Z. AKYÜREK: ONE-DIMENSIONAL MATHEMATICAL MODELING OF THE CAPSULE FLOW IN A HORIZONTAL PIPE
559–564
ONE-DIMENSIONAL MATHEMATICAL MODELING OF THE
CAPSULE FLOW IN A HORIZONTAL PIPE
ENODIMENZIONALNO MATEMATI^NO MODELIRANJE
PRETOKA LEDENIH KAPSUL V HORIZONTALNEM CEVOVODU
Zuhal Akyürek
*
Burdur Mehmet Akif Ersoy University, Faculty of Engineering and Architecture, Istiklal Campus, 15030 Burdur, Turkey
Prejem rokopisa – received: 2018-10-12; sprejem za objavo – accepted for publication: 2019-03-11
doi:10.17222/mit.2018.221
Horizontal hydraulic transport of ice particulates suspended in water is a well-known practice used for cooling systems in
industry. Many advantages of hydraulic transport include its environmentally friendly application and a relatively low cost of
maintenance and operation. This study presents a one-dimensional mathematical model, developed for simulating a pressure
drop of solid and water two-phase fluid flow inside a horizontal pipe. The impact of different parameters such as suspen-
sion-wall friction coefficients, suspension-viscosity models and solid volumetric concentrations on the pressure drop of the
capsule flow were evaluated. The results of the simulations were validated by the results of two different experiments. The first
experiment was performed with anomalous-shaped capsules, which had a higher density than the density of the liquid flow
while the second experiment was performed with capsules having a spherical shape and a density lower than the density of the
carrier liquid flow in the pipe. The measurements were carried out in the horizontal pipe. In both experiments, the carrier fluid
was water. The results of this modeling study revealed that the proposed model can be effectively used to predict the pressure
drop of ice and water two-phase flow with sufficient precision.
Keywords: two-phase flow, pressure drop, capsule flow, suspension viscosity
Horizontalni hidravli~ni transport delcev ledu v vodni raztopini je dobro znana praksa, uporabljena v industrijskih hladilnih
sistemih. Hidravli~ni transport je okolju prijazen in ima mnoge prednosti zaradi relativno nizkih stro{kov obratovanja in
vzdr`evanja. Avtorji predstavljajo {tudijo enodimenzionalnega matemati~nega modela, ki so ga razvili za simulacijo padca tlaka
v horizontalnem cevovodu med pretakanjem dvofaznega medija (suspenzija trdno/kapljevina). Ovrednotili so vpliv razli~nih
parametrov, kot so koeficienti trenja med steno in suspenzijo, modelov za viskoznost suspenzije in volumske koncentracije na
padec tlaka med pretokom ledu v obliki kapsule. Veljavnost rezultatov simulacij so ocenili s pomo~jo podatkov, dobljenih z
dvema razli~nima eksperimentoma. Prvi eksperiment so izvedli z nepravilno obliko kapsul, ki ima vi{jo gostoto kot pretok same
kapljevine. Drugi eksperiment pa je bil izveden s krogli~no obliko kapsul in ni`jo gostoto, kot jo ima dejanski nosilni medij v
cevovodu. Vse eksperimentalne meritve so bile izvedene v horizontalnem cevovodu. V obeh primerih je bila voda nosilni medij.
Rezultati modelnih {tudij so pokazali, da lahko predlagani model z zadovoljivo natan~nostjo u~inkovito uporabimo za napoved
padca tlaka pri pretoku dvofaznega (led in voda) medija.
Klju~ne besede: dvofazni pretok, padec tlaka, pretok kapsul, viskoznost suspenzije
1 INTRODUCTION
Hydraulic transport of capsules in pipelines have
become a subject of intense research in recent decades.
Capsule transportation through horizontal pipelines is
employed in many engineering processes.
1–9
Two-phase
flow of ice and water in cylindrical tubes is most widely
used in cooling systems in industry.
10
The use of ice-
water slurries in district cooling systems can be con-
sidered as one of the most crucial technical improve-
ments for the industrial development. In addition, glo-
bal-warming impacts of chemical refrigerants, especially
ozone-depletion concerns caused the refrigeration indus-
try to search for environmentally friendly refrigerant
options.
11
Moreover, ice-water slurry has a much higher
energy capacity compared to the conventional
chilled-water systems and it produces a much greater
cooling potential at the same pumping cost.
12
On the
other hand, using ice capsules is much more advanta-
geous compared to ice slurries due to a lack of dispersion
of the particles leading to a blockage in the pipes. The
cooling capacity in district systems can be improved by
using ice capsules instead of ice slurries due to their high
concentration and flow velocity. In the literature, several
studies focused on the hydrodynamic behavior of a two-
phase ice-water flow
1,4–10,13–16
and also on the transfer of
heat during the capsule flow.
12,17–20
Vast studies on two-phase ice-water flow experiments
are available in the literature; however, investigations on
the characterization of flow behavior are very limited.
This is mainly due to high complexity of the mathema-
tical modeling of two-phase flow such as ice and
water.
7,17,21,22
Intensive research is required to precisely
understand and be able to make prediction on the flow
behavior, and to search for the ways of improving their
processing. Therefore, the motivation behind this study
was developing a one-dimensional mathematical model
to simulate the pressure drop of the flow of an ice-water
Materiali in tehnologije / Materials and technology 53 (2019) 4, 559–564 559
UDK 519.711:622.685:621.643.2 ISSN 1580-2949
Original scientific article/Izvirni znanstveni ~lanek MTAEC9, 53(4)559(2019)
*Corresponding author's e-mail:
drzuhalakyurek@gmail.com

two-phase mixture inside a horizontal pipe. In the
modeling study, the parameters having a high impact on
the pressure drop of a capsule flow system such as
suspension-viscosity model, suspension-wall friction
coefficient and volumetric concentrations were investi-
gated. The results of the developed mathematical model
were validated with two different sets of experimental
data obtained from the literature. The first set of data
13
(A) referred to a capsule flow of an anomalous shape
having a higher density than the fluid, and the second set
of data
15
(B) was adopted from an experiment performed
with spherical-shaped capsules having a lower fluid
density.
2 MODEL DESCRIPTION
Modeling of two-phase flows such as ice and water is
a challenging process due to the fact that it depends on
many factors, resulting in a wide range of flow regimes.
Therefore, developing a simplified model, which can
describe the two-phase flow of an ice-water system with
a suitable precision is necessary. In this respect, a one-
dimensional mathematical model is developed and
solved with the Gauss-Seidel method using MATLAB.
The assumptions presented below were made to solve
the governing equations in the model for the capsule-
carrying pipeline:
Assumption 1: Homogenous flow
Assumption 2: Fully developed flow
Assumption 3: The influence of the heat transfer/
pressure drop along the pipe on the solid-phase con-
centration is negligible
Assumption 4: Buoyancy effects of the phases differ
Assumption 5: The suspension-wall friction is inde-
pendent of the water temperature
Figure 1 shows a schematic illustration of an ice-
water flow. As can be seen from the figure, a control vol-
ume is defined within the pipe where the ice-water
two-phase flow is modeled.
A flow chart of the model is presented in Figure 2.
The figure includes all the governing equations and cor-
relations for the steady-state one-dimensional homo-
geneous equilibrium flow in a horizontal pipe.
3 RESULTS AND DISCUSSION
The mathematical model of a homogenous two-phase
flow developed in this study was validated with the
published experimental data.
13,15
In the experiments,
two-phase capsule-flow measurements were carried out
in a horizontal straight pipe (plexiglass) where water was
the carrier fluid. In the first set of experiments,
13
the
accuracy of the liquid velocity was found to be about
2.5 %, whereas the accuracy of the capsule velocity was
1.5 %. During the pressure-drop measurements, the
accuracy of was about 3 %. Anomalous-shaped capsules
having their density greater than that of the carrier liquid
were used in these experiments. With the second data
set,
15
spherical capsules, fabricated from polypropylene
having their density lower than that of the water were
used. In this study, the flow rate was measured with
water flow meters having an average deviation of 0.5 %
(max. 0.1 %) from the actual values. The measurements
of the pressure drop were carried out on a 4-m section of
the pipe. The ratio of the capsule diameter to the pipe
diameter (d/D) was 0.8. The main parameters used in the
experiments and the conditions are given in Table 1.
Z. AKYÜREK: ONE-DIMENSIONAL MATHEMATICAL MODELING OF THE CAPSULE FLOW IN A HORIZONTAL PIPE
560 Materiali in tehnologije / Materials and technology 53 (2019) 4, 559–564
Figure 2: Flow chart of the modeling study
Figure 1: Schematic illustration of the spherical capsule in a pipe
flow: a) the side view of the pipe, b) the front view of the pipe and its
equivalent control volume

Table 1: Parameters used in experimental studies
Experiment A
13
Experiment B
15
Capsule geometry Anomalous Spherical
L (m) 2 4
D (m) 0.05 0.1
d/D (–) 0.8 0.8
 c
(kg/m
3
) 1200 870
C (–) 0.20-0.50 0.05-0.30
Reynolds number (–) 2.5 × 10
4
<Re<
1.5×10
5
1.2×10
4
<Re<
1.5×10
5
Fluid temperature (°C) 20 20
For the model validations, the suspension viscosity
and suspension-to-wall friction values were calculated
with the formulas given in Figure 2.
The comparison of the simulation results for the first
experimental data are shown in Figure 3. As can be seen
from the figure, for the capsule flow of an anomalous
shape and having a higher density than the carrier-liquid
density, the increasing average velocity of the carrier re-
sulted in an increase in the pressure drop. The simulation
results are in good agreement with the experimental re-
sults at capsule concentrations of (0.20, 0.30 and 0.50).
The simulation results of pressure-drop calculations
give a sufficient accuracy at low capsule concentrations.
The anomalous-capsule concentration is greater than
30 %, the solution diverges and the accuracy reduces. In-
teractions within the phases change the hydrodynamic
behavior of the flow. For the pressure-drop simulations
at higher concentrations, a separate flow model can be
applied. In this study, a homogenous-flow approximation
is used in order to have a vision about the variation in the
pressure drop of the capsule flow.
The comparison of the simulation results for experi-
mental data set B are given in Figure 4. In this set,
spherical capsules having a lower density than the water
were carried by the flowing fluid in the pipe. As it can be
seen from the figure, the simulation results are in good
agreement with the experimental data. The increase in
the carrier velocity resulted in a higher pressure drop at
capsule concentrations of (0.05, 0.10, 0.15 and 0.20).
The results of the simulation showed a small discrepancy
at capsule concentrations of (0.05 and 0.20) values.
In the two-phase flow of ice-water capsule system,
parameters such as suspension-to-wall friction coeffi-
cients and solid-phase concentrations ( /%) have a
significant impact on the analysis of the pressure drop.
Different suspension-viscosity models and suspension-
to-wall friction coefficients were applied in order to
investigate the change in the pressure drop of the
two-phase ice-water mixture. The cases analyzed (Case 1
and Case 2) are shown in Table 2. In the model valida-
tion, Case 2b was considered as the base case. Spherical
particles of ice with a 0.08 m diameter and water as the
carrier were used in the analyses. The temperature of the
carrier was taken as 20 °C for various Cases related to
Z. AKYÜREK: ONE-DIMENSIONAL MATHEMATICAL MODELING OF THE CAPSULE FLOW IN A HORIZONTAL PIPE
Materiali in tehnologije / Materials and technology 53 (2019) 4, 559–564 561
Figure 4: Comparison of simulation results for experimental data set
B at different concentrations: a) C = 0.05, b) C = 0.10, c) C = 0.15, d)
C = 0.20
Figure 3: Comparison of simulation results for experimental data set
A at different concentrations: a) C = 0.20, b) C = 0.30, c) C = 0.50

the suspension viscosity (Cases 1b and 2b) and to the
suspension-wall friction factors (Cases 1a and 1b) at
capsule concentrations within the range of 0.05–0.25.
Figure 5 presents the impact of the suspension-to-
wall friction factor on the pressure drop. As can be seen
from the figure, the pressure-drop variation, with respect
to the Reynolds number, follows a similar trend in both
Cases (1b and 2b). When the Blasius friction-factor
correlation was used in the pressure-drop calculations, a
deviation from the actual values was observed, especially
at high Reynolds-number values.
Table 2: Cases considered in the model
17,23,24–27
Case 1
Case 1a Case 1b
μ
m
μ
Liquid
(1 + 2.5C)
μ
Liquid e
51
3
() −  f
64
Re
m
,
Rem
< 2300
0 0791 .
Re
m
0.25
,
Rem
 2300
Case 2
Case 2a Case 2b
μ
m
μ
Liquid
(1 + 2.5C)
μ
Liquid e
51
3
() −  f
2
6565
⋅
Re
m
1.5
,
2800 < Rem < 15000
2
0 0395
⋅
.
Re
m
0.25
,
15000 < Rem < 32000
The pressure-drop variation, with respect to the
empirical suspension-viscosity formulations, is presented
in Figure 6 for Cases 1a and 1b. Both formulations
showed to have a similar impact on the pressure drop of
the ice-water two-phase mixture under consideration.
The maximum variation in the pressure drop amounted
to 10.7 %. This implies that for the ice-water mixtures
both formulations can be applied to homogeneous model
calculations. The impact of the suspension viscosity and
suspension-to-wall friction on the pressure drop was
found to be greater at high values of the Reynolds
number whereas no significant impact of these factors on
the pressure drop was observed at low values of the
Reynolds number.
The pressure-drop variation with the capsule concen-
tration of the ice-water two-phase mixture is shown in
Figure 7. As can be seen from the figure, higher capsule
concentrations resulted in higher pressure gradients. The
pressure drop also reached higher values with the in-
crease in the Reynolds number. The increase in the con-
centration raised the impact of the friction factor.
4 CONCLUSIONS
The modelling of two-phase flows is a challenging
and complex process, especially for liquid-solid flows.
Therefore, a simplified model has to describe the ice-
water two-phase flow characteristics with a satisfactory
accuracy. In this regard, in the present study, a one-
dimensional model was developed as the first approxi-
mation. In the simulation model, the pressure drop
during the ice-water two-phase-mixture flow inside the
pipe is described. The impact of different suspension-
viscosity models, suspension-wall friction coefficients
and volumetric capsule concentrations on the pressure
drop of the capsule flow were investigated throughout
Z. AKYÜREK: ONE-DIMENSIONAL MATHEMATICAL MODELING OF THE CAPSULE FLOW IN A HORIZONTAL PIPE
562 Materiali in tehnologije / Materials and technology 53 (2019) 4, 559–564
Figure 7: Effect of volumetric solid concentration on the pressure
gradient (base case, spherical capsules, d/D = 0.8)
Figure 5: Effect of suspension-wall friction coefficients on the
pressure gradient (spherical capsules, d/D = 0.8, C = 0.10)
Figure 6: Effect of suspension viscosities on the pressure gradient
(spherical capsules, d/D = 0.8, C = 0.10)

the study. The simulation results were also validated with
the results obtained from two different experimental
studies. One of the experiments was performed with
capsules having an anomalous shape and density greater
than the density of the carrier liquid and the other one
was performed with spherical-shaped capsules with a
density lower than the carrier liquid. In both experi-
ments, measurements were carried out in a horizontal,
straight plexiglass pipe and water was used as the carrier.
This study revealed that a steady-state, homogenous
model approach can be used with confidence for the
calculations of the pressure drop of the solid-liquid
two-phase mixture in a horizontal pipe. The model
results are reliable, especially for the spherical capsules
at a volumetric concentration of up to 25 % and for ano-
malous capsules at concentrations lower than 50 %. The
suspension-wall friction factor
17
and suspension-visco-
sity models
27
used in the calculations were found to be
applicable to the pressure-drop calculations for solid-
liquid two-phase mixtures.
Acknowledgement
This research did not receive any specific grant from
the funding agencies in the public, commercial or non-
profit sectors. The author would like to thank Professor
Dr. Afsin Güngör for his valuable comments.
5 REFERENCES
1
Y. Zheng, J. X. Zhu, J. Wen, S. A. Martin, A. S. Bassi, A. Margaritis,
The axial hydrodynamic behavior in a liquid-solid circulating
fluidized beds, Can. J. Chem. Eng., 77 (1999), 284–290,
doi:10.1002/cjce.5450770213
2
Q. Lan, A. S. Bassi, J. X. Zhu, A. Margaritis, Continuous protein
recovery with a liquid-solid circulating fluidized bed ion exchanger,
AIChE J., 48 (2002), 252–261, doi:10.1002/aic.690480209
3
D. G. Karamanev, T. Nagamune, I. Endo, Hydrodynamic and mass
transfer study of a gas-liquid-solid draft tube spouted bed bioreactor,
Chem. Eng. Sci., 47 (1992), 3581–3588, doi:10.1016/0009-2509
(92)85073-K
4
P. A. Shamlou, Hydraulic transport of particulate solids, Chem. Eng.
Commun., 62 (1987), 233–240, doi:10.1080/00986448708912062
5
M. S. Seki, R. Skalak, Asymmetric flows of spherical particles in a
cylindrical tube, Biorheology, 3 (1997), 155–169, doi:10.1016/
S0006-355X(97)00023-1
6
M. S. Seki, Motion of a sphere in a cylindrical tube filled with a
brinkman medium, Fluid Dyn. Re., 34 (2004), 59–76, doi:10.1016/
j.fluiddyn.2003.08.007
7
S. K. Mishra, C. Handra, A. Arora, Complex internal fluid flow
behavior, Int. J. Adv. Man. Tech. and Eng. Sci., 8 (2018), 1461–1467
8
T. Asim, A. Algadi, R. Mishra, Effect of capsule shape on hydro-
dynamic characteristics and optimal design of hydraulic capsule
pipelines, J. Pet. Sci. Eng., 161 (2018), 390–408, doi:10.1016/
j.petrol.2017.12.001
9
T. Asim, R. Mishra, S. Abushaala, A. Jain, Development of a design
methodology for hydraulic pipelines carrying rectangular capsules,
Int. J. Pres. Ves. Pip., 146 (2016), 111–128, doi:10.1016/j.ijpvp.
2016.07.007
10
D. Uluarslan, Experimental investigation of the effect of diameter
ratio on velocity ratio and pressure gradient for the spherical capsule
train flow, Eur. J. Mech. B/Fluids, 37 (2013), 42–47, doi:10.1016/
j.euromechflu.2012.05.004
11
P. W. Egolf, M. Kauffeld, From physical properties of ice slurries to
industrial ice slurry applications, Int. J. Refrig., 28 (2005), 4–12,
doi:10.1016/j.ijrefrig.2004.07.014
12
B. D. Knodel, D. M. France, U. S. Choi, M. W. Wambsganss, Heat
transfer and pressure drop in ice-water slurries, Appl. Therm. Eng.,
20 (2000), 671–685, PII: S1359-4311(99)00046-0
13
P. Vlasak, An experimental investigation of capsules of anomalous
shape conveyed by liquid in a pipe, Powder Technol., 104 (1999),
207–213, doi:10.1016/S0032-5910(99)00096-0
14
A. E. Vardy, J. M. B. Brown, Transient turbulent friction in fully
rough pipe flows, J. Sound Vib., 270 (2004), 233–257, doi:10.1016/
S0022-460X(03)00492-9
15
D. Ulusarslan, I. Teke, An experimental investigation of the capsule
velocity, concentration rate and the spacing between the capsules for
spherical capsule train flow in a horizontal circular pipe, Powder
Technol., 159 (2005), 27–34, doi:10.1016/j.powtec.2005.05.059
16
D. Ulusarslan, I. Teke, An experimental determination of pressure
drops in the flow of low density spherical capsule train inside
horizontal pipes, Exp. Therm. Fluid Sci., 30 (2006), 233–241,
doi:10.1016/j.expthermflusci.2005.07.003
17
R. V. Garic-Grulovic, Z. B. Grbavcic, Z. L. Arsenijevic, Heat transfer
and flow pattern in vertical liquid-solids flow, Powder Technol., 145
(2004), 163–171, doi:10.1016/j.powtec.2004.07.006
18
K. A. R. Ismail, J. R. Henriquez, Solidification of PCM inside a
spherical capsule, Energy Convers. Manag., 41 (2000), 173–187
19
A. V. Wilchinsky, S. A. Fomin, T. Hashida, Contact melting inside an
elastic capsule, Int. J. Heat Mass Transfer, 45 (2002), 4097–4106
20
Y. Huo, Z. Rao, Investigation of solid-liquid phase change in the
spherical capsule using axisymmetric lattice Boltzmann model, Int. J.
Heat Mass Transfer, 119 (2018), 1–9, doi:10.1016/j.ijheatmass-
transfer.2017.11.099
21
J. Ling, P. V. Skudarnov, C. X. Lin, M. A. Ebadian, Numerical inves-
tigations of liquid-solid slurry flows in a fully developed turbulent
flow region, Int. J. Heat Fluid Fl., 24 (2003), 389–398,
doi:10.1016/S0142-727X(03)00018-3
22
V. Gnielinski, Forced convection in ducts, Handbook of Heat
Exchanger Design, Begell House, New York 1992
23
R. B. Bird, W. E. Steward, E. N. Lightfoot, Transport Phenomena, J.
Wiley, New York 1960
24
D. G. Thomas, Transport characteristics of suspension, J. Colloid
Sci., 20 (1965), 267–277, doi:10.1016/0095-8522(65)90016-4
25
A. Kitanovski, A. Poredos, Concentration distribution and viscosity
of ice-slurry in heterogeneous flow, Int. J. Refrig., 25 (2002),
827–835, doi:10.1016/S0140-7007(01)00091-3
26
H. A. Barnes, An introduction to rheology, Elsevier, Amsterdam
1989
27
E. Barnea, J. Mizrahi, A generalized approach to the fluid dynamics
of particulate systems: I. General correlation for fluidization and
sedimentation of solid multiparticle systems, Chem. Eng. J., 5
(1973), 171–189, doi:10.1016/0300-9467(73)80008-5
Nomenclature
A Cross-sectional area of the pipe (m
2
)
f Suspension-wall friction coefficient (dimensionless)
C Volumetric concentration of solid phase (dimensionless)
d Capsule outer diameter (m)
D Pipe inner diameter (m)
l Capsule length for anomalous capsule shapes (m)
L Pipe length (m)
 m Mass flow rate (kg/s)
P Pressure (Pa)
Re Modified suspension Reynolds number (m)
U
m
Mean velocity of two-phase mixture (m/s)
 Volumetric flow rate (m
3
/s)
Z. AKYÜREK: ONE-DIMENSIONAL MATHEMATICAL MODELING OF THE CAPSULE FLOW IN A HORIZONTAL PIPE
Materiali in tehnologije / Materials and technology 53 (2019) 4, 559–564 563

Greek symbols
 Density (kg/m
3
)
 Radially averaged voidage in the pipe(m)
μ Viscosity (kg/ms)
 Shear stress (N/m)
Subscripts
acc Acceleration
fric Friction
ice Ice
m Mean
w Water
Z. AKYÜREK: ONE-DIMENSIONAL MATHEMATICAL MODELING OF THE CAPSULE FLOW IN A HORIZONTAL PIPE
564 Materiali in tehnologije / Materials and technology 53 (2019) 4, 559–564