Theoretical models of thermal conductivity and its relationship with electrical conductivity for compressed metal powder

TVE-K; 19004 , TVE-Q; 19003

Examensarbete 15 hp Juni 2019

Theoretical models of thermal conductivity and its relationship with electrical conductivity for compressed metal powder

Danilo de Souza V, Veronica Enblom Johanna Sjölund, Mattias Sjödin

Niklas Lindborg, Sam Tran

Teknisk- naturvetenskaplig fakultet UTH-enheten

Besöksadress:

Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0

Postadress:

Box 536 751 21 Uppsala

Telefon:

018 – 471 30 03

Telefax:

018 – 471 30 00

Hemsida:

http://www.teknat.uu.se/student

Abstract

Theoretical models of thermal conductivity and its relationship with electrical conductivity for

compressed metal powder

Danilo de Souza V, Veronica Enblom, Johanna Sjölund, Mattias Sjödin, Niklas Lindborg & Sam Tran

This Independent Project reviews literature about the effect of pressure and temperature on thermal conductivity in packed beds and its relationship with electrical conductivity. Exploring the relationships between thermal conductivity, porosity and pressure can give useful knowledge for further improvements in manufacturing processes in the field of powder metallurgy. The resulting theoretical models describing the effective thermal conductivity show that gas and contact conductance dominate at lower temperatures and that radiation gains dominance as the temperature increases. Modifications of the models covered in this report can be made in order to simulate the process of interest more accurately. It was also shown that Wiedemann-Franz law could be of interest when wanting to quantify the thermal conductivity in a powder compact. Furthermore, a lab manual for a future

Independent Project was developed.

ISSN: 1650-8297, TVE-K; 19004 , TVE-Q; 19003 Examinator: Peter Birch

Ämnesgranskare: Jens-Petter Palmquist, Emil Svensson Handledare: Jens-Petter Palmquist

Theoretical models of thermal conductivity and its relationship with electrical conductivity for compressed

metal powder

May 29, 2019

Project name: Thermal conductivity in pressed steel powder - Part 2: Material analysis and electrical conductivity

Contact person/Technical consultant at the company: Jens-Petter Palmquist Project supervisor: Emil Svensson

Student group: Danilo de Souza V, Veronica Enblom, Johanna Sjölund, Mattias Sjödin, Niklas Lindborg & Sam Tran

Abstract

This Independent Project reviews literature about the effect of pressure and temperature on thermal conductivity in packed beds and its relationship with electrical conductivity. Ex- ploring the relationships between thermal conductivity, porosity and pressure can give useful knowledge for further improvements in manufacturing processes in the field of powder metal- lurgy. The resulting theoretical models describing the effective thermal conductivity show that gas and contact conductance dominate at lower temperatures and that radiation gains dom- inance as the temperature increases. Modifications of the models covered in this report can be made in order to simulate the process of interest more accurately. It was also shown that Wiedemann-Franz law could be of interest when wanting to quantify the thermal conductivity in a powder compact. Furthermore, a method and a lab manual for a future Independent Project were designed in order to be able to empirically analyze the relationship between thermal conductivity and pressure in future projects.

Keywords: High-speed steel powder, Conductivity, Porosity & Weidermann-Franz law.

Contents

1 Introduction 5

2 Method 6

2.1 Literature study . . . . 6

2.2 Models. . . . 6

2.3 Development of future experiments . . . . 6

2.4 Limitations . . . . 7

3 Results 7 3.1 Thermal conductivity in packed beds . . . . 7

3.1.1 Thermal contact conductance . . . . 8

3.1.2 Heat transfer in gas . . . . 9

3.1.3 Conduction through radiation. . . . 10

3.2 Models for ETC . . . . 11

3.2.1 ETC model by Jingwen Mo and Heng Ban . . . . 11

3.2.2 ETC model by Zehner, Bauer and Schlünder (ZBS) . . . . 13

3.2.3 ETC model by Kunii and Smith . . . . 14

3.3 Illustration of models. . . . 14

3.4 Relationship between thermal and electrical conductivity. . . . 16

3.4.1 Deviations from Wiedemann-Franz law . . . . 16

3.4.2 Wiedemann-Franz law for porous compacts . . . . 18

3.5 Planned method for a future experimental study . . . . 20

3.5.1 Method for development of container for CIP . . . . 21

4 Discussion 24 4.1 Heat transfer as a function of temperature . . . . 24

4.2 Heat transfer as a function of pressure . . . . 25

4.3 Interplay between heat transfer mechanisms . . . . 25

4.4 Reliability of WFL . . . . 26

4.5 Future experimental work . . . . 26

5 Conclusions 28 References 29 6 Appendix 33 6.1 Lab manual - For a future Independent Project . . . . 33

6.1.1 Sample preparation . . . . 33

6.1.2 Cold Isostatic Pressing, CIP. . . . 33

6.1.3 Conductivity and porosity measurements . . . . 33

6.1.4 Analysis of data . . . . 34

Nomenclature

∆T Temperature gradient [Km⁻¹]

 Porosity

M Tap porosity

γ_a Adhesive force trough the particle sur- face energy

κ ratio between solid and gas conductivity ρ Resistivity [Ωm]

ρ0 Resistivity for fully sintered material [Ωm]

ρE Effective resistivity [Ωm]

σe Electrical conductivity [Sm⁻¹] σp Compressional stress [N m⁻²] θ_D Debye temperature [K]

ξ Surface fraction parameter (0 ≤ ξ ≤ 1 ) α Degree of accuracy for calculating heat

conducted through air and by radiation αT Thermal accommodation factor

βr Accommodation factor for micro-contact thermal resistance

βs Empirical value for close packed spheres

r Emissivity

γ Efficiency of the energy transfer at the interface between gas and solid

κ_G Gaseous conduction as a function of Knudsen number [Wm⁻¹K⁻¹]

κ_r Radiation parameter [Wm⁻¹K⁻¹] λ Mean free path of gas molecules [m⁻¹] µ Inverse molar mass ratio,1/mrMg/Ms

ν Poisson’s ration

ψ2 Heat transfer function for close packing ψ_t Heat transfer function of the numer of

contacts

σ Stefan-Boltzmann constant [5.6704×

10⁻⁸W m⁻²K⁻⁴]

σ₀ Electrical conductivity of the metal phase [Sm⁻¹]

σ₁ Electrical conductivity of the pores [Sm⁻¹]

θ Effective diameter of gas molecules [m]

θ0 Contact angle that is between 0 and θ1

θ1 Contact angle between adjacent parti- cles

θk Parameter equal to 1/n where n = 3√ 3 for close packed powder

Cp Isobaric heat capacity [J K⁻¹] dp Particle diameter [m]

k0 Thermal conductivity of the metal ma- trix [Wm⁻¹K⁻¹]

k₁ Thermal conductivity of the pores [Wm⁻¹K⁻¹]

k_b Boltzmann constant [1.38065 × 10⁻²³m²kgs⁻²K⁻¹]

kc Solid contact conductivity [Wm⁻¹K⁻¹] ke Electronic thermal conductivity

[Wm⁻¹K⁻¹]

k^g_e Heat transfer through gas conduction [Wm⁻¹K⁻¹]

k^r_e Heat transfer through radiation [Wm⁻¹K⁻¹]

k^s_e Heat transfer through solid conduction [Wm⁻¹K⁻¹]

ks Bulk thermal conductivity of the con- tacting material [Wm⁻¹K⁻¹]

k₀₁ Thermal conductivity through the skin effect [Wm⁻¹K⁻¹]

k_bed Thermal conductivity of the packed bed [Wm⁻¹K⁻¹]

k_{ef f} Thermal conductivity of the bed satu- rated with a stagnant fluid [Wm⁻¹K⁻¹] kf,ef f Enhenced fluid effective thermal conduc-

tivity [Wm⁻¹K⁻¹]

kgas Thermal conductivity of gas [Wm⁻¹K⁻¹] kg Thermal conductivity modified with respect to the Smoluchowski effect [Wm⁻¹K⁻¹]

krs Radiation in solid to solid [Wm⁻¹K⁻¹] krv Radiation in void to void [Wm⁻¹K⁻¹] ks,ef f Enhenced solid phase effective thermal

conductivity [Wm⁻¹K⁻¹]

L₀ Sommerfeld value 2.44 × 10⁻⁸ WΩ K⁻²

Lv Void size [m]

M ∗ Molecular mass of monoatomic gases [gmole⁻¹]

M_g Molecular mass of gas [gmole⁻¹] m_r M olarmassratio, M_s/M_g

M_s Molecular mass of solid [gmole⁻¹] q_contact Heat flow through solid-solid contact

[W]

q_gas Heat flow through gas [W]

q_rad Heat flow through radiation [W]

r_c Contact area radius [m]

r_p Particle radius [m]

r_c,hertz Contact radius defined by Hertz theory Rgas Thermal resistance of gas [KW⁻¹] Rmacro Effective macro-contact thermal resis-

tance [KW⁻¹]

Rrad Thermal resistance of radiation [KW⁻¹] Rsum Sum of parallel thermal resistances

[KW⁻¹] T0 273 K

Ts Solid surface temperature of the particle [K]

A Area [m²]

B Deformation parameter, B =1.364(1-

)/^1.055

b Lattice component of Wiedemann-Franz law

C Porosity dependent factor, C =2.812(1 −

)^−1/3f^2(1+f2)

D0 Density of bulk material [kg/m³] DA Apparent density [kg/m³] E Young’s modulus [Pa]

F Force [N]

f Porosity dependent factor, f = 0.07318 + 2.193 - 3.357² + 3.194³

H Height, defined as 2rsinθ0[m]

I Modified equation of mean free path Kn Knudsen number

k Thermal conductivity [Wm⁻¹K⁻¹] L Lorenz number [WΩK⁻²]

l Length [m]

N Parameter calculated by Equation22 p Relationship between porosity and the

percolation threshold (porosity at which the structure becomes fully permeable) q Heat flux density [Wm⁻²]

R Resistance [Ω]

T Temperature [K]

ζ Deviation of the mean free path of pho- tons from the typical void size

1 Introduction

An understanding of heat transfer in porous materials is of great importance for a variety of applications such as packed bed reactors, thermal insulators and powder metallurgy. In the field of powder metallurgy, alloy manufacturing is highly common for production of durable industrial materials. However, large scale production entails long sintering times because of a lowered thermal conductivity in powder compacts relative to a fully dense body. Therefore, mapping of the heat transfer and its mechanisms in packed beds would be beneficial for improving the efficiency of manufacturing alloys such as High-speed steels.

High-speed steel (HSS) is a type of highly alloyed steel manufactured through powder metallurgy to obtain the unique microstructure that make them superior to conventional carbon steels, in applications where hardness and strength is required.[1] In the manufacturing of the alloy, powder is first produced through atomization, a process where molten bulk metal is dispersed by jets of an inert gas, commonly nitrogen, and then rapidly cooled and solidified.[2] The powder is then collected in capsules and compacted through cold isostatic pressing (CIP) before being pre- heated and objected to hot isostatic pressing in order to reach full density.[3] The cold compacting pressure is crucial for the time required to densify the material since it is known that the thermal conductivity in packed beds strongly depends on compressional pressure.[4] Lowering the pressure during CIP would potentially reduce the maintenance cost of the press and increase its life span.

However, as it is unknown exactly how the reduction in pressure will affect the conductivity of the granular material, mapping of different heat transfer mechanisms depending on temperature and porosity is of interest for future process optimization in the manufacturing of HSS.

Apart from the temperature and pressure dependency, it is unclear how heat transfer in a porous compact is affected by its porosity. Additionally, thermal conduction in packed beds is generally hard to measure because of the systems complexity and its various heat transfer contributions, as opposed to the single system of a fully dense body. An alternative approach could therefore be to measure the electrical conductivity and calculate the thermal conductivity with an established relationship.[5]

This project will mainly focus on whether it is possible to map the heat transfer in compacted steel powder by plotting theoretical models of the effective thermal conductivity, and its contributing mechanisms, as functions of temperature and pressure. Furthermore, a proposal for a practical study is also of interest to empirically conclude how thermal conductivity behaves in cold isostatic pressing and pre-heating.

2 Method

This project consisted of a literature study that was mainly used to model the behaviour of thermal conductivity in powder metal compacts. An experimental method was also derived in order to empirically study the relationship between the thermal conductivity and pressure.

2.1 Literature study

The literature study began by searching for simpler terms by combining words associated with conductivity, deformation and pressure (left) with terms related to porous compacts (right).

• Thermal conductivity

• Electrical conductivity

• Plastic deformation

• Pressure dependency

• Heat transfer

• Electrical resistivity

• Compaction

• Cold isostatic pressing

• Porous material

• Metal powders

• Green bodies

• Packed beds

• Compact powder

• Compressed powder

• Granular materials

• Particle deformation

The literature found gave a basic understanding of the subject at hand and enabled more evolved searches by introducing relevant terminology in the field. Further specified key words was then used, whereof the most important queries are listed below.

• Thermal contact conductance

• Solid conduction in packed beds

• Gas conduction in packed beds

• Radiative conduction in packed beds

• Models for packed structures and effective thermal conductivity

• Radiative contribution of effective thermal conductivity in packed beds

• Wiedemann-Franz law in porous compacts

• Phonon-electron interactions thermal conductivity

Studies and other literature similar to this project, or that included the search terms and were valid for powder steel or metals in general, were labeled relevant. Abstract, discussion and conclusion were read to further determine their usefulness. For those that were relevant, the references served as a good source for further reading. Furthermore, two books, ’Thermal conductivity, theory properties and applications’ and ’Thermal contact conductance’, were used extensively and gave a more general understanding of the theory rather than of specific cases such as this project.

2.2 Models

Multiple models were acquired from the literature study and plotted to illustrate the thermal conductivity as a function of compressional stress and temperature.

2.3 Development of future experiments

In order to gain information on the impact of pressure on heat transfer in pressed steel powder and its relationship with electrical conductivity, data has to be gathered using appropriate mea- surements. The literature study served as a basis for the development of a future experimental study where the choice of parameters, relevant method and accessible instrumentation was made through comparisons of methods between previous similar studies.

2.4 Limitations

As a consequence of a shift in purpose, guidelines for a conventional systematic study were not followed. The resulting literature study has however benefited from the wide approach in the search for information. Another limitation of this project has been the availability of latest published literature as some studies required to be borrowed from distance which the time limitation of this project could not allow. However, many books were available for download and articles published throughout the 20th century until 2018 could be found in various databases.

3 Results

In section 3.1 the overall conductivity and its primary heat transfer mechanisms, conduction through radiation, gas and solid contact, are extensively described. Various models for the ef- fective thermal conductivity are presented in section3.2and plotted in section3.3

Section 3.4 describes the relation between electrical conductivity and thermal conductivity. It introduces Wiedemann-Franz law and its applicability in the context of determining the thermal conductivity of a metal powder compact. The validity of the law is supported with previous studies and theory and shows that it is only valid if and only if important factors could be controlled such as temperature, strength of the applied electrical field, applied pressure and pore size.

Section 3.5 relies on previous sections in order to formulate a method that can be used as a continuation of this project in order to empirically show the behaviour of thermal conductivity in metal powder compacts.

3.1 Thermal conductivity in packed beds

The heat transfer in a packed bed is commonly described as its effective thermal conductivity, k_bed, representing the sum of all heat transfer mechanisms that are present. As illustrated with Equation 1, there are three major contributing mechanisms to the single value. The first component, kef f, is the fundamental contribution to the thermal conductivity in all types of packed beds, namely the effective thermal conductivity (ETC) of the bed saturated with a stagnant fluid. Both of the latter components, kf,ef f and ks,ef f, are merely enhancements to the fundamental ETC.[6]

kbed= kef f + kf,ef f + ks,ef f (1)

The contribution from kf,ef f arises in the presence of a fluid flowing through the particles parallel to the walls and entails motionless particles. Turbulent mixing of the fluid in the interstices then enhances the ETC by exchanging energy while criss-crossing between the particles. When the particles are also in motion the third component, ks,ef f, becomes significant as energy is exchanged when the particles grind against each other. For systems where neither the fluid or solid particles are in motion, the two enhancing components can thus be neglected and the ETC of the packed bed solemnly be described by k_{ef f}.[6]

k_{ef f} = k^s_e+ k^g_e+ k^r_e, (2)

As seen in Equation2, the ETC in presence of a stagnant fluid (kef f) can be further split into three distinct heat transfer mechanisms; Solid conduction, k_e^s, accounting for heat transfer through the reduced contact area between particles as a result of surface roughness, heat transferred through gas conduction, k_e^g, and radiative conduction, k_e^r.[6] Which of the transferring mechanisms that dominates the overall conductivity in a packed bed at different temperatures and pressures has been shown to depend on a large number of parameters such as size and shape [7], surface roughness of the constituent particles[8], packing structure[9] and porosity[10].

Since several of the parameters affecting the ETC are interconnected with each other, mapping of the predominating heat transfer source has been proven difficult. However, according to Wei- denfeld et al. [11] the dominating heat transfer mechanism at low compressional pressures and temperatures is conduction through the ambient gas since low pressures usually coincide with large

enough gap thickness between particles for gas to move freely. Thus, Weidenfeld’s model also pro- poses that conduction through gas will decrease and give way for conduction through solid-solid contact to dominate at higher pressures since the gap thickness is then smaller and the contact area between the particles larger. In contrary to the gas conduction, the thermal contribution from radiation only becomes significant at higher temperatures[12].

3.1.1 Thermal contact conductance

At the surface of solid particles, irregularities such as microscopic surface roughness or macroscopic flatness deviations are usually present. The actual contact area between two adjacent particles is therefore smaller than that of the nominal one, as is illustrated in Figure 1, which give rise to a thermal resistance between the particles. Because of an increased deformation with increased pressure, the pressure at which the powder is compacted is of great importance for the contact area and thus the solid conductivity. Although, even for metallic surfaces at contact pressures as high as 10 MPa, the actual contact area relative to the nominal is merely 1 - 2 %[13].[8] The thermal contact conductance is of great importance when the solid conductivity is much higher than that of gas, which for a binary system is given by the ratio κ ≥ 10³ [6].

Figure 1: Exaggerated surface interface illustrating the finite contact points between adjacent par- ticles. [14]

Because of the low actual contact between the solids, thermal conductivity in powders is lower than that of bulk materials. For metal powders the usually surrounding oxide layer, which in general exhibits a lower conductivity than metals, further increases the thermal resistance obtained at the interface. As Figure 2 illustrates, the resistance is observable in the temperature profiles of two solid bodies in contact by a rapid drop in temperature where the surfaces meet.[8] Additionally, the surface irregularities weakens the adhesive force (Van der Waals force) between the particles which further affect the overall conductivity [15, 16]. The adhesive force has a larger impact on smaller particles, whereupon the solid conductivity in the presence of surface energy consequently decreases with increasing particle size. The solid conductivity is however independent of size when no surface energy exists, namely for perfectly smooth particles. [17].

Figure 2: Temperature profile for two solid bodies A and B.[18]

In 2017 Sakatani et al [19] presented a model for the solid conduction based on experimental studies of powdered materials in vacuum. The model, presented here as Equation3, is derived from the assumption of mono-sized spherical particles exposed to one-dimensional heat flow perpendicular to parallel planes of particles.

k_e^s= 4

π²ks(1 − )Cξrc

r_p, (3)

Apart from porosity , Sakatani’s model depends on the bulk thermal conductivity of the contacting material k_s, the particle radius r_p, the contact radius r_c, a function C given by Equation4 and a surface fraction parameter ξ which is 1 for perfectly smooth particles and less than 1 for rough particles.[19]

C =2.812(1 − )^−1/3

f²(1 + f²) , (4)

where f=0.07318 + 2.193-3.357²+3.194³.

The contact radius between two spheres compressed by an external normal force F follows Hertzian theory[20] and can thus be calculated from Equation5, where E is the Young’s modulus and ν Poisson’s ratio. The collinear force, F, is dependent on both the particle size and porosity according to Equation6, where σ_p corresponds to the compressional stress that is applied to the powdered media.[19]

rc,hertz= 3(1 − ν²) 4E F rp

1/3

, (5)

F = 2πr²_p

√

6(1 − )σp (6)

By taking the adhesive force into consideration trough the particle surface energy γa, Johnson et al expanded the Hertzian theory, giving Equation7[21]. However, in this report the adhesive force will be neglected for the representation of solid conduction since the surface energy of the steel powder is unknown.

rc=

"

3(1 − ν²) 4E

 F +2

3πγarp+ s

3πγarpF + 3

2πγarp
²

 rp

#1/3

(7)

The model for solid conductivity, Equation3, was plotted over the porosity range 0.4-0.9 at 300K in Sakatani’s article in order to study the dependency of porosity. Good correlation with experimental data was obtained and a clear depression of the solid conductivity was observed with increasing porosity, as was expected.[19]

3.1.2 Heat transfer in gas

Beyond heat transferring through solid conduction at the contacting joints between the particles, the thermal conduction in pressed powders greatly depends on the surrounding fluid or gas. In fact, the area of heat flow through the interstitial gaps is frequently of an order 2 to 4 magnitudes larger than that from the actual contact area, proving the importance of conductance through gaps.[8]

Poor solid conductors in particular cannot neglect the additional heat flow, especially when the interface medium is a good conductor.[19, 22]

The additional heat transfer from gas in porous media is built on the conductivity of the gas itself and generally occurs through convection along with an exchange of energy between the gas and solid. Convection in packed beds, which is the turbulent mixing of the fluid due to density variation arising from the temperature gradients, strongly depends on the interfacial gap thickness.

Typically the gap thickness between two particles is in the range of 1 µm, which is a range where the convection can be disregarded. Moreover, the heat transfer through convection has been shown

to be negligible for gap widths up to 6 mm for solid metal surfaces when the temperature is low (around 300 K). [23]

In 1938 Kennard et al [24] found a reduction in the conductivity of gas in the small interfacial gaps between particles due to an increasing pressure. The phenomenon was attributed to the so called Smoluchowski effect which describes the pressure dependency of the gas conductance in powders. Its significance is determined by the ratio between the gas molecules mean free path λ and the typical void size in the packed bed Lv, often referred to as the Knudsen number calculated from Equation8. As can be seen, the Knudsen number is dependent on the pressure P as well as temperature T, Boltzmann’s constant kb and θ the effective diameter of the gas molecules. [19]

Kn = λ Lv

= k_bT

√2πθ²P L, (8)

Bahrami et al. recently confirmed the depression of gas conductivity in granular materials with increasing pressure, but identified that it is limited to the region when Kn ≤ 10 [25]. When the void size becomes much smaller than the mean free path of the molecules, to an order where Kn > 10, the conduction through gas has reduced to a point where it becomes insignificant [19,26]. In other words, since θ = 3,7 · 10⁻¹⁰ m for a nitrogen molecule the gas conduction at a temperature around 300 K is negligible when the pressure P < 6,8 · 10⁻⁴/L_v[19]. Since the interfacial gap between two adjacent particles in granular materials is normally smaller than λ [27,28], the Smoluchowski effect is important to consider when modelling the conductivity. Equation9is therefore a modification of the gas conductivity kg with respect to the reduction in conductance at higher pressures. γ quantifies the efficiency of energy transfer at the interface between gas and solid and is given in terms of the molar mass ratio mr= Ms/Mg in Equation10.[9]

k⁰_g= kg

1 + 2γKn (9)

γ = 2(1 + m_r)²− 2.4mr

2.4mr

(10)

3.1.3 Conduction through radiation

As previously stated in Section 3.1 the contribution to the overall ETC from thermal radiation becomes significant at higher temperatures. Heat radiation is electromagnetic waves radiated due to thermal motion of particles in matter, commonly called gray body radiation. It mainly occurs between the solid surfaces and solid-gas interfaces and depends on particle as well as gas composition. Depending on composition of interest, earlier studies have shown that the gray body radiation in granular materials starts to become significant at temperatures around 300-500 ^◦C, unless the contact resistance is large.[8,29,30]

From a study in 2014 where they analyzed the thermal radiation in packed beds using a homogeni- sation method, it was concluded that the significance of radiation transfer not only depends on temperature, but also on the size of the constituent particles. Beds with larger particles had a higher contribution to the ETC from transfer through radiation than beds with small beads. More- over, the radiation could be neglected for temperatures ranging up to 1440^◦C when the particle size was less than 1 mm. For particles with diameters of 10 and 100 mm the radiation became significant already at 400 respectively 150^◦C.[12]

In addition to the model for solid conductivity mentioned in Section3.1.1, Sakatani et al.[19] also presented a model for the radiative heat transfer through the void spaces in packed powders. The equation,

k^r_c = 8 r

2 − _rσζ( 

1 − )^1/3r_p· T³, (11)

was modelled by one-dimensional thermal radiation between multiple infinitely-thin parallel planes.

As seen the radiative conduction depends on the emissivity _r, the Stefan-Boltzmann constant σ (= 5.67×10−8 Wm²K⁴), the temperature of the colder plane, T, and ζ which is a measure of the deviation in the mean free path of photons from the typical void size. ζ is fitted for the material

in consideration and ranged between 0.7 and 15 for their model material. The fitting parameter was also shown to increase in value with decreasing particle size.[19]

As with the solid conduction, even this model was plotted by Sakatani in the porosity range 0.4- 0.9 for 300 K. In contrary to the solid conductivity, the radiation contribution showed a positive correlation with increasing porosity. However, a large uncertainty was noted due to notably larger void spaces between aggregates instead of between individual particles.[19]

3.2 Models for ETC

3.2.1 ETC model by Jingwen Mo and Heng Ban

As mentioned earlier, ETC can be described by the three contributing factors k_e^s, k^g_e and k_e^r. The following models have adopted certain assumptions in their system in order to simplify the calculations. J. Mo and H. Ban (2017) approached the system by describing it as spherical particles stacked in parallel layers perpendicular to the direction of heat flux, see Figure3.

Figure 3: Representation of the unit cell model, where NAis the number of particles per unit area and NL

is the particles per unit length.[19]

This method is based on the unit cell concept, which is a suitable approach for high solid to gas conductivity ratios, κ ≥ 10. Hence, the assumption of one-dimensional heat conduction is valid and the effective thermal conductivity of the unit cell can be calculated with respect to each heat flow contribution, q:

k_{ef f} =q_gas+ q_rad+ (q_contact)H

∆T A = H

RsumA, (12)

where A = 4r²is the unit cell area perpendicular to the heat flow, ∆T is the temperature gradient from top to bottom, H = 2rsinθ₀ and R_sum is the sum of each parallel thermal resistance.

As mentioned in section3.1.2, the gas conduction in granular materials is dependent on the mean gap thickness between two adjacent particles. When J. Mo and H. Ban defined their heat flow through gas, they approached it by describing the mean gap thickness as a function of contact angles, θ0and θ1 between two particles in contact with each other, see Figure4.

Figure 4: Schematic of the relationship between two spheres and the three contributions of ETC as a function of the contact angles . [31]

The contact angles are defined as follows:

∆y = 2rp(sinθ0− sinθ) = λ

5, (13)

θ0= cos⁻¹(aL

r_p), (14)

dA = 2πr²_p· cosθ · sinθ · dθ. (15)

In Equation14, aL is defined as the contact radius, which can be observed in Figure4. However, in the models to come, the contact radius is referred to rc, therefore aL= rc and can be calculated by Hertz theory, Equation5. When the contact angles are defined, the thermal resistance of the gas can be calculated with the help of Equation13-14and Equation15is integrated and derived into the following equation:

R_gas=

"

k_gasπr



− sinθ₁+ sinθ₀· ln sinθ₀ sinθ0− sinθ1

#−1

, (16)

where kgas is the thermal conductivity of gas without being dependent on the gas pressure. The second contribution, thermal resistance of radiation, can be obtained by Equation 17 assuming that radiation only occurs through the voids of the unit cell.

R_rad=

4σT³2r_p²⁻¹

(17) The thermal resistance due to contact area is obtained by approaching the microscopic and macro- scopic irregularities. In the model it is assumed that the particles are smooth without micro-contact thermal resistance. Therefore there exists an accommodation factor β for this error in Equation 12. Rmacro is the effective macro-contact thermal resistance defined as:

Rmacro= (1 −_r^rc

p)^1.5

2k_sr_c , (18)

and r_c is the contact area radius which is obtained using Hertz theory from Equation5.

At last, Equation12can be derived to:

kef f =

"

α

 1 Rgas

+ 1

Rrad

 + βr

1 Rmacro

#

·sinθ0

2rp

, (19)

α in Equation 19 represents the physical process of conduction and radiation through gas. It is introduced in order to make the theoretical model more accurate for gas conduction and radiation, ideally α is equal to one. The closer α is to one the better the equation captures the relationship between heat conducted. βr is presented earlier as the accommodation factor for omitting the micro-contact thermal resistance. At very high pressures β_r will approach one since the contact pressure increases and the overall contact area will mostly be representative of the macroscopic resistance. [31]

3.2.2 ETC model by Zehner, Bauer and Schlünder (ZBS)

Furthermore, another model to describe ETC in packed beds was presented by Zehner, Bauer and Schlünder and is referred to the ZBS model (1978). Bauer and Schlünder improved the previous model done by Zehner and Schlünder by taking thermal radiation, the Smoluchowski and the surface fraction parameter ξ into consideration. ETC can be calculated as follows:

k_{ef f} kgas

= (1 −√ 1 − )

"

 − 1 + 1 κG

−1

+ κ_r

# +√

1 − ξκ + (1 − ξ)kc

 (20)

k_e^sis the effective solid contact conductivity and can be calculate by Equation21,

k^s_e= 2 N

"

B(κ + κr− 1) N²κG

· ln κ + κr

BκG+ (1+G)(+r)+B + 1 2B

 κ_r κG

− B(1 +1 − κG

κG

κr



−B − 1 N κG

# , (21) and N is a parameter calculated from,

N = 1 κG

1 + κ_r− BκG

κ
− B 1

κG

− 1
1 +κ_r

κ

(22)

B is the deformation parameter which is determined to be B = 1.364(1-)/^1.055 while κG is the gaseous conduction as a function of the Knudsen number and κr is the radiation parameter:

κr= 4σ

2

_r − 1T³ dp

k_gas, (23)

whilst ris the emissivity and dp is the particle diameter, whereas κG is defined as,

κG=

 1 + l

d_p

−1

, (24)

where l is a modified equation of λ:

l = 22 − a_T aT

2πR_gasT Mg

!^1/2

· k_gas

P (2C_p−^Rconst_M

g ) (25)

T is the absolute temperature, P is the gas pressure, Rconst is the universal gas constant and Mg

is the molecular mass of the gas. The thermal accommodation coefficient, aT, was definied by Bahrami et al. and it represents the exchange of kinetic energy in a collision with the surrounding obstacles [25]. It can be calculated by Equation26:

a_T = exp



− 0.57(Ts− T0

T₀ )

 M^∗ 6.8 + M^∗



+ 2.4µ (1 + µ)² ·



1 − exp − 0.57(Ts− T0

T₀ )



, (26) M* is defined as Mg for monoatomic gases, µ = Mg/Ms ratio between molecular mass gas and particle, T0 = 273 K and Ts is the solid surface temperature of the particle. In addition, Bauer and Schlünder pointed out that the surface fraction parameter, ξ is a function of quantities such as mechanical properties and external mechanical stress which is best obtained experimentally.

However, Bauer and Schlünder estimated that ξ = 0.01 is the most suitable contact parameter for the ZBS model, which have been concluded from previous experimental results [33]. [32,33]

3.2.3 ETC model by Kunii and Smith

Kunii and Smith (1960) presented another approach to describe ETC in packed beds. They also presented their model described as in Figure3but they introduced a factor for radiation in void to void, krvand solid to solid, krs, respectively. However, the model is only valid for porosity between 0.260 ≤  ≤ 0.476. They can be expressed as functions of temperature, porosity and emissivity.

[33,34]

krv=

 4σT³

1 +_2(1−)^{ (1−}_ ^r)

r



(27) and

k_rs= 4σT³

 r

(2 − _r)



(28) ETC can then be calculated by the following equation,

k_{ef f} kgas

= 



1 +β_sk_rvd_p kf



+ β_s(1 + )

1

1/ψt+krsdp/kf +^χ_κ
, (29) where β_sis an empirical value for close packing spheres which is set to 0.895, χ = 2/3 and ψ_tis a function dependent on the contact area according to:

ψ_t= ψ₂+ ψ₂( − 0.260

0.216 ), (30)

ψ2is dependent on the parameter θk= 1/n where n=3√

3 for close packed powder and ψ2is defined as:

ψ₂= 0.5(κ − 1/κ)²sin²θk

ln(κ − (κ − 1)cosθ_k) − (κ − 1/κ)(1 − cosθ_k)− 2

3κ (31)

3.3 Illustration of models

The three ETC models were plotted against the temperature to visualize how the models behave during the pre-heating process. In addition to this, the models were also plotted against the compressional pressure to simulate how k_{ef f} for the metal powder ASP 2030 is affected in the CIP process.

Table 1: Parameters that are used when calculating the ETC models [23,35].

Reference Parameters

J. Mo and H. Ban (2017): α = 0.90 βr= 0.723 Bauer and Schlunder

ZBS (1978) ξ = 0.01

Kunii and Smith (1960):

χ = 2/3 n = 4√

3 β = 0.895 (a) Parameters that are approximated for each model

Material Material properties (20^◦C)

ASP 2030

E - modulus = 240 GPa Cp = 420 J/KgK ks= 24 W/mK rp = 372.45 um Stainless steel r = 0.5

v = 0.30 Nitrogen gas kgas = 0.026 W/mK

Mg = 28 kg/kmol

(b) Material properties that are used in the models.

Table1presents the properties of the materials that are used in the three presented models. The values for emissivity and Poisson’s ratio were not known for ASP 2030, therefore the values for stainless steel were used instead. The variations of Young’s elasticity modulus and specific heat with the temperature are taken into account in the calculations. The emissivity also changes with temperature but is set as a constant in this case because no information on how the emissivity changes with temperature is available. The particle size of the powder steel is not uniform according to a powder analysis performed by Erasteel. However, the models are only valid for uniform particle sizes and they are therefore approximated to be uniform in the calculations. A value of the volume

change was needed in order to calculate the gas pressure throughout the CIP-process. The volume of the capsule was roughly approximated to change 10 % within the range of 0 - 2500 Bar and this assumption stems from the fact that the porosity is decreased with roughly 10% in the CIP process according to Erasteel, and it is therefore grossly assumed that the volume changes at the same rate.

The following graphs are the result of plotting the three ETC models for the temperature intervall 273 K≤ T ≤ 1500 K

(a) ETC model by J. Mo and H.

Ban (2017) plotted against the temperature.

(b) ETC versus temperature described by the ZBS model (1978).

(c) Kunii and Smith (1960) model for ETC versus temper- ature.

Figure 5: The three models describing ETC for packed powder versus temperature in the sintering process.

The porosity was constant at  = 0,1 and the particle diameter set to 372,45 µm.

ETC was also plotted against the cold isostatic pressure at 293 K, in order to see how ETC varies before the pre-heating process. Same models were plotted again but this time the porosity was changed with the pressures applied; 0.30 ≤  ≤ 0.20 in the interval of 1000 - 2500 Bar. The porosity interval is based on the porosities that Erasteel acquire before and after the CIP process that roughly gives a 10% decrease in porosity.

(a) J. Mo and H. Ban (2017) model for ETC versus applied compressive pressure.

(b) ZBS model (1978) describing the re- lationship between ETC and applied com- pressive pressure.

Figure 6: Two of the three models describing how ETC changes within the pressure range for the CIP process. The particle diameter was set constant to 372,45 µm, and the porosity ranged between 0.30 ≤  ≤ 0.20

In Figure6 only the models by J. Mo and H. Ban and ZBS were plotted because it gave a more realistic trend of how ETC changes against the applied pressure. Kunii’s and Smith’s (1960) model was not presented in this report because it gave an unrealistic trend opposite of what the other two models. The reason behind this is probably because this model is only valid in the range of 0.260 ≤  ≤ 0.476 meanwhile the porosity interval in the process is between 0.30 ≤  ≤ 0.20.

Lastly, Equations3and 11were plotted against the temperature in order to see how each contri- bution, k^s_e and k_e^rbehaves against temperature. Additionally, Equation 9were plotted to see how the gas conduction, k_e^g behaves compared to the other two contribution.

Figure 7: k^s_e and k_e^r from Sakatani’s model (2016). The equations are valid at vaccum, porosity is set to  = 0.1, surface fraction parameter is ξ = 0,4 and the emissivity changes from 0.5 ≤ r ≤ 0.64.

From Figure 7 one can see that the ETC contribution of solid, k_e^s, is almost linear when the temperature increases. This can be explained by observing that Equation3 does not depend on the temperature. Instead it is depended on the porosity and the contact radius, rc, which is strictly correlated to the applied compressive pressure. In contrast, the radiative parameter, k^r_e is however temperature dependent, which can be observed by Equation11. The modified gas conduction with respect to the Smoluchowski effect was not greatly impacted by the temperature. Equation9 is indirectly temperature dependent with respect to Knudsen number, Equation8, although it is not massively affected by the temperature which can be seen by the figure above.

3.4 Relationship between thermal and electrical conductivity

For metals the conduction of heat is partly attributed to the transfer of electrons which also carry electrical charge. Consequently, the ratio between the electronic contribution of the thermal con- ductivity has been found to be related to the electrical conductivity of a bulk metal and proportional to the temperature, T. This relation can be expressed by Wiedemann-Franz law (WFL),

ke

σe

= L0· T, (32)

where L0is the Sommerfeld value of the Lorenz number which is equal to 2.44 × 10⁻⁸W ΩK⁻²and is constant for low and high temperatures at which the heat and charge currents are carried by electrons.[36, 37] Deviations occur at finite temperatures and are attributed to two mechanisms, the phonon thermal conductivity and inelastic electron-phonon scattering. [38]

3.4.1 Deviations from Wiedemann-Franz law

WFL holds for bulk metals where the thermal and electrical conductivity depend on free electrons to transport energy. The relationship is expected to obey WFL when electrons primarily undergo elastic or elastic-enough scattering processes. [39] At high and very low temperatures, WFL is generally well obeyed, whereas intermediate temperatures lead to a failure of the law. Low tem- peratures are defined as temperatures below the Debye temperature, whereas high temperatures are above the Debye temperature. The deviation of L from the Sommerfeld value, L0, as a function of temperature over the Debye temperature can be described by Figure 3. [40]