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1997, Powder Technology

Journal of Babylon University/Engineering Sciences

EXPERIMENTAL INVESTIGATION OF TWO-PHASE GAS-LIQUID SLUG FLOW INCLINED PIPE2013 •

The experimental results of flow pattern and pressure drop of gas–liquid flow in inclined pipe are presented. The diameter of test section is 50 mm, and overall length of 4 m. The inclination angle of the test section is 30o. Air and water are used as working fluids. The gas superficial velocity and liquid superficial velocity are varied in a range of (1.358142-5.432568) m/s and (0.169764-1.527884) m/s, respectively. The pressure drop along the test section of the pipe is also measured. The characterization of flow patterns is achieved via visual observations and by analysis of local pressure measurements. The observed flow patterns are presented in terms of flow pattern maps for pipe inclination. The slug two-phase flow patterns are observed in the experiments. A video camera recording and pressure transducer sensor with interface are used to study flow regimes and pressure drop through test section. The following flow regimes, depending on the superficial liquid and gas velocities are observed. The flow regime and pressure fluctuating across pipe depending on superficial liquid and gas velocities. It noted that the pressure decreases with distance along pipe when gas superficial velocity increased and also increased liquid superficial velocity.

Romanian Journal of Petroleum & Gas Technology

Movement of Two-Phase Gas-Liquid Flow in Horizontal and Inclined Pipes2023 •

International Journal of Multiphase Flow

Experimental study of two and three phase flows in large diameter inclined pipes2003 •

2011 •

Open Journal of Fluid Dynamics

Experimental and Numerical Study of Dilute Gas-Solid Flow inside a 90° Horizontal Square Pipe Bend2013 •

Advanced Powder Technology

Investigation of gas–solid flow structure after a 90° vertical-to-horizontal elbow for low conveying gas velocities2005 •

2014 •

The main purpose of this work is to experimentally investigate the effect of pipe orientation on two phase flow phenomenon. Flow pattern, void fraction and two phase pressure drop is measured in a polycarbonate pipe with an inside diameter of 12.7mm for inclination angles ranging from -20<sup>o</sup> to +20<sup>o</sup> using air-water fluid combination. The experimental data covers all flow patterns and the entire range of void fraction typically observed in two phase flow. The effect of pipe orientation on void fraction and two phase pressure drop is justified with reference to the change in flow structure and two phase flow behavior. In addition to this, the top performing void fraction and two phase pressure drop correlations available in the literature are presented and their performance is assessed against the experimental data in the present study and that available in the literature.

POWDER
TECHNOLOGY
ELSEVIER
Powder Technology93 (1997) 253-260
A comparison of analytical and numerical models with experimental data
for gas-solid flow through a straight pipe at different inclinations
Avi Levy *, Thomas Mooney, Predrag Marjanovic, David J. Mason
Centrefor Industrial Bulk Solids Handliag, Department of Physical Sciences, GlasgowCaledonian University, CowcaddensRoad, GlasgowG40BA, UK
Received I 1 July 1996; revised 4 March 1997; accepted 27 May 1997
Abstract
An analytical model for gas-solid suspension flow through an inclined section of pipe was developed. This model predicts the ratio of the
total pressure drop in an inclined pipe to that of a horizontal pipe. The model has been used to predict the critical pipe angle, which is defined
as the angle at which the maximum pressure drop for a given solids flow rate is achieved. This angle differs from 90° (found in a single-phase
flow) and is directly proportional to the ratio between the gas superficial velocity and the particle terminal velocity. The three-dimensional
conservation equations for steady-state two-phase flow in an inclined pipe were solved numerically for constant solids and gas flow rates at
different pipe inclinations. This model was based on the continuum theory for describing the mass and momentum balance equations for the
fluid and solid phases. A packing model, describing the shear stress of the solid phase as a function of its volume fraction, is suggested in
order to limit the maximum value of the solid volume fraction. A new model for particle-wall interaction was developed taking into account
the angle of inclination of the pipe. The prediction of the numerical model was compared with experimental data obtained in a specially
designed test rig. In general, the agreement between the experimental data and the models was satisfactory. The results of the numerical
simulation also confirmed that the critical pipe angle for gas-solid flow is lower than 90°. The assumptions made during the development of
the models were assessed in order to explain the differences between the predicted and measured values of the flow parameters for different
flow regimes. © 1997 Elsevier Science S.A.
Keywords: Inclinedpipelines; Two-phase flow;Pneumatic conveying
c
1. I n t r o d u c t i o n
The designer of a pneumatic conveying system always has
to make a few decisions when designing the various components of a system. One o f these decisions relates to the
design of a pipeline layout which includes a change o f level.
T w o basic alternatives exist to reach point B from point A
(Fig. I):
(a) the level change can be designed as two sections ( A C
vertical, CB horizontal), or
(b) the level change can be designed as one inclined
section AB.
Considering that the whole pipeline lies in the vertical
plane, defined b,' points A, B and C, the first solution requires
two 90 ° bends (points A and C) and the second solution
requires two ( 1 8 0 - a ) ° bends (points A and B ). At the same
time it is obvious that the overall length o f the inclined pipeline is smaller than that of the vertical-horizontal solution.
* Corresponding author. Tel: +44 141 331 3711; fax: +44 141 331
3448; e-mail: a.levy@gcal.ac.uk
0032-5910/97/$17.00 © 1997 Elsevier Science S.A. All rights reserved
PIi S0032-591 O( 97 ) 03 280-4
A
I
:IV
B
I
A
Fig. I. Pipelnne layout.
For a single-phase flow the solution using an inclined section would be preferable as it results in a smaller energy loss
(shorter straight section, higher bend angle). However, the
nature o f two-phase flow, either gas-solid or liquid-solid,
influences energy loss in a way that, with inclined sections,
could turn out to be the worst possible scenario.
Although a number o f plant operating problems have been
reported (high energy losses, high fluctuations of all flow
parameters, even blockages o f pipelines), caused allegedly
254
A. Levy et aL / Powder Technology 93 (1997) 253-260
by inclined sections, only a few studies have been dedicated
to investigating the cause of this kind of performance. This
paper gives an indication of the possible reason, based on the
analytical analysis of the Laost influential forces which act on
the particulate phase in gas-solid flow, along with a numerical
and experimental investigation.
Most of the published work detailing methods for the calculation of energy loss in gas-solid flow either relates to
horizontal flow or to vertical flow. Examples of various
approaches are given in Ref. [ 1]. They normally start from
the assumption that the total energy loss is the sum of the
energy loss for gas flow and additional energy loss which
takes into account the influence of the presence of particles
in the gas stream. The determination of this additional energy
loss is the main problem and a number of proposed formulae
exist in the literature, e.g. Refs. [2,3]. Most, if not all, rely
heavily on one or a few empirical parameters, hence they are
not straightforwardly applicable to the design of a system
transporting a 'new' material. Very few experimental studies
of gas-solid flow in inclined pipes have been carded out. For
example, the study of flow regimes in vertical and inclined
(up to 18° from the vertical) pipelines is reported by Ginest
et al. [4]. This study gives a classification of various flow
regimes which can develop due to changes in pipeline inclination. Experimental results obtained for dolomite and
cement in a 120 mm bore inclined pipeline are reported by
Salt et al. [5]. In their study they measured the solids flow
rate at a constant pressure drop for different angles of
inclination.
One of the first detailed theoretical approaches, which
covers all pipe inclinations, is given by Muschelknautz [ 2,3 ].
This model considers only gas-solid flow at a low concentration of solid particles in a gas stream, i.e. low solids loading
ratio (SLR=n~Jtfig). This uses an analogy to Darcy's formula and the full model takes into account all forces which
act on a cluster of particles (gravity, buoyancy, aerodynamic
drag, aerodynamic lift, friction). Marjanovic [ 6] presents a
method for the comparative analysis of the influence of pipeline inclination on the energy loss in a pneumatic conveying
system. This method is based on the assessment of the angle
at which a solid particle moves in a pipeline and impacts on
the wall. This model includes, implicitly, only some of the
most influential terms. The results obtained, however, differ
by less than 5% from the full model of Muschelknautz and
Wojahn [3] for all conveying conditions. This simplified
model describes the overall pressure drop as the sum of two
terms. The first is the pressure drop for gas flow and the
second is the pressure drop caused by the presented solid
phase. This was written as
A p t = A p s + A p ~ Pi,-P~.....x
m~
L I
,
2p~ - Ag+
A -~p~v;
tively; pg and vg are the density and superficial velocity of the
gas, calculated at mean pressure p~; As is the well-known
parameter for calculating the pressure drop caused by gas
flow and the effective solids friction factor, A~,is an additional
parameter which takes into account all effects arising from
the presence of solid particles in the gas stream. This factor
is obtained through analysis of all the mentioned forces which
act on a cluster of particles:
A~=
A~ ÷ (vs/v~)Fr2
(2)
where W is the solid phase velocity, A* is a factor which takes
into account both impact and friction (to be determined
experimentally ), Fr ( - v~/ (gD) ' / 2) is the Froude number,
D is the pipe bore and/3 is a parameter which takes into
account the effects of the incline pipeline, defined by
fl = sin(a) + 14 cos(a)
v~
(3)
where t~ is the terminal velocity of a particle falling vertically,
and t~ is the angle of inclination of the pipe to the horizontal.
Determination of the slip velocity ratio represents a major
problem, particularly if the influences of particle-particle
impact and the gas velocity profile are to be taken into
account. The relationship given in this model is rather complex and a detailed analysis can be found in Refs. [ 2,3 ]. It
should be pointed out here that the slip velocity ratio depends
upon several parameters, e.g. vJv~=f(A*, 14, D, a, v~).
The terminal velocity 14 is often used in such correlations
since it is a function of the drag coefficient of the particle,
as is the slip velocity, and is rel:aively simple to measure.
Muschelknautz and Wojahn [ 3 ] tested a number of products
in order to determine the factor A* and the ratio between the
particle terminal velocity and the gas velocity. Their values
varied within the range 0.001-0.040 and 0.3-0.7, respectively, depending on both the material and the pipeline
characteristics, as well as the gas velocity. An empirical correlation for the velocity ratio was presented by Hinkle as a
function of the particle density and diameter [ 7 ].
Note that the range of the panicle terminal velocity to gas
velocity ratio can be estimated for differem particles. For
example, polyethylene pellets (mean size 3 ram, terminal
velocity 8.1 m/s) would normally be transferred in dilute
phase flow with an air superficial velocity between 14 and 35
m/s. This gives a velocity ratio between 0.23 and 0.58. Based
on the Hinkle [7] equation, the value of the velocity ratio for
3 mm polyethylene pellets was estimated as 0.66. Note that
the lower limit of the air superficial velocity range is the
minimum usually chosen in order to convey in a dilute phase
flow and the upper limit is the maximum air superficial
velocity necessary to minimise particle damage.
(l)
where Pi,, P~ and P8 are, respectively, the pressures at the
inlet and exit of the pipeline and their mean value; d~ and n~
are the mass flow rates of the gas and solid phases, respec-
2. Present study
In the present study, analytical, numerical and experimental investigations were conducted. The analytical model for
A. Levv et al. / Powder Technology 93 (1997) 253-260
gas-solid suspension flow through an inclined section of pipe
was developed. Unlike Marjanovic [6], whose analysis is
based on the forces acting on a single particle and the angle
at which the particle impacts the pipe wall, this model is
based on the dimensional analysis of the major forces acting
on a dispersed solids phase.
The three-dimensional conservation equations for steadystate two-phase flow in an inclined pipe were solved numerically for constant solids and gas flow rates at different pipe
inclinations. This model was based on the continuum theory
for describing the mass and momentum balance equations for
the fluid and solid phases. A packing model, describing the
shear stress of the solid phase as a function of its volume
fraction, is suggested in order to limit the maximum value of
the solid volume fraction. A new model for wall-particle
interaction was used during the numerical simulations.
In order to validate the model and the prediction of the
numerical simulations an experimental investigation was carried out. The prediction of the numerical model was compared
with experimental data obtained in a specially designed test
rig.
3. Analytical study
In order to determine the overall effect of the pipe's
angle of inclination, a, on the total pressure drop it was
decided to investigate Eq. ( 1). Note that the total pressure
drop in Eq. ( 1) differs by less than 5% from the full model
of Muschelknautz and Wojahn [ 3 ] for all conveying conditions, as noted by Marjanovic [61. Substituting Eqs. (2) and
(3) into Eq. ( I ) yields
LI
~
LI
,
+ r,p~ sin(a) L + r,p.,g cos(a) L t~
vs
(4)
where r, is the solid volume fraction. In order to find the
extreme value of the pressure drop, for given gas and particle
flow rates, as a function of the angle of inclination, tr, Eq.
(4) was differentiated with respect to a and compared with
zero.
c3pt 0p~
-----'~=r,p.,gcos(oO
L-r,p.,gsin(a) L t&---0
0a 0or
v~
(5)
This resulted in
tan(a¢~t) =v~
(6)
Differentiating Eq. (5) with respect to the angle of
inclination, a, gives
i~p, O2p.~
0tx,2=~---~= - r , o• , •g s i n ( a ) L - r ~
cos(or) L tIJg
t<0
(7)
255
For angles of inclination between 0 ° and 90° this second
derivative is always negative, hence a maximum value of the
pressure drop will be achieved in this range. The angle which
results in the maximum pressure drop, amax,can be estimated
for representative values of the particle density, volume fraction, and velocity ratio. For large particles in a dilute phase
flow the particle terminal velocity to gas velocity ratio varies
between 0.3 and 0.7. Hence, the maximum pressure drop will
occur at ~maxbetween 55 ° and 73 °. For similar particles conveyed in a dense phase mode of flow where the solid volume
fraction is much higher and the superficial gas velocity is
much lower the critical angle will be close to 0 °. Conversely,
a critical angle close to 90 ° will be achieved by small particles
conveyed in a dilute phase mode at very high velocities. From
this analysis it can be seen that the critical angle can vary
between 0 ° and 90° depending upon the particle characteristics (e.g. particle diameter, particle density, etc.) and the flow
regime.
4. Numerical study
A three-dimensional model based upon the concept of
interdispersed continua was used to model this flow. This
model solves the conservation equations for mass and
momentum for the gas and solid phases by using a finitevolume numerical method. The three-dimensional model
employed in this work is incorporated in the PHOENICS
software by CHAM, UK. The models and their solution are
described by Spalding [8]. This model uses a finite-volume
formulation of the conservation equations for mass and
momentum for two phases. PHOENICS provides solutions
to the discretised version of a set of differential equations
having the general form
O( riPA°i) t- V. ( ripiv~o~ + r~F,,V~o~) = r,S,,
Ot
(8)
where t is the evolution time, r~, p, and v~ are the volume
fraction, the density and the velocity of phase i, ~p~ is the
conserved property of phase i, such as enthalpy, momentum
per unit mass, etc., and F,, and S,, are the diffusion coefficient
and source term of ~o~.Since we are looking for a steady-state
solution the transient term in Eq. (8) was set to zero.
The concept of interpenetrating continua is used to obtain
the two-phase conservation equations by time and space averaging. These equations are solved using the Inter Phase Slip
Algorithm. This is an iterative method:
• starting with a guessed pressure field,
• the conservation equations are solved in the following
order: energy, mass, momentum;
• finally, a pressure correction equation is solved and the
velocities are corrected in order to ensure satisfaction of
the overall mass conservation equation.
This cycle is performed many times until the errors remaining
in all the equations are acceptably small.
256
A. Levy et aLI Powder Technology 93 (1997) 253-260
4.1. Inter-phase momentum transfer
Source terms in each of the conservation equations provide
the means of specifying the nature of the transfer of mass and
momentum between the phases. In this case the inter-phase
momentum is based upon that used for fluidisation processes,
since the range of solids concentrations experienced in pneumatic transport systems is similar. This has been employed
by both Patel and Cross [9] and Kuipers [ 10] for modelling
gas-solid fluidised beds.
The inter-phase friction force F is evaluated using
F = lpvVceH( Ug-- Us)
(9)
where Vceu is the volume of a control volume. For solids
concentrations greater than 0.2 the model computes the interphase friction coefficient. IpF. using the Ergun [ 1 1] equation:
Sk = - 2p~rgr~ 7E+k ¢p
(14)
S~ = - 2c~pjgr, ca + rp
(15)
where c~ is an empirical constant ( 1.0 in this case). ~-pand eE
are, respectively, time factors characterising the particle
response and large-scale turbulent motion:
Tp=pr,v,-- vJ I I
(16)
k
TE=CT-E
(17)
where the empirical constant CT= 0.35.
4.3. WaU friction model
lpF = 150 ~ ~2 + 1.75r~ ~ pgl ug-- vsl
rgd~
( l 0)
where/L is the dynamic viscosity of the gas phase and d~ is
the average diameter of the particles. For solids concentrations less than 0.2, the model computes the coefficient as
follows:
3r~ I2 rspgl vR- v,I
IpF = Core- 2.65 2d.
(11)
where the single-particle drag coefficient, Co, is given by
CD = max
( 1 + 0.15Re°6sT), 0.44
(12)
and is modified to take account of multiparticle effects
using the method of Richardson and Zaki [ 12 ]. The particle
Reynolds number is given by
Re=P~d~(rsl v s - v,I )
(13)
The friction force between the conveying gas and the pipe
wall was modelled by adding a source term to the gas phase
momentum equation for those control volumes adjacent to
the pipe wall. This term assumed no slip at the wall and a
logarithmic velocity profile, and calculated the wall friction
factor based upon a smooth pipe correlation.
The friction force between the solid phase and the pipe
wall was modelled by adding a source term to the solid phase
momentum equation for those control volumes adjacent to
the pipe wall. In order to take into account the inclined pipe
effect, a solid phase-wall friction source term was developed.
This was developed by usi:tg Eq. (4) and subtracting the
pressure drop caused by gravity from the pressure drop of the
solid phase, i.e.
Ap.~ - r~p~g sin(a) L
Ll
ffi A*r.~ -~ ~ p~v.~2 + r~p.~gcos(t~) L 14= ew
v~
(18)
As a consequence, the three-dimensional source term for
the solid phase-wall friction was written as
4.2. Turbulence model
S w = r ~ p , A w , , , ( k , f f g •.n p A*Iv,.
v.)
+ ~
The k-e turbulence model was used in this work to model
the gas phase turbulence, with additional source terms to take
account of the influence of the presence of particles. Several
extensions of the k-e turbulence model have been proposed,
and PHOENICS provides two options, the models of Chen
and Wood [ 13,14] and of Mostafa and Mongia [ 15]. The
effect of these source terms is to reduce the turbulence kinetic
energy k and the turbulence dissipation rate e. However,
depending on the relative extent of the reductions of k and e,
the turbulent viscosity may be either reduced or increased by
the presence of particles.
In this work the Mostafa-Mongia model of Ref. [ 15 ] was
used. The additional volumetric source terms in the k and e
equations are
( 19)
where Aw,o is the wall area of the control volume, kv ( - vt/
vg) is the particle terminal velocity to gas velocity ratio and
t~p is a unit vector perpendicular to Aw,u.
4.4. Other relationships
The gas was assumed to behave as a perfect gas, and the
flow was assumed to be isothermal. Thus the conservation
equations for energy were not solved. A fixed temperature
was assigned so that values such as the gas density could be
calculated.
Normally the mass conservation equations are solved for
both phases, which can lead to problems when the concen-
257
A. Levy et al. / Powder Technology 93 (1997) 253-260
tration of one of the phases is significandy less than the other.
If this is the case then only the equation for the dispersed
phase is solved, with the volume fraction of the continuous
phase determined from
In this study this second option was used with only the
solids mass conservation equation being solved. When solving the momentum conservation equation for one of the
phases, the code solves the conservation equations for the
velocity component in each coordinate direction. The turbulence model requires the solution of two conservation equations, one for the turbulence kinetic energy and one for the
turbulence dissipation rate. Thus a set of ten equations must
be solved for this three-dimensional two-phase problem ( nine
conservation equations and the pressure correction equation).
5. Experimental study
The influence of pipeline inclination on pressure gradient
for two-phase gas-solid flow was investigatedexperimentally
in order to validate the predictions of the physical model. The
experiments were conducted in an industrial-scale pneumatic
conveying system at the Centre for Industrial Bulk Solids
Handling in the Department of Physical Sciences at Glasgow
Caledonian University.
A schematic illustration of the experimental system is
shown in Fig. 2. This system was operated as a batch process.
Air was supplied from a Roots type blower capable of delivering up to 0.33 m3/s of free air when running in a positive
pressure mode. The bulk material was fed into the pipeline
via a variable-speed rotary valve from a 0.7 m 3 feed hopper.
The gas-solid mixture was then transported through a pipeline approximately 100 m long, with an internal diameter of
81 ram, to the receiving hopper which was mounted directly
above the feed hopper. This arrangement simplifies the task
of conveying a batch of material at a number of operating
conditions.
Receiving
F~t
Hopper
Hopper
Rotary Valve
InclinedSection
Material
Polymer pellets
Mean diameter
3 mm
Particle density
880 kg/m 3
Bulk density
550 k g / m 3
(20)
rcontmuou s "~- 1 - rdlsper,~ d
Filter
Table 1
Basic material properties
J ~
Fig. 2. Isometric sketch of pipeline loop.
The blower supplying air to the system operated at a fixed
speed, and so a bleed valve was used to control the air flow
rate to the system. Because of the way it was operated, air
also leaked through the solids feeder. Thus the air flow rate
was measured at the end of the pipeline after the filter using
a thermal mass flow meter capable of measuring up to 0.2
kg/s with an accuracy of + 2% of the reading. The solids
mass flow rate was also determined at the end of the system
by measuring the mass of solids collected in the receiving
hopper. This hopper was mounted independently on three
500 kg load cells with an accuracy of + 0 A kg. The solids
mass flow rate was determined by differentiatingthe load cell
transient. Pressure and temperature measurements were made
at a number of locations around the system. Pressure was
measured using single-ended transducers with a range of 02.5 bar and an accuracy of ±0.006 bar. Temperature was
measured using probes with a range of 0-100°C with an
accuracy of +0.5°C. Data from all the ~:~;msducers were
recorded by a computer-controlled data logger, usually at a
rate of I Hz.
Polyethylene pellets were used as the test material in this
programme. Table 1 lists the properties ofthesepellets. Pressure gradients were measured at 0 °, 45 °, 60°, 75 ° and 90 °
orientations, superficial gas velocities ranged from 10 to 20
m/s and solids throughputs from 0.25 to 0.5 kg/~. The experimental data for the pressure gradients data, as measured for
10 different angles of inclination and solids and air mass flow
rates, are presented in Table 2.
6. Comparison between the numerical prediction and
the experimental result and parametric investigation
In order to validate the physical model (described in Section 4), it was solved numerically for several different initial
conditions and angles of inclination and compared with
experimental results. A comparison between the predictions
of the numerical simulation and the analytical model with
experimentally determined pressure gradients at various initial conditions is presented in Table 3. From this it can be
seen t~mt the numerically predicted pressure gradients are up
m 20% lower than the experimental values. The predicted
pressure gradient is mainly influenced by models for the drag
force on a particle, the solids-wall friction, and the turbulence
reduction due to particles. A combination of one or more of
the following would result in an increase in the predicted
pressure gradient: an increase in the particle drag coefficient,
a higher coefficient of particle-wall friction, or a decrease in
the turbulence reduction. From this it is clear that some refinement of these models is required in order to predict the flow
more accurately. Since the difference between the predicted
258
A. Levy et al, / Powder Technology 93 (1997) 253-260
Table 2
Experimental data for the pressure gradients as measured tbr different angles of inclination and mass flow rates
Test no.
Air mass flow rate (kg/s)
Solids loading ratio
Angle of inclination ( deg. )
Experimental data for pressure gradient (mbar/m)
I
2
3
4
5
6
7
8
9
10
0.131
0.114
0.123
0.109
0.126
O.126
0.127
0.103
O.113
0.106
3.24
5.02
3.01
3.85
2.26
4.45
4.59
6.1
4.47
4.06
0
0
45
45
60
60
75
75
90
90
1.89
1.93
2.79
2.85
1.89
3.33
2.72
3.3
2.56
2.32
Table 3
Comparison between the predictions of the numerical simulation and the analytical model with the experimental results for the pressure gradient
Test no.
!
2
3
4
5
6
7
8
9
I0
Pressure gradient (mbar/m)
Error ( % )
Experimental data
Numerical prediction
Analytical prediction
Numerical prediction
Analytical prediction
1.89
1.93
2.79
2,85
1.89
3.33
2.72
3.3
2.56
2.32
1.97
1.86
2.24
2.75
1.72
2.70
2.94
2.69
2,65
2.22
I. I0
1.41
1.65
2.01
1.35
2.45
2.41
3.45
2.10
1.88
4.23
3.63
19.7%
3.51
8.99
18.99
8.09
18.48
3.52
4.31
41.9
27. I
40.9
29.6
28.6
26.4
I 1,3
4.5
18.1
19.2
and measured pressure gradients for horizontal and vertical
pipes was much lower than for the inclined pipes, the wall
friction model is probably the most likely cause of error.
Unlike the numerically predicted pressure gradient, the
error in the analytically predicted gradient is much greater
(up to 42%) for angles of inclination between 0° and 60 °. It
should be noted that, in calculating the pressure gradient, Eq.
(4) was used with the same values for the parameters as used
in the numerical study. One of the key parameters in the
analytical model is the particle velocity. A number of correlations for this parameter were examined and the correlation
of Muschelknautz and Wojahn [ 3] was selected, since it was
the only model which took into account the angle of inclination. One of the main reasons for the analytical model
underestimating the pressure gradient was that the particle
velocity was overestimated.
in general, the comparison showed satisfactory agreement
between the predictions of the numerical simulations and the
experimental data for the pressure gradients. Since it is not
possible to change the angle of inclination, or, continuously
during the experiments and to find the critical angle ~,,..,xfor
a constant set of initial conditions, it was decided to find the
critical angle numerically for different solids loading ratios.
The physical model was solved numerically for three different soiids loading ratios, 5, 10 and 20. This was done by
keeping a constant air flow rate, having an inlet superficial
velocity of 9.2 m/s and changing the mass flow rate of the
solids. Note that due to the compressibility of the air the
superficial velocity changes along the pipe. For different solids loading ratios the inter-phase exchange rates of momentum and energy change significantly. This results in a
different average superficial gas velocity and hence a change
in the critical angle am~x.
The difference between the total pressure gradient for an
inclined pipe and that for a horizontal pipe, under the same
flow conditions, as a function of the angle of inclination is
shown in Fig. 3. Data are presented for solids loading ratios
of 5, 10 and 20. It should be noted that the pressure gradient
for a horizontal flow is also dependent on the solids loading
ratio and grows continuously as the solids loading ratio
increases. From this figure it can be seen that by doubling the
solids loading ratio the maximum difference in the pressure
gradients increases by almost the same amount. Hence, one
could conclude that this maximum is proportional to the
solids mass flow rate.
Fig. 4 represents the normalised pressure gradient (pressure gradient for a particular angle of inclination divided by
that for the horizontal) as a function of the angle of inclination
for different solids loading ratios. This figure shows that the
normalised pressure gradient ranges from 1 for horizontal
flow to 1.5-1.6 at the critical angle C~m,,x. The normalised
pressure gradient increases when the solids loading ratio is
A. Levy et al. / Powder Technology 93 (1997) 253-260
3.5
8
]
•.-e-SLR = 5
3 :-4-SLR = 10
~
]
I
SLR = 20
2.5
i g 15
~
o.
0.5
0
0
10
20
30
40
50
60
70
80
90
Inclination Angle [Deg.]
Fig. 3. The total pressure gradient difference between that of an inclined
pipe and that of a horizontal pipe for different solids loading ratios (SLR)
as a function of the inclination.
_o
"0
1.6
-e-SLR = 5
~ " ~ 1 ~
]
8. List of symbols
~ 1.3
R,
I 1.2
I
shear stress of the solid phase as a function of its volume
fract[on, is suggested in order to limit the maximum value of
the solid volume fraction. A new model for particle-wall
interactions was developed, taking into account the angle of
inclination of the pipe.
The predictions of the nu~.rical model were compared
with experimental data obtained in a specially designed test
rig. In general, the agreement between experimental data and
the models was satisfactory. The results of numerical simulation also confirmed that the critical pipe angle for gas-solid
flow is lower than 90 °.
The analytical and numerical models developed in this
study are not restricted to a particular particle size, although
they have only been validated with experimental data for
relatively large particles. The authors plan further experiments with other bulk materials to extend the validation of
the models over a range of particle sizes.
1.5
0
-=
,,,,= 1.4
IA
o
10 20 30 40 50 60 70 80 90
Indinstion Angle [Dog.]
Fig. 4. The normalised pressure gradient (pressure gradient at an angle
divided by that at the horizontal) as a function of the angle of inclination
for different solids loading ratios (SLR).
A,,~.
CD
d~
D
F
Fr
g
IPF
k
wall area of control volume
drag coefficient
particle diameter
diameter of pipe section
inter-phase friction force
Froude number
gravity acceleration
inter-phase friction factor
turbulence kinetic energy
changed from 5 to 10. A further increase to 20 results in a
reduction of the normalised pressure gradient. This was due
to a significant increase in the predicted horizontal pressure
gradient at this solids loading ratio.
k~
q/v~
L
th
~p
p
r
7. Conclusions
S,
S~,
Sk
S~
v
V~.
x
length of pipe section
mass flow rate
unit vector perpendicular to Aw~,
pressure
volume fraction
Reynolds number
solid phase-wall friction source term
source term of q~
source term for particle effects on k
source term for particle effects on E
velocity
volume of control volume
axial distance
Re
An analytical model for gas-solid suspension flow through
an inclined section of pipe was developed. This model predicts the ratio of the total pressure drop in an inclined pipe to
that of a horizontal pipe. The model has been used to predict
the critical pipe angle which gives the maximum pressure
drop for given solids flow rate, or the minimum solids flow
rate for a given pressure drop. This angle differs from 90 °
( found in a single-phase flow) and is directly proportional to
the ratio between the gas superficial velocity and the particle
terminal velocity. Further experimental studies would be
required to confirm this result for other bulk materials.
The three-dimensional conservation equations for steadystate two-phase flow in an inclined pipe were solved numerically for constant solids and gas flow rates at different pipe
inclinations. This model was based on the continuum theory
for describing the mass and momentum balance equations for
the fluid and solid phases. A packing model, describing the
259
Greek letters
o[
O[ma X
/3
E
A~
A,
A.*
/x
inclination angle
critical pipe angle
defined in Eq. (3)
diffusion coefficient of ~Pi
turbulence dissipation rate
gas friction factor
Muschelknautz effective solids friction factor
Muschelknautz solids-wall friction factor
dynamic viscosity of gas phase
260
P
TE
l"w
A Levy et al. / Powder Techmdogy 93 (1997) 253-260
density
time factor for large-scale turbulent motion
time factor for particle response to turbulence
wall shear stress
conserved property of phase i
Subscripts
g
s
i
in
ex
t
gas phase property
solid phase property
phase i property
inlet property
exit property
total or terminal property
References
[ 1] R.D. Marcus, L.S. Leung, G.E. Klinzing and E Rizk, Pneumatic
Conveying of Solids, Chapman and Hall, London, UK, 1990.
[2] E. Mu~helknautz, VDI Forschungsh., 25 (1959) 476.
[3] E. Muscbelknautz and H. Wojahn, Chem.-Ing.-Tech., 46 (6) (1974)
223-236.
[4] A. Ginestet, J.F. Large, P. Guigon and J.M. Beeckmans, A new
classification of flow regimes in vertical and inclined pneumatic
transport, 2nd Int. Syrup. Reliable Flow of Particulate Solids, Oslo,
Norway, 1993, EFChE Publication Series No. 96, Powder Science and
Technology Research, Porsgrunn, Norway, pp. 685-71 I.
[5l P.E. Salt, S.M. Cohen and J.C. Whalen, Affect of inclined lines on
pneumatic conveying systems, Proc. 14th Int. Conf Powder and Bulk
Solids Handling and Processing, Rosemont, IL, USA, 1989, Cahners
Exposition Group, Des Plaines, IL, pp. 670-680.
[6] P. Marjanovic, Pneumatic conveying in an incline pipe: inappropriate,
unfortunate or wrong concept? Proc. Powder and Balk Solids Conf.,
Rosemont Convention Center, Rosemont, IL, USA, 1994, Reed
Exhibition Companies, Des Plaines, IL, pp. 235-251.
[ 7l B.L. Hinkle, Ph.D. Thesis, Georgia Institute of Technology, Atlanta,
GA, 1953.
[8] D.B. Spalding, Numerical computation of multi-phase fluid flow and
heat transfer, in C. Taylor and K. r~organ (eds.), Recent Advances in
Numerical Methods, Vol. 1, Pinefidge Press, Swansea, UK, 1980, Ch.
5, pp. 139-167.
[9] M.K. Palel and M. Cross, The modelling of fluidised beds for ore
reduction, in C. Taylor, P.N. Gresho, R.L Sani and J. Hanser (eds.),
Numerical Methods in Laminar and Turbulent Flow, Pineridge Press,
Swansea, UK, 1989, pp. 2051-2090.
[10] J.A.M. Kuipers, A two-fluid micro-balance model of fluidized
beds, Ph.D. Thesis, Twente University of Technology, Enschede,
Netherlands, 1990.
[ l i I S. Ergun, Fluid flow through packed columns, Chem. Eng. Prog., 48
(2) (1952) 89-94.
[12] J.F. Richardson and W.N. Zaki, Sedimentation and Fluidization,
Part I, Trans. Inst. Chem. Eng., 32 (1954) 35.
{ 13] C.P. Chen and P.E. Wood, A turbulence closure model for dilute
gas-particle flows, Can. J. Chem. Eng., 63 (1985) 349.
[ 14l C.P. Chen and P.E. Wood, Turbulence closure modelling of the dilute
gas-particle axisymmetric jet, AIChE J., 32 ( 1) (1986) 163.
[ 15] A.A. Mostafa and H.C. Mongia, On the interaction of particles and
turbulent fluid flow, Int. J. Heat Mass Transfer, 31 (10) ( 1988 ) 2063.

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