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The effect of lattice structure and porosity on thermal conductivity of additively-manufactured porous materials

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The effect of lattice structure and porosity on thermal conductivity of additively-manufactured porous materials

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Content THE EFFECT OF LATTICE STRUCTURE AND POROSITY ON THERMAL
CONDUCTIVITY OF ADDITIVELY-MANUFACTURED POROUS MATERIALS
by
BO-WEI WU
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
MASTER OF SCIENCE
(AEROSPACE AND MECHANICAL ENGINEERING
(COMPUTATIONAL FLUID AND SOLID
MECHANICS))
August 2023
Copyright 2023 BO-WEI WU
TableofContents
ListofTables iv
ListofFigures v
Nomenclature viii
Abstract x
Chapter1: Introduction 1
1.1 Motivation: Thermal Management . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Porous Material Thermophysical Properties . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Flow Resistance: Permeability and Forchheimer Coefficient . . . . . . . . 2
1.2.2 Effective Thermal Conductivity . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Porous Materials in Thermal Management Applications . . . . . . . . . . . . . . . 6
1.3.1 Metal Foams as a Heat Transfer Media . . . . . . . . . . . . . . . . . . . . 6
1.3.2 Additively Manufactured Materials as Heat Transfer Media . . . . . . . . 7
1.4 Objectives and Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Chapter2: ExperimentSetupandMethods 10
2.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Measurement Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Computation of Effective Thermal Conductivity . . . . . . . . . . . . . . . . . . . 13
2.4 Testing Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.5 Uncertainty Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.6 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Chapter3: Results 24
3.1 Isotropic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2 Anisotropic Materials with Different In-Plane Spacing . . . . . . . . . . . . . . . . 27
3.3 Anisotropic Materials with Identical In-Plane Spacing . . . . . . . . . . . . . . . . 28
Chapter4: Discussions 31
4.1 Effect of Porosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.2 Effect of Porous Material Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
ii
Chapter5: Conclusions 35
Bibliography 37
Appendices 39
Appendix A: CAD Renderings of Porous Materials . . . . . . . . . . . . . . . . . . . . . 39
Appendix B: Raw Data of Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . 42
iii
ListofTables
2.1 Equipment list. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Porous sample configurations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.1 Measurement results for isotropic samples. . . . . . . . . . . . . . . . . . . . . . . 24
3.2 Measurement results for anisotropic samples withS
x
̸=S
y
. . . . . . . . . . . . . . 27
3.3 Measurement results for anisotropic samples withS
x
=S
y
. . . . . . . . . . . . . . 28
iv
ListofFigures
1.1 Examples of thermal management systems. . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Open-celled foam structure used to develop forced convention model [9]. . . . . . 4
1.3 Hexagonal geometry used for model development; from [1]. . . . . . . . . . . . . 5
1.4 Example of AM material application for heat pipes [11]. . . . . . . . . . . . . . . . 8
2.1 Schematic showing experimental setup. Note that the copper plates were carved
with three slots to fit in thermocouples and patched with thermal paste covered
by kapton tape. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 CAD diagram of setup. To reduce thermal resistance, the bottom copper was
sealed into the cavity in (b), leaving no space for air bubbles form on the bottom
surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Photograph showing experimental setup, with different components highlighted.
Chilled water was circulated through the experimental setup to maintain a
near-constant temperature for the bottom plate. . . . . . . . . . . . . . . . . . . . 13
2.4 Measurement results for steel and PLA. Note that the setup was verified with
samples with known thermal conductivity. . . . . . . . . . . . . . . . . . . . . . . 14
2.5 Measurement results for solid resin cube of identical dimension to the porous
samples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.6 CAD profile of the testing samples. The white part represents the solid resin and
the black part is the void space. Note that the rod size (r) was fixed for all the
materials, but rod spacing (s) varied. . . . . . . . . . . . . . . . . . . . . . . . . . . 16
v
2.7 Rendering of the25 mm× 25 mm× 6 mm MHM sample with spacingsS
x
= 2
mm, S
y
= 3 mm, and S
z
= 2 mm. There are 25/2 = 12.5 unit cells in the x
direction,25/3 = 8.3 unit cells in they direction, and6/2 = 3 unit cells in thez
direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.8 Image showing 3D-printed samples. All the samples were washed using
isopropyl alcohol (IPA) and cured under an appropriate light source to ensure no
liquid resin residue prior to measurements. . . . . . . . . . . . . . . . . . . . . . . 18
2.9 Series and parallel illustration in lumped parameter models. . . . . . . . . . . . . 21
2.10 Unit-cell calculations for the mathematical model. In the unit cell, the rods (blue
parts) are the solid components with thermal conductivityK
s
and the porous
space (gray parts) is filled with fluid (water) with thermal conductivity K
f
. In
each unit cell, the top part has rods intersection in a cross-shaped geometry and
the bottom part has only one rod in the middle. . . . . . . . . . . . . . . . . . . . 22
3.1 Effective thermal conductivity measurements compared against upper- and
lower-bound estimates and model predictions. . . . . . . . . . . . . . . . . . . . . 25
3.2 Measurement results for isotropic samples. The power was limited such that the
temperature difference remained below 30
◦ C. . . . . . . . . . . . . . . . . . . . . 26
3.3 Effective thermal conductivity measurements, upper- and lower-bounds, and
model predictions for anisotropic samples with S
x
̸= S
y
. Measurements are
generally consistent with the empirical correlation in equation 1.5, which is
closer to the lower bound. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.4 Measurements and model predictions anisotropic samples with the sameS
x
and
S
y
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.1 K
e
/K
s
measurements for all materials, separated into isotropic cases (red
markers), anisotropic cases withS
x
̸= S
y
(green markers), and anisotropic cases
withS
x
=S
y
(black markers). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.2 Samples with different pore geometries: (a) isotropic HHH case, (b) anisotropic
HHL case, and (c) anisotropic HLH case. The geometries in (a) and (b) exhibit
higher thermal conductivity compared to the geometry in (c). . . . . . . . . . . . 33
5.1 CAD renderings for all porous samples. . . . . . . . . . . . . . . . . . . . . . . . . 41
5.2 Raw results for LML sample. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.3 Raw results for MLM sample. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
vi
5.4 Raw results for LHL sample. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.5 Raw results for MHM sample. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.6 Raw results for HLH sample. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.7 Raw results for HMH sample. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.8 Raw results for LLM sample. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.9 Raw results for MML sample. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.10 Raw results for LLH sample. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.11 Raw results for MMH sample. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.12 Raw results for HHL sample. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.13 Raw results for HHM sample. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
vii
Nomenclature
GreekSymbols
β coefficient of thermal expansion
∆ difference of a property
ϵ porosity
µ dynamic viscosity
ρ density of the resin
General
A area
AM additively manufactured
B empirical constant
c Forchheimer coefficient
D characteristic length scale
Da Darcy number
dP/dx pressure gradient
g gravity acceleration
Gr Grashof number
h height
K thermal conductivity
k permeability
m slope of regression line
Q heat input
R resistance of the heater
r rod size of the porous sample
viii
S spacing (rod size plus void space) of the
sample
T temperature
V voltage input to the heater
v superficial velocity
W mass of the sample
(L,M,H) low (1.5 mm), medium (2.0 mm), and
high (3.0 mm) spacing for a unit cell
Subscripts
1 material 1
2 material 2
e effective
f fluid
s solid
ix
Abstract
Thermal management is a critical aspect of many modern technological systems, as highly ef-
fective thermal transport is necessary for ensuring their proper functioning and longevity. As
a result, the design of thermal management systems has become increasingly crucial in a wide
range of fields, from electronics to aerospace to energy production. To meet the growing demand
for effective thermal management, researchers have been exploring new materials and technolo-
gies that can be used to improve heat transfer performance in these systems. Metal foams have
shown great potential for thermal management applications. Metal foams are highly porous ma-
terials that have a large surface area, which makes them ideal for dissipating heat in a system. The
use of metal foam in heat pipes and other systems has been shown to be highly effective. More
recently, the development of additive manufacturing (AM) technology has opened up new pos-
sibilities for designing and manufacturing porous media for thermal management. AM allows
for the creation of complex geometries and structures that would be difficult or impossible to
produce using traditional manufacturing methods. This technology could enable the creation of
efficient heat dissipation materials and revolutionize the design of thermal management systems.
This thesis presents an analytical and experimental investigation into the effective thermal
conductivity (K
e
) of 3D-printed porous resin samples with porosity ranging from 70% to 90%. The
x
overarching goal is to explore the effect of lattice structure and porosity on the thermal behav-
ior of the porous materials. An experimental setup was developed to measure steady-state heat
transfer across the porous samples fully-saturated with water. The porous material was sand-
wiched between two conductive copper plates. A patch heater was used to provide uniform heat
input at the upper plate and the lower plate was exposed to a water bath attached to a chiller.
Thermocouples were used to measure upper and lower plate temperature, and the vertical tem-
perature gradient was use to estimate effective thermal conductivity. Experimental data were
compared to predictions made using a lumped parameter model as well as an empirical formula
proposed in prior literature. The results demonstrate that porosity is a critical factor in deter-
mining the thermal conductivity of porous materials, with higher porosity resulting in higherK
e
due to the presence of voids for water, which has a higher thermal conductivity than the solid
resin used in the experiments. Tests with anisotropic pore structures showed that the specific
pore geometry also has an impact on the thermal conductivity of the materials. Empirical cor-
relations developed in previous literature for metal foams generated reasonable predictions for
the measurements made in this study. This suggests that similar empirical formulae may also
hold for additively-manufactured porous materials made of metal. However, the effect of the
specific pore geometry must be taken into account for more accurate predictions, especially for
anisotropic materials.
Overall, this study provides valuable insights into the thermal behavior of additively manu-
factured porous materials and highlights the need to consider the effect of lattice structures on the
thermal conductivity of such materials. Future research could focus on investigating the impact
of printing parameters and exploring new printing techniques and materials to achieve enhanced
thermal properties.
xi
Chapter1
Introduction
1.1 Motivation: ThermalManagement
Managing heat is crucial for various industrial applications, ranging from power generation to
electronics and automobiles. With more stringent power and packaging requirements increasing
temperature gradients, it becomes even more critical to manage the heat generated by such sys-
tems to prevent performance degradation or even failure. Many review articles related to thermal
management note the use of heat pipes for heat transfer. Indeed, heat pipes are frequently used
for thermal managements in robotic systems [13], electric vehicle batteries [16], solid oxide fuel
cell (SOFC) stacks [18], etc. Although heat pipes generally rely on phase transitions (i.e., evap-
oration and condensation) for effective thermal management, the efficacy of these systems also
depends on the internal structure through which the working fluid is transported between the
high- and low-temperature interfaces. This structure is often porous, and so the permeability and
effective thermal conductivity are important design parameters [7]. Even for conventional heat
exchangers that do not rely on phase changes, effective thermal conductivity and flow resistance
play a critical role in dictating performance.
1
(a) Heat pipe in electric vehicle batteries [16]. (b) Heat pipe in SOFC [18].
Figure 1.1: Examples of thermal management systems.
This work focuses on the use of porous materials in thermal management systems. A brief
review of porous material thermophysical properties (i.e., permeability and effective thermal con-
ductivity) and related modeling efforts is provided below. Typical porous materials used in ther-
mal management applications are discussed thereafter.
1.2 PorousMaterialThermophysicalProperties
1.2.1 FlowResistance: PermeabilityandForchheimerCoefficient
The permeability of porous materials is commonly expressed in dimensionless terms as the Darcy
number,Da =k/D
2
, whereD denotes a characteristic length scale andk represents the perme-
ability.
In the context of heat exchanger design, the permeability determines the pressure drop across
a section, which is a critical parameter that impacts both the required power and the overall
efficiency of the system [2]. To describe the behavior of viscous flows in porous media, Darcy’s
Law is used, which states that the pressure gradient across the media is directly proportional
2
to the superficial velocity and dynamic viscosity of the fluid, and inversely proportional to the
permeability of the porous medium, as shown in equation 1.1:
dP
dx
=− µ k
v (1.1)
Here
dP
dx
is the pressure gradient across the porous material for a fluid of dynamic viscosity µ flowing at a superficial velocity of v in the streamwise (x) direction. While this relationship
holds for viscous flows without inertial effects, at higher velocities, an additional Forchheimer
term is introduced, as shown in equation 1.2:
dP
dx
=− µ k
v+
c
√
k
v
2
. (1.2)
This Forchheimer correction term accounts for the inertial effects and is expressed as
c
√
k
v
2
, where
c is the Forchheimer coefficient. In summary, the overall flow resistance in heat exchangers
depends on the permeability as well as the inertial Forchheimer coefficient. For packed beds
of granular media, these quantities can be estimated using the established Carmen-Kozeny or
Ergun equations. Simple models have also been developed to predict these quantities for foam-
like porous materials and structured materials comprising lattice geometries [9, 2, 1]. In the
context of thermal management, Lu et al., [9] developed a simple model that estimated both the
heat transfer and pressure drop as a function of porous medium properties assuming the simple
pore geometry shown in Figure 1.2.
3
Figure 1.2: Open-celled foam structure used to develop forced convention model [9].
1.2.2 EffectiveThermalConductivity
The effective thermal conductivity of saturated porous materials depends both on the conductiv-
ity of the solid phase as well as the fluid phase. Simple lumped parameter models show that, for
one-dimensional heat transfer through a porous material, the upper bound and lower bound of
effective thermal conductivity can be represented using the relations shown in equation 1.3 and
equation 1.4 below:
K
e,max
=K
s
(1− ϵ )+K
f
ϵ, (1.3)
and
K
e,min
=

ϵ K
f
+
1− ϵ K
s

− 1
. (1.4)
Here,K
f
andK
s
are the fluid and solid thermal conductivity, and ϵ is the porosity of the medium.
Note that the above equations essentially represent thermal resistances from the fluid and solid
domains appearing in series or parallel configurations.
4
Figure 1.3: Hexagonal geometry used for model development; from [1].
More complex models that take into account porous medium geometry have also been devel-
oped. For instance, Calmidi and Mahajan [4] developed a physically-motivated analytical model
to estimate the effective thermal conductivity for hexagonal open-cell foam structures shown as
those shown in Figure 1.3, and experiments with Aluminum foams of high porosity (ϵ > 0.9)
were used to verify model predictions. The analytical model did reasonably well in predicting
effective thermal conductivity for a variety of aluminum and carbon foams, in both air and water.
However, generating accurate predictions still required the inclusion of an empirical geomet-
ric parameter. Moreover, subsequent work showed that a simpler empirical expression, which
represents a weighted average of the upper and lower bound predictions shown above, yielded
predictions that were just as accurate [1]:
K
e
=B(ϵK
f
+(1− ϵ )K
s
)+
1− B
(
ϵ K
f
+
1− ϵ Ks
)
. (1.5)
In the expression above, B is an empirical constant fitted to experimental data. For aluminum
foams,B = 0.35 led to good predictive capability across a range of porosities. Below, we evaluate
5
whether this expression yields useful predictions for the additively-manufactured materials tested
in this thesis.
1.3 PorousMaterialsinThermalManagementApplications
1.3.1 MetalFoamsasaHeatTransferMedia
Porous materials have been recognized for their potential to improve the thermal properties of a
system for many years. Through extensive research and experiment, it has been shown that the
unique structure of porous materials can provide efficient heat transfer with minimal pressure
drop. This has made them an attractive option for use in a wide range of applications, including
automotive, aerospace, and electronic cooling [12, 8, 10].
Much of this research has focused on the use of metal foams in heat transfer applications. As
noted in the previous section, several researchers have developed physically-motivated models
for spatially-periodic porous media [6, 4, 1]. Experiments carried out since the late 1990s and early
2000s have shown that open-cell metal foams have thermal resistance values that are significantly
lower than those for commercial heat exchangers. Efficient heat transfer with minimal pressure
drop can reduce the power needed to cool a system, making it more energy-efficient [3]. The rea-
son behind this performance enhancement is that metal foams provide a high specific surface area
to volume ratio and have a good thermally conducting solid phase, which enhances mixing and
promotes convective heat transfer. This unique combination of properties makes metal foam an
excellent material for compact heat exchangers, which are essential for high-power applications.
6
In recent years, metal foam-based solutions have been successfully used in a variety of applica-
tions, such as battery modules in electric vehicles, data centers, and high-power semiconductor
applications.
1.3.2 AdditivelyManufacturedMaterialsasHeatTransferMedia
Although metal foams have shown promise in thermal management applications, the manufac-
turing process for such media allows for limited control over porosity and pore or ligament size
distributions. In addition, it is difficult to generate materials with anisotropic pore structures.
Additive manufacturing (AM) is a process that enables the production of complex compo-
nent geometries directly from 3D computer-aided design (CAD) data, without requiring external
machining or molds. This makes it an ideal method for fabricating open-cell porous structures.
Unlike other manufacturing processes that involve applying porous materials to the surface of ex-
isting solid components, AM can produce integrated porous and solid materials simultaneously.
Moreover, AM allows for the creation of nearly identical porous structures with high repeatabil-
ity, while providing good control over key structural properties, including porosity, pore size, and
homogeneity. Work by Evans et al., [5] showed that 3D printed materials provide the benefits
of high-porosity and high surface area-to-volume ratio, similar to metal foams. However, they
also have the potential to customize pore structures for meeting heat transfer and pressure drop
requirements as needed. For instance, previous studies have designed multifunctional lattice-
frame materials for compact heat exchangers [14] and incorporated AM materials in heat pipe
7
Figure 1.4: Example of AM material application for heat pipes [11].
designs[11] (see e.g., Fig 1.4). With additive manufacturing, aerospace components with struc-
tural functions can also serve as thermal ducts without adding any additional weight, making
them highly attractive applications.
1.4 ObjectivesandOrganization
This study focuses on assessing the feasibility of predicting the thermal conductivity of addi-
tively manufactured porous materials. To achieve this, two approaches are employed. First, an
empirical correlation equation is employed, which considers the porosity of the material to esti-
mate its thermal conductivity. Second, a lumped parameter model is utilized, where the thermal
conductivity is estimated based on the geometry of the samples. Furthermore, this paper gen-
erates a database of thermal conductivity measurements for 3D-printed porous materials which
could meet different demands of heat transfer. Specifically, we measure thermal conductivity for
a range of cubic lattice structures with isotropic and anisotropic pore geometries. These materials
8
are manufatured using commercial stereolithographic (SLA) 3D printers (formlabs Form3). A lim-
ited number of measurements are also made for materials manufactured using fused deposition
techniques.
The remainder of this thesis is structured as follows. The second chapter describes the ex-
perimental techniques. This includes a description of the experimental setup and the porous
geometries tested, together with details on the manufacturing process for the porous materials.
The effective thermal conductivity for the samples was measured via steady-state heat transfer
experiments. For each sample, multiple repeat experiments were carried out to generate measure-
ments of the average effective conductivity and the corresponding uncertainty. This chapter also
presents a simple physically-motivated lumped parameter model that can be used to generate a
priori predictions for the thermal conductivity of the different porous geometries tested.
In the third chapter, measurement results are presented for the different samples. A total of 17
different datasets are presented for porous samples of varying geometry. These measurements are
categorized into three groups: isotropic materials, anisotropic materials with identical spacings in
thex− y plane perpendicular to the direction of heat flux (i.e., S
x
=S
y
), and anisotropic materials
with differing spacings ( S
x
̸= S
y
). Measurements are also compared against predictions made
using empirical relations proposed in previous literature, and with predictions from the physics-
based lumped parameter model.
The fourth chapter provides further discussion of the measurements, focusing mainly on the
effect of lattice structure on effective thermal conductivity. Specifically, the effective thermal
conductivity is discussed in the context of the three different structural groups mentioned above.
The fifth chapter summarizes the findings of this study and provides several suggestions for future
research.
9
Chapter2
ExperimentSetupandMethods
2.1 ExperimentalSetup
As noted above, steady-state heat transfer experiments were carried out to generate effective
thermal conductivity measurements for the porous samples. A schematic of the experimental
setup is shown in Figure 2.1.
Figure 2.1: Schematic showing experimental setup. Note that the copper plates were carved with
three slots to fit in thermocouples and patched with thermal paste covered by kapton tape.
10
Table 2.1: Equipment list.
Item Vendor Model
Patch Heater Omega KHLVA-101/10-P
Data Acquisition Omega TC-08
AC Power Supply Philmore 48-1205
Multimeter WeePro Vpro850L
Chiller S&A CW-5300
SLA Printer Formlabs Form 3
SLA Resin Formlabs clear V4
Thermocouples Omega 5TC-TT-T-36-72
Submersible Pump Little Giant 503103
FDM Printer Prusa i3 MK3
PLA Hatchbox 3D PLA-1KG1.75-WHT
The experimental setup consisted of three layers. The top layer featured a 1-inch square patch
heater (Omega KHLVA-101/10-P) attached to a 3 mm thick copper plate with three 0.8mm slots
for thermocouples (Omega 5TC-TT-T-36-72). The patch heater had a nominal power rating of
10W, though it was possible to exceed this value. The thermocouples had a resolution of 0.5
◦ C.
The middle layer was composed of porous materials surrounded by thermal insulation, including
styrofoam and a PLA wall with a heat conductivity of approximately 0.02 Wm
− 1
K
− 1
. The bottom
layer was another copper plate with three slots on the bottom side. This plate was in contact
with a water bath maintained at 10
◦ C. A chiller (S&A CW-5300) cooled the water in a source
tank to maintain a constant temperature. The bottom copper plate was sealed with silicone to
prevent water from penetrating into the middle layer. The entire setup was 3-D printed from
polylactic acid (PLA) using a fused deposition (FDM) printer (Prusa i3). A CAD diagram is shown
in Figure 2.2. The entire experimental setup is shown in Figure 2.3. Experimental items used are
listed in Table 2.1.
11
(a) (b)
Figure 2.2: CAD diagram of setup. To reduce thermal resistance, the bottom copper was sealed
into the cavity in (b), leaving no space for air bubbles form on the bottom surface.
2.2 MeasurementProcedure
Effective thermal conductivity measurements were generated as follows. The sample material
was attached between the copper plates using thermally conductive paste, and the patch heater
was used to provide near-uniform heat input, Q, to the upper copper plate. The temperature
difference across the samples, ∆ T , was measured by three thermocouples attached to the top and
bottom plates. The thermal conductivity of the sample was calculated based on the temperature
difference, the dimensions of the sample, and the power of heater applied as discussed below.
Prior to the thermal conductivity measurements, the porous samples underwent a soaking
process in water for over 3 hours to eliminate any air bubbles present in the porous space. Subse-
quently, the sample slot was filled with water, and the porous sample was put in place thereafter.
To ensure reduced thermal resistance in the desired heat transfer direction, it was imperative to
attain equilibrium at around 10
◦ C for the bottom plate temperature. Upon attaining equilibrium,
the heater was switched on and the data acquisition system (DAQ, Omega TC-08) started record-
ing. For each measurement, it took approximately 20-40 minute for the system to reach steady
state (i.e., near-constant temperature difference). Temperature measurements for the upper and
12
lower plates were then collected for a period of 10 minutes at 1 Hz (600 data points). During this
10 minute period, the temperature difference generally remained consistent with within ± 0.5
◦ C.
Figure 2.3: Photograph showing experimental setup, with different components highlighted.
Chilled water was circulated through the experimental setup to maintain a near-constant tem-
perature for the bottom plate.
2.3 ComputationofEffectiveThermalConductivity
To determine the effective thermal conductivity, K
e
, two calculation methods were employed.
The first method involved using the heat conduction equation directly:
Q =
K
e
A∆ T
h
, (2.1)
where Q represents the power output, A is the cross-sectional area of the sample (i.e., the area
over which conduction takes place),h is the length of the sample in the direction of heat transfer,
and ∆ T is the temperature difference between the upper and lower surfaces. The power input
13
(a)∆ T vsQ diagram for Steel cube (b)K
e
vs∆ T diagram for steel
(c)∆ T vsQ diagram for Solid PLA (d)K
e
vs∆ T diagram Solid PLA
Figure 2.4: Measurement results for steel and PLA. Note that the setup was verified with samples
with known thermal conductivity.
was controlled by controlling the voltage input,V , to the heater. An estimate ofQ was obtained
using Ohm’s law:
Q =
V
R
2
, (2.2)
whereR is the known resistance of the patch (78.4Ω ). For each sample, temperature differences
were measured for 3 to 6 different heat inputs, and the resulting effective thermal conductivity
was estimated. The overall effective thermal conductivity of the sample was obtained by averag-
ing the values acquired from each heat input.
14
The other method is to plot∆ T vsQ diagrams for the 3 to 6 different measurements, and pur-
suing linear regression analyses. The regression line slope,m, can be translated into an effective
conductivity estimate using:
K
e
=
h
Am
. (2.3)
The results of both methods were compared and were consistent with one another within uncer-
tainty.
Sample results are shown in Figure 2.4 for solid samples of stainless steel (K≈ 45 Wm
− 1
K
− 1
)
and PLA (K ≈ 0.25 Wm
− 1
K
− 1
). Panels on the left illustrate the second slope-based method for
estimating K
e
while panels on the right show the direct estimates obtained using equation 2.1.
Both methods yield results consistent with expectations and suggest that the experimental setup
and analysis procedure described above can be used to generate reliable measurements of effective
thermal conductivity over at least 3 orders of magnitude.
2.4 TestingSamples
Stereolithographic (SLA) 3D printing was used to manufacture the porous testing samples. Open-
SCAD was used to design the porous samples with a repeating cubic lattice microstructure. The
designed were fabricated using a formlabs Form3 printer from their proprietary V4 clear resin
material. To properly analyze the thermal properties of the printed samples, the thermal con-
ductivity of solid resin was also measured before testing. The thermal conductivity of the solid
sample was measured to be K
s
= 0.316± 0.023 Wm
− 1
K
− 1
, as shown in Figure 2.5. In the
sections below, the effective thermal conductivity is normalized by K
s
to present the results in
dimensionless form.
15
(a)∆ T vsQ diagram for solid resin (b)K
e
vs∆ T diagram for solid resin
Figure 2.5: Measurement results for solid resin cube of identical dimension to the porous samples.
Figure 2.6: CAD profile of the testing samples. The white part represents the solid resin and the
black part is the void space. Note that the rod size (r) was fixed for all the materials, but rod
spacing (s) varied.
An example CAD rendering for the 3D-printed materials is shown in Figure 2.6. This profile
provides a visual representation of the geometry and structure of the printed parts, which can be
useful for understanding their thermal behavior and for optimizing their design.
All 3D-printed porous material samples were designed to have a dimension of 25 mm× 25
mm× 6mm. The patch heater was attached to the largest side, with areaA = 25
2
= 625 mm
2
.
Heat transfer occurred along the shortest dimension of length h = 6 mm. For the rest of this
document, thex andy coordinates represent the in-plane directions while thez coordinate rep-
resents the direction of heat transfer. The rod size was fixed at r = 0.4 mm for all materials due to
16
print resolution limitations. The spacing (S) was varied systematically between 3 values: 1.5mm
(Low, L), 2mm (Medium, M), and 3mm (High, H). The highest spacing was close to the largest
unsupported length possible for the 3D-printed materials. Isotropic samples were constructed
with identical spacings in each direction (S
x
= S
y
= S
z
= S): these samples are termed LLL,
MMM, and HHH, respectively. Anisotropic samples were generated by varying the spacings in
each dimension, with the designations HHL, HHM etc. representing the spacings in thex,y, and
z directions, respectively. As an example, the material with medium spacings (M) in thex andz
directions (S
x
=S
z
= 2 mm), and high spacing (H) in they direction (S
y
= 3 mm) is designated
MHM as shown in Figure 2.7.
Figure 2.7: Rendering of the 25 mm× 25 mm× 6 mm MHM sample with spacings S
x
= 2 mm,
S
y
= 3 mm, andS
z
= 2 mm. There are25/2 = 12.5 unit cells in thex direction,25/3 =8.3 unit
cells in they direction, and6/2 =3 unit cells in thez direction.
The porosity of the 3D-printed samples was estimated by measuring the mass of the samples
and using the following relation:
ϵ = 1− W
ρAh
(2.4)
17
Here,ρ represents the density of the resin,Ah is the volume of the sample, andW represents the
mass of the sample. The dimensions and measured porosity of the testing samples are shown in
Table 2.2. Figure 2.8 shows 3D-printed samples of all the porous materials. CAD renderings for
all samples can be found in Figure 5.1 in Appendix A.
Figure 2.8: Image showing 3D-printed samples. All the samples were washed using isopropyl
alcohol (IPA) and cured under an appropriate light source to ensure no liquid resin residue prior
to measurements.
2.5 UncertaintyAnalysis
To generate uncertainty estimates for the measured conductivity and porosity values, the stan-
dard Klein-McClintock procedure [15] was employed. Since the thermal conductivity values were
obtained by combining the power readings with measurements of the sample dimensions and
temperature differences, the corresponding measurement uncertainties need to be considered.
The power input was calculated from voltage readings with an uncertainty of∆ V = 0.05 V. All
18
Table 2.2: Porous sample configurations.
spacings
configuration
S
x
(mm) S
y
(mm) S
z
(mm)
ϵ HHH 3 3 3 0.91±0.01
MMM 2 2 2 0.83±0.01 isotropic
LLL 1.5 1.5 1.5 0.72±0.01
HHL 3 3 1.5 0.88±0.01
HHM 3 3 2 0.89±0.01
MMH 2 2 3 0.86±0.01
MML 2 2 1.5 0.81±0.01
LLH 1.5 1.5 3 0.80±0.01
anisotropic
identical spacing inx− y plane
LLM 1.5 1.5 2 0.76±0.01
HLH 3 1.5 3 0.86±0.01
HMH 3 2 3 0.88±0.01
MHM 2 3 2 0.86±0.01
MLM 2 1.5 2 0.80±0.01
LHL 1.5 3 1.5 0.80±0.01
anisotropic
different spacing in x− y plane
LML 1.5 2 1.5 0.77±0.01
dimensions were measured using calipers with 0.01 cm resolution, and so ∆ h = 0.005 cm (and
the error in the area and volume estimates must be propagated appropriately). The precision
digital scale using to measure the mass of the porous samples had a resolution of 0.001 g, and so
∆ W = 0.0005 g.
Based on the preceding uncertainties in individual measurements, the total uncertainties in
the measurement of the effective thermal conductivity and porosity can be calculated using equa-
tion 2.5 and equation 2.6:
∆ K
e
K
e
=
s

∆ Q
Q

2
+

∆ h
h

2
+

∆ A
A

2
+

∆(∆ T)
∆ T

2
, (2.5)
and
∆ ϵ ϵ =
s

∆ h
h

2
+

∆ A
A

2
+

∆ W
W

2
. (2.6)
19
For typical values, the relatively uncertainties in K
e
and ϵ were 2% and 0.8%, respectively. The
largest contributor to these uncertainties was the measurement ofH, which had a relative uncer-
tainty of 0.8%. The uncertainty in the temperature difference ( ∆ T ) due to the combined effects
of thermocouple and data acquisition unit resolution was 0.75% for a temperature difference of
around 30
◦ C. The samples were all filled with water (with conductivity K
f
≈ 0.6 Wm
− 1
K
− 1
)
and contacted with top and bottom copper plates. Therefore, we neglect any thermal resistance
at the contacts. Furthermore, based on the thermal conductivity of the styrofoam and insulation
wall (around 0.02 Wm
− 1
K
− 1
), the heat loss through the insulation was estimated to be less than
0.5% of the heat input.
Finally, buoyancy effects can be estimated via the Grashof number in equation 2.7, which
represents the ratio of buoyancy forces to viscous forces in the porous material:
Gr =
gβ ∆ TD
3
ν 2
. (2.7)
In this study, the working fluid was water, which has a coefficient of expansion of β ≈ 0.21× 10
− 3
K
− 1
and kinematic viscosityν ≈ 10
− 6
m
2
/s. As noted below, the characteristic pore-scale length
scale isD≈ 1 mm and the temperature difference is expected to be ∆ T ≈ 10 K. The acceleration
due to gravity is g = 9.81 m/s
2
. For these conditions, we expect Gr ≈ 20, which is below the
threshold that gives rise to natural convection. In addition, all the experiments were carried out
with the heated surface on top, i.e., with stable density stratification. So, we do not anticipate
buoyancy-driven convection to arise in the experiments.
20
2.6 MathematicalModel
As shown in previous studies, [6, 17], one can estimate the effective thermal conductivity of an
object by dividing it into several parallel and series parts, as illustrated in Figure 2.9. For the
parallel parts, the thermal conductivity K
parallel
can be calculated using equation 2.8, while for
the series parts, the thermal conductivityK
series
can be estimated using equation 2.9:
K
parallel
A
total
D
total
=
A
1
K
1
D
+
A
2
K
2
D
, (2.8)
D
total
K
series
A
total
=
D
1
AK
1
+
D
2
AK
2
. (2.9)
In the equations above,A
total
andD
total
represent the total area and length of the samples, while
(A
1
,A
2
),(D
1
,D
2
), and(K
1
,K
2
) respectively represent the areas, lengths, and thermal conduc-
tivities for components 1 and 2 in the series or parallel configurations. These quantities are shown
in Figure 2.9.
Figure 2.9: Series and parallel illustration in lumped parameter models.
21
Figure 2.10: Unit-cell calculations for the mathematical model. In the unit cell, the rods (blue
parts) are the solid components with thermal conductivityK
s
and the porous space (gray parts)
is filled with fluid (water) with thermal conductivity K
f
. In each unit cell, the top part has rods
intersection in a cross-shaped geometry and the bottom part has only one rod in the middle.
A physically-motivated lumped parameter model for the porous geometry described in the
previous section was developed as follows. This model was developed using the unit cell geom-
etry shown in Figure 2.10. The unit cell was separated into two series components: and upper
portion in which the rods intersect in a cross-shaped geometry and a lower portion comprising
a single rod. In each of these components, the solid rods and the fluid can be assumed to act in
parallel. The thermal conductivity for the upper portion (K
1
) can be estimated as
K
1
A
total
=K
f
A
f1
+K
s
A
s1
(2.10)
where the total areaA
total
, fluid area A
f1
, and solid areaA
s1
are as followed:
A
total
=S
x
S
y
, (2.11)
A
f1
= 4
S
x
− r
2
S
y
− r
2
= (S
x
− r)(S
y
− r), (2.12)
22
and
A
s1
=S
x
r+S
y
r− r
2
. (2.13)
Therefore,K
1
becomes:
K
1
=
K
f
(S
x
− r)(S
y
− r)+K
s
(S
x
r+S
y
r− r
2
)
S
x
S
y
. (2.14)
Using similar arguments, the thermal conductivity for the bottom component (K
2
) can be shown
to be:
K
2
=
K
f
(S
x
S
y
− r
2
)+K
s
r
2
S
x
S
y
. (2.15)
Finally, assuming that the effective thermal conductivity in the direction of heat transfer ( K
e
,z)
can be estimated by combining theK
1
andK
2
contributions in series, we have
K
e,z
=
S
z
r
K
1
+
Sz− r
K
2
. (2.16)
This model can be used to generate a priori estimates for the effective thermal conductivity of
the different porous geometries tested in the experiment.
23
Chapter3
Results
This section presents thermal conductivity measurements for the different porous materials. Re-
sults are presented separately for the isotropic porous materials and anisotropic materials with
S
x
̸= S
y
andS
x
= S
y
. Experimental results are also compared with predictions made using the
empirical correlation developed by Calmidi et al., [1] (equation 1.5) and the lumped parameter
model described above (equation 2.16).
3.1 IsotropicMaterials
Table 3.1: Measurement results for isotropic samples.
Geometry porosity
K
e
/K
s
measured equation 2.16 equation 1.5
LLL 0.72±0.01 1.604±0.032 1.726 1.563
MMM 0.83±0.01 1.699±0.034 1.797 1.682
HHH 0.91±0.01 1.775±0.036 1.852 1.778
The measured thermal conductivities for the isotropic samples (LLL, MMM, HHH) are shown
in Table 3.1 and Figure 3.1. The raw measurements are shown in Figure 3.2. In general, these
24
Figure 3.1: Effective thermal conductivity measurements compared against upper- and lower-
bound estimates and model predictions.
measurements show that the normalized effective conductivity increases with increasing poros-
ity. This is consistent with expectations since the fluid medium (water, K
f
≈ 0.6 Wm
− 1
K
− 1
)
is more conductive than the solid phase in these experiments (resin, K
s
≈ 0.3 Wm
− 1
K
− 1
). For
instance, the dimensionless conductivity increases fromK
e
/K
s
= 1.604± 0.032 for the lowest
porosity LLL sample with ϵ = 0.72± 0.01 to K
e
/K
s
= 1.775± 0.036 for the highest porosity
HHH sample withϵ = 0.91± 0.01.
All measurements fall within the theoretical upper and lower bounds presented in the in-
troduction (equations 1.4 and 1.3). Interestingly, the measurements generally show reasonable
agreement with predictions from the empirical model from equation 1.5. The lumped parameter
model (equation 2.16) captures the overall trends reasonably well (i.e., increasing conductivity
with porosity) though there is some disagreement in magnitude. This can be attributed to dif-
ferences in geometry between the CAD specifications used in equation 2.16 and the as-printed
25
(a)∆ T vsQ diagram for HHH sample (b)K
e
vs∆ T diagram for HHH sample
(c)∆ T vsQ diagram for MMM sample (d)K
e
vs∆ T diagram for MMM sample
(e)∆ T vsQ diagram for LLL sample (f)K
e
vs∆ T diagram for LLL sample
Figure 3.2: Measurement results for isotropic samples. The power was limited such that the
temperature difference remained below 30
◦ C.
model. Specifically, the as-printed specimens generally had much lower porosity ( ϵ ≈ 0.7− 0.9)
compared to the designed geometries (ϵ ≈ 0.8− 0.95) due to printing tolerances (see Figure 3.1).
This contributes directly to reduced effective conductivity. The raw measurements in Figure 3.2
26
show consistent behavior across all power levels, indicating that the measurements themselves
are reliable.
3.2 AnisotropicMaterialswithDifferentIn-PlaneSpacing
Table 3.2: Measurement results for anisotropic samples withS
x
̸=S
y
.
Geometry porosity
K
e
/K
s
measured equation 2.16 equation 1.5
LML 0.77±0.01 1.627±0.033 1.752 1.615
MLM 0.80±0.01 1.649±0.033 1.776 1.648
LHL 0.80±0.01 1.617±0.032 1.778 1.648
MHM 0.86±0.01 1.753±0.035 1.818 1.717
HLH 0.86±0.01 1.699±0.034 1.821 1.717
HMH 0.88±0.01 1.731±0.035 1.837 1.741
Figure 3.3: Effective thermal conductivity measurements, upper- and lower-bounds, and model
predictions for anisotropic samples withS
x
̸= S
y
. Measurements are generally consistent with
the empirical correlation in equation 1.5, which is closer to the lower bound.
Next, we consider effective thermal conductivity for materials exhibiting in-plane anisotropy,
i.e., materials with S
x
̸= S
y
. The raw ∆ T and K
e
measurements for these materials can be
27
found in Figures 5.2-5.7 in Appendix B. The results are summarized and compared against model
predictions in Table 3.2 and Figure 3.3.
Again, all K
e
measurements fall between the upper and lower bounds predicted theoreti-
cally. In this case, the measurements cluster towards the lower bound estimate, indicating that
these geometries act more like the series configuration. The empirical correlation formula (equa-
tion 1.5) yields predictions closer to the measurements when compared to predictions made using
the lumped-parameter model (equation 2.16). The lower measured conductivity values can again
be attributed to the as-printed geometries having lower porosity.
The raw measurements shown in Figures 5.2-5.7 below show very consistent K
e
measure-
ments for all power readings. The exception to this trends is the LHL geometry shown in Fig-
ure 5.4, for which a couple of outliers at low∆ T (andQ) may have artificially lowered the average
K
e
measurement. Note that this is the only geometry for which the measuredK
e
is significantly
below the predictions made using equation 1.5: measured K
e
/K
s
= 1.617± 0.032 compared
with a predicted value ofK
e
/K
s
= 1.648 (see Table 3.2).
3.3 AnisotropicMaterialswithIdenticalIn-PlaneSpacing
Table 3.3: Measurement results for anisotropic samples withS
x
=S
y
.
Geometry porosity
K
e
/K
s
measured equation 2.16 equation 1.5
LLM 0.76±0.01 1.652±0.033 1.752 1.604
MML 0.81±0.01 1.646±0.033 1.776 1.659
LLH 0.80±0.01 1.699±0.034 1.779 1.648
MMH 0.86±0.01 1.731±0.035 1.819 1.717
HHL 0.88±0.01 1.741±0.035 1.822 1.741
HHM 0.89±0.01 1.750±0.035 1.837 1.753
28
Figure 3.4: Measurements and model predictions anisotropic samples with the sameS
x
andS
y
.
This section shows effective thermal conductivity measurements for anisotropic materials
withS
x
=S
y
̸=S
z
. In other words, the in-plane spacing is identical in both thex andy directions.
Figure 3.4 and Table 3.3 show summary results and comparisions with model predictions. The
raw measurements are provided in Figures 5.8-5.13 in Appendix B .
For these samples, the measurements are generally closer to the upper bound compared to the
results shown in the previous section for samples withS
x
̸=S
y
. As before, the empirical correla-
tion from Calmidi et al. [1] generates predictions more consistent with measurements compared
to the lumped parameter model. Interestingly, all the samples with lower spacings in thex− y
plane compared to the direction of heat transfer (i.e., LLM, LLH, MMH) show thermal conductiv-
ity values that are above the predictions made using equation 1.5. The remaining samples have
K
e
values very close to the predictions. This suggests that as-printed geometries with the largest
29
spacing in the direction of heat transfer (S
x
= S
y
< S
z
) yield larger thermal conductivity com-
pared to samples with identical spacings that are rotated such that the largest spacing is in-plane.
This is also consistent with predictions from the lumped parameter model, which suggests that
such geometries may show behavior that is more closely approximated by a parallel model.
30
Chapter4
Discussions
The preceding section showed that all thermal conductivity measurements fell within established
upper and lower bounds for porous materials. This is also evident in Figure 4.1, which compiles
together thermal conductivity measurements for all the materials, separating them into the 3
geometrical classes described previously: isotropic materials withS
x
= S
y
= S
z
(red markers);
anisotropic materials withS
x
̸=S
y
(green markers); and anisotropic materials withS
x
=S
y
̸=S
z
(black markers).
This section briefly evaluates the effects of porosity and porous microstructure on the thermal
behavior of the 3D-printed samples.
4.1 EffectofPorosity
As seen in the results presented in the previous section, porosity plays a critical role in dictat-
ing effectively thermal conductivity. In general, increasing porosity leads to increasing effec-
tively thermal conductivity for the particular material-fluid combination considered here, i.e.,
3D-printed samples made from resin and saturated with water such thatK
s
< K
f
. Indeed, the
31
Figure 4.1: K
e
/K
s
measurements for all materials, separated into isotropic cases (red markers),
anisotropic cases with S
x
̸= S
y
(green markers), and anisotropic cases with S
x
= S
y
(black
markers).
empirical correlation in equation 1.5, which depends only on porosity, generates very good pre-
dictions for the effective thermal conductivity across all the anisotropic and isotropic geometries
considered here.
This is consistent with observations made in prior literature, which show that the thermal
behavior of metal foams can be accurately predicted based only on the porosity of the samples
[1]. In that case, increasing porosity leads to decreasing effective thermal conductivity since the
solid phase has a higher thermal conductivity than the fluid phase. Nevertheless, the observation
that the same empirical model yields reasonable predictions for the present scenario indicates
that the underlying mechanism is consistent regardless of the different materials.
Note that the lumped parameter model (equation 2.16) does not yield predictions that are as
good as the empirical model. This is in part because the empirical model predictions make use
32
of the measured porosity values of the as-printed samples while the lumped parameter model
only makes use of the CAD-specified geometry. The as-printed samples had slightly different
spacing in the different dimensions compared to specifications, and so the samples had much
lower porosity values compared to the specified geometries (see ◦ vs. ∗ in Figures 3.1, 3.3, 3.4).
Moreover, the manufacturing tolerances may also depend on the print-direction.
4.2 EffectofPorousMaterialStructure
Despite the conductivity measurements showing strong porosity dependence, the specific mi-
crostructure does play a role. For example, the LHL and LLH samples both have an as-printed
porosity of ϵ = 0.80± 0.01 and nominally-identical unit cell geometry, that is simply rotated.
Yet, measured thermal conductivities are K
e
/K
s
= 1.617 ± 0.032 for the LHL sample and
K
e
/K
s
= 1.699± 0.034 for the LLH sample. Similarly, the LLM sample also shows a higher
effective conductivity ( K
e
/K
s
= 1.652± 0.033) compared to the nominally similar but rotated
LML sample (K
e
/K
s
= 1.627± 0.033).
(a) HHH sample (b) HHL sample (c) HLH sample
Figure 4.2: Samples with different pore geometries: (a) isotropic HHH case, (b) anisotropic HHL
case, and (c) anisotropic HLH case. The geometries in (a) and (b) exhibit higher thermal conduc-
tivity compared to the geometry in (c).
33
In general, most of the geometries withS
x
=S
y
show higher effective conductivity compared
to corresponding geometries withS
x
̸=S
y
. This is evident from a comparison between Tables 3.2
and 3.3), and from the observation that, for the same porosity, the black markers generally fall
above the green markers in Figure 4.1. Moreover, these cases withS
x
= S
y
generally fall on the
same trendline as the isotropic samples withS
x
= S
y
= S
z
(LLL, MMM, HHH). Together, these
observations suggest that the isotropic materials and anisotropic materials withS
x
=S
y
exhibit
thermal behavior that is more akin to the parallel model that yields higher effective conducitivity
(lower thermal resistance). In contrast, theS
x
̸=S
y
material exhibit behavior closes to the series
model. This behavior is also consistent with the lumped parameter model (equation 2.16) which
predicts (marginally) higherK
e
/K
s
values for theS
x
= S
y
geometries in Table 3.3 compared to
theS
x
̸=S
y
geometries in Table 3.2.
The CAD renderings shown in Figure 4.2 provide further insight into these trends. For the
HHL sample in Figure 4.2(b), the solid lattice and the fluid cavity effectively act like thermal
resistance components in parallel, leading to higher conductivity. In contrast, for the HLH sample
in Figure 4.2(c), the solid lattice and fluid space act more like thermal resistance components in
series, leading to reduced effective conductivity.
34
Chapter5
Conclusions
The primary objective of this study was to investigate the thermal properties of 3D-printed porous
materials and to determine whether porosity is the sole factor affecting the thermal conductiv-
ity, or whether the specific lattice structure of the material also plays a role. The study involved
the development of a simple analytical model and extensive experiments measuring the effec-
tive thermal conductivity of 3D-printed isotropic and anisotropic porous materials with different
porosity and lattice structure. A setup was developed to measure the effective thermal conduc-
tivity of the samples via steady-state heat transfer experiments. This setup was verified via tests
with solid steel and 3D-printed samples prior to the porous material tests. The results of the study
have provided several insights, which are summarized below, along with suggestions for further
research.
First, it was found that porosity is a critical factor in determining the thermal conductivity
of porous materials. For the specific 3D-printed materials (formlabs clear resin) and working
fluid (water) tested in this study, as porosity increases, the effective thermal conductivity of the
material increases as well. This can be attributed to the presence of greater void space for water,
which has higher thermal conductivity than the solid resin material used in the experiments.
35
Second, it was observed that the specific pore geometry and heat transfer direction can mean-
ingfully impact thermal behavior. In particular, the results show variations of over 5% in effective
thermal conductivity for nominally-identical anisotropic geometries that are rotated with respect
to the direction of heat transfer. This opens up the possibility of designing anisotropic porous
materials with directional thermal conductivity optimized for heat transfer applications.
Third, the effective thermal conductivity of the 3D-printed materials can be predicted reason-
ably well using the simple correlation formula from shown in equation 1.5 [1], which depends
solely on porosity. This formula was developed based on porous metal foam experiments in which
the solid phase had significantly higher conductivity compared to the fluid. This suggests that
similar empirical formulae may also hold for additively-manufactured porous materials made
of metal. However, the effect of the specific pore geometry must be taken into account for more
accurate predictions, especially for anisotropic materials. A simple lumped parameter model cap-
tured the geometry-dependent trends in effective thermal conductivity for anisotropic materials.
However, the predictions did not match quantitatively with the experimental data. This is in large
part because model predictions were based purely on the CAD-specified geometry, rather than
the geometry of the as-printed materials.
In light of these findings, future research in this field could focus on further investigating
the impact of printing parameters on the thermal behavior of additively-manufactured materials.
This could involve examining different printing orientations, as well as exploring the potential of
new printing techniques and materials to achieve enhanced thermal properties. Further studies
could also examine the impact of other factors on the thermal conductivity of porous materials.
This includes the shape and size of the pores as well as the relative conductivity of the solid and
fluid phases.
36
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38
AppendixA:CADRenderingsofPorousMaterials
CAD schematics for all porous samples are shown in Figure 5.1. Note that H, M, and L denote
dimensions of 3, 2, and 1.5 mm for the unit cell shown in Figure 2.10. For any given sample
designation (e.g., HHH), the first letter represents the total spacing inthe x-direction,S
x
, for the
unit cell. Similarly, the second and third letters representS
y
andS
z
, respectively.
(a) HHH sample. (b) MMM sample
(c) LLL sample. (d) HHM sample
39
(e) HHL sample. (f) HMH sample
(g) MMH sample. (h) HLH sample
(i) LLH sample. (j) LML sample.
40
(k) MML sample. (l) LHL sample
(m) LLM sample. (n) MLM sample
Figure 5.1: CAD renderings for all porous samples.
41
AppendixB:RawDataofMeasurements
Temperature difference versus power input curves and resulting K
e
estimates for the anisotropic
materials can be found in Figures 5.2-5.13 below.
(a)∆ T vsQ diagram for LML sample. (b)K
e
vs∆ T diagram for LML sample.
Figure 5.2: Raw results for LML sample.
(a)∆ T vsQ diagram for MLM sample. (b)K
e
vs∆ T diagram for MLM sample.
Figure 5.3: Raw results for MLM sample.
42
(a)∆ T vsQ diagram for LHL sample. (b)K
e
vs∆ T diagram for LHL sample.
Figure 5.4: Raw results for LHL sample.
(a)∆ T vsQ diagram for MHM sample. (b)K
e
vs∆ T diagram for MHM sample.
Figure 5.5: Raw results for MHM sample.
(a)∆ T vsQ diagram for HLH sample. (b)K
e
vs∆ T diagram for HLH sample.
Figure 5.6: Raw results for HLH sample.
43
(a)∆ T vsQ diagram for HMH sample. (b)K
e
vs∆ T diagram for HMH sample.
Figure 5.7: Raw results for HMH sample.
(a)∆ T vsQ diagram for LLM sample (b)K
e
vs∆ T diagram for LLM sample
Figure 5.8: Raw results for LLM sample.
(a)∆ T vsQ diagram for MML sample. (b)K
e
vs∆ T diagram for MML sample.
Figure 5.9: Raw results for MML sample.
44
(a)∆ T vsQ diagram for LLH sample. (b)K
e
vs∆ T diagram for LLH sample.
Figure 5.10: Raw results for LLH sample.
(a)∆ T vsQ diagram for MMH sample. (b)K
e
vs∆ T diagram for MMH sample.
Figure 5.11: Raw results for MMH sample.
(a)∆ T vsQ diagram for HHL sample. (b)K
e
vs∆ T diagram for HHL sample.
Figure 5.12: Raw results for HHL sample.
45
(a)∆ T vsQ diagram. for HHM sample. (b)K
e
vs∆ T diagram for HHM sample.
Figure 5.13: Raw results for HHM sample.
46

Abstract (if available)

Abstract Thermal management is a critical aspect of many modern technological systems, as highly effective thermal transport is necessary for ensuring their proper functioning and longevity. As a result, the design of thermal management systems has become increasingly crucial in a wide range of fields, from electronics to aerospace to energy production. To meet the growing demand for effective thermal management, researchers have been exploring new materials and technologies that can be used to improve heat transfer performance in these systems.
Metal foams have shown great potential for thermal management applications. Metal foams are highly porous materials that have a large surface area, which makes them ideal for dissipating heat in a system. The use of metal foam in heat pipes and other systems has been shown to be highly effective.
More recently, the development of additive manufacturing (AM) technology has opened up new possibilities for designing and manufacturing porous media for thermal management. AM allows for the creation of complex geometries and structures that would be difficult or impossible to produce using traditional manufacturing methods. This technology could enable the creation of efficient heat dissipation materials and revolutionize the design of thermal management systems.
This thesis presents an analytical and experimental investigation into the effective thermal conductivity (Ke) of 3D-printed porous resin samples with porosity ranging from 70% to 90%. The overarching goal is to explore the effect of lattice structure and porosity on the thermal behavior of the porous materials.
An experimental setup was developed to measure steady-state heat transfer across the porous samples fully-saturated with water. The porous material was sandwiched between two conductive copper plates. A patch heater was used to provide uniform heat input at the upper plate and the lower plate was exposed to a water bath attached to a chiller. Thermocouples were used to measure upper and lower plate temperature, and the vertical temperature gradient was use to estimate effective thermal conductivity. Experimental data were compared to predictions made using a lumped parameter model as well as an empirical formula proposed in prior literature.
The results demonstrate that porosity is a critical factor in determining the thermal conductivity of porous materials, with higher porosity resulting in higher $K_e$ due to the presence of voids for water, which has a higher thermal conductivity than the solid resin used in the experiments. Tests with anisotropic pore structures showed that the specific pore geometry also has an impact on the thermal conductivity of the materials. Empirical correlations developed in previous literature for metal foams generated reasonable predictions for the measurements made in this study. This suggests that similar empirical formulae may also hold for additively-manufactured porous materials made of metal. However, the effect of the specific pore geometry must be taken into account for more accurate predictions, especially for anisotropic materials.
Overall, this study provides valuable insights into the thermal behavior of additively manufactured porous materials and highlights the need to consider the effect of lattice structures on the thermal conductivity of such materials. Future research could focus on investigating the impact of printing parameters and exploring new printing techniques and materials to achieve enhanced thermal properties.

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Asset Metadata

Creator Wu, Bo-wei (author)

Core Title The effect of lattice structure and porosity on thermal conductivity of additively-manufactured porous materials

School Viterbi School of Engineering

Degree Master of Science

Degree Program Aerospace and Mechanical Engineering (Computational Fluid and Solid Mechanics)

Degree Conferral Date 2023-08

Publication Date 06/07/2023

Defense Date 05/25/2023

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag 3-D print,additively manufactured,effect of structure and porosity,OAI-PMH Harvest,porous materials,thermal conductivity,thermal management

Format theses (aat)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Luhar, Mitul (committee chair), Chen, Yong (committee member), Sadhal, Satwindar (committee member)

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