pressure drop of distilled water owing through helically
coiled tube in various pulsating parameters. Their results
revealed that with the increase of pulsating frequency, the
pressure drop decreases and the mean pressure drop in-
creases by 3–7 % compared with the steady ow. Wang et al
(Wang et al., 2014). experimentally studied the pulsating ow
characteristics induced by the rolling motion. They reported
that the time-averaged friction factor has no signicant dif-
ference between pulsating ow and constant ow. Song et al
(Song et al., 2021). numerically studied the ow behaviors of
pulsating intermittent liquid-solid uidized bed. Their re-
search found that with increasing the pulsation period and
particle diameter, the solid holdup rises in the middle part of
the uidized bed, and reduces at the top and bottom of the
uidized bed. Yang et al (Yang et al., 2018). numerically
analyzed the ow and heat transfer of pulsating ow in
ribbed channels. It was found that the pulsating ow causes
a lager pressure loss, and the friction factor ratio enhances
with the increase of pulsating frequency and amplitude.
Naphon et al (Tao et al., 2021). investigated the pulsating
thermal-uidic characteristics of TiO
2
/water nanouids in
the spirally coiled tubes. The results revealed that the friction
factor does not increase with increasing the pulsating fre-
quency. Yang et al (Yang et al., 2020). studied the liquid-
vapor two-phase pressure drop of evaporator under pul-
sating ow conditions. They found that the increasing rare of
pressure drop is related to the pulsating period. Shah et al
(Shah et al., 2017). analyzed the inuence of pulsating ow
on the uid dynamics of gas-solid two-phase ow in the
cold-ow riser. They pointed out that at high pulsating am-
plitude, the pulsating ow with 0.5 Hz leads to nearly double
solid content near the wall, while it remains unchanged in
the core region. From the above discussion, the ow char-
acteristics of pulsating ow are affected by various factors,
such as pulsating frequency, pulsating amplitude, uid
properties, geometric channel, etc. So far, the investigation of
pulsating ow behaviors of solid-liquid two-phase slurry is
relatively scarce, especially about the two-phase ice slurry.
Although the heat transfer characteristics of pulsating ow
of ice slurry in the horizontal pipe have been analyzed in our
previous study (Cai et al., 2022b), the ow resistance of ice
slurry with pulsating ow is still unclear. Therefore, it is
necessary to clarify the ow characteristics of ice slurry in
various pulsating ow conditions for strengthening its
practical applications.
In this present study, the Eulerian-Eulerian multiphase
model is applied to describe the ow behaviors of adiabatic
pulsating ow of ice slurry. The variation of instantaneous
face-averaged velocity and instantaneous frictional re-
sistance coefcient under different pulsating velocity am-
plitude, pulsating frequency, cycle-averaged velocity and ice
volume fraction is investigated. And the cycle-averaged re-
sistance coefcient is also evaluated in different pulsating
parameters and inlet ice volume fraction. In addition, the
prediction correlation for the cycle-averaged frictional re-
sistance coefcient of ice slurry in pulsating ow is proposed.
2. Numerical methodology
2.1. Governing equations
The Eulerian-Eulerian multiphase model based on the kinetic
theory of granular ow is employed to simulate the solid-
liquid two-phase ow of ice slurry. To simplify the calcula-
tion, ice particles are assumed to be smooth, inelastic and
spherical (Cai et al., 2022a). Table 1 shows the equations for
the numerical model.
In the Eulerian-Eulerian multiphase model, both the solid
and liquid phases are represented as interpenetrating con-
tinua. The continuity and momentum equations are solved
for both solid and liquid phases, and the interphase forces
consist of drag force, lift force and turbulent dissipation
force. The drag force is one of the most important forces in
solid-liquid two-phase ow, and the drag coefcient κ
sl
is
calculated by the Gidaspow model as shown in Eqs.(8)-(10).
The lift force, which results from the velocity gradient of li-
quid phase and pushed the particles toward the region with
higher velocity, is found signicant in the near wall region.
The lift coefcient C
L
is assigned to be 0.2 in this study (Cai
et al., 2022a). The turbulent dissipation force is important for
the uniform distribution of solid volume fraction. The tur-
bulent dissipation force coefcient C
TD
is taken as 1, the
dispersion Prandtl number σ
sl
is taken as 0.9.
The kinetic theory of granular ow is used to simulate the
uctuating motion of solid particles. Granular temperature
Θ
s
, representing the kinetic energy of the particles’ random
motion, is determined by Eqs.(11)-(15). The particle-particle
restitution coefcient e
ss
is taken as 0.9. α
s,max
is the max-
imum solid volume fraction, taken as 0.63.
The turbulence ow of ice slurry is modeled by the per-
phase k-ε turbulence model in this paper. The turbulence
kinetic energy (k) and its dissipation rate (ε) are calculated by
Eqs.(16) and Eqs.(17). The constants in the model are set to
C
1ε
= 1.42, C
2ε
= 1.92, C
3ε
= 1.2, C
μ
= 0.09, σ
k
= 1 and σ
ε
= 1.3.
At the inlet, the velocity is considered to be parallel uni-
form and periodically changes with time as u=u
ta
(1 + A
u
sinωt)
for pulsating ow. The solid volume fraction is assumed to be
uniform and constant. The outlet of the pipe is set to be
pressure outlet condition. All the walls are treated as no-slip
walls. The initial conditions such as the velocity and solid
volume fraction are set to be the same as the inlet condition.
2.2. Numerical solutions
The numerical calculation is performed by the commercial
CFD software Ansys Fluent. The horizontal circular pipes
with the diameter of D = 10 mm, 16 mm and 30 mm are
adopted in the present study. The length of pipes is large
enough so that the uid can achieve fully developed ow.
The wall thickness of the pipe is neglected to simplify the
computation. The mesh of pipe is created using the struc-
tured grid with hexahedron cells. Different grid sizes of
99840, 312000, 405600 and 548250 are used in the numerical
calculation to verify the grid independence. It is found that
the amount of 312000 cells is adequate to ensure the accu-
racy of the numerical results. A uniform time stepping
method is applied and the time step is T/64, which has been
proved ne enough to ensure the accuracy of the numerical
results without consuming too much computing time. The
487 Chemical Engineering Research and Design 186 (2022) 485–496