APPLICATION OF THE DISCRETE ELEMENT MODELLING IN AIR DRYING
OF PARTICULATE SOLIDS
Jintang Lia and David J. Masonb
a
Department of Chemical Engineering, UMIST, PO Box 88, Manchester M60 1QD,
UK. Email: jintang.li@umist.ac.uk
b
School of Engineering, University of Brighton, Brighton BN2 4AT, UK. Email:
d.j.mason@bton.ac.uk
Key Words and Phrases: gas-solids flow; mathematical simulation; momentum, heat
and mass transfer
ABSTRACT
The Discrete Element Method (DEM) has been widely used as a mathematical
tool for the study of flow characteristics involving particulate solids. One distinct
advantage of this fast developing technique is the ability to compute trajectories of
discrete particles. This provides the opportunity to evaluate the interactions between
particle, fluid and boundary at the microscopic level using local gas parameters and
properties, which is difficult to achieve using a continuum model. To date, most of
these applications focus on the flow behaviour. This paper provides an overview of
the application of DEM in gas-solids flow systems and discusses further development
of this technique in the application of drying particulate solids. A number of submodels, including momentum, energy and mass transfer, have been evaluated to
describe the various transport phenomena. A numerical model has been developed to
calculate the heat transfer in a gas-solids pneumatic transport line. This
implementation has shown advantages of this method over conventional continuum
approaches. Future application of this technique in drying technology is possible but
experimental validation is crucial.
INTRODUCTION
Particulates are employed in a wide variety of industrial processes, such as in
chemical, pharmaceutical and food engineering. In chemical plants, for instance, the
production of two thirds of products involves particulate flows (Roco 1993). Whereas
in the food production chain, almost all unit operations involve the handling of
powdered and bulk foodstuffs. One of the major process during the production and
utilisation of powdered and granular materials is the drying or cooling via a flowing
gas, usually air. Due to its convenience in practical implementation, air drying has
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1
been the most common drying technique for particulates in modern plants. Mujumdar
(2000) states that more than 85 percent of industrial dryers are estimated to be of this
type.
The drying mechanism was originally defined by Sherwood (1929) as the manner
in which moisture moves through a solid and thence out into the air during the drying
process. Extensive theoretical and experimental works have been conducted to
investigate the various mechanisms governing the coupled heat and moisture transfer
phenomena since the1980’s (e.g. Mujumdar 1980). Although basic drying theories
were well developed in the past two decades, Tsotsas (1998) stated that the design of
connective dryers were still based on the scale-up models derived from single particle
drying kinetics. Continuous static (steady state) operation is usually modelled in terms
of an imaginary, average particle, which would be right for ideal plug flow or linear
drying kinetics. However, most drying processes are highly non-linear and the
distribution of solids moisture content at the dryer outlet, a potentially important
aspect of product quality, are usually non-uniform. This may result in large errors
when using continuous steady state models. Therefore, a dynamic modelling
technique based on individual residence time and drying history would be of
considerable importance in the future.
Numerical techniques have been developed to solve the partial differential
equations governing the coupled heat and mass transfer in a single particle (e.g. Abuaf
and Staub 1986, Oliveira and Haghighi 1997 and Jumah et al. 1997). Although
Oliveira and Haghighi discussed the multi-particle system and introduced a method to
model the fluid flow around stationary particles (two soybean kernels), most
contemporary models, such as thin-layer and deep-bed drying analyses (Mhimid and
Nasrallah 1997 and Benet et al. 1997), are based on the assumptions of flow through a
porous medium. In these models, a continuous and homogeneous behaviour of the
medium has to be considered. This assumption may not accurately represent the
distinctive distribution of solids in an aerated particle assembly. More importantly, the
interactions between particles and surrounding fluid are difficult to take into account.
These interactions play an important role when determining the particle behaviour and
heat transfer in gas-solids pipe flows (Li and Mason 2000). Oliveira and Haghighi
(1997) state that the analysis of drying of a multi-particle system may provide
fundamental knowledge for understanding the effects of interaction on the
hydrodynamics and heat and mass transfer characteristics of closely spaced particles.
The mathematical modelling of gas-solids flow was originally built on the
theories of fluid flow, either homogenous flow (ie. Julian and Dukler 1965,
Michaelides 1984 and Chung et al. 1986) or two-phase flow (ie. Mason 1991 and Kuo
and Chiou 1988), where the mixture or each of the two phases is considered as a
continuum. Similar to the thin-layer and deep-bed drying analyses, in these models
the assumption of a continuous change of variables for the solids phase is not realistic,
particularly for large particles and flows with discontinuities in the concentration
gradient. It also introduces problems when modelling particle-particle and particleDrying Technology - An International Journal, ISSN 0737-3937
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2
wall interactions, which have a great influence on the particle motion and the flow
behaviour.
A recently developing category of multiphase flow models uses the so called
Lagrangian approach or the Discrete Element Method (DEM), which calculates the
motion of each individual particle separately. This method is able to take into account
the simultaneous occurrence of various kinds of movements and interactions of the
particles with each other and with the surfaces of the boundary. Cundall et al. (1979)
developed this method and used it in soil mechanics. It has been used to model a
number of applications in gas-solids systems. Tsuji et al. (1993) and Pritchett et al.
(1978) have examined gas-solids flow behaviours in fluidised beds and Langston et
al. (1995, 1996 and 1997) have modelled granular flows in hoppers. Particle motions
in the air stream and pneumatic pipelines have been simulated by Matsumoto and
Saito (1970), Ottjes (1978), Yamamoto (1986), Summerfeld (1992), Tsuji et al.
(1992), Frank et al. (1993), Peng et al. (1994 and 1996) and Salman et al. (1997).
However, to date little report of this application in heat and mass transfer of gas-solids
flows has been found in literature.
The paper addresses the DEM method and its application to the modelling of gassolids flows. The various physical mechanisms that occur in air drying, including
mass, momentum and heat transfer, are discussed and formulated as sub-models.
Although there is potential to apply DEM to liquid-solid slurry droplets (such as
Abuaf and Staub 1986 and Levi-Hevroni et al. 1995) and particle aggregation, this
work will focus on the drying of particulate solids with an outer crust such as a kernel
of cereal grain. A heat transfer model has been developed and simulations were
conducted to examine the heat exchanges in gas-solids flows within a pneumatic
pipeline. The modelled results are discussed in view of further implementation of this
technique in drying process where the incorporation of a mass transfer model is also
required.
THE DISCRETE ELEMENT METHOD
Pioneering work in the application of the Discrete Element Method was initially
carried out by Cundall and Strack (1979) to model the behaviours of dense solids
assemblies in soil mechanics. Their work concentrated on the use of ‘springs and
dash-pots’ to represent particle interactions, as shown in Figure 1.
This concept, which expresses the interaction with the use of a spring, dash-pot
and friction slider analogy, has been adopted in most of the current DEM applications
in particulate flows (e.g. Tsuji et al. 1992). Particles are usually assumed to be
cohesionless elastic bodies and the microscopic particle-particle and particleboundary interactions are calculated with the evolution of particle trajectories.
(Cohesive and inter-particle forces may also be considered as required by more
complex applications. See discussions by Thornton and Yin 1991.)
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The contact force between impacting particles is split into a normal force Fn and a
tangential force Ft:
(1)
Fn   K n  vn
Ft   K t  vt
(2)
where n and t are particle displacement in the normal and tangential directions, vn
and vt are the relative velocities, K is the stiffness of the spring and  is the coefficient
of viscous dissipation (Determination of these properties may be found in the
literature, e.g. Tsuji et al. 1992). If Ft is bigger than the limiting friction force, then
the particles slide over each other and the tangential force is calculated by:
(3)
Ft   f Fn
It is noted that Equations (1) and (2) show a linear relationship between the
contact forces and the displacements, which take the form of Hookes’s law. This
assumption is in contradictory to the Hertzian contact theory. Cundall and Strack
(1979) stated that a non-linear law, such as that of Hertzian contact, could equally
well be employed. Seville et al. (1997) summarise that the linear spring model is the
simplest mathematical form and is fairly widely used. The precise details of the
contact mechanics may not be necessary, since in most real granular flow systems,
particles are irregular and rough, it is unnecessary to model granular flow with ideal
Hertzian contact.
APPLICATION OF DEM IN GAS-SOLIDS FLOW
The initial application of DEM in particulate systems has focused on the
modelling of interactions between particles and between particles and boundary
surfaces. This has been extended to take into account the interactions with the
surrounding gases, such as Tsuji et al. (1992 and 1993) for plug flows and fluidised
beds, where the interstitial gas flow plays an important role for the solids behaviours.
In a gas-solids flow system, the particle motion and the flow structure are
dominated by the interactions between gas, particle and boundary. These interactions
are determined by the laws defining the mass, momentum and energy transfer
mechanisms. To model the flow, the various interactions need to be calculated.
(Details of these sub-models are discussed in the following sections.) Using DEM, the
trajectory of each particle is evaluated together with the evolution of the flow over
many time steps. Interactions between gas, particle and boundary wall are
subsequently calculated using the local particle and gas parameters, and these
interactions eventually determine the particle motion and the gas flow. Seville et al.
(1997) state that the basic advantage of this method over continuum techniques is that
it simulates effects at particle level. Individual particle properties, such as size and
shape variation, can be specified directly and the assembly response is a direct output
from the simulation. There is less need for global assumptions, such as uniform stress
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4
at a certain depth in the assembly. Certain phenomena such as particle size
distribution can be simulated directly, whereas in the continuum method this is far
more difficult.
In a DEM model, particle trajectories are computed by considering the various
forces acting on each particle. The gas field is solved simultaneously with the particle
movements considering the particle presence as a source of mass, momentum and
energy and as a change to its volume fraction (See Crowe et al. 1977. Very fine space
discretization for the gas field may be used to model fluid flow around particles but,
as discussed by Oliveira and Haghighi 1997, this requires much more computational
resources for multi-particle systems and may not complete a case in a reasonable time
scale with the power of current computers.). As the location of a particle in the fluid is
known all the time, this gives an opportunity to determine which cell in the fluid
domain contains the particle and, therefore, the interactions between gas and particles
can be computed using the local fluid conditions. This results in a more
comprehensive coupling between the gas phase and the discrete particles and hence a
more realistic gas and solids flow field. The importance of these coupling effects
(temperature of gas, gas properties, solids properties, solids behaviour and mass and
heat transfer between gas and solids) in the pneumatic drying of grains has been
addressed by Matsumoto and Pei (1984).
A variety of applications of DEM in gas-solids flow systems have been reported
in literature. These works may be categorised in accordance with their applications in
specific industrial processes, such as fluidised beds, hopper flows and pipe flows. The
application of DEM in fluidised beds by Tsuji et al. (1993) has focused on the flow
behaviours such as gas velocity distribution, particle flow patterns and bubbling and
pressure fluctuation across the bed. In Tsuji’s model, a finite difference approach has
been used to model the gas flow and a soft-contact model with reduced particle
stiffness has been adopted for the particle-particle impact. A similar model was
developed by Hoomans et al. (1996) but using a hard-sphere collision approach
(Wang and Mason 1992), which is commonly encountered in the field of molecular
dynamics. More recently, Li et al. (1999) developed a coupled CFD (Computational
Fluid Dynamics) and DEM model to simulate the gas-liquid-solids fluidisation
systems. A close-distance interaction model is introduced for the particle-particle
collision, which considers the liquid interstitial effects among particles. In the above
models, it has been shown that the fluid-particle interactions are usually modelled by
the fluid drag approach but the particle-particle interactions may be considered in
accordance with actual particulate properties.
Langston et al. (1995, 1996 and 1997) have carried out extensive works in
granular flows in hoppers. Their works have concentrated on the dynamic hopper
discharge rates and hopper wall stresses with different filling methods. The effects of
interstitial air flow have also been examined (Langston et al. 1996). It demonstrates
that the DEM method has shown certain transient and oscillatory features of the flow
field, which has not been produced by continuum theories. The simulated results have
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5
been compared with experimental data and good agreements have been reported
between the simulations and the imaging data from photography and gamma-ray
tomography.
There are wide spread applications of DEM in pneumatic pipelines. Matsumoto
and Saito (1970) examined the mechanisms of particle suspension using a Monte
Carlo simulation. Salman et al. (1997) discussed the simulation of particle movement
in dilute pneumatic conveying and Tsuji et al. (1992) presented a DEM model for the
study of plug flow in a horizontal pipe. Summerfeld (1992) modelled particle-wall
collisions and Peng et al. (1994) analysed the effect of particle collision and particle
rotation for the transport of coarse particles.
The applications of DEM in the various gas-solids systems have shown potentials
of this fast developing technique in analysing the flow behaviours and the interactions
within the mixture and with the boundary. However, the incorporation of a heat
transfer model in this technique has not been tackled until recent years. Assuming a
random motion of particles and neglecting the interstitial fluid flow and interactions
with boundary, Hunt (1997) developed the first DEM model to predict temperatures
and effective thermal conductivities for flows of granular materials (in an imaginary
two dimensional domain). Apart from this over simplified model, to date little report
of this application in heat transfer of gas-solids flows has been published.
EVALUATION OF MOMENTUM TRANSFER
Particles in the gas flow field may be subjected to forces due to gravity, fluid
drag, and collisions with other particles and the pipe wall. The buoyancy force can
usually be ignored since the densities of the particles are normally two or three orders
of magnitude larger than air. Inter-particle forces, such as van der Waals forces,
capillary forces and electrostatic forces may also be neglected if the particle size is
bigger than 100 m when those inter-particle forces become insignificant (Seville et
al. 1997).
The fluid drag force on a particle is given by:
FD  CD Ap
1
2
g ( u g  u s ) u g  us
(4)
where Ap is the projected area of the particle.
Extensive experiments have been conducted to correlate the drag coefficient CDs,
for a single particle in a finite fluid, with the particle Reynolds number Res. The
relation is divided into three regimes (not considering the critical range at Res
approaching 105):
i)
ii)
for Res > 1000, CD 0.44 and is roughly constant (Newton’s law);
in the intermediate range, 0.25 < Res < 1000, CDs is significantly
dependent on Res.
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for Res < 0.25, CD /Res. This is the ‘creeping flow’ regime and
known as Stokes’ law.
A general correlation was proposed by Schiller and Naumann (1933) for the three
regions:
iii)


 24

0.687
(5)
CDs  max 
1  0.15 Re s
,0.44
 Re s

For very dilute phase flow, this formula has given good prediction of the drag
force. However, when applying to particles in flows with high solids concentration it
was found that it underestimated the drag force. The presence of neighbouring
particles alters the local gas flow field and eventually influences the fluid drag on the
particle.
The drag coefficient CD for a particle in a group of particulate assembly is
computed using the value for a single particle CDs modified according to the presence
of neighbouring particles (Richardson and Zaki 1954). The Richardson-Zaki exponent
n was derived by equating the Richardson-Zaki relationship with Gibilaro’s equation
(Gibilaro et al. 1985), an improved Ergun equation (Ergun 1952). Thus,
CD  CDs  gn
(6)
1.8


 7 0.4  600 (1  0.4) 1

n  2.8  log 
 1 log  g 


 g Re s


 3 C Ds  7

where g is the gas volume fraction.
(7)
MODELLING OF HEAT TRANSFER
The main heat transfer mechanism in an air dryer is the heat exchange between
gas and particles. The contribution of conductive heat transfer between particles in
contact may affect the drying process to some extent depending the concentration of
particulate solids in the dryer (Sharma 1997). (Heat exchanges with boundaries, such
as between gas and boundary walls, may be modelled by boundary-layer models
according to practical boundary conditions, e.g. adiabatic or isothermal, or using
empirical equations such as formulae for pipe flows proposed by Wrangham 1961.)
A Particle in an Infinite Fluid
For a stationary particulate sphere in an infinite fluid, it is easily shown from the
heat conduction equation that:
Nus  hs d / k  2
(8)
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When there is a relative motion between the particle and fluid, the most
commonly used correlation is due to Ranz and Marshall (1952), which was developed
by experiments on evaporation from water droplets.
Nus  2  0.6 Re s1/2 Pr 1/3
(9)
This formula was found sufficiently accurate when applied to particles with small
Reynolds numbers (less than 200). However, when a higher Reynolds number is
encountered, such as at the solids feeding area of a gas dryer, where the relative
velocity between gas and particles is high (Mason and Li 2000a), the above equation
was found only adequate to cover a limited range (Li and Mason 2000). After
comparing with experiments using 3mm polyethylene pellets, a modified equation
was adopted for 200<Res<1500:
Nus  2  0.5 Re s1/ 2 Pr1/3  0.02 Re 0s .8 Pr1/3
(10)
When the particle Reynolds number is bigger than 1500, a significant
enhancement of heat exchange has been observed and another correlation, which is
similar to the equation of Frantz (1962) but with a higher exponent, has been
introduced:
Nus  2  0.000045 Re s1.8
(11)
It is noted that although this equation has been tested with a variety of flow
conditions for 3mm polyethylene pellets at Res > 1500, further experimental
validation may be necessary for different particle sizes.
A Particle in a Particle Assembly
In a similar manner to the fluid drag force, the heat transfer between gas and
particle is also affected by the presence of neighbouring particles. This effect has been
found more significant with the increase of solids concentrations. There are few
works in literature that examine this subject (Bandrowski and Kaczmarzyk 1978).
Using the similar concept as for the fluid drag, the following correlations have been
proposed with the modification by gas volume fraction g:
Nu s  2  0.6 gn Re s
1/ 2
Nu s  2  0.5 gn Re s
1/ 2
Pr 1 / 3
Pr 1 / 3  0.02 gn Re 0s .8 Pr 1 / 3
Nu s  2  0.000045 Re s
n
g
Res <= 200
(12)
200 < Res <= 1500
(13)
1.8
Res > 1500 (14)
The exponent n = 3.5 was found suitable for the 3mm polyethylene pellets when
modelling dilute flows. However, a temperature discrepancy was found at the solids
inlet in flows with higher solids concentrations. Again, this exponent needs to be reevaluated in the light of more sophisticated experiments.
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Gas-particle Heat Transfer Model
Boothroyd (1971) stated that, for calculation purposes, it is often sufficiently
accurate to use the steady-state heat transfer coefficient at all times if the suspension
is not too dense. A uniform particle temperature distribution has been assumed in the
present work. More sophisticated considerations, such as the assumptions of a crust
temperature and wet core temperature by Abuaf and Staub (1986), may be adopted for
the drying of slurry droplets or large particles. Then the rate of heat transfer qs from
the gas to the particle can be calculated by:
qs  hs A(Tg  Ts )
(15)
where A is the surface area of the particle.
The heat transfer coefficient hs can be determined by
hd
Nus  s s
kg
(16)
The amount of heat transferred in a time step t is
Qs  qs t
(17)
This heat transfer between gas and particle is considered as an energy source to
the gas phase, to be added to the source term of the energy conservation equation, and
a balance of the internal energy of the particle.
The thermal balance in a time step t for an evaporating droplet moving in the
gas stream can be written as:
(ms c ps  ml c pl )Ts  Qs  Lml
(18)
The terms on the left-hand-side of the equation express the heat capacity of both
liquid and solid portions, whereas the right-hand-side denotes the heat exchange with
the gas and the latent heat of liquid evaporation. The mass of evaporated liquid water
ml is calculated by the mass transfer model.
After heat exchange, the temperature change of the particle can be calculated by
Qs  Lml
Ts 
ms c ps  ml c pl
(19)
Particle-particle Heat Transfer Model
For solid particles, when two bodies impact each other or a moving particle
impacts on a stationary surface, heat conduction occurs through the contact area of the
particles. Based on both numerical and analytical solutions of the heat conduction
equation with time-dependent boundary conditions, Sun and Chen (1988) produced a
quantitative calculation of this conductive heat transfer due to particle impact. For
cases where the contact time is short, it is sufficient to use a simple one-dimensional
calculation for this problem. The interface heat flux is given by:
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9
q imp
(T2  T1 )t c1/ 2

( 1c ps1k1 ) 1/ 2  ( 2 c ps 2 k 2 ) 1/ 2
(20)
This enables the particle-particle and particle-boundary heat transfers be
evaluated at the resolution of a particle time step. As the time step used for the
calculation of particle motion in DEM is very small, which is only a fraction of the
total contact time (Li 2000), it is assumed that if particles or particles and surface
contact at a certain time step the contact will occur over the whole period of the time
step. Therefore in the above equation the contact time tc can be replaced by the length
of a time step t. Then the energy exchange in a time step can be computed by
Qimp 
(T2  T1 )rc2 t 1/ 2
( 1c ps1 k1 ) 1/ 2  ( 2 c ps 2 k 2 ) 1/ 2
(21)
where rc is the instantaneous contact radius considering the particle deformation as
shown in Figure 2, which can be computed by:
R12  rc2  R22  rc2  R1  R2  R12
rc 
(22)
4 R12 R22  (( R1  R2  R12 )2  R12  R22 )2
4( R1  R2  R12 )2
(23)
For particle and surface contact, the surface deformation may be neglected. rc can
be calculated by replacing R2 with infinity in the above equations.
R12  rc2  R1  R12
rc 
(24)
R12  ( R1  R1 ) 2
(25)
The total energy exchange during the entire contact time is the sum of energy
exchange for all time steps during the course of contact. Similar to the heat exchange
between gas and particle, the energy transfer on impact also contributes to the particle
internal energy, ie. a change of temperature.
MASS TRANSFER MODEL
One class of model describing the drying of a particulate solids is known as the
‘single-kernel drying model’ (Puiggali and Quintard, 1992). Most of these models
were developed based on the application of non-equilibrium thermodynamics theories
(Luikov 1966) in porous media. Drying models for generic hygroscopic porous media
were reviewed by Bories (1989), Puiggali and Quintard (1992) and Waananen et al.
(1993).
The evaporation rate of liquid in a moist particle is usually formulated as (Abuaf
and Staub 1986 and Levy and Mason et al. 1998):
dml
(26)
 hd d 02 (Cvg  Cvs )
dt
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10
where Cvg and Cvs are the water vapour concentrations at the outer diameter of the
particle and the ambient air respectively. They are given by the following expressions:
M p
(27)
Cvs  w vo
RT s
C vg 
M w p vg
(28)
RT g
The partial pressure of the water vapour in the ambient air, pvg, is determined by
the proportion of water vapour content in the air, which is calculated by the
continuum modelling of the gas and the water vapour phases. For a liquid-droplet or
at the constant-rate drying period of a slurry droplet (Mujumdar 1983), the water
vapour pressure pvo at the outside periphery of the droplet is straightforwardly
assumed to be the saturation pressure of the water vapour at the droplet temperature
Ts. However, when the drying approaches the falling-rate period, which is the case for
the drying of moist particulate solids, the process becomes much more complicated
and various assumptions have to be made.
The moisture transport process in a porous medium is usually much slower than
the heat transfer. It is common to assume a uniform temperature distribution inside a
small solid particle but this assumption is not applicable to moisture distribution.
Abuaf and Staub (1986) developed a model assuming that, during the falling-rate
stage of the drying process, a dry crust surrounding a wet core was formed. The
amount of liquid evaporated that diffuses through the dry porous crust is calculated by
a Stephen-type diffusion equation:
dml 1  1
p  pvi
1 D p
    v ln
(29)
dt 2  d 0 d i  RTave p  pvo
where di and d0 are the diameters of the wet core and the dry crust respectively.
Assuming a constant dry crust diameter, i.z. the moist particle diameter, d0, and a
uniform solids fraction through the particle, Abuaf and Staub derived the following
equation to calculate the change in the diameter of the receding wet core:
d (d i )
d0
dml
(30)

1/ 3 2 / 3
dt
dt
3ml 0 ml
In equation (29), the porosity of the dry crust, , can be determined by actual
measurement for the dry material. As a uniform temperature is assumed, here the
average temperature, Tave, shall be equivalent to the particle temperature Ts. The
partial pressure of the water vapour, pvi, in the wet core is assumed to be the saturation
pressure of water vapour at the particle temperature Ts. Combining equation (26) with
(29) to eliminate the partial pressure of water vapour in the dry crust, pvo, and obtain
an implicit equation for the water evaporation rate:
dml d i  d 0 2Dv p 
p  pvi

ln
2

dt
di d0
RT s
 p  ( RT s / hd d 0 M w )( dml / dt )  pvg Ts / Tg
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



(31)
11
The mass transfer coefficient hd is calculated from the well known Sherwood
number correlation:
h d
1/ 2
(32)
Sh  d  2  0.6 Re s Sc1 / 3
Dv
The inclusion of the Spalding number in the above equation was used by Abuaf
and Staub (1986). This effect has been found minor (error less than 5%) if the
temperature of the drying media is not significantly higher (within 100oC) than the
particle temperature.
SIMULATIONS FOR HEAT TRANSFER IN PIPE FLOWS
As part of a program investigating gas-solids flow and heat transfer in dilute and
dense phase pneumatic conveying both experimentally and numerically (Li 2000), a
two-dimensional transient model has been developed to describe heat transfer in gassolids pipe flows (Li and Mason 2000). In the model, the gas phase has been modelled
as a continuum, but the solid phase has been modelled as a set of discrete particles by
DEM. The influence of solid particles on the gas flow is considered as a source of
mass, momentum and energy and as a change to its volume fraction. Here a brief
introduction of the model is presented and details are referred to Li (2000).
Gas Phase Model
The gas flow is governed by the following mass, momentum and energy
conservation equations.

Mass conservation
 ( g g )

   ( g g ug )  Sm
t
(33)
where Sm is the mass source due to the mass transfer with solids and boundaries. Mass
transfer has been neglected in the current model.

Momentum conservation
 ( g g ux )

p
   (  g  g u g u x )    (  g  g (u x ))   g
 Sx
t
x
(34)
where Sx is the momentum source in the x direction, which is due to momentum
transfer with the particles, and pipe wall friction. A similar equation may be written
for the y direction.
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2002 vol.20, no.3, pp.255-282.
12

Energy conservation
 ( g g H )

   (  g  g u g H )    (g  g ( H ))  S e
t
where the diffusion coefficient is:
g  k g / cg
(35)
(36)
and the stagnation enthalpy:
H  Tg cpg  12 ug 2
(37)
The energy source term Se includes the heat transferred between gas and particles
and gas and pipe wall and the mechanical work done due to friction at the boundary
wall.

Numerical calculation
The gas flow is calculated using the SIMPLE (Semi-Implicit Method for
Pressure-Linked Equations) method developed by Patankar (1980), taking into
account the interactions with particles and boundary wall. The procedure of
calculation follows the physical process of experiment, i.e. it starts from the singlephase gas flow and, when a well converged gas field is obtained, the particles are
incorporated.

Gas turbulence
Because of the random nature of gas turbulence, now it is still difficult to model
the physical modes of turbulence (such as eddies) in a gas-solids flow. In this model,
the gas turbulence was modelled by employing an average effective turbulent
viscosity, which was obtained from running the same problems in a commercial CFD
package – PHOENICS, CHAM, UK. The numerical models and their solutions are
described by Spalding (1980). In PHOENICS the gas turbulence is modelled by k-
model. This effective viscosity was evaluated by comparing the pressure drop with
experimental data.
Solids Phase Model
Particles in the gas flow field may be subjected to forces due to gravity, fluid
drag, and collisions with other particles and the pipe wall. The buoyancy force is
ignored since the densities of the particles are significantly larger than air at the
pressures commonly used in pneumatic conveying systems. The Saffman lift and
Magnus forces are also neglected because they are of the order between 10-2 and 10-3
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13
of the fluid drag force for the particles modelled in the present work. In addition,
inter-particle forces, such as van der Waals forces, capillary forces and electrostatic
forces have been neglected since the particle sizes (3mm sphere) used in this study are
large and these inter-particle forces are insignificant (Seville et al. 1997). Fluid drag
forces and collision forces are calculated in accordance with the sub-models described
previously. Particle trajectories were computed by applying Newton’s second law of
motion to each particle. Heat transfer between the gas and the particle is computed
simultaneously with the evolution of particle trajectory.
Simulations
The numerical model was set up to simulate the gas-solids flow and heat transfer
in an 8m section of a 3in (80mm bore) pneumatic transport line (Figure 3). The
simulations used the same gas and particle properties and boundary conditions as
adopted in an experimental study (Mason and Li 2000b). Validation of the model and
comparisons with experimental data are presented in Li (2000). A common industrial
material, 3mm polyethylene pellets, has been used in the experiments. Details of the
pipe geometry and gas and solids properties are given in Table 1.
The boundary conditions used in the model were specified to match those in the
experimental system as illustrated in Figure 3 (See details in Li 2000). The gas was
supplied from a straight pipe of sufficient length to ensure that the flow was fully
developed. Particles were then introduced through the top of the pipeline, falling from
a rotary valve through a dropout box (with a length of approximately 235mm). At the
pipe outlet the pressure was maintained at a constant value. A steel pipeline with no
thermal insulation was used for the flow of the gas-solids mixture. Therefore, the
following boundary conditions were constructed:
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14
 Inlet
 Outlet
 Wall
Gas was introduced with developed turbulent velocity and
temperature profiles in the axial direction with a constant mass flow
rate (a fairly constant pipe Reynolds number at the inlet since the
change of gas density within the range of absolute pressure variation
was negligible), supplied by a set of choked nozzles and heated by
an electrical heater.
Particles were introduced with a small velocity in the direction
of gravity but zero axial velocity for a horizontal pipe. This velocity
was computed from the actual height of the dropout box. All
particles were assumed at ambient temperature (21oC).
Pressure at the pipe exit was specified.
A no slip condition was taken for the gas flow.
A no slip condition was taken for the point on the particle in
contact with the wall. This did not preclude particles from rolling
along the wall.
The surface temperatures of the pipe walls were assigned equal
to the fluid temperature adjacent to the wall. A free convection
thermal boundary at the wall was adopted.
This study focused on the microscopic motion and heat transfer of individual
particles and how these affect the gas-solids flow patterns and heat transfer. Figure 4
shows the solids distribution in the pipe at Reynolds number ReD = 79,400 and solids
loading ratio SLR = 2, a typical dilute phase pneumatic conveying. (It is noted that the
co-ordinates are not in scale in the radial and the axial direction.) Figure 5 traces a
particle motion and its temperature variation in relationship to the surrounding fluid
temperature. Due to the large thermal capacity of the 3mm particle, the temperature
change of such a large particle during the drying in such a low fluid temperature
(<50oC) is relatively slow. As the fluid temperature decreases along the pipeline, the
heat transfer rate with the particle falls. Although a higher gradient of particle
temperature change is observed at the particle inlet, the particle temperature shows a
steady increase after the accelerating region. A similar simulation was conducted for a
dense flow at ReD = 49,400 and SLR = 50. Figure 6 shows a much unsteady particle
motion due to frequent particle collisions. A sharp gas temperature decrease is
observed at the solids inlet, which indicates a rapid cooling of the gas due to the high
solids concentration. The particle temperature change is very small after passing
through the 8m long pipeline. This shows the difference of heat transfer mechanisms
between the air drying in a dilute phase solids flow and that in a dense solids
assembly or a moving particle bed.
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15
The above simulations show a slow temperature change for the 3mm pellets,
which means that to dry such a large particle needs a much higher gas temperature
and/or longer resident time. Using similar flow conditions (ReD = 79,400 and SLR =
2), computational experiments were also conducted for smaller particles: 1mm
polyethylene pellets. Although a more pronounced temperature gradient was observed
as plotted in Figure 7, there still remains a temperature difference between the particle
and the gas. This instant temperature difference determines the heat transfer rate
between the gas and the particle as shown in Figure 7, where more pronounced
particle temperature changes happen at higher gas temperatures. This phenomenon
can also be seen in Figure 5 for the 3mm pellet though not so obvious as in Figure 7
for the 1mm pellets. This reveals a quite different thermal behaviour of the large
particles in a gas compared with a previous work of the Authors (Li and Mason 1998)
on fine particles as shown in Figure 8. (A very small particle, 0.05mm aluminium
sphere, in a very dilute phase flow at ReD = 47,200 and SLR = 1.1 in a 2in, 50mm
bore, electrically heated pipe.) It demonstrates that, although there is always a
temperature difference between the particle and the gas except some critical points,
fine particles tend to follow the temperature of the gas. Therefore, a rapid temperature
change is usually expected when drying fine particles. The thermal behaviours of
particles with different sizes in a gas shall be of great interest to the study of drying of
particulates.
Figure 8 also clearly shows that by traversing across the pipe section and
exchanging heat with the gases, particles actually indirectly enhance the heat transfer
between the gas near the boundary wall and the gas in the middle of the pipe and,
consequently, alter the flow and thermal structure of the gas. The integration of
temperature difference between the particle and the fluid over time indicates the
energy transferred during a certain period of time in correspondence to the exact
particle trajectory. This is important in any gas-solids flow system where heat transfer
is of primary concern.
Although the present work only concerns the heat transfer in gas-solids flow
through pipes, Figures 4 to 8 show that the application of DEM provides an
opportunity to study the various flow and heat transfer phenomena in discrete particle
level, including the solids distribution, the particle trajectory and particle temperature,
the gas temperature, and the influences of particle diameter and particle concentration.
These are also key parameters for the drying of particulate solids since the moisture
transfer processes are largely dependent upon the conditions, temperature and
moisture distribution, in the flow field surrounding the particles. Further development
of the model to incorporate the mass transfer is envisaged.
Drying Technology - An International Journal, ISSN 0737-3937
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16
CONCLUDING REMARKS
The application of the Discrete Element Method in gas-solids flow systems has
been discussed in detail. A numerical model has been developed to simulate heat
transfer in gas-solids flows through pipes. The implementation of this method for heat
transfer in gas-solids systems has shown advantages over traditional continuum
models. Further application of this technique in the drying of particulate solids has
been evaluated. This evaluation shows the potential of using this method to model the
coupled heat and mass transfer process in drying but validation by experiment is
crucial. Future work is expected to focus on the implementations of the mass transfer
model and its validation procedure.
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17
NOMENCLATURE
Area (m2)
Vapour concentration (kg/ m3)
Fluid drag coefficient (-)
Specific heat capacity (J/kg K)
Diameter of pipe(m); Diffusion coefficient of water
vapour in air within pore (m2/s)
ds
Particle diameter (m)
F
Force (N)
f
Coefficient of sliding friction (-)
H
Stagnation enthalpy (J/kg)
h
Heat transfer coefficient (W/m2 K)
hd
Mass stagnation enthalpy (J/kg)
K
Stiffness (N/m)
k
Thermal conductivity (W/m K)
L
Latent heat of water evaporation
(J/kg)
M
Molecular weight (-)
m
Mass (kg)
p
Pressure (N/m2)
Q
Energy (J)
q
Heat transfer rate (W)
R
Gas constant (J/kg K)
R
Particle radius (m)
r
Radius (m)
S
Source term (Mass: kg/m3; Momentum: N/m3;
Energy: W/m3.)
SLR Solids Loading Ratio (kg/kg)
T
Temperature (K)
t
Time (s)
u
Velocity (m/s)
v
Relative velocity (m/s)
A
C
CD
cp
D
Greek letters
Density (kg/m3)

Dynamic viscosity (kg/m s)

diffusion coefficient (kg/m s)

Volume fraction (m3/m3); Porosity (m3/m3)

Coefficient of viscous dissipation (kg/s)

Displacement (m)

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18
Subscripts:
0
Initial condition
c
Contact
D
Fluid drag; pipe
l
Liquid
g
Gas
n
Normal direction
s
Solids
t
Tangential direction
w
Water
Dimensionless groups:
Pipe Reynolds number
Re D   g Du g / 
Particle Reynolds number
Re s   g d s | u g  u s | / 
Particle Nusselt number
Nus  hs d s / k g
Prandtl number
Pr  c g / k g
Schmidt number
Sc   /  g Dv
Sherwood number
Sh  hd d / Dv
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19
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Pipe:
Pipe internal diameter (m)
Pipe length (m)
Gas:
Gas used
Density (kg/m3)
Thermal conductivity (W/m K)
Specific heat capacity (J/kg K)
Particles:
Material
Shape
Diameter (mm)
Particle density (kg/m3)
Stiffness (N/m)
Coefficient of friction (-)
Coefficient of restitution (-)
Thermal conductivity (W/m K)
Specific heat capacity (J/kg K)
0.080
8.0
Air
By ideal gas law
0.02564
1006.1
Polymer
Spherical
3.0
880
2E+5
0.3
0.8
0.17
2300
Table 1: Data used by model
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25
dash-pot
spring
spring
slider
slider
dash-pot
(a)
(b)
FIGURE 1. Models of contact forces: (a) normal force; (b) tangential force.
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R1
R2
rc
R12
FIGURE 2. Contact between two particles
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27
Cold solids in
235mm
Un-insulated pipe
Hot gas in
80mm
8m
FIGURE 3. Gas/Solids loading and flow boundaries
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Particle 3mm
0.08
0.07
Radial (m)
0.06
0.05
0.04
0.03
0.02
0.01
0
0
1
2
3
4
5
6
7
8
Axial (m)
FIGURE 4. Solids distribution in pipe
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0.08
320
Particle trajectory
Particle temperature
315
Fluid temperature
310
0.04
305
300
0.02
Temperature (K)
Radial (Pipe dia. m)
0.06
295
0
290
0
2
4
Axial Distance (m)
6
8
FIGURE 5. Particle (3mm) trajectory and temperature in a dilute phase flow
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Particle trajectory
Particle temperature
Fluid temperature
330
0.06
320
0.04
310
0.02
300
0
Temperature (K)
Radial (Pipe dia. m)
0.08
290
0
2
4
Axial Distance (m)
6
8
FIGURE 6. Particle (3mm) trajectory and temperature in a dense phase flow
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0.08
325
Pronounced
increases
0.06
320
315
310
0.04
305
300
0.02
Slow increases
Temperature (K)
Radial distance (m)
Particle trajectory
Particle temperature
Fluid temperature
295
0
290
0
1
2
3
4
5
Axial distance (m)
6
7
8
FIGURE 7. Particle trajectory and temperature for a 1mm pellet
Drying Technology - An International Journal, ISSN 0737-3937
2002 vol.20, no.3, pp.255-282.
32
297
Particle Trajectory
Particle Temperature
Fluid Temperature
0.04
0.03
294
291
0.02
288
0.01
0
285
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Axial distance (m)
FIGURE 8. Particle trajectory and temperature of a 0.05mm aluminium sphere
Drying Technology - An International Journal, ISSN 0737-3937
2002 vol.20, no.3, pp.255-282.
33
Temperature (K)
Radial distance (m)
0.05