International Journal of Heat and Mass Transfer 171 (2021) 121070

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International Journal of Heat and Mass Transfer

journal homepage: www.elsevier.com/locate/hmt

A new pool boiling heat transfer correlation for wetting dielectric

fluids on metal foams
Leonardo Lachi Manetti a,b, Ana Sofia Oliveira Henriques Moita c,d, Elaine Maria Cardoso a,e,∗
a
UNESP – São Paulo State University, School of Engineering, Post-Graduation Program in Mechanical Engineering, Av. Brasil, 56, 15385-000, Ilha Solteira, SP,
Brazil
b
IFMS - Federal Institute of Education, Science and Technology of Mato Grosso do Sul, Campus Campo Grande, Campo Grande, MS, Brazil
c
IN+, Dep. Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Lisbon, Portugal
d
CINAMIL, Military Academy Research Center, Department of Exact Sciences and Engineering, Portuguese Military Academy, Lisbon, Portugal
e
UNESP – São Paulo State University, Campus of São João da Boa Vista, São João da Boa Vista, SP, Brazil

a r t i c l e i n f o a b s t r a c t

Article history: Porous structures, as metal foams, can enhance the heat transfer performance. For a safe industrial ap-
Received 24 November 2020 plication, a predictive model for both heat transfer coefficient and maximum heat flux is required. But,
Revised 16 January 2021
there is no correlation for dielectric fluids on metal foams available in the literature. This work aims to
Accepted 5 February 2021
develop a correlation based on dimensional analysis for metal foam surfaces in pool boiling with dif-
Available online 19 February 2021
ferent dielectric fluids. The model takes into account the porous heating surface characteristics (porosity,
Keywords: pore diameter, and thickness), the working fluid thermophysical properties, and its interaction. The model
Pool boiling was developed based on the experimental database obtained by the authors and validated with the open
Metal foam literature database. Two metal foams with different characteristics were used for carrying out the pool
Dielectric fluid boiling tests with two different working fluids: HFE-7100 and ethanol. The newly developed correlation
Predictive model predicted well the database with an average error equal to 10.8% where 93.8% within the error range of
Immersion cooling
± 30%. To the maximum heat flux, the average error was 13.6% where 100% within the error range of
± 30%. The pore diameter and thickness play an important role in both models. The porosity and solid-
phase thermal conductivity from the metal foam change the porous medium thermal conductivity, which
influences the heat transfer coefficient (HTC). Finally, the properties of the working fluid also influence
the predictive model, mainly the latent heat of vaporization, liquid thermal conductivity, and saturation
temperature.
© 2021 Elsevier Ltd. All rights reserved.

1. Introduction the coolant must be dielectric [4–6]. Thus, although water pos-
sesses a higher boiling heat transfer coefficient, it is not compatible
Continuous improvement of the performance of instruments with electronic devices by using direct immersion cooling due to
and equipment, mainly of electronics, supercomputers, and data- the problem of electrical short-circuit; on the other hand, dielectric
centers has led researchers to seek methods for cooling compo- fluids such as fluorochemical liquids are not electrically conductive
nents with higher heat flux and heat density. Two-phase cooling and it has been proved to be highly suitable as a liquid medium
systems have been widely studied for thermal management. Pool for cooling [7]. However, these fluids have relatively poor thermo-
boiling is a low-cost technique due to the buoyancy of the bub- physical properties and extremely high wettability with most of
bles, which creates a passive cooling without requiring pumping the surfaces, which requires large superheat to initiate the boil-
power or moving parts in the system [1–3]. It can be used in two ing process - commonly referred to as ‘incipience excursion’ [8].
configurations: indirect and direct immersion cooling; the former Therefore, to meet the cooling requirements of modern electronic
allows the use of many working fluids because there is an interface devices by using dielectric fluids, the use of surface enhancement
material between the heat source and the coolant; the latter elim- techniques have been widely applied to reduce boiling incipience
inates the interface material and so increases the power density superheat and improve both nucleate boiling heat transfer coeffi-
and energy efficiency in cooling high power electronics. However, cient (HTC) and critical heat flux (CHF) - the highest heat flux in
the nucleate boiling regime. According to Hendricks et al. [9], the
∗
heating surface can be modified by using three factors: (i) exis-
Corresponding author.
E-mail address: elaine.cardoso@unesp.br (E.M. Cardoso).
tence of random micro- or nano-size crevices and surface irregu-

https://doi.org/10.1016/j.ijheatmasstransfer.2021.121070
0017-9310/© 2021 Elsevier Ltd. All rights reserved.
L.L. Manetti, A.S.O.H. Moita and E.M. Cardoso International Journal of Heat and Mass Transfer 171 (2021) 121070

Nomenclature foam Metal foam property

l or liqSaturated liquid fluid condition Saturated liquid fluid
Alphabetic condition
a cell diameter [m] max Maximum point
A Area [m2 ] Ni Nickel property
ai Multiple linear regression exponent [-] s Metal foam Solid-
asf Specific area [m2 /m3 ] phase material
bi Multiple linear regression exponent [-] sat Saturation fluid con-
cp Specific heat capacity [J/kg•K] dition
Csf Surface-fluid coefficient [-] sp Sintered particle
df Fiber diameter [m] square Square heater cross-
dp Pore diameter [m] section
G Shape factor, given by Eq. (39) v or vap Vapor saturated fluid
h Heat transfer coefficient [W/m2 •K] condition
hlv Latent heat of vaporization [J/Kg] w Surface wall
Ja∗ Modified Jacob number [-]
k Thermal conductivity [W/m•K]
keff Effective thermal conductivity [W/m•K] larities for bubble nucleation; (ii) porous surface structure that al-
kM Thermal conductivity from the porous medium lows fluid inflow to keep nucleation sites active; and, (iii) surface
given by Eq. (10) [W/m•K] protrusions that enlarge the boiling surface area. Porous surface
L Copper block distances [m] structures have been widely reported to enhance heat transfer per-
Lc Characteristic length [m] formance due to their interconnected porous, which increase the
m Mass [kg] wetted area and the nucleation site density [10]. The porous thick-
Nu Nusselt number [-] ness and pore size are the most important parameters of a porous
p Pressure [Pa] surface, and their optimal values mainly depend on the fluid prop-
pint Boiling chamber internal pressure [Pa] erties [11].
Pr Prandtl number [-] For a safe industrial application, a predictive model for both
Preff Effective Prandtl number [-] HTC and CHF is required. According to Wu et al. [12], mechanistic
q  Heat flux [W/m2 ] models and empirical or semi-empirical correlations have achieved
q0 Reference heat flux [W/m2 ] some success in predicting boiling HTCs, mainly for smooth sur-
q  measured Heat flux measured at copper block [W/m2 ] faces, although some of them may predict experimental data well
T Temperature [K] while failing for other data sets. The former is based on the heat
u Uncertainty flux partitioning model (RPI Model), which addresses various boil-
ing heat transfer mechanisms calculated separately [13,14]. The to-
Greek symbols tal boiling heat flux is attributed to three heat transfer mecha-
μ Absolute viscosity [kg/m•s] nisms: (1) natural convection in the region of the surface not in-
γ Ratio of the ligament node radii to the ligament fluenced by the bubbles and microconvection due to the bubble
length given by Eq. (28) [-] motion; (2) evaporation, where the vapor is formed by the evap-
γ 20 Parcel of data predicted within an error band of ± oration of the thin liquid layer under a growing bubble and/or of
20 [-] the superheated liquid to the bubble through the bubble wall (a
γ 30 Parcel of data predicted within an error band of ± term related to the latent heat of evaporation); and, (3) so-called
30 [-] quenching due to the inflow of cold fluid on the heating surface
δ Foam thickness [m] and subsequent thermal boundary layer re-formation after bubble
T Temperature difference [K] departure [15]. However, the mechanistic models require many pa-
ε Porosity [-] rameters information, like bubble departure diameter and its fre-
ξ Dimensionless thickness [-] quency, and active nucleation site density. These parameters are
k Dimensionless number from Buckingham π theo- difficult to obtain, especially the active nucleation site densities,
rem [-] which restricts the general validation of mechanistic models. Be-
ρ Density [kg/m3 ] sides, the mechanistic models normally eliminate bubble interac-
ρr Relative density [-] tions, and thus, they are not valid at high heat fluxes [16].
σ Surface tension [N/m] As a workaround, several authors proposed modeling nucleate
Dimensionless porosity [-] boiling via dimensional analysis, avoiding the use of bubble diam-
ω Dimensionless PPI [-] eter, frequencies, and nucleation sites that must be obtained ex-
perimentally [17–20]. The empirical or semi-empirical correlations
Subscripts require critical analysis to find the main phenomena, properties,
1 to 4 Thermocouples posi- and dimensionless numbers to describe the pool boiling data set.
tion The classical Rohsenow correlation [17] was developed based on
atm Atmospheric condi- a physical analogy between single-phase convection and nucleate
tion boiling where Reynolds, Prandtl, and Stanton numbers are corre-
calc Calculated/predicted lated [21]. Stephan and Abdelsalam [18] used regression analysis
circular Circular heater cross- to derive the Nusselt number from other dimensionless variables.
section According to Stephan [22], the regression analysis showed that cer-
Cu Copper property tain non-dimensional quantities prove to be important for some
exp Experimental substances, but unimportant for others. A disadvantage of these
correlations is the fact that the composition of the heating surface

2
L.L. Manetti, A.S.O.H. Moita and E.M. Cardoso International Journal of Heat and Mass Transfer 171 (2021) 121070

has not been taken into consideration. Some of them address the Table 1
Metal foams images from SteREO and SEM microscopy (adapted from
effect of surface topography by considering the surface roughness;
[31]).
however, this parameter is not representative of the geometric fea-
tures of the surface pattern and so it is weakly correlated with the
quantities describing nucleation and bubble dynamics [19].
Teodori et al. [19] proposed a new correlation based on an em-
pirical method to predict the pool boiling HTC on regular micro-
patterned heating surfaces. Two correlations were obtained by di-
mensional analysis, which was slightly different in terms of the
phenomenological description of the boiling process. Both of them
were able to predict the heat transfer coefficient within an er-
ror of ± 30%. Also, these correlations could predict fluids behav-
ior with significantly different thermophysical properties. Recently,
Liang et al. [23] studied pool boiling enhancement on micro-pit
surfaces by using water as the working fluid; to predict the data,
they modified the Rohsenow correlation [17] by incorporating the
pit depth. The correlation showed a good agreement with the ex-
perimental data, especially in the nucleate boiling region with rel-
atively high HTCs. On the other hand, for the relatively low HTCs,
partial data deviate slightly from the 20% error line due to the
single-phase HTCs included in the database. Despite this, the cor-
relation estimated the HTC prior to CHF, with a mean absolute
percentage error (MAPE) of 8.3%. However, according to Lin and
Kedzierski [11], the boiling models for the structured surface are
not directly applicable to the porous surface due to the random-
ness of the porous surface geometry (it is more difficult to de-
Fig. 1. Open-cell foam structure.
velop a mechanistically based model for the boiling heat transfer
on a porous surface); consequently, few studies predicting the heat
transfer on porous surfaces were found in the open literature.
In this way, this work aims to develop a correlation, based
Nishikawa and Ito [24] developed an empirical correlation
on dimensional analysis, for metal foams surfaces in pool boil-
based on their pool boiling data of R-11, R-113, and benzene at sat-
ing with a dielectric fluid, taking into account the porous heating
uration temperature and atmospheric pressure on the porous sur-
surface characteristics (porosity, pore diameter, and thickness), the
faces coated by copper or bronze particles, whose diameter ranged
working fluid thermophysical properties, and the interaction be-
from 100 to 10 0 0 μm. The correlation incorporates the effects of
tween them. The correlation is developed based on our experimen-
porosity – ranged from 0.38 to 0.71 – and thickness of the coat-
tal database (some of them were based on Manetti et al. [31,32])
ing layer – ranged from 0.5 to 5 mm, particle diameter, and fluid
and validated with the open literature database.
properties. Xu et al. [25] used the Rohsenow correlation [17] to de-
termine the correlation for a highly porous surface – metal foams
– by adding a modification factor and exponent. The effects of pore 2. Experimental setup and foams samples
density, porosity, and thickness were included in the modified fac-
tor and exponent. For water pool boiling heat transfer, approxi- 2.1. Foams parameter
mately 80% of the predicted data were within an error band of
± 30%. Besides, Righetti et al. [26] modified an exponent from Xu The metal foams used in this work are the same as Manetti
et al. [25] model to predict the boiling of water on thicker alu- et al. [31]. Two metal foams with different parameters were used
minum foams and obtained an absolute deviation of 12.1%. Re- for carrying out the pool boiling tests, namely a copper foam
cently, Hu et al. [27] also developed a correlation based on the (Cu foam), and nickel foam (Ni foam). Table 1 shows the SteREO
Rohsenow correlation [17] by introducing a metal foam influence and SEM images from each one; they are open-cell metal foams
factor. They incorporated the effect of area density, pore density, (Fig. 1) fabricated by using the metal deposition process as de-
porosity, and advancing contact angle. Besides, they used their tailed by Ashby et al. [33] and Banhart [34]. They were acquired
database with boiling of water on uncoated and hydrophobic metal from Nanoshel R
in 500 × 500 mm2 panels with 3 mm thick. To
foam covers and data from Shi et al. [28] and Xu et al. [29] for fit- use the foam in pool boiling tests, they were cut in a square sec-
ting the coefficients. The predicted values of the new correlation tion of 16 × 16 mm2 by using a wire electrical discharge machin-
agreed with 95% of the experimental data within a deviation of ± ing (wire-EDM). After, the number of pores per inch (PPI) - defined
20%, and the average deviation was 10.2% as the number of pores along a straight test line related to the line
Based on the literature review, immersion cooling with porous length - were measured by using the optical images; at least seven
heating surfaces has the potential for industrial applications but, lines in each direction – horizontal and vertical, were traced and
before this, it is advisable to determine the pool boiling heat trans- the number of porous intercepted by the lines was counted and
fer performance. Correlations obtained from the dimensional anal- divided by the line length – in inches; so, the average from ra-
ysis are more suitable for the entire nucleate boiling curve. As re- tios yields the PPI of each open-cell metal foam. For Cu and Ni
ported, some researchers focus on correlations by using refrigerant open-cell metal foam, the average PPI values were 31.75 ± 6.2 and
fluids with a sintered porous layer while others by using water on 62.72 ± 12.85, respectively.
metal foams; but until now, there is no correlation for dielectric The metal foams porosity (ε ) was obtained by weighing sam-
fluids on metal foams. Moreover, the predictive model should be ples of the same size in a precision balance and comparing the
validated with a database from literature, i.e., data that were not foam density, ρ foam = mfoam /Vfoam , with the solid density,
used in the coefficients/exponent regression to test the model and ρfoam
avoid overfitting [30]. ε = 1 − ρr = 1 − (1)
ρs
3
L.L. Manetti, A.S.O.H. Moita and E.M. Cardoso International Journal of Heat and Mass Transfer 171 (2021) 121070

Table 2
Metals foams characteristics (adapted from [31]).

Foam mfoam × 103 (kg) ρ foam (kg/m³) ρ r (%) ε (%) a (mm) dp (mm) df (mm)

Cu 0.697 ± 0.022 908.1 ± 28.63 10.0 ± 0.32 90.0 ± 0.32 1.08 ± 0.24 0.46 ± 0.25 0.13 ± 0.04
Ni 0.106 ± 0.010 138.0 ± 14.12 1.6 ± 0.15 98.4 ± 0.15 0.46 ± 0.10 0.25 ± 0.09 0.07 ± 0.02
1
Solid density: ρ Cu = 8960 kg/m³; ρ Ni = 8900 kg/m³ [38].

Fig. 2. Steps of the foam granulometry characterization by using iMorph.

Also, X-ray microcomputed tomography (μCT) images were

taken from a Skycan 1272 at a resolution of 15 μm (100 kV X-
ray source voltage) to precisely characterize the foam granulom-
etry - cell, porous, and fibers diameters (a, dp and df , respec-
tively) as shown in Fig. 1. First, the μCT virtual slices were in-
put in the iMoph software [35] and the gray-level threshold value
was selected such that the porosity of the reconstructed 3D vol-
ume matched with the measured foam porosity (Table 2); next,
iMorph uses the ‘marching cubes’ algorithm to extract the inter-
face between the porous and solid phases by creating a triangu- Fig. 3. Metal foam (δ = 3 mm) soldered on the plain surface: (a) Cu foam; and
(b) Ni foam. (For interpretation of the references to colour in this figure legend, the
lated surface mesh that is rendered to form a solid. Finally, cells
reader is referred to the web version of this article.)
were individualized (represented by a color) by using cell segmen-
tation that was obtained after maximal ball identification and wa-
tershed transform [36,37]. The pores (throat) are the contact re- (3 mm, 2 mm, and 1 mm) while four different thickness was tested
gion between two individualized cells that can be observed after for the nickel foam (3 mm, 2 mm, 1 mm, 0.5 mm). The last thick-
the cell extraction. Therefore, the cell diameter and fibers diam- ness for both foams is relative to the cell mean diameter, i.e., the
eters were obtained from the maximal ball identification and the lowest thickness must be close to the cell diameter in order to not
pore diameter was obtained from the area of cell contacts by con- mischaracterize the foam geometry.
sidering the diameter of a circle. Fig. 2 shows a process flowchart
from the steps in iMorph and Table 2 shows the average and stan-
dard deviation of the porosity (ε ), cell diameter (a), pore diameter 2.2. Pool boiling experimental facilities
(dp ), and fiber diameter (df ) for each foam.
The metal foams with the original thickness were soldered on The database for developing the new correlation was obtained
the copper block with a plain and square plate on the upper sur- by boiling two different working fluids (both dielectric fluids):
face (16 × 16 mm2 ) of the copper cylinder (Fig. 3) following the HFE-7100 (3 M Novec) and ethanol (96% v/v) on the metal foams
procedure reported by Manetti et al. [31] that provides a much previously described. Moreover, the experimental tests were car-
lower thermal contact resistance - in the order of 10−6 m2 •K/W. ried out in two different research laboratories with different facili-
Furthermore, other foam thicknesses were used in this work. The ties by using a similar methodology. First, pool boiling of HFE-7100
thickness level variation was carried out by using the electric dis- (at local atmospheric pressure) on the metal foams were carried
charge machining process (EDM) as explained by Manetti et al. out at the School of Engineering (UNESP, Brazil). Next, pool boil-
[32]. Three different thickness (δ ) was tested for the copper foam ing of ethanol (at local atmospheric pressure) on the metal foams

4
L.L. Manetti, A.S.O.H. Moita and E.M. Cardoso International Journal of Heat and Mass Transfer 171 (2021) 121070

Fig. 4. Schematic of the experimental facility: (1) Data acquisition; (2) DC power
source; (3) VARIAC; (4) Pool boiling chamber; (5) thermal bath; (6) Datalog-
ger/Computer. (adapted from [31,32]). Fig. 5. Pool boiling chamber for ethanol boiling: (1) cooper block; (2) lower flange;
(3) tank with windows; (4) upper flange; (5) pressure transducer; (6) vacuum/feed
valve and connection to the condenser; (7) auxiliary heater; (8) heating surface (9)
were carried out at the IN+ (IST, Portugal). The following subsec- main heater.
tions describe both facilities.

uration temperature with the aid of an auxiliary cartridge heater

2.2.1. HFE-7100 pool boiling facility and data reduction
fed by a VARIAC transformer and the pressure inside the boiling
The pool boiling tests by using HFE-7100 were performed in
chamber was maintained close to the local atmospheric pressure,
the apparatus in Fig. 4, which consists of a boiling chamber with
patm = 100.5 kPa, during the boiling tests. The lower part is closed
a rectangular glass section (120 × 100 mm²) with a thickness of
with a circular flange where the test section is coupled. The test
5 mm and 200 mm height. A thermal bath was used to control the
section consists firstly of the main cartridge heater fed by another
internal condenser coil temperature located at the top of the boil-
VARIAC transformer; next, a copper block was used to measure the
ing chamber. An auxiliary heater, controlled with a variable trans-
heat flux; finally, the heating surface is located at the top. The ther-
former (VARIAC), submerged in the working fluid was used to de-
mal insulation in the radial and axial directions of the test section
gas the fluid before the beginning of each test and maintain the
consisted of PTFE. Moreover, to decrease the contact resistance be-
liquid temperature near the saturation point during the test. Two
tween the copper block and the heating surface, a thermal paste
K-type thermocouples, Tliq , and Tvap , located in the working fluid
(arctic mx-2) was applied.
and vapor, respectively, allowed monitoring the test condition tem-
Three K-type thermocouples (T2 , T3 , and T4 ) with 0.5 mm diam-
perature. The pressure inside the boiling chamber was measured
eters were used to measure the heat flux (q”measured ) in the copper
by a pressure transducer and maintained close to the local atmo-
block and a fourth one (T1 ), located on the heating surface after
spheric pressure, patm = 98 kPa, during the boiling tests.
the contact resistance, was used to estimate the wall temperature
Three K-type thermocouples (T1 , T2 , and T3 ) with 0.5 mm diam-
(Tw ). A data acquisition system (Data Translation R
DT9828) was
eters were used to measure the heat flux (q”measured ) and estimate
used to acquire the data signals from the thermocouples, a Na-
the wall temperature (Tw ). A cartridge resistance heated the bot-
tional Instruments R
BNC-2120 acquired the signal from the pres-
tom part of the copper block; the power was supplied by a stabi-
sure transducer and a digital multimeter Tektronix R
DMM 4020
lized variable DC power source. The thermal insulation of the test
measured the power applied. Then, all signals were registered in a
section consisted of polytetrafluoroethylene (PTFE). A data acquisi-
personal computer using a LabView routine.
tion system (Agilent 34970A) was used to acquire all the data sig-
Before the boiling tests, a vacuum was created in the tank to
nals (power, pressure, and temperature) and, then, they were reg-
feed it with ethanol. After feeding, the auxiliary heater boiled the
istered in a personal computer using the Agilent Benchlink Data
working fluid for 1 hour for degassing it. So, the heating effect was
Logger. The heating effect was imposed by increasing the electrical
imposed by increasing the electrical power according to heat flux
power according to heat flux steps until a condition close to the
steps, starting close to 35 kW/m2 , until a condition close to the
CHF. For each metal foam test, the experiment was carried out at
CHF, characterized by the non-stability of the heating surface tem-
least twice under similar conditions to ensure that the results were
perature. For each metal foam test, the experiment was carried out
repeatable. More details about the experimental setup and data re-
at least twice under similar conditions in a period of 24 h to check
duction of that subsection can be found in Manetti et al. [31,32].
the effect of aging on the results obtained to ensure that the re-
sults were repeatable.
2.2.2. Ethanol pool boiling facility and data reduction
The heat flux was measured between the three first thermo-
The pool boiling tests by using ethanol were performed in
couples, where the middle one was used just to confirm the lin-
the apparatus in Fig. 5. The boiling chamber was projected to
ear temperature profile [32]. So, the measured heat flux by using
be heated by Joule effect in a stainless-steel foil as presented by
Fourier’s law of conduction through the more spaced thermocou-
Pontes et al. [39,40]; however, it was modified to a copper block
ples in the copper metering block was calculated,
heating for this work. The core component is a cylindrical metallic
tank with three glass windows for visualization. The upper part is Acircular T24
closed with a rectangular flange where an absolute pressure trans- qmeasured = · kCu · (2)
Asquare L24
ducer (0 – 2 bar) is located, together with two K-type thermocou-
ples, Tliq , and Tvap , a valve for filling the chamber, and a connec- where the area ratio is due to the square cross-section (16 × 16
tion to an external condenser. Ethanol is maintained at the sat- mm2 ) at the surface upper level and the circular area of the cop-

5
L.L. Manetti, A.S.O.H. Moita and E.M. Cardoso International Journal of Heat and Mass Transfer 171 (2021) 121070

per block (25 mm in diameter) at the lower level; L24 is the ther- the boiling regimes change to fully developed nucleate boiling. In
mocouples distance equal to 14 mm as shown in Fig. 5. this range of heat fluxes, there is a competition of the convection
The HTC was calculated using Newton’s law of cooling given by: area (wetted area) and vapor bubble flow resistance to outlet the
foam structure. Therefore, for each heat flux range, there is an op-
qmeasured q timum thickness as showed by Manetti et al. [32].
HTCexp = = measured (3)
Tw − Tsat ( pint ) Tsat Another parameter on the foam heat transfer phenomenon is
the pore diameter, which is correlated to the specific area (area
where Tsat (pint ) corresponds to the saturation temperature of the
density) as reported by Calmidi and Mahajan [42]; a lower pore
ethanol, at pressure inside de boiling chamber and Tw is the wall
diameter can result in a better heat transfer performance at low
temperature given as follows:
heat fluxes; however, with high thickness, it could increase the va-
qmeasured por bubble flow resistance decreasing the HTC. So, the thicker and
Tw = T1 − L1w (4)
kCu smaller the pore, the larger the liquid-vapor counter-flow, since the
where the second term is the linear temperature profile in the liquid replenishment is inhibited, leading to a lower heat trans-
square section (Lsw = 5 mm). fer coefficient and an earlier occurrence of dryout - defined as the
The experimental uncertainties (u) were calculated by using the condition at which the HTC presents a maximum value [31].
method described by Moffat [41] where the uncertainty in the re- In addition, the foam material properties, specifically the ther-
sult R is a function of the independent variables Xi as follow: mal conductivity, ks , also play an important role in the thermal
behavior; the higher the thermal conductivity of the foam mate-
⎡ 2 ⎤ 12 rial the higher the effective foam thermal conductivity and, con-

n
∂R
uR = ⎣ u ⎦ (5) sequently, the higher the foaming efficiency (fin efficiency) as re-
i
∂ Xi Xi ported by Manetti et al. [31].
Finally, fluid thermophysical properties influence HTC perfor-
Therefore, the relative uncertainty for the heat flux between the mance. Table 3 shows the main thermophysical properties of both
thermocouples 4 and 2 was given by: fluids tested. It can be seen that ethanol has a higher latent heat of
1/2 vaporization, thermal conductivity, specific heat, and absolute vis-
uq uT42 2 uL42 2
measured
= + (6) cosity; the higher the fluid properties the higher the HTC.
q
measured
T42 L42
3.1. Prediction of the experimental database by using known
where the differential uncertainty of the K-type thermocouples, correlations
uT , was ± 0.3 °C (corresponds to the thermocouples uncertain-
ties after the calibration); the uncertainty in the position of the As cited in the Introduction Section, Nishikawa and Ito [24] cor-
thermocouple junction was estimated to be ± 0.03 mm, and the related their data to predict the Nusselt number as follow:
wall superheat uncertainty was given by:  0.0284  0.560  0.593
h·δ σ 2 hlv δ q dsp
 2 1/2 Nu = = 1 × 10−3
L1w 2
qmeasured kM q 2 δ 2 dsp ε hlv μv
uTsat = (uT )2 + − u  + − uL1w  −0.708
kCu qmeasured kCu
kM ρl 1.67
× (9)
(7) kl ρv
Finally, the HTC uncertainty was given by: where σ , hl v , kl , ρl , ρv , and μv represent the fluid thermophysical
 2 1/2 properties: surface tension [N/m], latent heat of vaporization [J/kg],
uh uTsat 2 uq thermal conductivity [W/m.K], liquid and vapor density [kg/m3 ],
= + measured
(8)
h Tsat qmeasured and absolute viscosity [kg/m.s], respectively. Besides, dsp represents
the sintered particle diameter (solid phase of the porous medium)
Therefore, the experimental uncertainty of the heat transfer co- and kM is the thermal conductivity of the porous medium given by
efficient is higher in low heat fluxes while it decreases as heat the parallel model:
fluxes are increased. k M = ε k l + ( 1 − ε )k s (10)
Before the tests with metal foams, we carried out a test on
To evaluate the capability of the previous correlation, Eq. (9),
a plain surface to validate the apparatus and have a reference to
to predict HTC results for pool boiling, the current experimental
compare the experiments. The results obtained with the plain sur-
database was compared with the values given by Eq. (9) (calcu-
face and the comparative analysis between them and those ob-
lated Nusselt). Thus, to apply our database in Nishikawa and Ito
tained with the metal foam can be found in the Supplementary
[24] correlation, we changed dsp to the foam fiber diameter, d f ,
Data.
because they are both the solid phase of the porous medium. In
3. Database addition, the data after the highest HTC of each curve, Fig. 6, was
removed because the predictive model does not consider the dry-
The first run of the experimental database – HTC for both metal out phenomenon. Fig. 7 shows the calculated Nusselt number by
foams: Cu and Ni; and, both fluids, HFE-7100 and Ethanol – are the previous correlation versus the experimental database. Also,
presented in Fig. 6. The results for HFE-7100 on Ni foam with a it presents the statistical analysis between experimental and pre-
3.0 mm thickness are from Manetti et al. [31] and on Cu foam with dicted data, including the parcel of data predicted within an error
different thicknesses are from Manetti et al. [32]. band of ± 20%, γ20 , and ± 30%, γ30 ; and, the mean average per-
centage error, MAPE, defined as follows:
One may observe in Fig. 6 that the thickness level variation  
plays an important role in the boiling curves. Higher thickness N  Nucalc −Nuexp 
i=1  Nuexp 
has a greater wetted area, which can improve the heat transfer at MAPE = (11)
the first heat fluxes while some nucleation sites are activated and N
other regions stayed on natural convection. As the heat flux in- As can be seen, the Nishikawa and Ito [24] correlation over pre-
creases, more vapor bubbles emerge from the foam structure and dicted most of the database. It is expected that the experimental

6
L.L. Manetti, A.S.O.H. Moita and E.M. Cardoso International Journal of Heat and Mass Transfer 171 (2021) 121070

Fig. 6. HTC curves of metal foam with different thickness and fluids: (a) HFE-7100 on Cu foams (δ = 1 to 3 mm) [32]; (b) HFE-7100 on Ni foams (δ = 0.5 to 3 mm [31]);
(c) Ethanol on Cu foams (δ = 1 to 3 mm); (b) Ethanol on Ni foams (δ = 0.5 to 3 mm).

Table 3
Thermophysical properties from HFE-7100 and ethanol at local atmospheric pressure.

Fluid patm (kPa) Tsat ( °C) ρ l (kg/m³) ρ v (kg/m³) 106 × μl (Pa.s) cp, l (J/kg.K) hlv (kJ/kg) kl (W/m.K) σ (mN/m) Lc (mm)
HFE-7100 98∗ 60.3 1420.7 9.47 431 1253.6 111.9 0.062 10.26 0.86
Ethanol 100.6† 78.1 737.2 1.66 514 3111.0 849.4 0.157 17.62 1.56
∗
Atmospheric pressure at Ilha Solteira.
†
Atmospheric pressure at Lisbon.

database does not fit well with the Nishikawa and Ito [24] correla- Fig. 8 shows the Eq. (12) left-hand, i.e., the calculated heat flux
tion mainly due to the surface characteristics. The sintered particle versus the experimental database. Also, it presents the statistical
layer tested by Nishikawa and Ito [24] had a porosity range from analysis between experimental and predicted data.
0.38 to 0.71 while our metal foams samples are from 0.9 to 0.98. Unlike the Nishikawa and Ito [24] correlation, the Xu et al.
Moreover, the particle layer had a higher particle diameter than [25] model modified by Righetti et al. [26] under-predicted our ex-
the metal foam fiber diameter. perimental data. It is because the data used to develop the model
Another model tested was from Xu et al. [25] modified by was based on water pool boiling tests, which has wettability and
Righetti et al. [26] as follow: thermophysical properties higher than the dielectric fluids tested
  b in the current work.
q σ c p,l Tsat
=c (12)
μl hlv g ( ρl − ρv ) Cs f hlv Prl 4. Results and discussion

where b and c are parameters fitted by Righetti et al. [26], 4.1. New HTC predictive model formulation

b = 1.53ψ −0.5124 ω0.01926 ξ 0.1793 (13) The newly developed correlation was obtained based on the
Buckingham π theorem [43] to formulate the independent funda-
mentals physics dimensions, r, chosen to represent the dependent
⎧ 5.5949ψ −0.2323 ω0.003588 ξ 0.025 −5.506
⎨ 10( ), 0 < q ≤ 250 kW/m2 parameters. The application of this theorem first requires a deci-
c = 10(5.5949ψ ω ξ −5.4059 ) sion on which of the parameters play roles on the boiling HTC (h)
−0.2323 0.003588 0.025
, 250 < q ≤ 490 kW/m2
⎩ (5.5949ψ −0.2323 ω0.003588 ξ 0.025 −5.3089) of wetting dielectric fluids on metal foams:
10 , 490 < q ≤ 1460 kW/m2
(14)
• first, the boiling HTC is dependent on the heat flux, q”. Note
that the heat flux is the imposed variable; thus, the wall super-
and, ω = PPI/5; ξ = δ /5 mm; = ε /0.9; and, Csf = 0.0165. heating (Tsat ) must not be chosen to avoid redundancy.

7
L.L. Manetti, A.S.O.H. Moita and E.M. Cardoso International Journal of Heat and Mass Transfer 171 (2021) 121070

length), Lc ; the latent heat of vaporization, hlv ; specific heat of

the liquid, cp ,l ; and absolute viscosity of the liquid, μl ; also, are
important parameters for the HTC predictive model.

Finally, the following functional relation can be written:

 
h = f q , Lc , keff , c p,l , Tsat , μl , hlv , δ, dp (15)

The n = 10 variables are listed in the dimension MLT [44] that

implies in r = j = 4 repeating variables (q”, Lc , keff , cp ,l ) resulting in
k = n – j = 6 dimensionless variables (k ).
The first dimensionless number obtained is the Nusselt number,

h · Lc
1 = Nu = (16)
keff

which is similar to the Nusselt number obtained by Teodori et al.

[19] and Kiyomura et al. [20]; however, it incorporates the effective
thermal conductivity from the porous medium as Nishikawa and
Ito [24] correlation.
The second dimensionless number,
q · Lc
2 = (17)
keff Tsat
is the only one where the imposed variable, q”, appears. That di-
mensionless number is associated with the effect of liquid agita-
tion near the wall. A similar dimensionless number was obtained
by Stephan and Abdelsalam [18] and Teodori et al. [19].
Fig. 7. Comparison between Nusselt number from the experimental database and
Nusselt number calculated by Nishikawa and Ito [24] correlation. The third dimensionless number is a porous media Prandtl
number [45,46], or also called of effective Prandtl number [47] be-
cause it incorporates the effective thermal conductivity from the
porous media as follow:
c p,l μl
3 = Preff = (18)
keff
The effective Prandtl number is important because the heat
transfer occurs directly from the porous heating surface to the ad-
jacent liquid, adapting a single-phase forced convection heat trans-
fer model to nucleate pool boiling. As reported by Gerardi et al.
[14], Thiagarajan et al. [15], and Teodori et al. [19], the quenching
effect, i.e., the heat transfer due to the re-formation of the ther-
mal boundary layer, plays an important role on the nucleate boil-
ing heat transfer.
The fourth dimensionless number is the modified Jakob num-
ber, which accounts the sensible heat and the latent heat of vapor-
ization – one of the most relevant property for phase-change heat
transfer, as follow:
c p,l Tsat
4 = Ja∗ = (19)
hlv
where the wall superheating was changed by the saturation tem-
perature to avoid two imposed variables in the same model, as
previously described.
Up to now, the foam properties only appear in the effective
thermal conductivity. The following numbers take into account the
characteristic dimension of the metal foam; the effect of the thick-
ness,
Fig. 8. Eq. (12) left-hand comparison between the experimental database and those
calculated by the model from Xu et al. [25] modified by Righetti et al. [26]. δ
5 = (20)
Lc
• next, as previously mentioned, the boiling HTC on metal foams which is important due to the effect of thickness in the HTC, as
is a function of surface characteristics; thus, the proposed cor- shown in Fig. 6. Also, the effect of the pore diameter,
relation takes into account the effect of foam thickness (δ ), pore
dp
diameter (dp ) and porous medium thermal conductivity or ef- 6 = (21)
Lc
fective thermal conductivity (keff ).
• moreover, the fluid thermophysical properties such as the sat- These dimensionless numbers represent the influence of the
uration temperature, Tsat ; the characteristic length (capillary wetted area and the vapor flow resistance into the foam structure.

8
L.L. Manetti, A.S.O.H. Moita and E.M. Cardoso International Journal of Heat and Mass Transfer 171 (2021) 121070

Once all the  groups were identified, the new correlation for
the pool boiling HTC on metal foams can be written as,
 a1  a3  a4  a5
h · Lc q · Lc c p,l μl a2 c p,l Tsat δ dp
= C1
keff keff Tsat keff hlv Lc Lc
(22)
where C1 and the exponent ai are constant, except a4 , defined by
the authors as a function of the heat flux,
  a
a4 = f q = −e (23)
b + exp (c · q − d )
that is an inverse ‘S-shaped’ curve, which takes into account the
effect of the thickness in the boiling curve of metal foams. At low
heat fluxes, thicker foams result in higher HTC and vice-versa due
to the balance of heat transfer area and vapor bubble flow resis-
tance as reported by Manetti et al. [32].
Fig. 9. Exponent a4 variation with the imposed heat flux.
The thermophysical properties of both working fluids used in
the regression analysis are listed in Table 3 (corresponds to the liq-
uid saturated conditions on the average local atmospheric pressure
where the tests were carried out).
For the metal foams effective thermal conductivity, keff , it was
used the model from Yao et al. [48],
γ 1 − 2γ γ −1
keff = + + (24)
kA kB kC
where kA , kB e kC are defined as:
√  √ 
5 2 5 2
kA = π γ (3 − 4γ )ks + 1 − π γ ( 3 − 4γ ) kl (25)
27 27
√  √ 
5 2 10 2
kB = π γ 2 ks + 1 − π γ 2 kl (26)
27 9
√  √ 
5 2 5 2
kC = π γ ks + 1 −
2
π γ kl
2
(27)
27 27
and γ is the ratio of the ligament node radii to the ligament
length, which is a function of the porosity and can be obtained
by,
√
5 2
ε =1− π γ 2 ( 3 − 5γ ) (28)
8
The model from Yao et al. [48] was chosen due to the absence
of an empirical factor for adjusting data with the experimental
data from the same authors. Moreover, it was tested by Amani
et al. [49] who also found a good agreement of the Yao et al.
[48] model with the experimental and numerical simulation data.
Based on the regression analysis of the experimental database Fig. 10. Comparison between predicted and experimental Nusselt number. (For in-
by using the least-squares method implemented on SciPy, a terpretation of the references to colour in this figure legend, the reader is referred
to the web version of this article.)
Python-based library, the following correlation was obtained with
an R-square = 0.992:
 0.615  −0.118
h · Lc q Lc c p,l μl 0.322 c p,l Tsat Table 4
Nu = = 19.9 Statistical analysis between experimental and predicted data for
keff keff Tsat keff hlv each surface used in the current study.

  f (q )  −0.200  Surface Fluid MAPE γ 20 γ 30

δ dp
× (29) Cu foam 3.0 mm HFE-7100 15.4% 72.7% 84.8%
Lc Lc Cu foam 2.0 mm HFE-7100 15.4% 78.6% 89.3%
Cu foam 1.0 mm HFE-7100 12.5% 86.1% 97.2%
where, Cu foam 3.0 mm Ethanol 6.0% 96.3% 100.0%
  5.92 Cu foam 2.0 mm Ethanol 6.2% 100.0% 100.0%
f q = − 0.0366 (30) Cu foam 1.0 mm Ethanol 11.1% 90.5% 90.5%
25.3 + exp(0.0314 × 10−3 q − 0.362 ) Ni foam 3.0 mm HFE-7100 16.7% 95.0% 95.0%
which returns a positive exponent for low heat fluxes and a nega- Ni foam 2.0 mm HFE-7100 15.2% 85.0% 90.0%
Ni foam 1.0 mm HFE-7100 10.6% 100.0% 100.0%
tive exponent for high heat fluxes as shown in Fig. 9.
Ni foam 0.5 mm HFE-7100 10.6% 87.9% 90.9%
Fig. 10 shows the Nusselt number calculated by the Eq. (29), Ni foam 3.0 mm Ethanol 9.2% 96.6% 100.0%
Nucalc , versus the experimental Nusselt number, Nuexp . The MAPE Ni foam 2.0 mm Ethanol 9.5% 88.9% 88.9%
was 10.8% with 89.9% of the data fitted within the error range of Ni foam 1.0 mm Ethanol 11.0% 80.6% 87.1%
± 20% and 93.8% within the error range of ± 30%. Table 4 shows Ni foam 0.5 mm Ethanol 6.3% 100.0% 100.0%

the statistical analysis of each surface used in the current work.

9
L.L. Manetti, A.S.O.H. Moita and E.M. Cardoso International Journal of Heat and Mass Transfer 171 (2021) 121070

following parameters play role in the qmax for wetting dielectric

fluids on metal foams:
• the first parameter is the reference heat flux, q0 =
ρv0.5 hlv [σ g(ρl − ρv )]1/4 , which is presented in most of the
dimensionless heat flux models, as reported by Liang and
Mudawar [53,54];
• the following two parameters represent the metal foam charac-
teristics: the thickness (δ ) and the pore diameter (dp );
• the last two parameters are related to the working fluid: satu-
rated liquid density (ρl ) and saturated vapor density (ρ v ).

Thus, the following functional relation can be written:

   
qmax = f q0 , δ, dp , ρl , ρv (31)
The n = 6 variables are listed in the dimension MLT [44] that
implies in r = j = 3 repeating variables (q0 , dp , ρ l ) resulting in
k = n – j = 3 dimensionless variables (k ).
The first dimensionless number obtained is,
qmax
1 = (32)
q0
which captures the established dependence of maximum heat flux
on the fluid properties and gravity [55].
The second dimensionless number is the metal foam length ra-
tio,
δ
2 = (33)
dp
which means the balance between the foam thickness and pore
diameter/area density.
Fig. 11. Comparison between the predicted Nusselt number and the experimental The third dimensionless number is the working fluid density ra-
Nusselt number from the literature. (For interpretation of the references to colour
in this figure caption, the reader is referred to the web version of this article.)
tio,
ρv
3 = (34)
ρl
4.1.1. HTC predictive model validation
The validation step by using data from literature is to check if which is associated with the fluid expansion due to the phase-
the proposed correlation is overfitted. So, the performance of the change.
proposed correlation was evaluated by comparing it with 113 ex- Once all the  groups were identified, the model which pre-
perimental data points obtained by other authors, namely by Ath- dict the maximum heat flux to pool boiling on metal foams can be
erya et al. [50], Moghaddam et al. [51], and Xu et al. [52]. Ath- written as,
erya et al. [50] used aluminum foams on boiling of FC-72 while  b 1 
qmax δ ρv b2
Moghaddam et al. [51] used copper foams on boiling of FC-72; Xu = C2 (35)
et al. [52] used copper foams on boiling of acetone as the work- q0 dp ρl
ing fluid. Even though the working fluids are different from our
where C2 and the exponent bi are constants.
experimental database, they were considered due to the wetting
The experimental maximum heat flux, qmax, exp , is given by the
fluid behavior and dielectric characteristics. Fig. 11 shows the pre-
heat flux in which the experimental HTC is maximum. Such values
diction of the independent experimental data from the literature.
are strongly dependent on the applied heat flux and these max-
The MAPE was 19% with 58.6% of the data within an error range of
imum values may not match the true value. So, instead of using
± 20% and 79.5% within an error range of ± 30%.
qmax, exp , it was carried out a polynomial regression on each ex-
We are aware that there are more works in the literature using
perimental HTC curve and the maximum heat flux was obtained
metallic foams and dielectric fluids, however, only works that had
by the polynomial curve turning point, qmax,pol (Fig. 12).
the sufficient reported characteristics of the metallic foam (poros-
ity, pore diameter, and thickness) could be checked by the new In this way, we tested second and third-degree polynomials
correlation. regression and realize that the regression by using third-degree
polynomial, y = ax3 + bx2 + cx + d, obtained an R-square coefficient
4.2. Maximum heat flux predictive model formulation greater than the second degree in all cases. Besides, the third-
degree polynomial regression curves presented more consistency
The model in Eq. (29) can predict the HTC but it does not have with the experimental boiling HTC curve. Table 5 shows the max-
a limit to the imposed heat flux. As previously discussed, the metal imum heat flux from the experimental HTC curve and the polyno-
foams can increase the boiling HTC; on the other hand, it can de- mial curve fit turning point, as well as, its R-square.
crease or increase the dryout heat flux. So, there is a balance be- Based on the regression analysis of the experimental database
tween the thickness and pore diameter due to the wetted area for by using the least-squares method implemented on SciPy, the
heat transfer and vapor flow resistance. Therefore, to determine following correlation was obtained with an R-square = 0.820:
the maximum heat flux, qmax , corresponding to the heat flux value  −0.487 
in which the HTC turning point occurs, we developed a second cor- qmax δ ρv 0.300
= 1.684 (36)
relation that was based on the Buckingham π theorem [43]. The q0 dp ρl

10
L.L. Manetti, A.S.O.H. Moita and E.M. Cardoso International Journal of Heat and Mass Transfer 171 (2021) 121070

Table 5
Maximum heat flux, experimental and polynomial, for all conditions tested in the current study.

Surface Fluid qmax, exp (kW/m2 ) qmax, pol

(kW/m2 ) R-square

Cu foam 3.0 mm HFE-7100 223.38 231.34 0.991

Cu foam 2.0 mm HFE-7100 264.95 267.49 0.991
Cu foam 1.0 mm HFE-7100 344.97 330.83 0.998
Cu foam 3.0 mm Ethanol 321.37 372.84 0.982
Cu foam 2.0 mm Ethanol 485.81 471.20 0.998
Cu foam 1.0 mm Ethanol 663.11 638.70 0.994
Ni foam 3.0 mm HFE-7100 120.55 132.94 0.989
Ni foam 2.0 mm HFE-7100 118.31 127.12 0.968
Ni foam 1.0 mm HFE-7100 171.47 183.26 0.994
Ni foam 0.5 mm HFE-7100 302.03 295.95 0.997
Ni foam 3.0 mm Ethanol 308.25 325.60 0.998
Ni foam 2.0 mm Ethanol 368.72 403.27 0.998
Ni foam 1.0 mm Ethanol 439.38 401.46 0.989
Ni foam 0.5 mm Ethanol 842.16 883.29 0.999

Fig. 12. Comparison of the maximum heat flux for the HTC experimental curve and
the third-degree polynomial curve fit.

Table 6
Absolute percentage error between the maximum heat flux calculated
by Eq. (36) and the polynomial obtained by fitting the experimental
curve.

Surface Fluid APE∗ Fig. 13. Maximum heat flux heat comparison between experimental data and the
Cu foam 3.0 mm HFE-7100 22.4% predicted data by the proposed model. (For interpretation of the references to
Cu foam 2.0 mm HFE-7100 18.2% colour in this figure legend, the reader is referred to the web version of this ar-
Cu foam 1.0 mm HFE-7100 7.4% ticle.)
Cu foam 3.0 mm Ethanol 7.6%
Cu foam 2.0 mm Ethanol 3.8%
Cu foam 1.0 mm Ethanol 7.2% error range of ± 30%. Table 6 shows the absolute percentage error
Ni foam 3.0 mm HFE-7100 0.1% (APE) of each surface used in the current work.
Ni foam 2.0 mm HFE-7100 27.5%
Ni foam 1.0 mm HFE-7100 23.9%
4.2.1. Maximum heat flux predictive model validation
Ni foam 0.5 mm HFE-7100 7.5%
Ni foam 3.0 mm Ethanol 8.7%
Data from the literature were used to validate the maximum
Ni foam 2.0 mm Ethanol 10.2% heat flux model. The experimental HTC curves from the litera-
Ni foam 1.0 mm Ethanol 26.4% ture were fitted by the third-degree polynomial curve with an R-
Ni foam 0.5 mm Ethanol 19.5% square > 0.9. Fig. 14 shows the heat flux ratio calculated by the
∗
APE =
|qmax,calc − qmax,pol |
qmax,pol
. Eq. (36) versus the polynomial, qmax,pol /q0 , obtained from the lit-
erature experimental data. The MAPE was 52.2% with 40.0% of the
data fitted within the error range of ± 20% and 50.0% within the
error range of ± 30%.
Fig. 13 shows the heat flux ratio calculated by the Eq. (36) ver- The APE, for Atherya et al. [50] case, was higher than the HTC
sus the polynomial, qmax,pol /q0 , obtained based on the current model validation because 5 PPI metal foam (the red outline point
work experimental data. The MAPE was 13.6% with 71.4% of the in Fig. 14) had a pore diameter higher than the pore diameters
data fitted within the error range of ± 20% and 100% within the used to fit the model. However, for the HTC predictive model, data

11
L.L. Manetti, A.S.O.H. Moita and E.M. Cardoso International Journal of Heat and Mass Transfer 171 (2021) 121070

ductivity, ks - the thermal conductivity from the solid phase of the

metal foam - the HTC also increases; the values used in Fig. 15(d)
are from stainless steel, nickel, aluminum, and copper, respectively,
which are the most used materials for metal foams. Finally, differ-
ent wetting and dielectric working fluids are shown in Fig. 15(e);
among the four fluids selected, HFE-7100 and FC-72 have simi-
lar thermophysical properties; thus, the HTC and maximum heat
flux are similar as well. Besides, acetone has an HTC performance
close to the HFE-7100 and FC-72 because the combination of 2
and 3 from Eq. (29) while the maximum heat flux is higher
mainly due to the latent heat of vaporization and surface ten-
sion. Ethanol was the one that showed better performance in both
correlations mainly due to the higher thermophysical properties,
such as the latent heat of vaporization, specific heat capacity, ab-
solute viscosity, and capillary length as also reported by Pastuszko
et al. [56].

5. Final remarks

In this work we developed a correlation, based on dimensional

analysis, for metal foams surfaces in pool boiling with a dielectric
fluid, taking into account the porous heating surface characteristics
(porosity, pore diameter, and thickness), the working fluid thermo-
physical properties, and the interaction between them. The corre-
lation was developed based on our experimental database and val-
idated with the open literature database. The following conclusion
can be drawn:

• For the newly HTC predictive model, Eq. (29), the MAPE is
Fig. 14. Maximum heat flux heat comparison between the predicted data by the 10.8% with 89.9% of the data fitted within the error range of
proposed model and data from the literature. (For interpretation of the references ± 20% and 93.8% within the error range of ± 30%. By compar-
to colour in this figure legend, the reader is referred to the web version of this ing it with experimental data points obtained by other authors
article.)
from the literature, the MAPE is 19% with 79.5% within an error
range of ± 30%.
• To complement the HTC predictive model, we also develop a
from Atherya et al. [50] with 5 PPI showed a good coherency prob- model to predict the maximum heat flux (qmax ), Eq. (36), corre-
ably due to the HTC model be lesser sensible to the pore diameter sponding to the heat flux value in which the HTC turning point
than the maximum heat flux model. occurs. To the maximum heat flux model, for the comparison
with the experimental database, the MAPE is 13.6% with 100%
4.3. Model sensibility within the error range of ± 30%.
• The Bhattacharya et al. [57] prediction model for the metal
To check the behavior of both new models obtained in the cur- foam pore diameter can be applied in our models - HTC pre-
rent work we coupled them; so, the Eq. (29) is valid between the dictive model, Eq. (29), and in the maximum heat flux model,
range of heat flux from 0 to qmax given by Eq. (36). Fig. 15 shows Eq. (36) - in those cases that the dp is unknown.
the sensibility of coupled models for each characteristic of metal • The thickness level variation plays an important role in the
foam: (a) pore diameter, (b) thickness, (c) porosity, and (d) metal boiling curves as well as pore diameter. Both can result in a
foam solid-phase thermal conductivity. Moreover, the last figure (e) better heat transfer performance when well combined. At low
shows the sensibility with different working fluids. heat fluxes values, thicker foams can increase the HTC due to
The pore diameter plays a role in both models since, as it in- the wetted area; however, at high heat fluxes, it decreases the
creases, the heat flux range increases and the HTC decreases. On HTC due to the vapor bubble flow resistance and vice-versa. For
the other hand, the thickness has an opposite behavior in the heat the pore diameter, it is observed the opposite effect: smaller
flux range, i.e., as it increases the heat flux range decreases; for the pores increase the HTC at low heat fluxes values and decrease
HTC, the thickness influence is quite similar. However, it is pos- it at high heat fluxes. So, the thicker and smaller the pores, the
sible to note the influence of Eq. (30): as the thickness increases larger the liquid-vapor counter-flow since the liquid replenish-
for low heat fluxes, the HTC increases up to 175 kW/m2 and, after ment is inhibited, leading to a lower HTC and an earlier occur-
that, thinner foams have better performance (similar results were rence of dryout.
reported and the mechanisms were already explained in previous • The metal foam porosity and solid-phase thermal conductivity
works [7, 32]). also play an important role in the HTC model. The lower the
Another variable that plays a role in the model is the poros- porosities the higher the HTC, due to the higher effective ther-
ity, which does not appear explicitly in Eq. (29) but it is included mal conductivity. In the same way, as the solid-phase thermal
in the effective thermal conductivity given by Eq. (24) to Eq. (28). conductivity increases the HTC also increases.
Lower porosities have better HTC performance since they have • The model is sensitive to the metal foams characteristics how-
higher effective thermal conductivity, which is a variable in the ever, the influence of the working fluid thermophysical proper-
Nusselt number. In the same way, increasing the solid thermal con- ties on the HTC predictive model is more pronounced.

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L.L. Manetti, A.S.O.H. Moita and E.M. Cardoso International Journal of Heat and Mass Transfer 171 (2021) 121070

Fig. 15. Coupled models sensibility: (a) pore diameter variation [mm]; (b) thickness variation [mm]; (c) porosity variation; (d) solid material thermal conductivity variation
[W/m.K]; and, (f) working fluids variation.

Declaration of Competing Interest acknowledge FCT for partially financing the work through project
JICAM/0 0 03/2017.
The authors declare that they have no known competing finan-
cial interests or personal relationships that could have appeared to Supplementary materials
influence the work reported in this paper.
Supplementary material associated with this article can be
CRediT authorship contribution statement found, in the online version, at doi:10.1016/j.ijheatmasstransfer.
2021.121070.
Leonardo Lachi Manetti: Methodology, Validation, Investiga-
tion, Data curation, Formal analysis, Writing - original draft, Writ- Appendix A. Pore diameter determination from PPI and
ing - review & editing. Ana Sofia Oliveira Henriques Moita: Con- porosity
ceptualization, Supervision, Project administration, Methodology,
Formal analysis, Resources, Writing - original draft, Writing - re- To develop the previous models we used the pore diameter as
view & editing. Elaine Maria Cardoso: Conceptualization, Funding one of the main dimensions of the metal foam. The dimension was
acquisition, Project administration, Supervision, Methodology, For- obtained from image analysis as presented in Section 2. However,
mal analysis, Writing - original draft, Writing - review & editing. the commercial metal foams are usually found in terms of PPI and
porosity (or relative density); thus, the pore diameter value can be
Acknowledgments hard to find and, consequently, the model application is unsolved.
So, a solution is to use the geometric model available in the litera-
The authors are grateful for the financial support from the ture, which is a function of both PPI and porosity, to solve the pore
PPGEM – UNESP/FEIS, from CAPES, from the National Council of diameter and fiber diameter. The Calmidi [58] model improved by
Technological and Scientific Development of Brazil (CNPq grant Bhattacharya et al. [57] yields,
number 458702/2014-5) and FAPESP (grant number 2013/15431- 1
7; 2017/13813-0; 2019/02566-8, 2019/15250-9). The authors also PPI = (37)
dp + df

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and, Heat Mass Transf. 141 (2019) 818–834, doi:10.1016/j.ijheatmasstransfer.2019.07.

 036.
df 1−ε 1 [17] W.M. Rohsenow, A method of correlating heat transfer data for surface boiling
= 3.39 · (38) of liquids, Trans. ASME – J. Heat Transf. 74 (1952) 969–976.
dp 3π G [18] K. Stephan, M. Abdelsalam, Heat-transfer correlations for natural convection
boiling, Int. J. Heat Mass Transf. 23 (1980) 73–87, doi:10.1016/0017-9310(80)
where, 90140-4.
[19] E. Teodori, A.S. Moita, A.L.N. Moreira, Empirical and modeling-based correla-
1−ε tions for pool boiling on microstructured surfaces, Interf. Phenom. Heat Transf..
G = 1 − exp − (39)
0.04 2 (2014) 273–292, doi:10.1615/interfacphenomheattransfer.2015011663.
[20] I.S. Kiyomura, T.S. Mogaji, L.L. Manetti, E.M. Cardoso, A predictive model
These equations can be easily solved since the PPI and porosity for confined and unconfined nucleate boiling heat transfer coefficient, Appl.
are known. We calculated the pore diameter by using Eq. (37) and Therm. Eng. 127 (2017) 1274–1284, doi:10.1016/j.applthermaleng.2017.08.135.
[21] J.M. Jabardo, E. Silva, G. Ribatski, S.F. de Barros, Evaluation of the Rohsenow
Eq. (38) using as input the PPI and porosity presented in Section 2; correlation through experimental pool boiling of halocarbon refrigerants on
the predicted pore diameters were 0.58 mm and 0.29 mm for Cu cylindrical surfaces, J. Braz. Soc. Mech. Sci. Eng. 26 (2004) 218–230.
foam and Ni foam, respectively. [22] K. Stephan, Heat Transfer in Condensation and Boiling, Springer, Berlin Heidel-
berg, Berlin, Heidelberg, 1992, doi:10.1007/978- 3- 642- 52457- 8.
To check the validity of predicted pore diameters, the experi- [23] G. Liang, Y. Chen, H. Yang, D. Li, S. Shen, Nucleate boiling heat transfer and
mental database was tested in the HTC predictive model, Eq. (29), critical heat flux (CHF) from micro-pit surfaces, Int. J. Heat Mass Transf. 152
and in the maximum heat flux model, Eq. (36), by using the pre- (2020) 119510, doi:10.1016/j.ijheatmasstransfer.2020.119510.
[24] K. Nishikawa, T. Ito, Augmentation of nucleate boiling heat transfer by pre-
dicted pore diameters instead of those presented in Table 2. The pared surfaces, Heat Transf. Energy Probl. (1982) 111–118.
HTC predictive model presented a MAPE equal to 10.8%, i.e., the [25] Z.G. Xu, Z.G. Qu, C.Y. Zhao, W.Q. Tao, Experimental correlation for pool boil-
same as Fig. 10. The maximum heat flux model presented a MAPE ing heat transfer on metallic foam surface and bubble cluster growth behavior
on grooved array foam surface, Int. J. Heat Mass Transf. 77 (2014) 1169–1182,
equal to 16.3%, which is just 2.7% higher than that in Fig. 13. There-
doi:10.1016/j.ijheatmasstransfer.2014.06.037.
fore, the Bhattacharya et al. [57] prediction model for the metal [26] G. Righetti, L. Doretti, H. Sadafi, K. Hooman, Water pool boiling across low
foam pore diameter can be applied in our models in those cases pore density aluminum foams, Heat Transf. Eng. 0 (2019) 1–10, doi:10.1080/
01457632.2019.1640464.
that the dp is unknown.
[27] H. Hu, Y. Zhao, Z. Lai, C. Hu, Experimental investigation on nucleate pool
boiling heat transfer characteristics on hydrophobic metal foam covers, Appl.
References Therm. Eng. 179 (2020) 115730, doi:10.1016/j.applthermaleng.2020.115730.
[28] J. Shi, X. Jia, D. Feng, Z. Chen, C. Dang, Wettability effect on pool boiling heat
[1] S. Fan, F. Duan, A review of two-phase submerged boiling in thermal man- transfer using a multiscale copper foam surface, Int. J. Heat Mass Transf. 146
agement of electronic cooling, Int. J. Heat Mass Transf. 150 (2020) 119324, (2020) 118726, doi:10.1016/j.ijheatmasstransfer.2019.118726.
doi:10.1016/j.ijheatmasstransfer.2020.119324. [29] Z.-.G. Xu, Z.-.G. Qu, D.-.G. Li, C.-.Y. Zhao, W.-.Q. Tao, T.-.J. Lu, M.-.Q. Wang, Ex-
[2] L. Duan, B. Liu, B. Qi, Y. Zhang, J. Wei, Pool boiling heat transfer on silicon chips perimental study of pool boiling on surface of metal foam with open cells,
with non-uniform micro-pillars, Int. J. Heat Mass Transf. 151 (2020) 119456, Kung Cheng Je Wu Li Hsueh Pao/J. Eng. Thermophys. 30 (2009) 1713–1716.
doi:10.1016/j.ijheatmasstransfer.2020.119456. [30] K.P. Burnham, D.R. Anderson, Model Selection and Multimodel Inference: A
[3] A. Mehralizadeh, S.Reza Shabanian, G. Bakeri, Effect of modified surfaces Practical Information-Theoretic Approach, 2nd ed., Springer-Verlag, New York,
on bubble dynamics and pool boiling heat transfer enhancement: a review, 2002, doi:10.1007/b97636.
Therm. Sci. Eng. Prog. 15 (2020) 100451, doi:10.1016/j.tsep.2019.100451. [31] L.L. Manetti, G. Ribatski, R.R. de Souza, E.M. Cardoso, Pool boiling heat trans-
[4] P. Tuma, Evaporator/boiler design for thermosyphons utilizing segregated hy- fer of HFE-7100 on metal foams, Exper. Therm. Fluid Sci. 113 (2020) 110025,
drofluoroether working fluids, in: Twenty-Second Annual IEEE Semiconductor doi:10.1016/j.expthermflusci.2019.110025.
Thermal Measurement And Management Symposium, IEEE, 2006, pp. 69–77, [32] L.L. Manetti, A.S.O.H. Moita, R.R. de Souza, E.M. Cardoso, Effect of copper foam
doi:10.1109/STHERM.2006.1625209. thickness on pool boiling heat transfer of HFE-7100, Int. J. Heat Mass Transf.
[5] M.S. El-Genk, Nucleate boiling enhancements on porous graphite and microp- 152 (2020) 119547 b, doi:10.1016/j.ijheatmasstransfer.2020.119547.
orous and macro-finned copper surfaces, Heat Transf. Eng. 33 (2012) 175–204, [33] M.F. Ashby, T. Evans, N.A. Fleck, J.W. Hutchinson, H.N.G. Wadley, L.J. Gibson,
doi:10.1080/01457632.2011.589305. Metal foams: a Design Guide, Elsevier, 20 0 0.
[6] V. Abreu, M. Harrison, J. Gess, A.S. Moita, Two-phase thermosiphon cooling [34] J. Banhart, Manufacture, characterization and application of cellular metals and
using integrated heat spreaders with copper microstructures, in: 2018 17th metal foams, Prog. Mater. Sci. 46 (2001) 559–632.
IEEE Intersociety Conference on Thermal and Thermomechanical Phenomena [35] E. Brun, J. Vicente, F. Topin, R. Occelli, IMorph: a 3D morphological tool to fully
in Electronic Systems (ITherm), IEEE, 2018, pp. 645–652, doi:10.1109/ITHERM. analyze all kind of cellular materials, Cell. Metals Struct. Funct. Applica. (2008).
2018.8419644. [36] J. Vicente, F. Topin, J.-.V. Daurelle, Open celled material structural proper-
[7] K.K. Wong, K.C. Leong, Saturated pool boiling enhancement using porous lat- ties measurement: from morphology to transport properties, Mater. Trans. 47
tice structures produced by Selective Laser Melting, Int. J. Heat Mass Transf. (2006) 2195–2202, doi:10.2320/matertrans.47.2195.
121 (2018) 46–63, doi:10.1016/j.ijheatmasstransfer.2017.12.148. [37] J. Mollicone, F. Ansart, P. Lenormand, B. Duployer, C. Tenailleau, J. Vicente,
[8] G. Liang, I. Mudawar, Review of pool boiling enhancement by surface Characterization and functionalization by sol-gel route of SiC foams, J. Eur. Ce-
modification, Int. J. Heat Mass Transf. 128 (2019) 892–933, doi:10.1016/j. ram. Soc. 34 (2014) 3479–3487, doi:10.1016/j.jeurceramsoc.2014.05.030.
ijheatmasstransfer.2018.09.026. [38] Online Materials Information Resource - MatWeb, (n.d.). https://apps.lib.umich.
[9] T.J. Hendricks, S. Krishnan, C. Choi, C. Chang, B. Paul, Enhancement of pool- edu/database/link/8677 (accessed September 1, 2020).
boiling heat transfer using nanostructured surfaces on aluminum and copper, [39] P. Pontes, R. Cautela, E. Teodori, A. Moita, Y. Liu, A.L.N. Moreira, A. Nikulin,
Int. J. Heat Mass Transf. 53 (2010) 3357–3365, doi:10.1016/j.ijheatmasstransfer. E.P. del Barrio, Effect of pattern geometry on bubble dynamics and heat trans-
2010.02.025. fer on biphilic surfaces, Exper. Therm. Fluid Sci. 115 (2020) 110088, doi:10.
[10] M. Shojaeian, A. Koşar, Pool boiling and flow boiling on micro- and nanos- 1016/j.expthermflusci.2020.110088.
tructured surfaces, Exper. Therm. Fluid Sci. 63 (2015) 45–73, doi:10.1016/j. [40] P. Pontes, R. Cautela, E. Teodori, A.S. Moita, A. Georgoulas, A.L.N.M. Mor-
expthermflusci.2014.12.016. eira, Bubble dynamics and heat transfer on biphilic surfaces: experiments
[11] L. Lin, M.A. Kedzierski, Review of low-GWP refrigerant pool boiling heat trans- and numerical simulation, J. Bionic. Eng. 17 (2020) 809–821, doi:10.1007/
fer on enhanced surfaces, Int. J. Heat Mass Transf. 131 (2019) 1279–1303, s42235- 020- 0064- x.
doi:10.1016/j.ijheatmasstransfer.2018.11.142. [41] R.J. Moffat, Describing the uncertainties in experimental results, Exper. Therm.
[12] Z. Wu, Z. Cao, B. Sundén, Saturated pool boiling heat transfer of acetone and Fluid Sci. 1 (1988) 3–17.
HFE-7200 on modified surfaces by electrophoretic and electrochemical depo- [42] V.V. Calmidi, R.L. Mahajan, Forced convection in high porosity metal foams, J.
sition, Appl. Energy 249 (2019) 286–299, doi:10.1016/j.apenergy.2019.04.160. Heat Transf. 122 (20 0 0) 557–565, doi:10.1115/1.1287793.
[13] R.W. Bowring, Physical model, Based On Bubble detachment, and Calculation [43] E. Buckingham, On physically similar systems; illustrations of the use of di-
of Steam Voidage in the Sub-Cooled Region of a Heated Channel, Institutt for mensional equations, Phys. Rev. 4 (1914) 345.
Atomenergi (Norway). OECD Halden Reaktor Prosjekt, 1962. [44] F.M. White, Fluid Mechanics, 6th ed, McGraw-Hill, New York, NY, 2009.
[14] C. Gerardi, J. Buongiorno, L. wen Hu, T. McKrell, Study of bubble growth in [45] C.W. Somerton, The Prandtl number effect in porous layer convection, Appl.
water pool boiling through synchronized, infrared thermometry and high- Sci. Res. 40 (1983) 333–344.
speed video, Int. J. Heat Mass Transf. 53 (2010) 4185–4192, doi:10.1016/j. [46] V. Kathare, J.H. Davidson, F.A. Kulacki, Natural convection in water-saturated
ijheatmasstransfer.2010.05.041. metal foam, Int. J. Heat Mass Transf. 51 (2008) 3794–3802, doi:10.1016/j.
[15] S.J. Thiagarajan, R. Yang, C. King, S. Narumanchi, Bubble dynamics and nucleate ijheatmasstransfer.2007.11.051.
pool boiling heat transfer on microporous copper surfaces, Int. J. Heat Mass [47] N. Dukhan, K.-.C. Chen, Heat transfer measurements in metal foam subjected
Transf. 89 (2015) 1297–1315, doi:10.1016/j.ijheatmasstransfer.2015.06.013. to constant heat flux, Exper. Therm. Fluid Sci. 32 (2007) 624–631, doi:10.1016/
[16] Z. Cao, Z. Wu, B. Sundén, Heat transfer prediction and critical heat flux mech- j.expthermflusci.20 07.08.0 04.
anism for pool boiling of NOVEC-649 on mic-roporous copper surfaces, Int. J.

14
L.L. Manetti, A.S.O.H. Moita and E.M. Cardoso International Journal of Heat and Mass Transfer 171 (2021) 121070

[48] Y. Yao, H. Wu, Z. Liu, A new prediction model for the effective thermal con- [54] G. Liang, I. Mudawar, Pool boiling critical heat flux (CHF) – Part 2: assessment
ductivity of high porosity open-cell metal foams, Int. J. Therm. Sci. 97 (2015) of models and correlations, Int. J Heat Mass Transf. 117 (2018) 1368–1383 b,
56–67. doi:10.1016/j.ijheatmasstransfer.2017.09.073.
[49] Y. Amani, A. Takahashi, P. Chantrenne, S. Maruyama, S. Dancette, E. Maire, [55] H. O’Hanley, C. Coyle, J. Buongiorno, T. McKrell, L.-.W. Hu, M. Rubner, R. Cohen,
Thermal conductivity of highly porous metal foams: experimental and image Separate effects of surface roughness, wettability, and porosity on the boiling
based finite element analysis, Int. J. Heat Mass Transf. 122 (2018) 1–10. critical heat flux, Appl. Phys. Lett. 103 (2013) 024102, doi:10.1063/1.4813450.
[50] B.P. Athreya, R.L. Mahajan, S. Sett, Pool boiling of FC-72 over metal foams: ef- [56] R. Pastuszko, R. Kaniowski, T.M. Wójcik, Comparison of pool boiling perfor-
fect of foam orientation and geometry, in: 8th AIAA/ASME Joint Thermophysics mance for plain micro-fins and micro-fins with a porous layer, Appl. Therm.
and Heat Transfer Conference, 2002, pp. 1–10. Eng. 166 (2020) 114658, doi:10.1016/j.applthermaleng.2019.114658.
[51] S. Moghaddam, M. Ohadi, J. Qi, Pool Boiling of Water and FC-72 on Cop- [57] A. Bhattacharya, R.L. Mahajan, Finned metal foam heat sinks for electronics
per and Graphite Foams, in: 2003 International Electronic Packaging Technical cooling in forced convection, J. Electron. Packag. Trans. ASME. 124 (2002) 155–
Conference and Exhibition, Volume 2, ASME, 2003, pp. 675–680, doi:10.1115/ 163, doi:10.1115/1.1464877.
IPACK2003-35316. [58] V.V. Calmidi, Transport phenomena in high porosity fibrous metal foams Ph.D.
[52] J. Xu, X. Ji, W. Zhang, G. Liu, Pool boiling heat transfer of ultra-light copper Thesis, University of Colorado, 1998.
foam with open cells, Int. J. Multiph. Flow 34 (2008) 1008–1022, doi:10.1016/
j.ijmultiphaseflow.20 08.05.0 03.
[53] G. Liang, I. Mudawar, Pool boiling critical heat flux (CHF) – Part 1: review
of mechanisms, models, and correlations, Int. J. Heat Mass Transf. 117 (2018)
1352–1367 a, doi:10.1016/j.ijheatmasstransfer.2017.09.134.

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