
and
(36)
4E∗ a3max
.
(37)
3Fmax
A closer inspection reveals, however, that this model may lead to nonphysical results for the case when the compression terminates in the elasticplastic range. This can be seen in the following way. Express Fmax using
Eq. (8) and amax using Eq. (7). Then for the ratio δr /δmax one obtains:
−1
1/2 δmax
1
δmax
δmax
δr
= 2
−1
1+
ln 2
−1
,
(38)
δmax
δy
3.3
δy
δy
R̄∗ =
19
where Eq. (36) and the relation a2y = δy R∗ were used. If δmax /δy is less than
∼ 5.0, something that occurs at the beginning of the elastic-plastic range, the
ratio δr /δmax becomes greater than 1.0, i.e. the indentation during recovery
is larger than the obtained indentation during compression.
The conclusion from the exploration of this modification to the Johnson
recovery model is thus that this model is not suitable for simulating collisions
where the indentation terminates in the beginning of the elastic-plastic range.
We come back to this issue later when discussing some selected simulation
results.
6. Comparison between models: simulation results
Model A is the model compiled by Stronge (2000) and it differs from
Model B (developed by Johnson (1985)) in that collision dynamics in all the
three regimes (elastic, elastic-plastic and fully plastic deformation) are taken
into account (the model by Johnson assumes that the deformation in the
contact region is fully plastic). Also the rebound process is modelled using
different assumptions in models A and B.
For the compression, one may therefore expect model A and B to converge
for collisions where fully plastic deformation dominates, i.e. for higher impact
velocities. It is, however, less clear whether this is also the case for the
recovery period.
Some simulation results for a selected set of physical parameters are shown
below. The selected parameters are as follows. Particle radii: R1 = R2 =
2 mm, particle densities: ρ1 = ρ2 = 6000 kg/m3 , effective Young’s modulus:
E∗ = 100 · 109 Pa, heat capacities: c1 = c2 = 400 J/(kg K), conductivities:
k1 = k2 = 20 W/(m K) and temperature difference: ∆T = 100 K. Further
the parameters b1 and b2 were selected to be 1.1 and 2.8, respectively. The
yield stress was 300 MPa in all the simulations. The results were computed
for various values of the impact velocity.
Focussing first on the compression process: this was calculated using
analytical relations for model B (see Section 3.1.2) and numerically for model
A (see Section 3.1.1).
For the selected parameters the elastic-plastic deformation in the contact
region starts at a velocity vy = 0.0055 m/s (i.e. a very low velocity considering various practical applications of the model). The elastic-plastic range
continues until the velocity vp = 1.123 m/s was reached and then the fully
20
plastic region begins (see also Appendix where vy and vp are defined and
determined).
Figure 2 shows the duration of the compression period. Models A and
C lead to the same results since they use the same approach during this
period. For high velocities all the models converge to the same value, since
the deformation in the contact region is dominated by the fully plastic range,
the range constituting the basis for model B. It is also useful to notice that the
compression duration for model B does not depend on the impact velocity,
see Eq. (15).
Figure 2: Compression duration for particle-particle collisions as a function of impact
velocity computed using different models. For large velocities all the models lead to the
same results since the fully plastic deformation in the contact region dominates.
Similarly, Fig. 3 shows the maximum indentation that was reached during
the compression process. The results differ between the models, even though
the difference is not very large. The curves should again converge for high
impact velocities, but do so relatively slowly: a convergence is not clearly
seen in Fig. 3 (the objective was to emphasize the region where the impact
velocity was relatively low and therefore the x-axis of the graph does not
exceed 1.0 m/s). Therefore the ratio of maximal deformations using models
B and A (and C), respectively:
r1 =
δmax,M odelB
δmax,M odelA
21
(39)
is plotted against the impact velocity as shown in Fig. 4. The main conclusion
is that the approximate model B, although, as expected, becoming better
at high impact velocities still show some deviation from the more accurate
models even when the deformation extends into the fully plastic region (as
mentioned, for the chosen parameters this occurs for impact velocities larger
than 1.15 m/s).
Figure 3: The maximum indentation as a function of impact velocity. Models A and C
lead to the same results. The results for model B converge to the results obtained using
models A and C: this trend is, however, not visible in the figure.
Figure 5 shows how the residual deformation that remains after the collision, δr,S used in model A and δr,J used in model B, vary as a function of
impact velocity (recall that model C uses the same approach for recovery as
model B). The obvious conclusion is that the models lead to quite different
results, something that can also be seen by comparing Eq. (19) and Eq. (26).
This can be also investigating analytically in the following way. Rewrite
Eq. (26) by making use of Eq. (6):
b2
(2δy δmax )1/2
(40)
δr,J =
b1
and then find that the ratio between δr,S , expressed by Eq. (19), and δr,J :
1/2
b1 2δmax − δy
r2 =
.
(41)
b2
2δmax
22
Figure 4: The ratio of maximal deformations using models B and A (and C) as a function
of impact velocity. The ratio converges to 1.0 for high velocities as also discussed in Fig. 3.
For the case where δmax ≫ δy , which occurs for relatively high impact velocities, this ratio reduces to b1 /b2 that in the current simulations
is 1.1/2.8 = 0.393. This value can also be seen in the simulations: Figure 6
shows how the ratio r2 (the thick line) varies with impact velocity. The
parameter converges to 0.393, as expected.
Another difference between models A and B was the choice of the effective
radius during recovery R̄∗,S and R̄∗,J . These radii are shown in Fig. 7. Also
here the ratio between the effective radii defined by Eqs. (16) and (24),
respectively, was calculated. This leads to:
2(b1 /b2 )2
r3 =
2 − δy /δmax
1/2
,
(42)
where Eq. (40) is also used.
For δmax >> δy , the ratio converges to b1 /b2 , i.e. the same as for r2 . The
value of r3 as a function of the impact velocity is shown in Fig. 6 (the thin
line). As before it approaches the value 0.393.
Finally we focus on the duration of the recovery. The results for model
B are shown in Fig. 8. The results for model A turns out to be very similar
with a slight difference for lower velocities. Figure 9 shows the ratio of the
two, r4 , as a function of impact velocity.
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Figure 5: δr,S (model A) and δr,J (model B) as a function of impact velocity. As discussed
in the text, the models lead to different results.
Figure 6: The ratios r2 and r3 (defined in the text) as a function of impact velocity. The
results converge to 0.393.
24