Exploring flow boiling characteristics on surfaces with various micro-pillars using the lattice Boltzmann method

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RESEARCH ARTICLE | MA RC H 06 2 02 4
Exploring flow boiling characteristics on surfaces with
various micro-pillars using the lattice Boltzmann method
Chao Luo (罗超) ; Toshio Tagawa (⽥川俊夫)
Physics of Fluids 36, 033312 (2024)
https://doi.org/10.1063/5.0195765
10 March 2024 12:05:12

Exploring flow boiling characteristics on surfaces
with various micro-pillars using the lattice
Boltzmann method
Cite as: Phys. Fluids 36, 033312 (2024); doi: 10.1063/5.0195765
Submitted: 4 January 2024 .Accepted: 11 February 2024 .
Published Online: 6 March 2024
Chao Luo (罗超), and Toshio Tagawa (田川俊夫)
a)
AFFILIATIONS
Department of Aeronautics and Astronautics, Tokyo Metropolitan University, Tokyo 191-0065, Japan
a)
Author to whom correspondence should be addressed: tagawa-toshio@tmu.ac.jp
ABSTRACT
In this study, the lattice Boltzmann method is utilized to simulate flow boiling within a microchannel featuring a micro-pillar surface. This
investigation aims to explore the impacts of micro-pillar shape and quantity on the flow boiling characteristics across various superheats and
Reynolds numbers (Re). A systematic examination is conducted on three types of micro-pillars, five quantities of micro-pillars, four Re values,
and 18 superheat levels. The mechanisms contributing to enhanced heat transfer in flow boiling are elucidated through a comprehensive anal-
ysis of bubble dynamics, temperature and velocity fields, local and transient heat fluxes, and boiling curves. Moreover, the critical heat fluxes
(CHF) of all surfaces are evaluated to identify the superior micro-pillar configurations. The findings revealed that microchannels with micro-
pillar surfaces induce more vortices compared to those with smooth surfaces, attributable to the combined effects of bubble dynamics and
micro-pillars. Bubble patterns and boiling curves demonstrated the significant impact of micro-pillar geometrical shapes on the boiling
regime and heat transfer performance. As flow boiling progressed, an increase in micro-pillar quantity and Re can mitigate the fluctuation
and decline rate in transient heat flux, respectively. Among the three types of micro-pillar surfaces, the circular shape exhibited the highest
flow boiling performance, followed by the triangular and rectangular shapes. For all surfaces, the CHF increased with Re, and each micro-
pillar type displayed an optimal quantity for achieving maximum CHF, with the highest increase reaching 45.2%. These findings are crucial
for optimizing microchannel designs to enhance flow boiling heat transfer efficiency.
Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0195765
NOMENCLATURE
a,b,R,xParameters in P–R equation of state
c
s
Lattice sound speed
c
v
Specific heat at constant volume
DBubble departure diameter
e
a
Lattice velocity vector
f
a
Density distribution function
FTotal force
F
ads
Adhesion force
F
g
Gravitational force
F
m
Interparticle interaction force
gGravitational acceleration
GFluid–fluid interaction strength
G
w
Fluid–solid interaction strength
h
co
Heat transfer coefficient
h
fg
Specific latent heat
H
mp
Height of micro-pillar
H
th
Thickness of heater
Ja Jacob number
l
0
Characteristic length
L
mp
Length of micro-pillar
Lx Length of heater
Ly Height of heater
MOrthogonal transformation matrix
NNumber of micro-pillars
Nu Nusselt number
pPressure
P
mp
Pitch of micro-pillars
q
ave
Space- and time-averaged heat flux
q
local
Local heat flux
q
s
Space-averaged heat flux
R
0
Droplet radius
Re Reynolds number
Phys. Fluids 36, 033312 (2024); doi: 10.1063/5.0195765 36, 033312-1
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SForcing term in the moment space
tTime
t
0
Characteristic time
t

Dimensionless time
TTemperature
T
b
Temperature on heater bottom surface
T
c
Critical temperature
T
s
Saturation temperature
u
0
Characteristic velocity
U
in
Inlet velocity
vReal fluid velocity
w
a
Weighting coefficient
W
mc
Width of microchannel
x,yCoordinates
xPosition
Greek Symbol
hContact angle
qDensity
KDiagonal matrix
dLattice spacing
-Modify factor
wPseudopotential
sRelaxation time
rSurface tension
kThermal conductivity
vThermal diffusion coefficient
Subscripts or Superscripts
aLattice direction
lLiquid
vVapor
x,yDirections
I. INTRODUCTION
Effective thermal management is essential for the safe operation
of high-performance devices, which generate considerable amounts of
heat. Although several advanced devices provide enhanced power and
efficiency, they impose constraints on weight and size. Traditional
single-phase flow proves inadequate for fulfilling the thermal manage-
ment requirements of sophisticated equipment, rendering two-phase
flow a viable alternative owing to its superior heat dissipation capabil-
ity and maintenance of lower operating temperatures.
1–4
As an effi-
cient cooling technology, flow boiling emerges as an optimal solution
for handling high heat loads and achieving exceptional thermal perfor-
mance across various applications. Moreover, the trend toward further
miniaturization of microelectronic components has increased the sig-
nificance of flow boiling in microchannels.
5
Specifically, flow boiling in
microchannels offers a compact system structure, reduced pump
power consumption, elevated heat transfer coefficients, and phase
change latent heat benefits, along with diminished flow require-
ments.
6,7
Nonetheless, flow boiling inherently presents a highly com-
plex and multidimensional multiphase flow phenomenon, with
numerous factors influencing bubble generation, growth, and motion.
8
Klausner et al.
9
highlighted the complexity of flow boiling in a
microchannel by illustrating the forces acting on a bubble during
detachment in a horizontal channel, including bubble growth force,
quasi-steady drag force, buoyancy force, surface tension, shear lift
force, and the forces due to contact and hydrodynamics. Therefore, the
presence of these forces indicates the complexity of flow boiling in a
microchannel.
In recent decades, extensive experimental studies on flow boiling
inmicrochannelshavebeenconducted.
10–15
Wang et al.
16
observed
the evolution of flow patterns during flow boiling in microchannels,
from bubble formation through restricted bubbly flow to slug-annular,
annular, and wispy-annular flows. Li et al.
17
introduced a novel micro-
channel model with triangular cavities, which enhanced the heat trans-
fer efficiency by up to 9.88 times and reduced the pressure loss by
50.3% compared to conventional microchannels. Such experiments
have clarified mechanisms to enhance the flow boiling performance in
microchannels.
18
However, several challenges persist in observing and
analyzing the flow boiling phenomena within specific microchannels
owing to their microscopic scale. The advancement in computational
techniques and increased computational power have made numerical
simulations an indispensable tool for exploring microscale
phenomena.
19
Through simulations, additional details can be obtained that will
provide deeper insights into flow boiling. Consequently, numerous
simulation studies have complemented experimental research. Ling
et al.
20
integrated the volume of fluid (VOF) with a level-set approach
to investigate the dynamics of flow boiling in a 3D microchannel with
a rectangular cross section. Their findings demonstrated that the intro-
duction of growing bubbles within the microchannel significantly
enhances heat transfer and promotes the acceleration of fluid flow
both around and downstream. Mukherjee et al.
21
employed the level-
set technique to accurately capture the liquid–vapor interface for simu-
lating flow boiling in a microchannel with a square cross section. Their
research reported that heat transfer increases with wall superheat but
remains nearly constant despite variations in fluid flow rate. Lin et al.
22
examined flow boiling in two different microchannel designs using a
numerical model that integrates the VOF and a phase change model.
They observed that, in comparison with a smooth microchannel, the
microchannel with micro-pillars displayed a 61.92% increase in heat
transfer and a 12.74% increase in pressure loss, whereas the micro-
channel with microcavities exhibited increases of 17.16% in heat trans-
fer and 4.36% in pressure loss. Guo et al.
23
explored the effect of a
microchannel with a randomly rough surface on the thermodynamic
performance of flow boiling using a VOF model coupled with the Lee
model. Their results highlighted that with increased relative roughness,
both the Nusselt number (Nu) and friction factor Reynolds number
(fRe) initially increased and then declined, indicating superior thermo-
dynamic performance of the rough surface.
The mesoscopic lattice Boltzmann method (LBM) has attracted
sustained interest for its application in multiphase flow studies
24–26
because of its advantages including simplicity in handling complex
geometries,
27
facile treatment of liquid–vapor interfaces, incorporation
of micro/mesoscale physics, and suitability for parallel computing.
Wang et al.
28
conducted a study on the impact of bubble coalescence
on flow boiling using both experimental and LBM approaches. Their
discovery revealed that bubble merging at the central nucleation site
flanked by two other sites with reduced heat transfer effectiveness.
Chen et al.
29
combined experimental methods and LBM to study flow
Physics of Fluids ARTICLE pubs.aip.org/aip/pof
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boiling, observing that bubbles within channels typically exhibited an
inclined angle and a reduced contact diameter ratio. Zhang et al.
30
applied the LBM to investigate flow boiling in vertical microchannels
with single and multiple cavities at various Re, concluding that heat
transfer efficiency improves with increased Re due to a higher bubble
release rate and stronger forced convection. Utilizing the LBM, Zhang
et al.
31,32
conducted simulations on flow boiling within both serpentine
and corroded microchannels. The findings from an LBM simulation
by Gong and Cheng
33
revealed that superheated bubbles on hydro-
philic surfaces exhibit enhanced heat conductivity and a reduced ther-
mal boundary layer, leading to improved nucleate boiling heat transfer
in horizontal microchannels. Sun et al.
34
employed the LBM to accu-
rately replicate various flow patterns including slug flow, bubble flow,
and departure from nucleate boiling in vertical channels, while exam-
ining the impact of factors such as gravity, wettability, superheating,
and Re on flow boiling. Similarly, Mukherjee et al.
35
investigated the
influence of different initial conditions on bubble dynamics and pat-
terns in a horizontal microchannel using the LBM. In contrast to uni-
form hydrophilic micro-pillar surfaces in horizontal microchannels,
Chen et al.
36
introduced a biphilic micro-pillar surface, resulting in a
significant 105.8% increase in flow boiling heat transfer. Shah et al.
37
conducted LBM simulations on flow boiling in microchannels with
counter-flow and co-flow configurations, noting that the counter-flow
design exhibited superior heat transfer performance.
The literature review spanning both experimental and simulation
studies of microchannel flow boiling assesses the impact of microchan-
nel surface modifications on bubble dynamics and consequent
improvements in heat transfer performance. Although micro-pillar
surfaces are known to promote boiling efficiency,
38,39
a comprehensive
understanding of the mechanisms affecting flow boiling characteristics
on intricate surfaces is crucial for further advancements. By employing
LBM simulations, we can effectively capture and analyze the complete
processes of nucleation, growth, and departure of bubbles during the
flow boiling, as well as their consequential effects on temperature and
flow fields. Moreover, a quantitative assessment of heat transfer perfor-
mance across microchannels with varied complex surfaces aids in opti-
mizing heat transfer processes and designing more effective thermal
management systems. Therefore, this work utilizes the LBM to simu-
late flow boiling in microchannels with diverse micro-pillar surfaces,
offering an exhaustive analysis of bubble dynamics and heat transfer
phenomena. This analysis includes an examination of flow boiling pro-
cesses, temperature and velocity fields, local, transient, and averaged
heat fluxes, as well as the effects of micro-pillar shape and quantity on
flow boiling performance under different superheat levels and Re.This
research contributes significantly to the understanding of the mecha-
nisms and effects of complex micro-pillar surfaces on flow boiling
performance.
II. PHYSICAL MODEL AND BOUNDARY CONDITIONS
In this investigation, the calculation domain of the horizontal
microchannel is segmented into three sections: the upstream adiabatic
section, heated section (highlighted in red), and downstream adiabatic
section, as illustrated in Fig. 1(a). The fluid enters through the left inlet,
flows through the adiabatic upstream segment, proceeds to the heated
segment, continues through the adiabatic downstream segment, and
exits via the right outlet. The inner surface of the heater is designed to
be replaceable. Figure 1(b) depicts a micro-structured surface equipped
with micro-pillars. The height of the microchannel including the
heater is designated as Ly ¼100 lattice units. The length of the heater
is Lx ¼600 lattice units, while the upstream and downstream segments
extend 2Ly and 22Ly, respectively. The width of the microchannel for
fluid flow is established at W
mc
¼80 lattice units, wi th the thickness of
the heater set at H
th
¼20 lattice units. The different micro-pillar shapes
comprising the rectangle, triangle, and circle are presented in Fig. 1(c).
The base length (L
mp
)andheight(H
mp
) of each micro-pillar type are
fixed at L
mp
¼2H
mp
¼20 lattice units. The dimensions of L
mp
and
H
mp
are uniform for all the micro-pillars on the surface. The pitch
(P
mp
) of the micro-pillars is determined based on their number (N).
This study examined five different Nvalues:6,11,16,21,and26.
Employing the aforementioned physical model, this study
explored saturated flow boiling within a microchannel featuring vari-
ous micro-pillar surfaces, under different superheats and Re values.
The specific boundary conditions (BCs) applied within the calculation
zone included: initially setting the entire calculation zone to the satura-
tion temperature (T
s
). The velocity at the inlet (U
in
) is regulated using
the Zou–He scheme,
40
with T
s
uniformly maintained. Fully developed
temperature BC and convective BC with a second-order scheme are
applied at the outlet, as outlined in Refs. 31,41,and42.Thetop
boundary and bottom walls of both the upstream and downstream
segments are subjected to adiabatic BC. In the heating segment, a con-
stant heating temperature (T
b
) is applied to the bottom of the heater,
and conjugate heat transfer BC is utilized at the solid–fluid inter-
face.
25,26,42
Thus, by modulating the heating temperature T
b
and inlet
velocity U
in
, the study aims to establish the desired conditions for vary-
ing superheats and Re.
In this study, the BCs are established for four Re and 18 superheat
levels for exploring three micro-pillar shapes, five quantities of micro-
pillars, and smooth surfaces. A total of 2 10
6
computational steps are
allocated for each case. To conserve computational resources and
enable an exhaustive comparison of flow boiling performance, a subset
of cases is selected. Additionally, the graphics processing unit (GPU)
computing is employed, utilizing an NVIDIA GeForce RTX 4090
equipped with 16 384 cores and 24 GB of GDDR6X memory. The
numerical program is compiled under the Windows 11 operating sys-
tem using CUDA version 12.2, facilitating the execution of simulations
FIG. 1. Schematic of microchannel: (a) calculation zone, (b) micro-pillars surface,
and (c) micro-pillar shapes.
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on the GPU. Notably, considering the variances in flow characteristics
between two-dimensional and three-dimensional models,
43
future
studies will expand the current model to three-dimensional scenarios
utilizing GPU calculations.
III. MATHEMATICAL FORMULATION
The study’s 2D model, addressing both flow and temperature
fields, is solved using the multiple-relaxation-time (MRT) pseudopo-
tential LB model
30–32,44,45
and finite difference method (FDM),
46
respectively. The equation of state (EOS) is important for linking
the mutual coupling between these fields. Detailed descriptions of the
MRT-LB model and FDM are available in Appendix A. Moreover, the
numerical model is validated against several classical benchmark
cases,
47–49
as outlined in Appendix B.
The basic characteristic parameters such as length (l
0
), velocity
(u
0
), and time (t
0
) are defined following standard conventions:
42
l0¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
rst
gqlqv
ðÞ
r;u0¼ffiffiffiffiffiffi
gl0
p;t0¼l0
u0
:(1)
The dimensionless superheat Jakob number (Ja)andthe
Reynolds number (Re) are defined as follows:
42
Ja ¼cp;lTbTs
ðÞ
hfg
;(2)
Re ¼UinWmc
vl
;(3)
where c
p,l
denotes the liquid specific heat at a constant pressure, h
fg
representsthespecificlatentheat,
42
and v
l
indicates the liquid kine-
matic viscosity.
The local heat flux q
local
can be derived as follows:
25,26
qlocal ¼ksolid
@T
@yy¼0
dx;(4)
where the local temperature gradient can be determined as follows:
37
@T
@yy¼0¼4Tjy¼1Tjy¼23Tjy¼0
2dy
:(5)
The space-averaged heat flux (q
s
) can be evaluated as follows:
33
qs¼ksolid
Lx ðLx
0
@T
@yy¼0
dx:(6)
The space- and time-averaged heat flux (q
ave
) can be calculated as
follows:
33
qave ¼ 1
t2t1ðt2
t1
qsdt;(7)
where t
1
and t
2
denote the beginning and end of the time interval,
respectively. The dimensionless time (t

¼t/t
0
) is defined in this study.
Given that LB simulations are customarily performed in lattice
units for simplicity, this study adhered to such convention. However,
note that unit conversions may lead to varying outcomes depending
on different methodologies.
50
Following Refs. 24,51–53, a saturation
temperature T
s
¼0.86T
c
¼0.0941 in the lattice unit, which corre-
sponds to a real-world temperature of 556 K. This working condition
has been frequently employed in numerous prior studies.
24–27,35,51–53
The conversion between lattice and real-world units is elucidated in
Table I, with a comprehensive methodology for unit conversion
detailed in Refs. 24 and 29.
IV. RESULTS AND DISCUSSION
Upon validating the numerical approach, this section delves into
the study of flow boiling heat transfer within microchannels featuring
micro-pillar surfaces. Unless stated otherwise, all simulations are con-
ducted under conditions of saturated liquid–vapor system at a temper-
ature of 556 K and a pressure of 7 MPa, with the specific parameters
for lattice units in the present LBM simulations detailed in Table I.
Moreover, the surfaces of all models are hydrophilic, with a contact
angle of 53.3.
A. Influence of micro-pillar shapes on the flow boiling
process
Toexplorehowvariousmicro-pillarshapesonmicrochannelsur-
faces influence the flow boiling process, three distinct micro-pillar
shapes are constructed. Figure 2 presents snapshots of flow boiling on
a smooth surface and three micro-pillar surfaces at a Re ¼10, N¼16,
and Ja ¼0.1721. The heat conduction from the bottom of the heater to
its top surface induces a phase change within the microchannel. For
the smooth microchannel, uniform heating at the bottom of the heater
initiates the formation of a thin bubble film on the top surface. This
leads to the generation of bubble columns driven by surface tension.
As illustrated in Fig. 2(a), these bubbles move downstream under the
influence of the bulk fluid flow at t

¼70.93, with a noticeable ten-
dency for all the bubble columns to tilt toward the right. This tilting
behavior of the bubbles primarily emerges from the synergistic effects
of drag force, surface tension, and buoyancy force.
54
As the heating
continues, an increasing number of bubbles form on the surface, with
the growing bubbles tilting toward the flow direction due to fluid iner-
tia. These bubbles detach from the surface when buoyancy and drag
become the dominant forces, as illustrated in Fig. 2(a) at t

¼165.51.
Once released from the high-temperature heater surface, the bubbles
TABLE I. Unit conversion.
Parameters Lattice unit Real-world unit
Saturation temperature (T
s
) 0.0941 556 K
Liquid density (q
l
) 6.5 745.17 kg/m
3
Vapor density (q
v
) 0.38 34.77 kg/m
3
Surface tension (r) 0.0846 18.33 10
3
N/m
Gravitational acceleration (g) 3.0 10
5
9.80 m/s
2
Liquid kinematic viscosity (v
l
) 0.1 3.79 10
5
m
2
/s
Vapor kinematic viscosity (v
v
) 1.6667 6.31 10
5
m
2
/s
Thermal diffusivity (v) 0.05 1.89 10
5
m
2
/s
Characteristic length (l
0
) 21.47 1.61 10
3
m
Characteristic velocity (u
0
) 0.0254 0.1257 m/s
Characteristic time (t
0
) 845.89 0.0128 s
Mesh size (Dx) 1 7.51 10
5
m
Time steps size (Dt) 1 1.52 10
5
s
Wall superheat degree (DT) 9.41 10
3
55.56 K
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are surrounded by lower temperatures, leading to their condensation.
Moreover, the concerted action of forced convection and buoyancy
leads to the emergence of floating bubbles of various sizes. As flow
boiling progresses, a certain distance is maintained between the adja-
cent bubbles on the smooth surface. On the smooth surface, bubble
nucleation sites are randomly distributed, in contrast to the micro-
pillar surfaces where nucleation sites are fixed, as depicted in Figs. 2
(b)–2(d). The bubbles initially form in the grooves between the adja-
cent micro-pillars, thereby inducing more bubble nucleation sites on
the micro-pillar surfaces compared to smooth surfaces. At the onset of
bubble nucleation, bubbles grow independently within each groove.
Upon exceeding the micro-pillar height, the bubbles are pressed down-
ward by the mainstream fluid and become deformed. The fluid tem-
perature downstream of the heated surface is higher than upstream,
resulting in faster growth of downstream bubbles. This is because the
surface tension decreases as the temperature increases.
55
Additionally,
bubble mergers are observed on micro-pillar surfaces but not on the
smooth surface. This phenomenon occurs because micro-pillars create
stable, independent nucleation sites, and the proximity of these sites
facilitates the coalescence of adjacent bubbles, further aided by forced
convection. Nevertheless, the incidence of coalescence diminishes as
flow boiling evolves. Among the three micro-pillar surface types, surfa-
ces with rectangular micro-pillars consistently exhibit residual bubbles
in the grooves between micro-pillars, which does not occur on surfaces
with triangular and circular micro-pillars, potentially because of the
lower drag force experienced within rectangular grooves. Thus, the
bubble dynamics on micro-pillar surfaces are significantly influenced
by the geometrical shape of the micro-pillars.
The temperature fields for various microchannels, corresponding
to Fig. 2 at t

¼118.22, are illustrated in Fig. 3, where the bubble out-
lines are highlighted in the dark blue, and the partial velocity vectors
along the vertical lines are uniformly spaced in the xdirection. As evi-
dent that the phase change significantly alters the velocity profile
within the microchannel, diverging from the initial parabolic configu-
ration. The variations in temperature at the liquid–vapor interface
cause changes in surface tension, which consequently affect the veloc-
ity around the bubble.
55
The most significant alterations in flow occur
within the bubble, indicating that fluid flow near the heater surface is
affected by the phase change occurring at this surface. The temperature
within the bubble is elevated due to phase change heat transfer, and
the areas above the bubbles exhibit higher temperatures due to
thermal convection and buoyancy-induced changes in the flow field.
FIG. 2. Snapshots of flow boiling on the
surfaces with various micro-pillars shapes
at Re ¼10, N¼16, and Ja ¼0.1721: (a)
smooth surface, (b) rectangle, (c) triangle,
and (d) circle.
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Thispatternisconsistentacrosseachbubbleinthesmoothmicrochan-
nel, where the temperature gradient is higher on the upstream (left)
side of the bubble compared to the lower gradient on the downstream
(right) side. This phenomenon occurs as the cooler fluid approaches
the bubble, whereas the warmer fluid recedes from it. As illustrated in
Fig. 3(a), the velocities near the bubble sides exhibit a downward slope
on the left and an upward slope on the right. Additionally, the thermal
boundary layer between adjacent bubbles gradually thins from the first
to the second bubble due to fluid flow distortion. On the micro-pillar
surfaces, the bubble release process decreased because of bubble coa-
lescence, leading to an increase in the thickness of the thermal
boundary layer on the heater surface, as shown in Figs. 3(b)–3(d).
Meanwhile, the temperature inside the heater is higher in the regions
where the heater surface is covered by bubbles. Compared to the sur-
face featuring rectangular micro-pillars, those with triangular and
circular micro-pillars exhibit lower temperatures inside the heater.
This is because the triangular and circular micro-pillars prevent the
occurrence of flow dead zones and consequently enhance boiling
heat transfer.
Figure 4 presents the velocity field in various microchannels, corre-
sponding to Fig. 2 at t

¼118.22, including a smooth microchannel for
comparison. For improved visualization of fluid flow near the micro-
pillar surface, a region from x¼200 to 400 in the middle section of the
heater is selected. In the smooth microchannel, fluid flow undergoes
significant changes at the bubble location, resulting in a pronounced vor-
tex on the lower-right side of the bubble. According to Ref. 56, bubble
growth and detachment during pool boiling are associated with the for-
mation of counter-rotating vortices at the nucleation site. Additionally,
in pool boiling, the absence of transverse flow effects allows symmetrical
vorticesoneithersideofthebubbletointensifyasthebubbleascends
due to buoyancy. Conversely, in transverse flow boiling, the influence of
the flowing fluid weakens the upstream vortex, leaving only a down-
stream vortex, which is propelled downstream by the bulk fluid. This
vortex formation and movement pattern alongside the bubble has been
documented in Refs. 35,54,and57. Micro-pillars in a single-phase
flow disrupt the flow, disturbing the flow boundary layer and promoting
vortex formation, resulting in smaller vortices at locations without
phase changes, as illustrated in Fig. 4(c). The increase in nucleation
sites leads to more bubbles and vortices. Consequently, micro-pillar
microchannels exhibit more vortices compared to smooth micro-
channels. As bubbles grow and tilt downstream along the micro-pillars,
FIG. 3. Temperature field for surfaces with various micro-pillars shapes at Re ¼10,
N¼16, Ja ¼0.1721, and t

¼118.22: (a) smooth surface, (b) rectangle, (c) trian-
gle, and (d) circle.
FIG. 4. Velocity field for surfaces with various micro-pillars shapes at Re ¼10, N¼16, Ja ¼0.1721, and t

¼118.22: (a) smooth surface, (b) rectangle, (c) triangle, and (d)
circle.
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the vortices within the bubbles move downstream, as depicted in
Figs. 4(b)–4(d).
Figure 5 displays the temperature field for different microchan-
nels, corresponding to Fig. 3 at various superheats. With the superheat
increasing to Ja ¼0.2410, nucleation boiling intensifies, resulting in a
higher rate of bubble generation. On the smooth surface, a thin bubble
film forms behind the bubble column and remains connected to it, as
illustrated in Fig. 5(a).Theliquid–vapor interface of the bubble is
observed to expand during its growth under the influence of buoyancy.
The bubble eventually disconnects from the bubble column and is
released, driven by an increase in the shearing effect around the bub-
ble. Therefore, the bubble detaches when the drag force exceeds the
surface tension.
58
Additionally, the bubble released tends to adopt a
circular shape due to surface tension, a detachment process similarly
reported in Ref. 57.AtJa ¼0.2410 on micro-pillar surfaces, there is a
notable increase in bubble columns, with the positioning of these col-
umns on different micro-pillar surfaces exhibiting consistency.
However, the temperature within the grooves between adjacent rectan-
gular micro-pillars is observed to be higher than that between triangu-
lar and circular micro-pillars. As the superheat increases to Ja ¼0.3787
[Fig. 5(b)], a bubble film encompasses the smooth surface, indicating a
transition to the film boiling regime. Moreover, the temperature inside
the heater with a smooth surface exceeds that of micro-pillar surfaces.
Among the micro-pillar surfaces, the highest internal heater tempera-
ture is recorded with rectangular micro-pillars, where the temperature
within the grooves between rectangular micro-pillars surpasses that
between the triangular and circular micro-pillars owing to the thicker
bubble film on the rectangular micro-pillar surface. Furthermore, tri-
angular micro-pillar surfaces demonstrated incomplete bubble film
coverage over certain vertices, suggesting incomplete film boiling on
this surface. Hence, the superheat required to achieve complete film
boiling varies across the different micro-pillar surfaces.
Figure 6 illustrates the velocity field in various microchannels,
corresponding to Fig. 5(a) at Ja ¼0.2410, including a smooth micro-
channel for comparison. In the smooth microchannel, the bubbles
detach from the bubble column and move downstream because of a
significant velocity, as depicted in Fig. 6(a). The velocity around the
bubble varies during the phase change and results in vortex formation,
which has been reported in Ref. 54. The formation of a vortex is evi-
dent on the lower-right side of the bubble column. In pool boiling, two
symmetrical vortices form during bubble growth and ascend with the
detachment of the bubble. However, in flow boiling, vortices induced
by the growth and departure of the bubble are affected by the bulk
fluid flow.
54
With an increase in inlet velocity, the influence of the bulk
fluid on bubble-induced vortices becomes more significant during flow
boiling, limiting the development and formation of vortices during
bubble shedding. Simultaneously, the presence of a thin bubble film on
the heater surface alters velocity, resulting in the formation of weak
vortices near the bubble film. Furthermore, in the thickest portion of
the bubble film, weak vortices induced an upward velocity. Driven by
the bulk fluid, the bubble column and film merge, facilitating interac-
tions among the vortices surrounding these bubbles. Across all micro-
channels with micro-pillar surfaces, nucleation boiling is intensified
with an increase in superheat. On surface with rectangular micro-
pillars, despite varied bubble patterns within individual grooves, dis-
tinct vortices are noted on the right side of each bubble, as illustrated
in Fig. 6(b). The growth of bubbles in the narrow spaces between adja-
cent rectangular micro-pillars leads to the formation of vortices on the
top of the micro-pillars. In the smaller bubbles within the groove cre-
ated by adjacent rectangular micro-pillars, lower velocities enter from
both sides of the groove, ascending from its center. This upward veloc-
ity mirrors the buoyancy-driven growth direction of the bubbles.
Nevertheless, at the base of the groove, the presence of stagnant fluid
with low velocity impedes effective heat transfer. Transitioning the
rectangular micro-pillars to triangular ones alters the surface velocity,
as shown in Fig. 6(c), where vortices became evident within the
grooves, thereby enhancing heat transfer between adjacent triangular
micro-pillars. Similarly, in Fig. 6(d), the circular micro-pillars demon-
strated a comparable improvement in fluid flow within the grooves,
contributing to enhanced heat transfer.
FIG. 5. Temperature field for different superheats and surfaces with various micro-
pillar shapes at Re ¼10, N¼16, and t

¼118.22: (a) Ja ¼0.2410 and (b)
Ja ¼0.3787.
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Figure 7 illustrates the velocity field in various microchannels,
corresponding to Fig. 5(b) at a Ja ¼0.3787. In the smooth microchan-
nel, a bubble film distinctly covers the heater surface. Despite the low
velocity near the heater, a weak vortex is still present near its surface,
as depicted in Fig. 7(a). On surfaces with rectangular micro-pillars, as
depicted in Fig. 7(b), a thick bubble film layer completely envelops the
entire surface, with smaller velocities within the grooves and tiny vorti-
ces. Conversely, on surfaces with triangular micro-pillars, as portrayed
in Fig. 7(c), the bubble film is notably thinner at the apexes of the
micro-pillars, accompanied by several pronounced vortices near the
surface, indicating that boiling on these surfaces did not achieve a sta-
ble film boiling regime. In Fig. 7(d), despite the bubble film covering
FIG. 6. Velocity field for surfaces with various micro-pillars shapes at Re ¼10, N¼16, Ja ¼0.2410, and t

¼118.22: (a) smooth surface, (b) rectangle, (c) triangle, and (d)
circle.
FIG. 7. Velocity field for surfaces with various micro-pillars shapes at Re ¼10, N¼16, Ja ¼0.3787 and t

¼118.22: (a) smooth surface, (b) rectangle, (c) triangle, and (d)
circle.
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the surface with circular micro-pillars, vortices near the surface are
clearly visible, suggesting that the micro-pillar shape significantly
impacts flow patterns and alters the boiling regime.
Figure 8 presents the local heat flux q
local
in various microchan-
nels at different superheats, corresponding to boiling moments at
t

¼118.22 in Figs. 3 and 5.AtJa ¼0.1721 in Fig. 8(a),thevaluesof
q
local
consistently decreased at the front of the heater in all microchan-
nels owing to the gradual generation of the thermal boundary layer.
On the smooth surface, as revealed in Fig. 3(a), the alignment of five
bubbles corresponds with five peaks in q
local
.Thesepeakvaluesofq
local
are located near the three-phase contact line on the upstream sides
of the bubbles, whereas the lower q
local
values are located beneath
the bubbles. This observation aligns with findings reported in Ref. 59.
The variation in q
local
on the smooth surface is primarily influenced by
the phase change, whereas on micro-pillar surfaces, the variation in
q
local
is driven not only by the phase change but also by the presence of
micro-pillars, which alter fluid flow. Therefore, a notable variation in
local heat flux q
local
is observed at the position of the first micro-pillar
compared to the smooth surface. Specifically, the q
local
value for the
surface with rectangular micro-pillars is lower than that for other
micro-pillar surfaces. This is demonstrated in Fig. 3(b), where a higher
temperature is noted inside the heater of the surface with rectangular
micro-pillars. On micro-pillar surfaces, reduced q
local
values occur in
areas where bubbles coalesce. Conversely, the heater surface with trian-
gular micro-pillars exhibits a higher q
local
value associated with the
thinnest thermal boundary layer, a result of bubble mergers and com-
plete downstream transport by the bulk fluid.
Similarly, in Fig. 8(b) at Ja ¼0.2410, the peak values of q
local
are
located upstream of the bubble columns on the smooth surface,
whereas trough values are focused within the bubble columns where
temperatures are elevated, as depicted in Fig. 5(a). Fluctuations in q
local
are more pronounced at Ja ¼0.2410 compared to Ja ¼0.1721 on
micro-pillar surfaces. However, the q
local
values on the surface with
rectangular micro-pillars remain lower than those on surfaces with tri-
angular and circular micro-pillars. This diminished heat transfer
within the groove of adjacent rectangular micro-pillars is linked to
higher temperatures and lower velocities in the groove, as analyzed in
discussions pertaining to Figs. 5 and 6.Moreover,Fig. 5(a) indicates
comparable temperature fields and bubble patterns on surfaces with
triangular and circular micro-pillars, leading to similar variations in
q
local
. As superheat increases to Ja ¼0.3787, there is an increase in the
temperature gradient at the heater’s front, resulting in a significant
decrease in q
local
in front of all microchannels, as illustrated in
Fig. 8(c).ForJa ¼0.3787 on a smooth surface, Fig. 5(b) reveals the for-
mation of a bubble film with a thick center and thin sides beneath the
detached bubble. This generated a temperature distribution with high
temperatures in the middle and lower temperatures on both sides,
aligning with the film boiling results reported by Dong et al.
60
Consequently, q
local
reaches lower values in regions of high tempera-
ture, whereas higher values are observed on both sides. At Ja ¼0.3787,
q
local
valuesformicro-pillarsurfacesshowanincreasecomparedtothe
smooth surface. As depicted in Fig. 7, the groove between rectangular
micro-pillars exhibits smaller velocities and vortices, whereas surfaces
with triangular and circular micro-pillars feature higher velocities and
more pronounced vortices, leading to the lowest heat transfer perfor-
mance observed with rectangular micro-pillars. Moreover, q
local
values
for the triangular micro-pillar microchannel are higher than those for
the circular micro-pillar microchannel at Ja ¼0.3787, as discussed in
relation to Fig. 7.
Figure 9 displays the transient space-averaged heat flux q
s
in vari-
ous microchannels at different superheats. As shown in Fig. 9(a) at
Ja ¼0.1721, q
s
values decrease as t

increases in all the microchannels
FIG. 8. Local heat flux q
local
in various microchannels at different superheats: (a)
Ja ¼0.1721, (b) Ja ¼0.2410, and (c) Ja ¼0.3787.
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during the initial stages, with the smooth microchannel exhibiting the
smallest q
s
.Thisisattributedtothegreaternumberofnucleationsites
on micro-pillar surfaces. Additionally, the circular micro-pillar surface
demonstrates a higher q
s
value, suggesting its circular shape enhances
heat transfer during the early stages of bubble nucleation. With an
increase in t

,q
s
for the smooth surface suddenly rises due to bubble
formation and detachment, whereas q
s
for micro-pillar surfaces con-
tinues to decline due to bubble coalescence, as observed in Fig. 2 at
t

¼118.22. As heating persists, leading to a gradual increase in the
fluid temperature within the microchannel, q
s
for the smooth surface
initially rises then slowly decreases. Additionally, as bubbles detach
from micro-pillar surfaces, q
s
values on these surfaces begin to
increase, then fluctuate during the boiling process. Among the three
types of micro-pillar surfaces, the circular shape exhibits the highest q
s
,
followed by the triangular shape, with the rectangular shape having the
lowest. Increasing the superheat to Ja ¼0.2410 significantly intensifies
nucleate boiling, causing q
s
values for all surfaces to rapidly rise to their
peak, as illustrated in Fig. 9(b).Theq
s
value for the smooth surface
increases earlier and reaches its peak before that of the micro-pillar
surfaces. On micro-pillar surfaces, all q
s
values follow a similar pattern:
initially decreasing with the onset of phase change, then rising during
bubble detachment, and finally gradually declining as flow boiling pro-
gressed. With the evolution of flow boiling on the surface with circular
micro-pillars, bubbles consistently form at fixed nucleation sites within
the grooves between micro-pillars, coalescing between adjacent sites
and influenced by forced convection. Additionally, as the fluid flows
around the circular micro-pillars, it undergoes changes, resulting in
the formation of vortices. Thus, owingto the combined effects of phase
change and forced convection, heat transfer on the surface with circu-
lar micro-pillars exhibits slight oscillations. The q
s
values for surfaces
with triangular and circular micro-pillars reach their peak earlier and
are higher than those for the smooth and rectangular micro-pillar sur-
faces. The difference in the highest q
s
value among the various micro-
pillar surfaces reaches 30.3% at Ja ¼0.2410. All q
s
values decrease dur-
ing sustained nucleate boiling, but all micro-pillar surfaces contribute
to improved heat transfer compared to the smooth surface, with circu-
lar micro-pillars showing the most efficient heat transfer, and rectan-
gular micro-pillars the least. When Ja ¼0.3787, as depicted in
Fig. 9(c), except for the triangular micro-pillar surface, the other surfa-
ces are entirely enveloped in a thick bubble film, leading to less fluctua-
tion in q
s
.AtJa ¼0.3787, q
s
for the triangular micro-pillar surface
exceed those of the other surfaces. As indicated in Figs. 5,7,and8,
boiling on the triangular micro-pillar surface had not yet reached a
fully stable film boiling regime at Ja ¼0.3787.
Figure 10 illustrates the boiling curves for various microchannels
at Re ¼10 and N¼16, including a smooth microchannel for compari-
son. The boiling curves demonstrate an initial increase, followed by a
decrease, and then a subsequent rise. Before the onset of nucleation
boiling at Ja 0.15, the heat transfer performance of all micro-pillar
surfaces exceeds that of the smooth surface. This is due to the micro-
channel initially being dominated by single-phase forced convection
before bubble nucleation, where micro-pillars disrupt the flow bound-
ary layer, causing intensified fluid disturbance and enhanced heat
transfer. However, in the nucleation boiling regime, the heat transfer
performance of the rectangular micro-pillar surface falls below that of
the smooth surface. This is explained by the flow boiling process
including bubble coalescence, with the presence of residual bubbles
and the reduced velocity within the grooves, which collectively dimin-
ish heat transfer on the rectangular micro-pillar surface. As Ja
increases, marking the transition to transition boiling on both smooth
and rectangular micro-pillar surfaces, the heat transfer on the rectan-
gular micro-pillar surface consistently exceeds that on the smooth
FIG. 9. Transient space-averaged heat flux q
s
in various microchannels at different
superheats: (a) Ja ¼0.1721, (b) Ja ¼0.2410, and (c) Ja ¼0.3787.
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surface. Heat transfer on surfaces with triangular and circular micro-
pillars significantly outperforms that on the smooth and rectangular
micro-pillar surfaces, with the circular micro-pillar surface displaying
the highest heat transfer efficiency. Additionally, the Ja values corre-
sponding to the CHF for surfaces with triangular and circular micro-
pillars are elevated, indicating these surfaces require higher Ja values to
enter the film boiling regime. Consequently, utilizing rectangular
micro-pillars may not necessarily improve flow boiling heat transfer in
the nucleate boiling regime. However, the use of triangular and circular
micro-pillars has proven to be effective in enhancing flow boiling heat
transfer across all boiling regimes, also delaying the onset of film
boiling.
B. Influences of micro-pillars number and Re
on the flow boiling process
An analysis is conducted to examine the impact of varying the
number of micro-pillars on the flow boiling process, compared to a
previously studied surface featuring circular micro-pillars with N¼16.
Figure 11 showcases the temperature field for surfaces with different
numbers of micro-pillars at Re ¼10, Ja ¼0.2410, and t

¼118.22.
With a constant width of the heater, the pitch of the micro-pillars
varies according to the number of micro-pillars, resulting in increased
spacing between adjacent micro-pillars as Ndecreases and vice versa.
On surfaces with fewer micro-pillars (N¼6), the bubble pattern
resembles that of a smooth surface, where a thin bubble film forms
behind the bubble column, adversely affecting boiling heat transfer
and resulting in larger high-temperature regions on the heater.
Compared to N¼16, at N¼11, the reduced number of bubble nucle-
ation sites and enlarged spacing between micro-pillars facilitate bubble
coalescence within this space, extending the bubble departure period.
Additionally, bubble columns are slightly tilted due to the surface ten-
sion holding the bubbles in place against the drag from the bulk
flow.
21,57
The temperature inside the heater beneath the flat surface is
notably higher than beneath the micro-pillar surface at N¼11, attrib-
uted to bubble growth within the wider spacing between adjacent
micro-pillars. Increasing the number of micro-pillars from N¼16 to
21 leads to more bubble nucleation sites and reduced spacing between
micro-pillars, enhancing bubble coalescence. At N¼21, the tempera-
ture inside the heater is slightly lower than at N¼16. With the maxi-
mum number of micro-pillars at N¼26, the micro-pillars appear
seamless between adjacent pillars, increasing the number of nucleation
sites and bubble coalescence. At N¼26, bubbles coalesce over a
broader region, with the presence of additional small residual bubbles
on the surface. Furthermore, the temperature difference between
N¼21 and N¼26 is small. Thus, an increased number of micro-
pillars leads to a higher number of bubble nucleation sites, thereby pro-
moting bubble coalescence.
Figure 12 demonstrates the velocity field for surfaces with varying
numbers of micro-pillars, aligning with the temperature field depicted
in Fig. 11. For enhanced observation of fluid dynamics near the micro-
pillar surface, the focus is placed on the region spanning x¼200 to
400 within the central portion of the heater. As the fluid passes the
micro-pillar, it undergoes transformations resulting in vortex forma-
tion. Additionally, velocity changes are noted at the phase change loca-
tions. At N¼6, significant vortices are predominantly seen on both
sides of the micro-pillar due to the extensive spacing between adjacent
micro-pillars. A weak vortex near the bubble film indicates lower
velocity, reducing forced convection heat transfer. With an increase in
the number of micro-pillars, the quantity of vortices on the heater sur-
face also increases. Furthermore, as the gap between the adjacent
micro-pillars narrows, the vortices within the substrate between
micro-pillars are progressively pushed upward, thereby enhancing the
interaction between vortices.
Given the motion of the fluid during flow boiling heat transfer,
forced convection is a key factor affecting heat transfer efficiency.
Thus, Re significantly influences the strength of forced convection.
Figure 13 displays the temperature field for different Re values at
N¼21, Ja ¼0.2410, and t

¼118.22. With an increase in Re,thequan-
tity of bubble columns decreases, and the residual bubbles in the
FIG. 10. Boiling curves for different microchannels.
FIG. 11. Temperature field for surfaces with different micro-pillar number Nat
Re ¼10, Ja ¼0.2410, and t

¼118.22, (a) N¼6, (b) N¼11, (c) N¼16, (d)
N¼21, and (e) N¼26.
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grooves between micro-pillars become notably smaller. This is attrib-
uted to the bulk fluid that is cooler in temperature and consistently
sweeps across the liquid–vapor interface, thereby continuously cooling
the higher-temperature bubbles on the surface. This promotes the heat
dissipation as well as limits the bubble expansion.
61
Surface tension
decreases with temperature increases, which is crucial for retaining
bubbles at the nucleation site and preserving their spherical
shape.
21,57,61
Consequently, multiple bubbles expand and more bubble
columns are created on the surface at Re ¼5. However, the develop-
ment of bubble columns diminished at Re ¼20. Owing to the constant
buoyancy force and the growing impact of forced convection with
increasing Re, the tilt of the bubble columns becomes more pro-
nounced as Re increases. Moreover, due to the extended presence and
growth of numerous bubbles on the surface at Re¼5, regions with ele-
vated temperatures within the microchannel at Re ¼5 are significantly
larger than those at higher Re.
Figure 14 displays the velocity field for varying Re, aligning with
the temperature field shown in Fig. 13, focusing on the section between
x¼200 and 400 within the central region of the heater. With an
increase in Re, the velocity within the microchannel rises, while the
influence range of individual vortices diminishes, attributed to the
FIG. 12. Velocity field for surfaces with different micro-pillars number Nat Re ¼10, Ja ¼0.2410, and t

¼118.22, (a) N¼6, (b) N¼11, (c) N¼16, (d) N¼21, and (e)
N¼26.
FIG. 13. Temperature field for different values of Re at N¼21, Ja ¼0.2410, and
t

¼118.22, (a) Re ¼5, (b) Re ¼10, (c) Re ¼15, and (d) Re ¼20.
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restraining effect of higher bulk fluid velocity on vortex development.
Nonetheless, an increased bulk fluid velocity promotes vortex intensi-
ties and forced convection, leading to higher velocities directed toward
the heater surface at Re ¼20.
Figure 15 depicts the transient space-averaged heat flux q
s
in micro-
channels with different Nand Re values at Ja ¼0.2410. Figure 15(a) com-
pares q
s
for circular micro-pillar surfaces with varying Nat Re ¼10
against a smooth surface for reference. As t

increases, a consistent
trend emerges where all q
s
values peak and then decline. The peak q
s
values for micro-pillar surfaces exceed those of the smooth surface,
although surfaces with different Nnumbers peak at distinct t

values.
At N¼6, the t

value at which the q
s
peak occurred is nearly identical
to that of the smooth surface. Increasing Nfrom 6 to 11 delays the
onset of bubble release from the heater surface, causing q
s
values at
N¼11 to first dip to a minimum before sharply rising to their peak
during the initial stages of flow boiling. The q
s
variation for N¼16
resembles that for N¼11. However, the t

at which q
s
reaches its maxi-
mum faster at N¼16. This is because the bubbles are released on the
surfaces earlier at N¼16 than at N¼11. As Nis further increased to
21 and 26, the onset of bubble release within the microchannel signifi-
cantly advances. Consequently, the corresponding q
s
values exhibit a
discernible incremental trend at an earlier stage. Furthermore, when
N16, both the maximum value of q
s
and the corresponding q
s
values
at t

are similar. All q
s
valuesforthemicro-pillarsurfacesdeclinefrom
their peak values. During the decrease in q
s
, an initial increase followed
by a decline is observed with an ascending N. For lower Nvalues such
as N¼6 and 11, q
s
fluctuations are pronounced with N¼11 showing
a wider range and higher frequency of q
s
fluctuations. The fluctuation
range in q
s
narrows for N16, attributed to the increased number of
micro-pillars, which leads to more bubble nucleation sites and facili-
tates bubble release, as depicted in Figs. 11 and 12. This increase in
bubble detachment frequency enhances heat dissipation from the
heater surface. Notably, q
s
values at N¼21 are the highest, indicating
that N¼21 optimally supports surface bubble nucleation and detach-
ment. Figure 15(b) presents the q
s
values for different Re values at
N¼21 and Ja ¼0.2410. Apart from the q
s
value at Re ¼5, which
shows significant fluctuations before peaking, all other q
s
values follow
a stable and consistent upward trend until reaching their peaks. This
pattern occurs because, at Re ¼5, bubble formation and detachment
are more profoundly affected by buoyancy than by the bulk fluid flow.
As the temperature within the microchannel gradually increases, heat
transfer decreases, leading to a subsequent reduction in q
s
as t

increases. Furthermore, with an increase in Re, the rate of decrease in
q
s
values slows down, suggesting that higher heat transfer is achievable
by elevating Re. This observation aligns with the findings from flow
boiling studies by Guo et al.
23
and Zhang et al.,
30
reinforcing that
higher Re enhances heat transfer efficiency.
Figure 16 showcases the CHF values across various Nand Re in
both smooth and different micro-pillar microchannels. Specifically,
Fig. 16(a) demonstrates the correlation between CHF and Nat a fixed
Re ¼20, including the CHF of the smooth microchannel for compari-
son. At equal Nvalues,theCHFishighestforthecircularmicro-pillar
surface, followed by the triangular micro-pillar surface, the rectangular
micro-pillar surface displays the lowest CHF. All CHF values initially
increase then decrease with an increase in N. Interestingly, the CHF
for surfaces with fewer rectangular micro-pillars declines for N¼6
and 11 compared to the CHF of the smooth surface. Even with an
optimal number of rectangular micro-pillars, the CHF reveals a mod-
est increase in only 9%. For all examined Nvalues, the CHF values for
surfaces with triangular and circular micro-pillars surpass those of the
smooth surface. The highest CHF values on surfaces with triangular
and circular micro-pillars are observed at N¼16 and 21, showing
increases of 21.6% and 36.4%, respectively, over the smooth surface.
Figure 16(b) illustrates the CHF relationship with Re, maintaining a
FIG. 14. Velocity field for different values of Re at N¼21, Ja ¼0.2410, and t

¼118.22, (a) Re ¼5, (b) Re ¼10, (c) Re ¼15, and (d) Re ¼20.
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consistent N¼21. As Re increases, CHF values increase for both
smooth and micro-pillar surfaces, with the smooth surface consistently
exhibiting the lowest CHF and the circular micro-pillar surface achiev-
ing the highest CHF. The greatest difference in CHF between the
smoothandmicro-pillarsurfacesisobservedatRe ¼10. However, as
Re surpasses 10, the gap in CHF values between these surfaces begins
to narrow. The largest disparity in CHF between the smooth and circu-
lar micro-pillar surfaces reaches 45.2%.
V. CONCLUSIONS
In this study, the LBM is utilized to explore flow boiling on struc-
tured surfaces equipped with micro-pillars, covering a variety of
micro-pillar shapes and numbers, alongside a spectrum of superheats
and Re, to evaluate heat transfer effectiveness. The identification of the
optimal micro-pillar surface emerged from an extensive analysis
encompassing the flow boiling process, temperature and velocity fields,
local and transient space-averaged heat fluxes, alongside time- and
space-averaged heat flux. The principal findings from this study are
summarized as follows:
(i) Bubble nucleation sites are randomly distributed on smooth
surface, whereas micro-pillars establish fixed nucleation sites
on heated surfaces. The phase change prompts variations in
the velocity field, leading to vortices formation around the bub-
bles. Microchannels with micro-pillar surfaces generate more
vortices compared to smooth microchannel, influenced by
both bubble dynamics and micro-pillars.
(ii) Bubble coalescence on micro-pillar surfaces slows bubble
release due to reduced distances between adjacent nucleation
sites, resulting in a thinner thermal boundary layer. The micro-
pillar shape significantly impacts bubble dynamics during flow
boiling. Rectangular micro-pillars retain residual bubbles
within their grooves, largely because of limited shear force aid-
ing bubble detachment. Transitioning to triangular and circular
micro-pillar shapes promotes the formation of vortices within
grooves, thus improving heat transfer in these regions.
(iii) An increase in the number of micro-pillars leads to increased
bubble coalescence and interactions between vortices. As Re
increases, the forced convection intensifies and the surface
temperature decreases. Consequently, surface tension becomes
FIG. 15. Transient space-averaged heat flux q
s
in microchannels with various
micro-pillars number Nand Re at Ja ¼0.2410, (a) Nand (b) Re. FIG. 16. Variation in CHF for different Nand Re: (a) Nand (b) Re.
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Published under an exclusive license by AIP Publishing
10 March 2024 12:05:12

dominant, resulting in a reduction in the number of bubble
columns on the surface.
(iv) Transient heat flux variations during flow boiling evolution dif-
fer significantly across microchannel types owing to the signifi-
cant impact of micro-pillar shape and number on the onset of
bubble detachment. The fluctuation in transient heat flux post-
peak is largely dictated by the number of micro-pillars, whereas
the decline rate is chiefly affected by Re.
(v) Among the surfaces with different micro-pillars, circular
micro-pillar achieves the highest flow boiling heat transfer per-
formance, followed by triangular and rectangular types. The
CHF values for all surfaces increase with Re. However, the
CHF on micro-pillar surfaces initially increases then decreases
with an increasing number of micro-pillars. An optimal micro-
pillar number maximizes heat transfer enhancement, realizing
a 45.2% CHF increase compared to the smooth surface.
ACKNOWLEDGMENTS
This work was supported by the Tokyo Human Resources
Fund for City Diplomacy Scholarship Program, the Grant for
Scientific Research from the Tokyo Metropolitan Government, and
JSPS KAKENHI (Grant No. 23K03715).
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
Chao Luo: Conceptualization (equal); Data curation (lead); Formal
analysis (lead); Funding acquisition (equal); Investigation (lead);
Methodology (lead); Resources (equal); Software (lead); Validation
(lead); Visualization (lead); Writing—original draft (lead). Toshio
Tagawa: Conceptualization (equal); Funding acquisition (equal);
Project administration (lead); Resources (equal); Supervision (lead);
Writing—review & editing (lead).
DATA AVAILABILITY
The data that support the findings of this study are available from
the corresponding author upon reasonable request.
APPENDIX A: MRT-LBM AND FDM
A D2Q9 pseudopotential LB model with MRT is employed
and the evolution equation of the density distribution function
(DDF) is expressed as follows:
30–32,44,45
faxþeadt;tþdt
ðÞ
¼fax;t
ðÞ
M1KM
ðÞ
ab fbfeq
b

þdtF0a;
(A1)
where f
a
and feq
adenote the DDF and its equilibrium, respectively; t,
d
t
,x, and e
a
are the time, time step, position, and discrete velocity,
respectively; F0aindicates the forcing term;
44
Mand Krepresent the
transformation and diagonal matrices, respectively. The evolution
equation can be transformed using Mmatrix as follows:
m¼mKðmmeqÞþdtIK
2

S;(A2)
where m¼Mf,m
eq
¼Mf
eq
,Sdenotes the forcing term in moment
space, and Iindicates the unit tensor. The equilibrium of the DDF
m
eq
and diagonal matrix Kwith relaxation times are stated as
follows:
42,44
meq ¼q1;2þ3jvj2;13jvj2;ux;ux;uy;uy;u2
xu2
y;uxuy

T;
(A3)
K¼diag s1
q;s1
e;s1
f;s1
j;s1
q;s1
j;s1
q;s1
v;s1
v

;(A4)
where vrepresents the macroscopic velocity determined by the
velocity components u
x
and u
y
, namely, jvj¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u2
xþu2
y
q. In Eq.
(A4),s
q
¼s
j
¼s
e
¼s
f
¼1.0, s
q
¼1.0/1.1, and s
v
¼3vþ0.5, v
denotes the viscosity. A modified forcing term Sis adopted for
mechanical stability,
44
as follows:
S¼
0
6uxFxþuyFy
ðÞ
þ-jFmj2
w2dtse0:5
ðÞ
6uxFxþuyFy
ðÞ
-jFmj2
w2dtsf0:5
ðÞ
Fx
Fx
Fy
Fy
2uxFxuyFy
ðÞ
uxFyþuyFx
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
;(A5)
where F
m
indicates the intermolecular force, wdenotes the pseudopo-
tential, and -represents the correction factor for the mechanical stabil-
ity condition, which is set as 1.18. F
x
and F
y
denote the components of
thetotalforce(F), which comprises F
m
,F
ads
,andF
g
. Here, F
ads
and F
g
denote the solid–fluid adhesion and gravitational forces, respectively.
The expressions for F
m
,F
ads
,andF
g
are expressed as follows:
Fm¼GwðxÞX
a
wawðxþeaÞea;(A6)
Fads ¼GwwðxÞX
a
xawðxÞsðxþeaÞea;(A7)
Fg¼ðqqaveÞg;(A8)
where Grepresents the interaction strength between the particles,
and G
w
indicates the interaction strength between the fluid and
solid. x
a
¼w
a
/3, where w
a
denotes the weight with the values of
w
14
¼1/3 and w
58
¼1/12. s(xþe
a
) indicates a control function
that distinguishes solids from fluids, where solids are indicated by
one, whereas fluids are indicated by zero. Parameter grepresents
gravitational acceleration, and q
ave
denotes the averaged density.
The local pseudopotential w(x) is expressed as
45
wðxÞ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ðpEOS qc2
sÞ
Gc2
r;(A9)
where the Peng–Robinson EOS is employed
25,26
Physics of Fluids ARTICLE pubs.aip.org/aip/pof
Phys. Fluids 36, 033312 (2024); doi: 10.1063/5.0195765 36, 033312-15
Published under an exclusive license by AIP Publishing
10 March 2024 12:05:12

pEOS ¼qRT
1bqaaðTÞq2
1þ2bqb2q2;(A10)
where aðTÞ¼½1þð0:37464 þ1:54226x0:26992x2Þ1
ð
ffiffiffiffiffiffiffiffiffiffiffi
T=Tc
pÞ2,x¼0.344, a¼0:45724R2T2
c=pc,b¼0:0778RTc=pc,
and p
c
and T
c
denote the critical values of pressure and temperature,
respectively. The values of a,b, and Rare selected as 3/49, 2/21, and
1, respectively, based on Refs. 25 and 26. Thus, T
c
and p
c
are 0.1094
and 0.0896, respectively.
The macroscopic qand vare obtained as follows:
q¼X
a
fa;qv¼X
a
eafaþdt
2F:(A11)
Simulation of the phase change process is achieved by utilizing
the energy equation. When viscous heat dissipation is excluded, the
energy equation takes the following form:
46
@T
@t¼vrTþ1
qcvrðkrTÞ T
qcv
@pEOS
@T

qrv;(A12)
where c
v
and krepresent the specific heat at constant volume and
the thermal conductivity, respectively. By defining the right side of
Eq. (A12) as K(T) and solving it by applying the FDM with the
fourth-order Runge–Kutta scheme
Ttþdt¼Ttþdt
6h1þ2h2þ2h3þh4
ðÞ
;(A13)
h1¼KT
t
ðÞ
;h2¼KT
tþdt
2h1

;
h3¼KT
tþdt
2h2

;h4¼KT
tþdth3
ðÞ
:
(A14)
APPENDIX B: NUMERICAL MODEL VALIDATION
The numerical model utilizes benchmark examples pro-
posed in the literature to validate its accuracy, including
Laplace’s law, contact angle, analytical solution of the film boil-
ing heat transfer, and bubble departure diameter under various
gravity conditions.
Based on the Laplace’s law, the surface tension (r) can be
determined by the equation Dp¼r/R
0
, where Dpand R
0
are the
pressure difference and droplet radius, respectively. To validate this
benchmark, a liquid droplet is placed at the center of a computa-
tional region with 200 200 lattice surrounded by periodic BC
under saturated temperature conditions. Moreover, the droplet is
surrounded by vapor. The variations in the values of Dpwith
respect to the change in 1/R
0
are presented in Fig. 17(a) under vari-
ous saturation temperatures T
s
of 0.82T
c
, 0.86T
c
, and 0.90T
c
. All
values of Dpexhibit a linear relationship with 1/R
0
at different T
s
,
enabling the determination of the corresponding r. Thus, the valid-
ity of Laplace’s law is confirmed in this study. The liquid and vapor
densities are examined at the three T
s
, and the mechanical stability
is satisfactory. Unless specified otherwise, this study employs an
environment with a saturation temperature set at T
s
¼0.86 T
c
, the
corresponding ris 0.0846, and the liquid and vapor densities are
6.5 and 0.38, respectively. In Fig. 17(b), the static droplet contact
angle (h) is plotted against the solid–fluid strength (G
w
). The hvalue
increases with G
w
, resulting in a transition of surface wettability
from hydrophilic to hydrophobic. Therefore, the value of G
w
can be
precisely controlled to manipulate h, which is consistent with the
validation reported in Refs. 30 and 42.
Berenson’s
47
analytical solution serves as a benchmark for vali-
dating the capability of the model to simulate film boiling heat
transfer on a smooth horizontal surface within a pool boiling sys-
tem. The expressions for time- and space-averaged heat transfer
coefficients (h
co
) are derived as follows:
hco ¼q
TwTs¼0:425 k3
vgqvqlqv
ðÞ
h0fg
lvTwTs
ðÞ
"#
1=4
rst
gqlqv
ðÞ

1=8
;
(B1)
where l
v
denotes the dynamic viscosity and h0
fg
represents the spe-
cific latent heat, with h0
fg
¼h
fg
þ0.5 c
p,v
(T
w
T
s
), and the specific
heat capacity c
p,v
is set as six. The time- and space-averaged Nusselt
number (Nu
exp
) can be derived from h
co
as follows:
42,48
FIG. 17. Validation of Laplace’s law and the contact angle.
Physics of Fluids ARTICLE pubs.aip.org/aip/pof
Phys. Fluids 36, 033312 (2024); doi: 10.1063/5.0195765 36, 033312-16
Published under an exclusive license by AIP Publishing
10 March 2024 12:05:12

Nuexp ¼0:425 2ffiffiffi
3
ppgqvqlqv
ðÞ
h0fg
kvlvTwTs
ðÞ
"#
1=4rst
gqlqv
ðÞ

3=8
:
(B2)
In LBM simulations, the time- and space-averaged Nu
LBM
is
determined as follows:
42,48
NuLBM ¼2ffiffiffi
3
ppl0
TwTsksolid
kv1
Lx ðLx
0
@T
@yy¼0
dx;(B3)
where T
w
denotes the temperature at the solid–fluid interface, k
solid
and k
v
represent the thermal conductivities of solid and vapor,
respectively.
A pool boiling heat transfer model utilizing a computational
domain of 200 600 lattices is implemented to validate Eq. (B2).
This setup included a heater at the bottom with saturated liquid and
vapor above it. Simulation parameters from Table I are used, and
the wall surface is treated as hydrophilic with a contact angle (h)of
53.3. Designed to simulate film boiling on a horizontal surface, the
model specified both left and right boundaries as periodic BCs. The
application of higher superheat to the bottom of the heater starts
the film boiling, resulting in a vapor film entirely covering the
heater surface, as illustrated in the upper right of Fig. 18(a), captur-
ing a moment of the film boiling process near the heater at
t

¼118.68. The heater, vapor, and liquid are represented by red,
blue, and green regions, respectively. During film boiling, consistent
emission of bubbles from the vapor film leads to fluctuations in the
Nu value, as depicted in Fig. 18(a), with similar fluctuations docu-
mented in Refs. 42 and 48. Discrepancies between Nu
LBM
and Nu
exp
remained below 4% for both Ja, affirming the model’s applicability
in boiling heat transfer studies.
Experimental findings by Fritz,
49
widely referenced for numeri-
cal model validation, explore the bubble departure diameter (D)
under different gravity (g) conditions within a pool boiling frame-
work, suggesting the dependency of Don g, specifically D/g0:5.
For this validation, a pool boiling system is simulated using
200 300 lattices, with a microheater at the center of the bottom
surface to form bubbles, whereas maintaining the remainder of the
bottom surface adiabatic. A non-slip boundary is applied at the top
and periodic BC on the side boundaries. Snapshots portraying the
bubble detachment and corresponding diameters for various gval-
ues are obtained, as depicted in Fig. 18(b). These snapshots revealed
that bubble size diminishes with an increase in g. Fitting specific D
and gvalues into correlation expressions, a derived relationship
shows that Dis inversely proportional to the 0.4957 power of g.
This finding indicates the effectiveness of the LB model in replicat-
ing boiling phenomena. Therefore, successful validation of both
analytical expressions and experimental correlations demonstrates
the effectiveness of the LB model in flow boiling research.
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