Fernando Fernandes

Angeliki Santos

Olivier Kritikakou

Sentieys

Reliability evaluation of Convolutional Neural Network's basic operations on a RISC-V processor

Thanks to RISC-V open-source Instruction Set Architecture, researchers and developers can efficiently propose new solutions at a low-cost and low-power consumption. RISC-Vbased architectures can, then, be customized to run Machine Learning (ML) algorithms efficiently and be inserted on safety and mission-critical domains, where the execution must be reliable. However, a fault in the hardware resources can compromise the system's ability to operate correctly. Thus, it is necessary to characterize the ML applications' vulnerabilities on RISC-V processors, as it is not as thoroughly characterized as other accelerators. This work is the first to evaluate the neutroninduced error rate of Convolutional Neural Network (CNN) basic operations running on a RISC-V processor, GAP8. Our results show that executing the algorithm in parallel increases performance, and memory errors are the major contributors to the device error rate. We show that the error rate of the GAP8 unprotected memories is one order of magnitude higher than the CNN basic operations.

I. INTRODUCTION

Recent advances in Machine Learning (ML) algorithms, such as quantization, reduced precision training, and weight pruning, enabled tiny Convolutional Neural Networks (CNNs) models to be adopted in low-power consumption processors and accelerators, such as TPUs, tiny ARM CPUs, and RISC-V SoCs. Compared to other architectures, RISC-V processors have the advantage of implementing an open-source Instruction Set Architecture (ISA), allowing designers to propose new customized architectures with smaller non-recurring engineering costs. RISC-V processors are today adopted in several domains, including end-user applications [START_REF] Lu | Design and implementation of a smart home system based on the risc-v processor[END_REF], High-Performance Computing (HPC) [START_REF] Ficarelli | Meet monte cimone: Exploring risc-v high performance compute clusters[END_REF], and safety-critical applications [START_REF] Santos | A low-cost fault-tolerant risc-v processor for space systems[END_REF], [START_REF] Ruospo | On-line testing for autonomous systems driven by risc-v processor design verification[END_REF]. Such a market trend became a promising option for CNNbased safety-critical applications where power consumption, real-time execution, and reliability are mandatory.

Researchers have been overly focused on improving CNN's performance and power consumption on recent RISC-V architectures [START_REF] Flamand | Gap-8: A risc-v soc for ai at the edge of the iot[END_REF], [START_REF] Garofalo | Pulp-nn: accelerating quantized neural networks on parallel ultralow-power risc-v processors[END_REF]. However, for RISC-V processors to be employed on safety-critical applications, their reliability must be thoroughly characterized to define how faults can impact programs and prevent the system from fulfilling the mandatory constraints. Faults that disrupt a system's operation can be generated by different events, such as environmental perturbations, ionizing radiation, software errors, process, temperature, or voltage variations [START_REF] Laprie | Dependable computing and fault tolerance : Concepts and terminology[END_REF], [START_REF] Nicolaidis | Time redundancy based soft-error tolerance to rescue nanometer technologies[END_REF]. Ionizing radiation is particularly dangerous for safety-critical applications as it leads to very high error rates [START_REF] Baumann | Soft errors in advanced computer systems[END_REF]. Terrestrial neutrons can perturb the state of a transistor or change memory cells' values, leading to soft errors. Note that the device is not permanently damaged (a new operation or memory write will replace the faulty value), but the error may propagate to the output.

To evaluate a device radiation-induced error rate, researchers perform radiation experiments with CPUs [START_REF] Dávila | Identifying the most reliable collaborative workload distribution in heterogeneous devices[END_REF], GPUs [START_REF] Goncalves De Oliveira | Evaluation and mitigation of radiation-induced soft errors in graphics processing units[END_REF]- [START_REF] Ito | Analyzing due errors on gpus with neutron irradiation test and fault injection to control flow[END_REF], FPGAs [START_REF] Quinn | Radiationinduced multi-bit upsets in sram-based fpgas[END_REF], [START_REF] Lins | Register file criticality and compiler optimization effects on embedded microprocessor reliability[END_REF], and Tensor Processing Units [START_REF] Rech Junior | High energy and thermal neutron sensitivity of google tensor processing units[END_REF]. The error rate of soft RISC-V processors synthesized on FPGAs, or ASICs, has been measured when exposed to heavy ions and neutrons [START_REF] Wilson | Neutron radiation testing of fault tolerant risc-v soft processor on xilinx sram-based fpgas[END_REF]- [START_REF] Santos | Experimental evaluation of neutron-induced errors on a multicore risc-v platform[END_REF]. In this paper, we go above the stateof-the-art by not only evaluating the error rate of an ASIC RISC-V processor but also performing a detailed reliability evaluation of the most basic CNNs operations on a Parallel Ultra Low Power (PULP) RISC-V System on Chip (SoC), GAP8 from GreenWaves [START_REF] Flamand | Gap-8: A risc-v soc for ai at the edge of the iot[END_REF]. GAP8 includes an 8-cluster core that supports standard RISC-V instructions and a set of Single Instruction Multiple Data (SIMD) instructions. SIMD instructions are extremely interesting for CNNs as they can improve performance without increasing power consumption.

We expose the GAP8 on a neutron beam and measure its error rate while running the main CNNs operations (Convolution, Fully connected Linear, and Maxpool). We report findings from recently completed neutron beam testing, which represents a total of more than 130,000 years of operation in the terrestrial environment. As CNNs are both computationallyand memory-demanding and GAP8 does not have memory error protection, we evaluate the error rate and criticality of the main GAP8's memories. We discuss how memory errors can impact the main CNNs operations error rate.

The remainder of this paper is as follows. In the next Section, we present the background on radiation effects on electronic and RISC-V-based processors. Section III describes the methodology used to measure GAP8's realistic error rate. The experimental measured error rate and main results are presented in Section IV. Section V concludes the paper.

II. RADIATION EFFECTS ON RISC-V DEVICES

Terrestrial high-energy neutrons that interact with the hardware may generate soft errors in the system. The striking particle can generate single or multiple-bit flips on a memory resource, such as caches, registers, or buffers, or even corrupt the value of a functional unit inside a processing core. If the value is used as part of the algorithm computation, the incorrect value will be propagated by the code running on the device. At the application level, the failure may manifest as the following outcomes: [START_REF] Lu | Design and implementation of a smart home system based on the risc-v processor[END_REF] No effect on the program Prior works have studied the reliability of RISC-V cores synthesized on FPGAs with neutrons and heavy ions experiments [START_REF] Wilson | Neutron radiation testing of fault tolerant risc-v soft processor on xilinx sram-based fpgas[END_REF]- [START_REF] Santos | Characterization of a risc-v system-onchip under neutron radiation[END_REF]. As many RISC-V processors share an open-source concept, developers can modify the architecture to apply Full or Partial Modular Redundancy to increase the processor reliability and evaluate the hardened architecture on beam experiments [START_REF] Wilson | Neutron radiation testing of fault tolerant risc-v soft processor on xilinx sram-based fpgas[END_REF], [START_REF] Wilson | Neutron radiation testing of risc-v tmr soft processors on sram-based fpgas[END_REF]. Recently, a commercial RISC-V processor's error rate and criticality have been characterized on a neutron beam experiment to determine the impact of neutron-induced faults on a small set of applications [START_REF] Santos | Experimental evaluation of neutron-induced errors on a multicore risc-v platform[END_REF]. However, none of the prior works have focused the analysis on crucial ML algorithms for CNNs. In this paper, we point to two significant contributions: [START_REF] Lu | Design and implementation of a smart home system based on the risc-v processor[END_REF] We evaluate the most basic CNN operations running on an ASIC commercial multicore RISC-V platform (GAP8) on a neutron beam and extract the realistic cross-section; (2) We measure the memory cross-section and fault model of the main GAP8 memories and correlate the results with the cross-section observed on CNNs operations. This analysis provides valuable insights into the contributions of memory errors in the application error rate running on commercial RISC-V processors.

III. METRICS AND EVALUATION METHODOLOGY

In this Section, we describe the beam experiments methodology for measuring GAP8 RISC-V SoC's cross-section, the codes we evaluate, and the metrics we use to support the evaluation in the results section.

A. Device under test and evaluated codes

Device Under Test (DUT): For the beam experiments, we consider a commercial RISC-V SoC named GAP8. GAP8 is a multicore RISC-V platform from GreenWaves technologies based on PULP architecture. GAP8 is built with 55nm TSMC 55LP CMOS technology, and it consists of a cluster of 8 RISC-V cores connected by a Logarithmic Interconnect and a RISC-V Fabric Controler (FC) processor to manage the cluster. The FC core operates at 250MHz, while each cluster core operates at 175MHz. The FC core also has 16KB of data and 1KB of instruction cache. Both the FC and the cluster can access an L1 memory of 64KB and an L2 memory of 512 KB. None of the SoC memories have an Error Correction Code (ECC) nor protection against Single Events Upsets. GAP8 supports integer and fixed-point arithmetic. During our experiments, we set the voltage and frequencies of GAP8 SoC to the maximum optimal values according to the GAP8 manual (1.2 volts with 250MHz for the FC and 175MHz for the cluster). This configuration set has a power consumption of 75mW [START_REF] Flamand | Gap-8: A risc-v soc for ai at the edge of the iot[END_REF].

Evaluated codes: We focus our analysis on the reliability of the core operations of CNNs. We perform two types of experiments where we characterize the memories and the most common operations in a CNN.

1) Memory microbenchmark:

As CNNs are known for being resource-demanding in terms of memory and computational resources, and as GAP8's main memories do not have ECC, we investigate the GAP8's memory cross-section and correlate it with the results for CNN operations. We design a microbenchmark to measure L1 and L2 memories cross-section. Figure 1 depicts a behavioral code for the memory microbenchmark. The code consists of a loop that runs while no memory errors are observed (line 2). The L1 or L2 memories of GAP8 are populated with a specific pattern (chunks of 4 bytes with 0xAA) (lines 4 to 6). After the writing, the code sleeps for 1s (line 7). Finally, the microbenchmark reads all the memory values and logs on the host if there are errors (lines 9 to 16). The memory microbenchmarks use the maximum amount of memory possible, 87.5% of L1 SoC memory and 96.8% of L2 SoC memory.

2) Common CNN operations:

-Convolution layer is the main operation performed on a CNN. The selected code consists of a 5 × 5 filter convoluted into a matrix of 112 × 112. The Convolution layers also have the characteristics to be the most resource-demanding procedure of a CNN. Consequently, convolution layers have been demonstrated to be one of the most critical parts of CNNs [START_REF] Li | Understanding error propagation in deep learning neural network (dnn) accelerators and applications[END_REF]- [START_REF] Libano | How reduced data precision and degree of parallelism impact the reliability of convolutional neural networks on fpgas[END_REF]. As discussed in Section IV, the convolution operation has the highest error rate.

-Fully connected Linear layer is the class of algorithm that performs the last step of the inference on a CNN. The standard -Maxpool layer is a specific ML algorithm explicitly created to avoid over-fitting on CNNs. Maxpool is placed after a certain number of layers to reduce the amount of data by filtering the values based on a filter block. We selected a maxpool layer that applies a 2 × 2 filter to a matrix of size 112 × 112. For each 2 × 2 block, only the highest value is propagated. As proposed by prior works [START_REF] Li | Understanding error propagation in deep learning neural network (dnn) accelerators and applications[END_REF]- [START_REF] Libano | How reduced data precision and degree of parallelism impact the reliability of convolutional neural networks on fpgas[END_REF], maxpool is expected to mask at least 3/4 of the errors.

The selected codes are specially developed for GAP8 by Flamand et. al. [START_REF] Flamand | Gap-8: A risc-v soc for ai at the edge of the iot[END_REF]. The codes are designed to extract the best performance from the SoC and to compare the different implementations of the same algorithm. Each configuration changes regarding the number of cores and the usage or not of special SIMD vector instructions. In this work, due to beam time limitations, we select two configurations for each code, i.e., the least performance (using only one core of the GAP8) and the highest performance (using all eight cores and SIMD vector instructions).

B. Beam Experiment Setup

In order to effectively evaluate the reliability of GAP8, we expose it to a neutron beam and measure its crosssection. Similar to [START_REF] Flamand | Gap-8: A risc-v soc for ai at the edge of the iot[END_REF], [START_REF] Santos | Experimental evaluation of neutron-induced errors on a multicore risc-v platform[END_REF], we use the GreenWaves Software Development Toolkit (SDK) to build and run the experiments. Our experiments are performed at the ChipIR facility of the Rutherford Appleton Laboratory, UK. The available neutron flux was about 3.5 × 10 6 n/(cm 2 /s), ∼8 orders of magnitude higher than the terrestrial flux at sea level [START_REF] Jedec | Measurement and Reporting of Alpha Particle and Terrestrial Cosmic Ray-Induced Soft Errors in Semiconductor Devices[END_REF]. The facility delivers a beam of neutrons with a spectrum of energies that resembles the atmospheric neutron one [START_REF] Cazzaniga | Progress of the scientific commissioning of a fast neutron beamline for chip irradiation[END_REF]. As a metric to characterize the error rate, we calculate the cross-section by dividing the number of observed errors (#errors) by the received particle fluence (η), i.e., Equation 1. The fluence is obtained by multiplying the average neutron flux of the test facility (neutrons/(cm 2 • s) by the effective code execution time in seconds.

σ[cm 2 ] = #errors η (1)
The cross-section (cm 2 ) represents the circuit area that will generate an output error (SDC or DUE) if hit by a particle. The higher the number of computation resources, the higher the cross-section, and the higher the probability of an impinging particle generating an error.

Figure 2 shows an overview of the experiments. We created a software watchdog consisting of Python scripts that runs on the setup server computer outside the beam room. The watchdog controls GAP8 by monitoring, executing the programs, logging events, and recovering from device hangs. The program is killed and relaunched if it stops responding in a predefined interval. We set each code timeout individually depending on the code execution time, up to 3× the expected execution time. The code inside GAP8 executes the same kernel for a predetermined number of iterations. We set the maximum number of internal SoC iterations to 65KB for a good tradeoff between time wasted with communication and proper kernel executions. After each iteration inside the SoC, the output of the kernel is compared with a constant golden value. If there is a mismatch between both outputs, the main process will log the useful information and restart again. It is worth noting that errors only errors that happen on the kernel output are considered for the error rate analysis.

IV. GAP8 CROSS-SECTION

We start our analysis by presenting the error rate observed on the memory microbenchmark. It paves the evaluation for the CNNs-based operation codes. We then compare the crosssection of various CNN operations and the differences between sequential and parallel algorithms. All results (CNN common operations and memory benchmark) are reported with 95% confidence intervals considering a Poisson distribution.

A. Memory cross-section

Table II shows the L1 and L2 microbenchmarks SDC cross sections observed on the beam experiments. We present the total cross-section that considers the cross-section of the entire evaluated memory (based on the memory sizes from Table I) and the cross-section per byte.

The memory cross-section per byte is similar to values measured in prior works with similar technologies [START_REF] Dixit | The impact of new technology on soft error rates[END_REF], [START_REF] Xilinx | Device reliability report second half 2021[END_REF]. L1 cross-section per byte is 1.28×10 -14 and L2 cross-section TABLE II: Memory SDC Cross Section observed on the neutron beam experiments. L1 and L2 memories are evaluated.

Total cross-section

Byte cross-section L1 8.13 × 10 -9 ± 7.4 × 10 -10 1.28 × 10 -13 ± 1.2 × 10 -14 L2 4.21 × 10 -8 ± 3.9 × 10 -9

9.18 × 10 -14 ± 8.6 × 10 -15 per byte is 9.18×10 -15 . Although L1 and L2 are similar memories, the L2 cross-section per byte is 29% smaller than the L1 cross-section. This is because they are differently organized on the SoC and, consequently, have different latencies and bus connections. More precisely, L1 is a shared memory between the cluster, while L2 is a larger memory divided into 4 blocks.

Figure 3 depicts the Single-Bit Upsets (SBUs) and Multiple-Bit Upsets (MBUs) for both L1 and L2 memories. The results show that the SBUs on L1 and L2 memories are the most common events observed in the experiments, i.e., 76.4% and 65.3%, respectively. Contrarily, MBUs have a lower occurrence when compared to SBUs. For L1, 12.6% for 2 MBUs and 11.1% for 3 or more MBUs. For L2, 21.3% for 2 MBUs and 13.4% for 3 or more MBUs.

We observe different MBU patterns for the two GAP8 levels of memory. In 0.9% of the cases for L1 memory, we observe large MBUs with 10 to 18 bits. In 1.1% of the cases for L2, the MBUs have 7 or 9 bits. For L1 and L2, 18 and 9 are the maximum number of MBUs observed, respectively. GAP8 uses a Tightly-Coupled Data Memory (TCDM) to increase performance and save energy, and, as observed in prior works, the MBUs can be spatially random for some memories types (i.g., caches) depending on the cell organization and the access type [START_REF] Dixit | The impact of new technology on soft error rates[END_REF]. A single particle can, in fact, lead to corruption on many bits of a memory array.

While less frequent, MBUs are much harder to be masked on computation and can increase the fault propagation probability by at least 50% [START_REF] Chatzidimitriou | Multi-bit upsets vulnerability analysis of modern microprocessors[END_REF]. Additionally, it has been shown that the patterns of the MBUs can range in the number of bits flipped [START_REF] Dixit | The impact of new technology on soft error rates[END_REF], [START_REF] Suh | Macau: A markov model for reliability evaluations of caches under single-bit and multi-bit upsets[END_REF]. That is, the same particle may modify the value of an entire word (16 bits) in the memory. Based on the high cross-section and high criticality observed on the GAP8's memory reliability analysis, we expect memory errors significantly impact the overall code cross-section. In the next section, we show how the code's cross-section changes based on the code's memory usage. For all CNN operations, the SDC cross-sections of sequential algorithms are similar to parallel ones. The SDC rate differences of sequential and parallel versions are 9.2% and 20.1% for Convolution and Maxpool, respectively. Note that, even if the different implementations of the same algorithms have a significant performance difference (Table I the parallel version is 35% higher than the sequential one.

B. CNN operations cross-section

The parallel version of the linear Layer is performed entirely by vector instructions, stressing the functional units of GAP8, which leads to an increasing error rate for those units.

The SDC rate results show that the higher the memory resource usage, the higher the SDC rate. As the Convolutional layer is the most resource-demanding, it has the highest error rate. The same happens for Linear and Maxpool operations. Maxpool layers also can mask most of the faults, as 3/4 of the values are discarded as expected (details Section III).

On the contrary, the DUE cross-sections are much lower than the SDC ones. The SDC cross-section is on average 2.1 × 10 -9 , while the DUE cross-section is on average 3.4 × 10 -10 . DUEs are caused mainly by events unrelated to arithmetic calculations, e.g., illegal instructions, hangs due to a fault-induced infinity loop, fault-induced deadlocks, incorrect addresses for jump and branch instructions, and illegal memory accesses. Not surprisingly, the highest DUE cross section is the Maxpool operations (5.1 × 10 -10 ), which consists of multiple max instructions on memory blocks of the input.

Finally, the presented cross-sections and evaluations are essential to understand how faults impact a CNN layer on a RISC-V-based processor. Knowing the cross-section of each operation on a CNN allows us to calculate the overall crosssection on a CNN once the probability of the fault propagating is available, i.e., the Architectural/Program Vulnerability Factor. As the GAP8 is based on an open-source SoC (PULP RISC-V), it allows researchers to perform fault injection at different levels of abstraction, such as system and microarchitectural levels. In future works, we will demonstrate the impact of each CNN operation based on the fault propagation from fault injection and apply it to realistic CNN models.

V. CONCLUSIONS

We have discussed the cross-section of CNNs core operations executing on an ultra-low-power RISC-V SoC exposed to a beam of neutrons. For each CNN operation, we evaluated the sequential and parallel versions of the algorithms. Additionally, to investigate the impact of unprotected memories on the code's reliability, the error rate of the main memories of GAP8 RISC-V is evaluated. Although parallel and sequential versions of the CNNs operations have remarkable differences in performance, the error rate is still majorly driven by the error rate of the memories. We plan to deeply study fault propagation through CNN operations and realistic CNN models by performing system and microarchitectural level fault injection.

ACKNOWLEDGMENT

This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 899546 with the support of the Brittany Region and partially funded by ANR FASY (ANR-21-CE25-0008-01) and ANR RE-TRUSTING (ANR-21-CE24-0015-02). ChipIR provided neutron beam time (DOI 10.5286/ISIS.E.RB2220502). We acknowledge the researchers that helped with neutron experiments, Dr. Christopher Frost, Maria Kastriotou and Dr. Carlo Cazzaniga.

1 / 3 / 5 M 6 } 7 sleep f or 1s 8 / 13 / 18 Fig. 1 :

 13567813181 Fig. 1: Behavioral algorithm for L1 and L2 memory test.

Fig. 2 :

 2 Fig. 2: ChipIR setup overview. The GAP8 is exposed to the neutron beam. The communication is performed through JTAG/USB by a fast USB replicator. Outside the beam room, a software watchdog server monitors the events inside the SoC.

Figure 4

 4 Figure 4 depicts the experimentally measured SDC and DUE cross sections. The y-axis shows the cross-section, and the x-axis shows each combination of operations (Convolution, Linear, & Maxpool) by execution type (sequential & parallel).For all CNN operations, the SDC cross-sections of sequential algorithms are similar to parallel ones. The SDC rate differences of sequential and parallel versions are 9.2% and 20.1% for Convolution and Maxpool, respectively. Note that, even if the different implementations of the same algorithms have a significant performance difference (TableI), the memory error rate significantly contributes to the error rate. The highest difference comes from the Linear operation, in which

Fig. 3 :

 3 Fig. 3: Distribution of observed Single-Bit Upsets (SBUs) and Multiple-Bit Upsets (MBUs) on L1 and L2 microbenchmarks.

Fig. 4 :

 4 Fig. 4: Experimentally measured Silent Data Corruption (SDC) and Detectable Unrecoverable Errors (DUE) Cross Sections.

TABLE I :

 I Characteristics for the evaluated microbenchmarks on GAP8 RISCV (memory and CNN).

		SoC usage	Cycles [KB]	Mem size
	Linear	Sequential Parallel+Vector	57.6 2.2	1024x16 bytes
	Maxpool	Sequential Parallel+Vector	25.7 3.8	112x112 bytes by 2x2/2 filter
	Convolution	Sequential Parallel+Vector	913.4 22.8	112x112 bytes by 5x5 filter
	Mem. L1	Sequential	-	62KB
	Mem. L2	Sequential	-	448KB
	output: The fault is masked, the program output is not affected
	or the circuit functionality is not modified. (2) Silent Data
	Corruption (SDC): The program finishes, but the output is not
	correct, and neither flag nor indication is raised. (3) Detected
	Unrecoverable Error (DUE): The system stops working,
	forcing it to be rebooted or power cycled. A DUE can result
	from uncorrectable memory events, crashes, or an error that
	generates an infinite loop (program hangs).

  ), the memory error rate significantly contributes to the error rate. The highest difference comes from the Linear operation, in which

	Single Bit Upset	2-Multiple Bit Upsets	≥3-Multiple Bit Upsets
	L1		L2
	11.1%		13.4%
	12.6%
		21.3%
			65.3%
		76.4%

Fernando Fernandes dos Santos, Angeliki Kritikakou, and Olivier Sentieys are with Univ. Rennes, INRIA, Rennes, France Contact e-mail: {fernando.fernandes-dos-santos, angeliki.kritikakou, olivier.sentieys}@inria.fr


Jordan Letessier

Aïmen E Gheribi

Jean-Mathieu Vanson

Christelle Duguay

Fabrice Rigollet

Nathalie Ehret

Jerôme Vicente

Jean-Laurent Gardarein

Thermal transport-porosity-microstructural characteristics: unpicking the relationship in ultra-porous 𝛼-Al 2 O 3 powder

Keywords:

grain size measurement transient hot plate thermal quadrupole method inverse problems X-ray tomography scanning electron microscopy theoretical model percolation theory

A B S T R A C T

This study deals with the effective thermal conductivity of porous and ultra-porous unconnected solid phase media composed of 𝛼-alumina in air at low temperature (<500K), and at ambient pressure. A multi-scale model is developed, starting from the quantum scale, to describe the conductivity of the single crystal, passing through a microscopic scale to describe the particles (grain size), a mesoscopic scale to take into account the intraparticle porosity and a macroscopic scale with the interparticle porosity. The model is based on percolation theory to describe the strong decrease in conductivity with porosity in this type of medium. This translates into the breaking of contacts between particles past a certain porosity, the critical porosity 𝑝 𝑐 . This original approach makes it possible to consider this microstructural factor, which accounts for the thermal contacts between particles. These parameters are theoretically deduced or measured for certain samples using an X-ray micro-tomograph (µCT). Combining all these parameters, our model is able to predict 99.7% of the thermal conductivity decay from the pure crystal to the particle bed. Our model proves to be more consistent than current existing models on the same subject. Its validation range extends over the entire porosity range from 0 to 1.

Introduction

The Loss of Coolant Accident (LOCA) is one of the postulated accidents for Pressurized Water Reactors (PWR). In such a case, the nuclear pellets can fragment and relocate in its cladding, leading to a degradation of the thermal properties of the fuel rod [START_REF] Oberlaender | Overview of Halden reactor LOCA experiments (with emphasis on fuel fragmentation) and plans[END_REF], [START_REF] Boegli | Measured effective thermal conductivity of uranium oxide powder in various gases and gas mixtures[END_REF]). The material is then similar to a granular media with an unconnected solid phase (such as a bed of powder with several particle sizes or packed bed). To understand the behavior of the whole system, it is fundamental to be able to predict the evolution of the thermal properties of this new material. Moreover, as measurements are more expensive and more difficult to carry out in a nuclear environment, we use model materials to perform our parametric study of powder properties. The wide-ranging morphological and granulometric characteristics of 𝛼-alumina make it an adequate non-radioactive model material for investigation [START_REF] Boegli | Measured effective thermal conductivity of uranium oxide powder in various gases and gas mixtures[END_REF], [START_REF] Giraud | Investigation of a granular Bond number based rheological model for polydispersed particulate systems[END_REF]). Furthermore, the study of this material also has advantages for other types of applications.

In fact, the arising interest in ultra-porous alumina materials (i.e. with a porosity greater than 65%) in fields linked to insulation, catalysts, filters or antennae is mainly due to their interesting thermal and mechanical properties: very low density, very resistant to heat, very insulating and possessing large specific surfaces (several m 2 /g). They are therefore used as refractory materials, in the manufacture of furnaces and as insulators [START_REF] Huang | Thermal conductivity measurements of alumina powders and molded Min-K in vacuum[END_REF]), as a catalyst support [START_REF] Van Norman | Using atomic layer deposited tungsten to increase thermal conductivity of a packed bed[END_REF]), or in applications related to filtering. They have also shown their usefulness in new applications such as plane antennae, thanks to their electromagnetic properties [START_REF] Stepanenko | Ultra-porous alumina for microwave planar antennas[END_REF]). Moreover, alumina is a cheap material, which makes it a ceramic of choice for industrial applications. Within these ultra-porous materials, a distinction can be made between materials where the solid phase is connected and those where it is not. In the first category, we find foams [START_REF] Sciamanna | Mechanical properties and thermal conductivity of porous alumina ceramics obtained from particle stabilized foams[END_REF], [START_REF] Shimizu | Thermal conductivity of high porosity alumina refractory bricks made by a slurry gelation and foaming method[END_REF]), sintered ceramic fibres [START_REF] Sun | Calculation of Percolation Thresholds in High Dimensions for FCC, BCC and Diamond Lattices[END_REF]) and materials manufactured by combining processes such as "the gel-casting process with the poreforming agent technique", which are processes normally suitable for manufacturing denser materials with porosities less than 65% [START_REF] Li | Effect of starch addition on microstructure and properties of highly porous alumina ceramics[END_REF]). The thermal conductivity of this type of material is rather well described by EMT (Effective Medium Theory) or Maxwell-Hashin-Shtrikman models [START_REF] Bruggeman | Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielektrizitätskonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen[END_REF], [START_REF] Hashin | A Variational Approach to the Theory of the Effective Magnetic Permeability of Multiphase Materials[END_REF]). At equal porosities, connected solid phase materials can have an effective thermal conductivity that is an order of magnitude higher than that of unconnected solid phase alumina. In this study, we therefore focus on the later type of material in order to develop a thermal conductivity model that differs from the usual models used.

In this study, we focus on the description of heat transfer in unconnected solid phase 𝛼-alumina with a model based on percolation theory and using microstructural parameters. The thermal conductivity of porous materials depends on numerous microstructural parameters such as size and distribution size of pores, grains and particles. The presence or absence of connections between pores and how contacts between particles are defined are also important details to know. However, knowing all these parameters is sometimes impossible or requires a lot of resources. According to the present approach the thermal conductivity of bed of spherical particle can be predicted as a function of porosity via a functional which consider four key microstructural parameters: (i) interparticle porosity (𝑝 𝑖𝑛𝑡𝑒𝑟 ), (ii) intraparticle porosity (𝑝 𝑖𝑛𝑡𝑟𝑎 ), (iii) critical porosity (𝑝 𝑐 ) and (iv) mean grain size (⟨𝑑⟩). The interparticle porosity can be experimentally determined via X-ray micro-tomograph (µCT) which has been done in this work for some of alumina powder samples. The intraparticle porosity is deduced via the relationship between interparticle porosity and the total powder bed porosity (𝑝 𝑒 ) given by:

(1 -𝑝 𝑒 ) = (1 -𝑝 𝑖𝑛𝑡𝑒𝑟 )(1 -𝑝 𝑖𝑛𝑡𝑟𝑎 ) (1)
Where the total powder bed porosity is given by:

𝑝 𝑒 = 1 -𝜌 𝑒 ∕𝜌 𝑡ℎ𝑒𝑜 (2)
Where 𝜌 𝑒 is the effective density of the powder bed and 𝜌 𝑡ℎ𝑒𝑜 is the theoretical density of the solid (𝜌 𝑡ℎ𝑒𝑜 = 3.97 g.cm -3 for alumina). Grain size is deduced from supplier data and grain size measurements. The critical porosity is deduced from the results given by the model and from other studies on the subject, described below.

The effective thermal conductivity of particle beds has been thoroughly studied in the literature. Many of these studies are based on a simplified representation of an elementary mesh of the studied porous system, on which the heat transfer mechanisms are composed in parallel or in series [START_REF] Hadley | Thermal conductivity of packed metal powders[END_REF], [START_REF] Masamune | Thermal Conductivity of Beds of Spherical Particles[END_REF], [START_REF] Yagi | Studies on effective thermal conductivities in packed beds[END_REF], [START_REF] Zehner | Wärmeleitfähigkeit von Schüttungen bei mäßigen Temperaturen[END_REF]). Some of them are based on the effective medium theory (EMT), making it possible to obtain the average values of the physical characteristics of the medium [START_REF] Bruggeman | Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielektrizitätskonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen[END_REF], [START_REF] Polamuri | Experimental determination of effective thermal conductivity of granular material by using a cylindrical heat exchanger[END_REF]). All these models do not consider the real microstructure. This is a problem for describing and understanding the critical behavior of thermal conductivity when porosity increases. The universal value of these models is therefore questionable. Our approach, based on percolation theory to model the critical thermal conductivity behavior as a function of porosity of porous materials with unconnected solid phase, claims to be more universal. Models using percolation theory have already been studied in the literature. For example, [START_REF] Mclachlan | An equation for the conductivity of binary mixtures with anisotropic grain structures[END_REF] proposes the coupling of the (EMT) model of [START_REF] Bruggeman | Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielektrizitätskonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen[END_REF] with the theory of percolation to model the electrical conductivity in binary mixtures. Later, this model was used to describe the thermal conductivity of sintered nickel [START_REF] Deprez | The measurement and comparative analysis of the electrical and thermal conductivities and permeability of sintered nickel[END_REF]). [START_REF] Ghanbarian | Thermal conductivity in porous media: Percolation-based effective-medium approximation[END_REF]) use this model and develop it to model the thermal conductivity in sediments, rocks and soil as a function of water saturation. In [START_REF] Gheribi | Evidence of second order transition induced by the porosity in the thermal conductivity of sintered metals[END_REF] and Gheribi et al. (2015) the authors also use the percolation theory to model the thermal conductivity in sintered metals. The critical behavior of the thermal conductivity as a function of porosity is described and interpreted as a second order transition of the continuous phase of the metal. We use this thermodynamic interpretation to establish a model for the thermal conductivity of alumina powders in air. In the present study, the measurements (made on our powders as well as on other ones in literature) cover a wide range of values of mean particle diameters (from 11 nm to 0.4 mm) and the porosity values are quite high (from 42% to 83%), providing different scales of pore sizes in the samples. Our measurements were carried out by a bench using the unsteady hot plane method. The heat transfer mechanisms considered in our study are limited to conduction in the solid phase and in the gas phase. Heat transfer by radiation is neglected because the study temperatures are low (<500 K). Convection will also be neglected, as the pore size of the studied samples is small enough to guarantee a Grashof number below 1000. We are restricting ourselves to the study of the alumina pore system in air.

Experimental methods: Thermal conductivity measurement

Thermophysical properties have been measured in our laboratory using a "home-made" experimental setup based on the hot plane method (figure 1) [START_REF] Maillet | Analysis of the estimation error in a parsimonious temperature-temperature characterization technique[END_REF], [START_REF] Lahoori | Measurement of the thermal properties of unsaturated compacted soil by the transfer function estimation method[END_REF]). This experimental bench has been used previously at our laboratory to characterize the thermal properties of different grades of concrete whose composition included different proportions of recycled materials [START_REF] Sosoi | Experimental Investigation on Mechanical and Thermal Properties of Concrete Using Waste Materials as an Aggregate Substitution[END_REF]). A heat flux is imposed on the front face sample during approximately 2 min and the temperatures are measured at both sides. The hot plane is powered by a DC generator, with a power of about 14 W. The temperature measured at the front face is imposed as the Dirichlet condition in the thermal model. This calculation gives the temperature on the rear face. It is then compared with the measured rear face temperature and the thermal parameters are estimated by minimizing the difference between these two signals. The use of the measured front face temperature as the Dirichlet condition eliminates the need to know exactly the time shape and the exact value of the flux delivered by the power supply. It is this temperature evolution recorded at the front face (𝜃 𝑚𝑒𝑎𝑠 (0, 𝑡)) that serves as the input excitation for the model (8), it is called "the transmittance-based model" [START_REF] Maillet | Analysis of the estimation error in a parsimonious temperature-temperature characterization technique[END_REF], [START_REF] Lahoori | Measurement of the thermal properties of unsaturated compacted soil by the transfer function estimation method[END_REF]). The use of thermal grease between the thermocouples and the base plates limits the thermal contact resistance. A fibrous insulator is placed around the device to promote 1D transfer between the two faces. The 3 thermograms measured on the front and rear faces allow us to ensure the uniformity of the temperature on each face (figure 2). Once this verification is established, the thermogram measured by the central thermocouple is injected into the model. The standard deviation of the measurement noise is approximately 0.01 • C for a heating of 2 • C on the rear face and 7 • C on the front face. We model the powder bed as a homogeneous medium in perfect contact with a semi-infinite medium on the rear face. Using the thermal quadrupole formalism [START_REF] Maillet | Solving the heat equation through integral transforms[END_REF]) to relate, in Laplace space, the heating and fluxes between the front face ( θ(0, 𝑝 𝐿𝑝 ), φ(0, 𝑝 𝐿𝑝 )) and the rear face ( θ(𝑒 + 𝑒 𝑠 , 𝑝 𝐿𝑝 ), φ(𝑒 + 𝑒 𝑠 , 𝑝 𝐿𝑝 )), the thermal model equation can be written as:

( θ(0, 𝑝 𝐿𝑝 ) φ(0, 𝑝 𝐿𝑝 ) ) = 𝐶𝐵 ( θ(𝑒 + 𝑒 𝑠 , 𝑝 𝐿𝑝 ) φ(𝑒 + 𝑒 𝑠 , 𝑝 𝐿𝑝 ) ) (3) With, 𝐵 = ( 1 0 𝑥 4 𝑝 𝐿𝑝 1 ) and, ( 4
) 𝐶 = ( cosh ( √ 𝑥 1 𝑝 𝐿𝑝 ) 1 𝑥 2 √ 𝑝 𝐿𝑝 sinh ( √ 𝑥 1 𝑝 𝐿𝑝 ) 𝑥 2 √ 𝑝 𝐿𝑝 sinh ( √ 𝑥 1 𝑝 𝐿𝑝 ) cosh ( √ 𝑥 1 𝑝 𝐿𝑝 ) ) (5)
𝑥 1 = 𝑒 2 ∕𝛼 and, 𝑥 2 = 𝑏 with, 𝑝 𝐿𝑝 the Laplace variable (s -1 )

The matrix 𝐵 represents a copper plate of thickness 𝑒 𝑠 . Its small thickness and high conductivity allow it to be modelled as a purely capacitive layer. Matrix 𝐶 represents the powder bed as an equivalent homogeneous material of thickness 𝑒, diffusivity 𝛼 and effusivity 𝑏. The boundary condition on the PVC block, assumed to be semi-infinite (characteristic diffusion time 𝑡 𝑃 𝑉 𝐶 𝑑𝑖𝑓 𝑓 ≈ 10 4 s for a measurement time of about 800 s), allows to relate the flux and the rear face heating by:

φ(𝑒 + 𝑒 𝑠 , 𝑝 𝐿𝑝 ) = 𝑥 3 √ 𝑝 𝐿𝑝 θ(𝑒 + 𝑒 𝑠 , 𝑝 𝐿𝑝 ) (6)
By expanding equation (3) and applying it for a unit Laplace heating on the front face θ(0, 𝑝 𝐿𝑝 ) = 1, the resulting Laplace heating on the back face corresponds to the transmittance of the system θ𝑖𝑚𝑝 (𝑒 + 𝑒 𝑠 , 𝑝 𝐿𝑝 ):

θ𝑖𝑚𝑝 (𝑒 + 𝑒 𝑠 , 𝑝 𝐿𝑝 ) = [ cosh (√ 𝑥 1 𝑝 𝐿𝑝 ) + ( 𝑥 4 𝑥 2 √ 𝑝 𝐿𝑝 + 𝑥 3 𝑥 2 ) sinh (√ 𝑥 1 𝑝 𝐿𝑝 ) ] -1 (7)
The inverse Laplace transform (performed with De Hoogs algorithm [START_REF] De Hoog | An improved method for numerical inversion of Laplace transforms[END_REF])) of the system transmittance ( θ𝑖𝑚𝑝 (𝑒 + 𝑒𝑠, 𝑝 𝐿𝑝 )) then provides the impulse response of the modelled system.

𝜃 𝑚𝑜𝑑 (𝑒 + 𝑒 𝑠 , 𝑡) =  -1 { θ𝑖𝑚𝑝 (𝑒 + 𝑒 𝑠 , 𝑝 𝐿𝑝 )} ⊛ 𝜃 𝑚𝑒𝑎𝑠 (0, 𝑡) (8)
The modelled rear face heating versus time 𝜃 𝑚𝑜𝑑 (𝑒 + 𝑒 𝑠 , 𝑡), is then calculated by time convolution (denoted ⊛) between this impulse response and the measured front face thermogram (eq: ( 8)). In equation ( 7), it can be seen that the model transmittance of the system depends on only 4 parameters: 𝑥 1 [s] and𝑥 2 [Jm -2 K -1 s -1∕2 ] are the apparent diffusion time and the equivalent effusivity of the particle bed respectively, 𝑥 3 [Jm -2 K -1 s -1∕2 ] is the effusivity of the PVC block and 𝑥 4 [JK -1 m -2 ] is the surface capacitance of a copper sole. The orders of magnitude for the values of these parameters are:

𝑥 1 ≈ 500 s, 𝑥 2 ≈ 400 Jm -2 K -1 s -1∕2 , 𝑥 3 ≈ 518 Jm -2 K -1 s -1∕2 and 𝑥 4 ≈ 3400 J K -1 m -2 .
A sensitivity study shows that the surface capacitance of the soles 𝑥 4 is a low-sensitivity parameter of the model, so it will be fixed at its nominal value. In addition, the effusivity of the rear PVC sample holder has been measured by our device (with other sample holders) and by two subcontractors on the other hand, so it will also be fixed. Only the two parameters 𝑥 1 and 𝑥 2 remain to be estimated. Solving the problem by the inverse method involves adjusting 𝑥 1 and 𝑥 2 to minimize the RSS (Residual Sum of Squares) that measures the discrepancy between the measured 𝜃 𝑚𝑒𝑎𝑠 (𝑒 + 𝑒 𝑠 , 𝑡) and modelled 𝜃 𝑚𝑜𝑑 (𝑒 + 𝑒 𝑠 , 𝑡) rear face heating. To solve this Ordinary Least Squares problem, we use the Gauss-Newton algorithm. The figure 2 shows the measured thermograms (front face and rear face) as well as the optimal modelled thermogram on rear face, computed with the identified parameters. The residuals between measured and modelled thermograms are also shown and their statistics (mean and standard deviation) are close to those of measurement noise, which gives confidence to the model. In addition, some of the measurements were verified with other experimental setups (Transient Hot Bridge by Linseis and hot disc method, both methods are suitable for measurements on powders [START_REF] Lager | Thermal conductivity measurements of calcium oxalate monohydrate as thermochemical heat storage material[END_REF], [START_REF] Predeep | Effective thermal conductivity and thermal diffusivity of some rare earth oxides[END_REF]). The values of 𝑥 1 and 𝑥 2 obtained after minimization give us 𝛼 and 𝑏 and we can deduce the equivalent bed conductivity

𝜆 = 𝑥 2 √ 𝑒 2 ∕𝑥 1 = 𝑏 √ 𝛼.
For each estimated parameter, an uncertainty is calculated, corresponding to the sum of errors due to measurement noise and to the propagation of uncertainties of the fixed parameters assumed to be known [START_REF] Rigollet | Front Face Thermal Characterization of Materials by a Photothermal Pulse Technique[END_REF], [START_REF] Aster | Parameter estimation and inverse problems[END_REF]), which is estimated to be 10%. It can be seen that the overall uncertainty for a parameter corresponds essentially to the propagation of the uncertainties of the fixed parameters.

Theoretical description of the thermal transports within ultra-porous materials

The thermal conductivity of powder beds depends on thermodynamic parameters such as temperature, and microstructure-related parameters such as the size, size distribution of pores and grains and particles. The presence of phase compositions due to impurities or mixtures of chemical compounds can also significantly affect the effective thermal conductivity value of powder beds. These parameters variations can lead to an order of magnitude variation in the thermal conductivity of the same dense material. Measuring each microstructural parameter and knowing the variations of the conductivity as a function of these parameters is very complex. We assume that the powders studied consist of porous particles composed of grains (figure 3). We have therefore defined a limited set of parameters which are, according to the literature, the most impactful [START_REF] Yagi | Studies on effective thermal conductivities in packed beds[END_REF], [START_REF] Francl | Thermal conductivity: IX, experimental investigation of effect of porosity on thermal conductivity[END_REF], Gheribi and Chartrand (2015)). Firstly, we have chosen to distinguish intraparticle 𝑝 𝑖𝑛𝑡𝑟𝑎 from interparticle 𝑝 𝑖𝑛𝑡𝑒𝑟 porosity. Most studies do not make this distinction and thus omit an important feature of the microstructure of the media, especially in the case of ultra-porous powders where high porosity cannot be explained without considering these two scales of porosity. Our model is a multi-scale description of the effective thermal conductivity of a powder bed. Starting from the thermal conductivities of the microscopic elements that make up a powder bed, we can trace the thermal conductivity of the whole at a macroscopic scale. The conductivity can be described as follows:

𝜆 𝑚𝑜𝑑 𝑒 (𝑇 , ⟨𝑑⟩, 𝑝 𝑖𝑛𝑡𝑟𝑎 , 𝑝 𝑖𝑛𝑡𝑒𝑟 , 𝑝 𝑐 , 𝑛 𝑐 ) = 𝜆 𝑠𝑐 (𝑇 )𝑀(𝑇 , ⟨𝑑⟩, 𝑝 𝑖𝑛𝑡𝑟𝑎 ) ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ 𝜆 𝑝 𝑁(𝑝 𝑖𝑛𝑡𝑒𝑟 , 𝑝 𝑐 , 𝑛 𝑐 ) ⏟⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏟ macrostructural factor + 𝑝 𝑒 𝜆 𝑎𝑖𝑟 (𝑇 ) ⏟⏞⏟⏞⏟ contribution of air (9)
The conductivity of the medium is considered as a parallel model, the first term corresponds to heat transfer in the solid phase (considering the contacts between particles) and the second term to heat transfer in the gas phase. The function 𝑀 reflects the reduction of the thermal conductivity of a particle 𝜆 𝑝 with respect to that of the single crystal 𝜆 𝑠𝑐 , as a function of temperature 𝑇 , the average grain size ⟨𝑑⟩ and the intraparticle porosity 𝑝 𝑖𝑛𝑡𝑟𝑎 . Usually, ceramics have a polycrystalline solid phase and therefore their conductivity is different from that of the single crystal (𝜆 𝑠𝑐 ). The difference between the conductivity of a polycrystal and that of a single crystal depends directly on its grain size. The smaller the grain size is, the greater the specific amount of grain boundaries is, which increases the thermal resistance and degrades the thermal conductivity. Thus, the conductivity of the solid phase of a dense powder particle 𝜆 𝑔 can be described as in equation ( 10).

𝜆 𝑔 (𝑇

, ⟨𝑑⟩) = 𝜆 𝑠𝑐 (𝑇 ) [ 1 -𝑢 arctan (1∕𝑢) ] ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ 𝑔(𝑇 ,⟨𝑑⟩) ( 10
)
Where 𝑔 is a function that considers this degradation as a function of the average grain size (⟨𝑑⟩, eq:(10), Gheribi and Chartrand (2015)). 𝑢 2 = 𝜎(𝑇 )∕⟨𝑑⟩, where 𝜎 is a characteristic length of the material which represents the reduction of the mean free path of phonons, due to grain boundaries. 𝜆 𝑠𝑐 represents the thermal conductivity of the pure crystal (W.(mK) -1 ). Its expression as a function of temperature (eq: ( 11) valid from room temperature to melting point) has been determined in a purely theoretical manner [START_REF] Gheribi | Application of the CALPHAD method to predict the thermal conductivity in dielectric and semiconductor crystals[END_REF], [START_REF] Julian | Theory of heat conduction in rare-gas crystals[END_REF]). It can be written as:

𝜆 𝑠𝑐 (𝑇 ) = 𝐴. M𝜃 3 𝐷 𝛿(𝑇 ) 𝛾 2 𝑇 𝑛 2∕3 (11)
where M is the average atomic mass (in amu), 𝛿 3 is the volume per atom (𝛿 in Å), 𝜃 𝐷 is the Debye temperature in K, 𝑛 is the number of atoms per primitive cell and the constant 𝐴 = (2.43 × 10 -8 )∕(1 -0.514∕𝛾 + 0.228∕𝛾 2 ).

The thermal conductivity of a dense particle is therefore at most equal to 𝜆 𝑔 = 𝜆 𝑠𝑐 𝑔(𝑑 𝑝 ), with 𝑑 𝑝 being the diameter of the particle. Depending on the manufacturing process, the powder particles can be dense or porous. For example, in the case of aggregates, the particles have an internal porosity which we call 𝑝 𝑖𝑛𝑡𝑟𝑎 . Assuming that intraparticle porosity is homogeneously distributed, we describe the conductivity 𝜆 𝑝 of a porous particle as in equation ( 12). This difference between intraparticle and interparticle porosity (𝑝 𝑖𝑛𝑡𝑒𝑟 ) allows us to consider one aspect of the microstructure of the particules themselves, beyond the macroscopic porosity associated to their arrangement into the powder. +𝑝 𝑒 𝜆 𝑎𝑖𝑟 (𝑇 )

𝜆 𝑝 (𝑇 , ⟨𝑑⟩, 𝑝 𝑖𝑛𝑡𝑟𝑎 ) = 𝜆 𝑠𝑐 (𝑇 ) 𝑔(𝑇 , ⟨𝑑⟩) (1 -𝑝 𝑖𝑛𝑡𝑟𝑎 ) ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ⏟ 𝑀(𝑇 ,
(13)
By multiplying equation ( 12) by (1 -𝑝 𝑖𝑛𝑡𝑒𝑟 ), we obtain the maximum thermal conductivity of the solid phase of the medium. However, the conductivity of an unconnected solid phase porous medium is often far from this value. This is due to a second order transition of the conductivity with the porosity. This effect is expressed by the factor 𝑓 𝑐 (𝑝 𝑐 , 𝑝 𝑖𝑛𝑡𝑒𝑟 , 𝑛 𝑐 ), corresponding to the critical function that will be detailed later. More over, in order to take into account the heat transfer in air, we consider it as a parallel mechanism to the transfer in the solid phase. The latter is equal to the effective porosity of the bed 𝑝 𝑒 , multiplied by the conductivity of air 𝜆 𝑎𝑖𝑟 . The expression of 𝜆 𝑎𝑖𝑟 as a function of 𝑇 is given by a third order polynomial fitted to the measurements of [START_REF] Lemmon | Viscosity and Thermal Conductivity Equations for Nitrogen, Oxygen, Argon, and Air[END_REF].

The thermal conductivity of an unconnected solid phase porous medium 𝜆 𝑚𝑜𝑑 𝑒 , is then written as in equation ( 13). Figure 3 gives an order of magnitude of the conductivity values according to the different scales of observation. The 𝑓 𝑐 function is a critical function inspired by the one used by [START_REF] Inden | Project Meeting Calphad V. Ch[END_REF] for second order ferromagnetic transitions. In our case, it gives information on the connection between the particles and translates the critical behavior of the diffusivity of the solid phase (𝛼 𝑠 ) as a function of porosity. 𝑓 𝑐 is defined as follows:

𝑓 𝑐 (𝑝 𝑐 , 𝑝 𝑖𝑛𝑡𝑒𝑟 , 𝑛 𝑐 ) = 1 + ∫ 𝑝 𝑖𝑛𝑡𝑒𝑟 0 𝜒(𝑝 𝑐 , 𝑝, 𝑛 𝑐 ) 𝑑𝑝 (14) 𝜒(𝜏, 𝑛 𝑐 ) = ⎧ ⎪ ⎨ ⎪ ⎩ 𝐴 ln ( 1+𝜏 3 1-𝜏 3 ) , 0 ≤ 𝑝 < 𝑝 𝑐 𝐴 ( ln ( 1+𝜏 -𝑛 𝑐 1-𝜏 -𝑛 𝑐 ) -ln ( 1+𝑝 𝑛 𝑐 𝑐 1-𝑝 𝑛 𝑐 𝑐 ) ) , 𝑝 𝑐 < 𝑝 ≤ 1 (15)
Note that 𝜏 = 𝑝∕𝑝 𝑐 . The function 𝜒 corresponds to the derivative of the diffusivity of the solid phase 𝛼 𝑠 with respect to the porosity 𝑝, 𝜒 = 𝜕𝛼 𝑠 ∕𝜕𝑝, it is normalized and its integral on correspond to the percolation threshold (critical porosity) and to a critical exponent of the function, which depends on the material used (we describe thereafter how we fixed it for 𝛼-alumina). The critical exponent for the first part of the 𝜒 function is fixed at 3, corresponding to a thermodynamic phase transition [START_REF] Inden | Project Meeting Calphad V. Ch[END_REF]). The percolation threshold (identified as critical porosity in the present study) depends on the composition of the filling gas (Air, He, N 2 , Ar) and its pressure and several microstructural parameters related with both porosity and particle: pore shape, pore size, pore distribution, inclusion shape, particle shape. In the present work, it is intended to describe the percolation effect upon the thermal transport from an effective point of view i.e the critical porosity is not only defined through the definition of the percolation threshold of a given geometrical configuration considering the particles coordinations (1 𝑠𝑡 nearest neighbor and beyond). If the interparticle porosity is lower than 𝑝 𝑐 , the effective thermal connections between the particles are numerous. If it is higher, the number of effective thermal contacts decreases sharply, the particles are no longer thermally connected to each other. The thermal resistance across particle-particle boundaries is the major factor in the dumping of the geometrical porosity threshold. Several experimental studies on the thermal transport within spherical particle-based sintered metal show a drastic decrease in thermal conductivity (diffusivity) around a porosity level lying between 0.11 and 0.24, with an average value of 0.16 and a relatively small standard deviation [START_REF] Gheribi | Evidence of second order transition induced by the porosity in the thermal conductivity of sintered metals[END_REF], Gheribi et al. (2015)). For instance, the critical porosity of AISI 316L stainless steel is 0.11, that of sintered copper is 0.16, and about 0.24 for sintered tungsten samples. These values are very close to site percolation thresholds of 3D systems. For site percolation in a 3D system with a coordination number of 8, 𝑝 𝑐 = 0.245 (Van Der Marck (1998)). By varying the coordination number to 14, i.e. taking second neighbors into account, the site percolation threshold decreases to 0.168 [START_REF] Jerauld | Percolation and conduction on the 3D Voronoi and regular networks: a second case study in topological disorder[END_REF]). An accurate value for the effective thermal coordination number is therefore one of the important parameters to know in order to best estimate the effective thermal conductivity of a powder bed.

Identification of the critical exponent

The value of the critical exponent 𝑛 𝑐 in equation ( 15) is linked to the slope of 𝑓 𝑐 in the neighborhood of 𝑝 𝑐 . The larger this exponent is, the faster 𝜒 tends towards infinity, and therefore the steeper is the decrease in 𝑓 𝑐 . It reflects the behavior of the thermal conductivity at the transition from the connected state of the particles to an unconnected state. The value of 𝑛 𝑐 is specific to the used material, in our case 𝛼-alumina. We can therefore, thanks to the various data in the literature listed in this paper, determine its value. To do this, we fixed all the parameters of the model presented in equation ( 13) with the values given in table 1. We have fixed the parameter 𝑝 𝑐 to 0.168. 𝑝 𝑖𝑛𝑡𝑒𝑟 varies between 0.33 and 0.42 when its value is not known. This range was chosen considering that the porosity of a bed of randomly arranged dense particles lies in this range [START_REF] Dullien | Porous media: fluid transport and pore structure[END_REF], [START_REF] Van Norman | Using atomic layer deposited tungsten to increase thermal conductivity of a packed bed[END_REF]). The value of 0.168 corresponds, as explained above, to the site percolation threshold of a cubic system centered on 14 neighbors. We calculate the optimal value of 𝑛 𝑐 to minimize the squared deviation between the measurement points and the model for each study, and then we take the average value of all the studies. The results are shown in figure 4a. We find that the average value of 𝑛 𝑐 lies between 4.48 and 5.56. We can see that the results are close for the different studies, except for [START_REF] Laubitz | Thermal conductivity of powders[END_REF] and [START_REF] Van Norman | Using atomic layer deposited tungsten to increase thermal conductivity of a packed bed[END_REF], having 𝑛 𝑐 values closer to 3. This difference may be due to different factors. For example, the arrangement may be different and the fact that 𝑝 𝑐 is fixed adds an uncertainty that is not considered in the vertical estimation bars. The interparticle porosity values mentioned by the authors may also be different from reality when measuring conductivities. By doing the opposite, i.e. by imposing the critical exponent 𝑛 𝑐 at 5.02, we looked at the optimal values of 𝑝 𝑐 for each study (figure 4b). The variations are slight and centered around 0.17. In [START_REF] Laubitz | Thermal conductivity of powders[END_REF] and [START_REF] Van Norman | Using atomic layer deposited tungsten to increase thermal conductivity of a packed bed[END_REF], 𝑝 𝑐 rather seems to be around 0.25. This value corresponds to a cubic arrangement with a coordination number of 8, which is quite feasible. The small variation of these parameters shows the consistency of our statement that 𝑛 𝑐 is fixed and depends only on the material properties, and that the uncertainties on its determination come from the uncertainties on the coordination of the studied powders. In any case, the observed values of critical exponents correspond to typical values of thermodynamic phase transitions [START_REF] Inden | Project Meeting Calphad V. Ch[END_REF]).

Results and discussion

Our data

Our study started with thermal conductivity measurements in ultra-pure (>99.9%) 𝛼-alumina powders. These samples were chosen for their different microstructural properties, dependent on the manufacturing processes used. A first family of powders is derived from the spray dried process and a second one from the Bayer process (BP). For each family, we studied several samples of powders with average particle sizes ranging from about 1 µm to about 70 µm. The powder data is presented in table 1. The first column corresponds to the average value of the grain size (⟨𝑑⟩) making up the particles. From the specifications given by the suppliers and the analysis of the SEM images, we can approximate the grain size around 0.1 µm and 5 µm, for the alumina powders of our study. For measurements where ⟨𝑑⟩ is not given, it is assumed that the grain size is equal to the particle size (𝑑 𝑝 ). This corresponds to the maximum conductivity, that of a single crystal of size 𝑑 𝑝 . For our samples, the mass and volume measurements introduced into the experimental device give us the effective density of the sample 𝜌 𝑒 . And allows us to determine 𝑝 𝑒 (see equation ( 2)).

The granulometry was measured by laser diffraction (wet dispersion method), with at least 2 measurements per powder. The values presented in table 1 were obtained without the application of ultrasound. Measuring the interparticle porosity of powders is complicated, because the particle size is small and the pore size in the particles is too large to be measured by the BJH method (the pores of the particles are larger than 50 nm). This method consists of determining the pore size distribution of a medium by studying the variations between the adsorption and desorption of a gas in the medium. We therefore measured 𝑝 𝑖𝑛𝑡𝑒𝑟 , for two samples, using an X-ray micro-tomograph (µCT) EasyTom XL 150-160 from Rx-Solution, with a HAMAMATSU microfocus X-Ray source Modèle L 10711-23 (current= 47 A, voltage = 127 V, voxel size 1.5 µm). Due to the constraints of the µCT measurement, the sample holder is different from the one used for the conductivity measurement. This creates an uncertainty on the announced interparticle porosity values because it is not measured directly on the same part of the sample than the conductivity measurements. However, total porosity measurements were carried out in different containers. These measurements showed a small variation in the overall porosity, indicating that the arrangement of the particles does not vary much depending on the container. The container is considered as a representative volume of the medium. The measurement and processing of the images obtained with the tomograph give an uncertainty on 𝑝 𝑖𝑛𝑡𝑒𝑟 of about 1 % due to segmentation. Recall that 𝑝 𝑖𝑛𝑡𝑒𝑟 corresponds to the interparticle or inter-aggregate porosity (in the case where the powders are composed of aggregates, i.e. porous particles with 𝑝 𝑖𝑛𝑡𝑟𝑎 ≠ 0). 𝑝 𝑖𝑛𝑡𝑒𝑟 is measured for samples 5 and 6 from the spray dried process. Both samples have a similar particle size, with a monomodal distribution centered at about µm. However, sample 5 is composed of spherical particles with homogeneous porosity and sample 6 of particles with pores of several micrometers in their centre. Their appearance is hollow. Both aspects were observed by µCT and scanning electron microscopy (SEM, see figure 5). 6, we can see the hollows in the centre of the particles. In the SEM images we can see that, for most of the particles of powder 6, the hollow communicates with the outside. These hollows favor the entanglement of the particles with each other, which also favors thermal transfer. The interparticle porosity measured for powders 5 and 6 is 0.405 and 0.37 respectively. Figure 6 shows the measured thermal conductivities for the different SD and BP powders and their error bars. It also shows the range of possible interparticle porosities predicted by the model (red diamonds for SD powders and blue diamonds for BP powders). Interparticle porosity measurements made at the µCT are displayed by green diamonds. The model parameters are 𝑛 𝑐 = 5.02 and 𝑝 𝑐 = 0.168. We can observe that the range of 𝑝 𝑖𝑛𝑡𝑒𝑟 values predicted by the model for sample 6 excludes the value measured at the µCT (0.37 ∉ [0.325, 0.345]). This difference can be explained by the fact that we assume the critical porosity to be 0.168. However, the µCT and SEM images (figure 5) show that the particles of powder 6 are hollow and open on the outside. This can lead to entanglements and therefore to a smaller effective thermal coordination number and a larger 𝑝 𝑐 . The measured value would correspond to a 𝑝 𝑐 = 0.188. Furthermore, the grain size distribution influences this parameter and it can be observed that it is not the same for both samples. In general, we observe that the thermal conductivity predicted by the model is very sensitive to the interparticle porosity. This highlights, once again, the importance of the microstructural parameters 𝑝 𝑐 and 𝑝 𝑖𝑛𝑡𝑒𝑟 .

Other thermal conductivity data of non connected solid phase alumina

Figure 7 presents the results obtained for the prediction of our model of thermal conductivity of alumina powders for different studies. As before, the critical porosity and critical exponent are fixed and equal to 0.168 and 5.02, respectively. The vertical bars correspond to the range of possible values, to which 𝜆 𝑚𝑜𝑑 𝑒 belongs for 𝑝 𝑖𝑛𝑡𝑒𝑟 between 0.33 and 0.42. We can see that the model is sensitive to this parameter. The average sensitivity of the model for these data is about 0.015 W.(mK) -1 per percent of interparticle porosity. Knowledge of the microstructural parameters for predicting the effective conductivity is essential. The horizontal error bars correspond to the uncertainty of the measurement. The closest prediction is for [START_REF] Laubitz | Thermal conductivity of powders[END_REF] measurement (𝑛 • 12 in table 1). However, this is a point where the interparticle porosity is known. This discrepancy, as mentioned in the previous section, may be due to the fact that 𝑝 𝑐 was assumed to be constant, whereas it also depends on the microstructure. For the same reason, the measurement of [START_REF] Van Norman | Using atomic layer deposited tungsten to increase thermal conductivity of a packed bed[END_REF] is also above the prediction of our model. Moreover, the measurements of [START_REF] Laubitz | Thermal conductivity of powders[END_REF] were made at temperatures of 473 𝐾. It is possible that radiative transfer increases the effective thermal conductivity compared to that predicted by our model, which does not consider this mode of transfer. The possible values are given in the range plus or minus 0.1 W(mK) -1 . An exact knowledge of the microstructure would allow us to be more precise and to have a better estimate of 𝑝 𝑐 , making the model more consistent.

Other models

In this section we compare our model to models often cited in the literature, the expressions of the models are presented in table 2. It is important to distinguish experimental correlations from theoretical models. In the first category, we are interested in the models of [START_REF] Kunii | Heat transfer characteristics of porous rocks[END_REF] and [START_REF] Hsu | A Lumped-Parameter Model for Stagnant Thermal Conductivity of Spatially Periodic Porous Media[END_REF], two models with localized parameters (the heat transfer is described from an elementary cell representing the elementary mesh of the medium). In [START_REF] Kunii | Heat transfer characteristics of porous rocks[END_REF] the cell is 2D reduced to two spheres in contact and for [START_REF] Hsu | A Lumped-Parameter Model for Stagnant Thermal Conductivity of Spatially Periodic Porous Media[END_REF] the cell is 3D cubic, with an unidirectional heat flow. The latter gives the most satisfactory results of the three models presented in his paper. A parameter, 𝛾 𝑐 , represents the ratio between the contact width between two solid elements and the width of the solid cube, allowing us to have an expression of the model in the case of unconnected solid particles, when 𝛾 𝑐 tends towards 0. These models are calibrated by the authors with experimental measurements and have a good agreement with the conductivity of different particle beds. Models from effective media theory are also often cited in this field. We choose to compare our model with the model of [START_REF] Bruggeman | Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielektrizitätskonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen[END_REF]. Figure 9 represents the thermal conductivity as a function of porosity 𝑝 𝑒 given by the different models for 𝛼-alumina at 293 K. The models quoted depend on the porosity 𝑝 𝑒 and the conductivities of the two phases. For our model, we have considered dense particles, i.e. without intraparticle porosity, and with a maximum grain conductivity equal to that of the crystal. This corresponds to the curve given for 𝑝 𝑖𝑛𝑡𝑟𝑎 = 0 in figure 8. It can be seen that the predictions of our model fall between the correlation models and the EMT model, when 𝑝 𝑒 is less than 0.4. The conductivities predicted by [START_REF] Hsu | A Lumped-Parameter Model for Stagnant Thermal Conductivity of Spatially Periodic Porous Media[END_REF] are not realistic when 𝑝 𝑒 < 0.2. The conductivity decreases sharply with 𝑝 𝑒 , even for porosities close to 0, this has no physical correlation because, for these porosities, the links between particles are numerous, conduction is mainly through the solid phase and not through the fluid phase. The model of [START_REF] Kunii | Heat transfer characteristics of porous rocks[END_REF] is defined for 0.26 < 𝑝 𝑒 < 0.476 and is approximated on the other intervals (this explains the slight discontinuities at the ends of the interval). The approximation proposed by [START_REF] Kunii | Heat transfer characteristics of porous rocks[END_REF] for low porosities can be questioned. Indeed, the conductivity when 𝑝 𝑒 = 0 does not tend towards the value of the conductivity of the solid phase. The conductivity according to [START_REF] Bruggeman | Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielektrizitätskonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen[END_REF] decreases linearly until about 𝑝 𝑒 = 0.65, this approach is also in contradiction with the experiments. For example, assuming that the particles in the powder bed of [START_REF] Shapiro | Characterization of powder beds by thermal conductivity: effect of gas pressure on the thermal resistance of particle contact points[END_REF] with 𝑝 𝑒 = 0.42 are dense (i.e. in the case where the thermal conductivity would be the highest), we have 𝜆 𝑚𝑒𝑎𝑠 𝑒 = 0.381 W(mK) -1 and the EMT model predicts 𝜆 𝐸𝑀𝑇 𝑒 = 14.7 W(mK) -1 . More than an order of magnitude separates these two values. Of all these models, our model seems to be the most consistent with the data reported in the literature, whatever the considered porosity. It can be noted that our model and the model of [START_REF] Hsu | A Lumped-Parameter Model for Stagnant Thermal Conductivity of Spatially Periodic Porous Media[END_REF] (unconnected particles) are very close when 𝑝 𝑒 > 0.6. This is in agreement with the theory of our model which predicts a break in the connections between particles as soon as the total porosity exceeds the critical porosity 𝑝 𝑐 . The models of [START_REF] Hsu | A Lumped-Parameter Model for Stagnant Thermal Conductivity of Spatially Periodic Porous Media[END_REF] (connected particles) and [START_REF] Kunii | Heat transfer characteristics of porous rocks[END_REF] assume a constant exchange surface between the particles, whatever the porosity, over the interval [0, 1] and [0.476, 1]. These models are not suitable for modelling the conductivity of ultra-porous powders, where this area decreases with increasing porosity.

Figure 8 is a 2D map of the values predicted by our model for alumina at 𝑇 =293K, ⟨𝑑⟩=∞, 𝑝 𝑐 =0.168 and 𝑛 𝑐 =5.02 as a function of 𝑝 𝑖𝑛𝑡𝑒𝑟 and 𝑝 𝑖𝑛𝑡𝑟𝑎 . The grey lines are thermal iso-conductivity and the pink to yellow lines are total isoporosity. Our model gives a different weight to the different porosities; the thermal conductivity varies linearly with the intraparticle porosity and varies critically with the interparticle porosity. The thermal conductivity can therefore be very variable with this pair even if the total porosity is constant. It can be seen that, for a total porosity of 0.7, the conductivity can vary from about 1 W(mK) -1 to about 0.06 W(mK) -1 , depending if 𝑝 𝑖𝑛𝑡𝑒𝑟 is equal to 0.25 or 0.5. This representation illustrates the importance of these two microstructural parameters (𝑝 𝑖𝑛𝑡𝑒𝑟 and 𝑝 𝑖𝑛𝑡𝑟𝑎 ) on the conductivity in a particle bed. The interval [0.25,0.5] is chosen because it frames the values of the measurements we have studied. Furthermore, it corresponds to the interparticle porosity values of systems composed of mono-dispersed dense spheres with arrangements ranging from simple cubic (𝑝 𝑒 = 0.48) to facecentered cubic (𝑝 𝑒 = 0.26).

Conclusion

In our study, we measured thermal conductivities in different 𝛼-alumina powders at ambient pressure and in air, using an unsteady hot plane setup. We also took into account other values from the literature. These experimental data allowed us to verify the validity of the effective thermal conductivity model we developed for powder beds. In this model, we interpret the variations of the thermal diffusivity of a powder bed as a function of porosity as a second order transition. This model is a multi-scale description of the thermal conductivity in this type of material. Starting from the quantum scale, a purely theoretical value of the thermal conductivity of the pure compound (𝛼-alumina) is given. Then, the microstructure is taken in the following set of parameters:

• Grain size, ⟨𝑑⟩ • Intraparticle porosity, 𝑝 𝑖𝑛𝑡𝑟𝑎 • Interparticle porosity, 𝑝 𝑖𝑛𝑡𝑒𝑟 • Critical porosity, 𝑝 𝑐 This allows us to be more consistent with the physical reality of porous systems with unconnected solid phases. This model allows us to predict the thermal conductivity of a powder bed to the nearest ±0.1 W(mK) -1 assuming that the interparticle porosity varies between 0.33 and 0.42. It forecasts the degradation of the bed conductivity to within 0.3%, compared to the theoretical conductivity of the pure crystal (ratio between the measurements and the model for measurements found in literature (part 4). Assuming a dense particle bed (𝑝 𝑖𝑛𝑡𝑒𝑟 = 𝑝 𝑒 ), we compared our model to existing classical models, which do not take into account the distinction between intraparticle and interparticle porosity. Whether they are theoretical models or experimental correlations, they are not consistent over the whole porosity range ([0, 1]), unlike ours. The conductivity of ultra-porous powders is therefore better described by our model. The values chosen for the critical porosity 𝑝 𝑐 and for the critical exponent 𝑛 𝑐 can be discussed, since they were fixed without exact knowledge of the microstructural arrangement of all the beds studied. However, the values were not chosen randomly and seem to be in agreement with almost all measurements. The critical porosity is set by analogy with those obtained in previous studies on sintered material based on spherical particles. It corresponds to the site percolation threshold of centered cubic arrangement i.e with an effective coordination number equal to 14 (8 first and 6 second neighbors). Overall, the critical porosity can be considered as an adjustable parameter of the proposed model provided that sufficient information on the microstructural parameters of the solid system is available. We therefore agreed that most of the beds studied have a critical porosity of 0.168 and that the critical exponent in alumina is 5 (which can be compared to second order phase transitions). The original approach used by our model to describe heat transfer in porous media with unconnected solid phase seems to be validated. We plan to apply this method in a broader framework, for measurements carried out under more varied conditions (different gases and pressures).

Figure 1 :

 1 Figure 1: Schematic and photo of the experimental diffusivity measurement bench (photo taken without fibrous insulation)

Figure 2 :

 2 Figure 2: Top : typical thermograms provided by the Hot Plane set-up and its modelization : front face and rear face measured heatings (respectively red and black dashed lines), rear face model (blue line), rear face noise (grey line). Bottom : rear face noise (grey) and residuals (black).

)Figure 3 :

 3 Figure 3: Thermal conductivity of 𝛼-alumina powder as a function of the scale of study. The thermal conductivity values are the typical values issued from our measurements.

Figure 4 :

 4 Figure 4: (a) Plot of the critical exponent 𝑛 𝑐 for different studies with 𝑝 𝑐 fixed at 0.168 and 𝑝 𝑖𝑛𝑡𝑒𝑟 varying from 0.33 to 0.42 except for values where 𝑝 𝑖𝑛𝑡𝑒𝑟 is known (see table 1). (b) Plot of the critical porosity 𝑝 𝑐 for different studies with 𝑛 𝑐 fixed at 5.02 and 𝑝 𝑖𝑛𝑡𝑒𝑟 varying from 0.33 to 0.42 except for values where 𝑝 𝑖𝑛𝑡𝑒𝑟 is known (see table 1)

  Figure 5.a corresponds to sample 6 and figure 5.b to sample 5. The pictures on the left correspond to a slice of the µCT measurement stack for the two samples, the images in the centre were obtained after segmentation and after filtering of the original images by a median filter 3 voxels in width. Then, the picture on the right shows SEM images of the corresponding powders. For the µCT slice of powder

Figure 5 :)Figure 6 :

 56 Figure 5: Original µCT images of slices, images after segmentation and after applying a median filter and SEM images for two different samples (a) corresponds to the 𝑛 • 6 SD sample with 𝑑 50 = 65.8 µm and (b) corresponds to the 𝑛 • 5 SD sample with 𝑑 50 = 56.4 µm.

  with 𝑝 𝑐 = 0.168, 𝑛 𝑐 = 5.02 and 𝑝 𝑖𝑛𝑡𝑒𝑟∈ [0.33,0.42]

Figure 7 :

 7 Figure 7: Correlation of experimental and predicted thermal conductivity of 𝛼-alumina powder with 𝑛 𝑐 fixed at 5.02. The horizontal error bar corresponds to the measurement error and the vertical bars represent the values taken by 𝜆 𝑚𝑜𝑑 𝑒 when 𝑝 𝑖𝑛𝑡𝑒𝑟 varies from 0.33 to 0.42

)Figure 8 :

 8 Figure 8: Thermal conductivity predicted by our model as a function of interparticle and intraparticle porosity. 𝜆 𝑎𝑖𝑟 = 0.025 W(mK) -1 , 𝜆 𝑠𝑐 = 36.6 W(mK) -1 , and 𝑝 𝑐 = 0.168, 𝑛 𝑐 = 5.02. The grey lines are iso conductivity, the other lines (yellow, pink) are iso total porosity.

  𝜆 𝑚𝑜𝑑 𝑒 ∕𝜆 𝑎𝑖𝑟 = 𝑝 𝑒 + 𝛽(1 -𝑝 𝑒 )∕(𝜓 𝑡 + 𝛾∕𝜅), 𝜅 = 𝜆𝑠𝑐∕𝜆𝑎𝑖𝑟 𝑡 = 𝜓 2 for 𝑝 𝑒 < 0.26 𝛽 = 1, 𝑛 1 = 1.5, 𝜓 𝑡 = 𝜓 1for 𝑝 𝑒 ≥ 0.476 𝛽 = 0.486(𝑝 𝑒 -0.26) + 0.895, 𝜓 𝑡 = 𝜓 2 + (𝜓 1 -𝜓 2 )(𝑝 𝑒 -0.26)∕0.216 for 0.26 ≤ 𝑝 𝑒 ≤ 0.476 𝜓 1𝑜𝑟2 = 0.5 * ((𝜅 -1)∕𝜅) 2 sin 2 𝜃 0 ln(𝜅 -(𝜅 -1) cos 𝜃 0 ) -(𝜅 -1)∕𝜅 (1 -cos 𝜃 0

  𝑝 𝑐 , 𝑛 𝑐 , 𝑝 𝑖𝑛𝑡𝑒𝑟 , 𝑝 𝑖𝑛𝑡𝑟𝑎 , 𝑝 𝑒 ) = 𝜆 𝑝 (𝑇 , ⟨𝑑⟩, 𝑝 𝑖𝑛𝑡𝑟𝑎 ) × (1 -𝑝 𝑖𝑛𝑡𝑒𝑟 ) 𝑓 𝑐 (𝑝 𝑐 , 𝑝 𝑖𝑛𝑡𝑒𝑟 , 𝑛 𝑐 ) ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ 𝑁(𝑝 𝑖𝑛𝑡𝑒𝑟 ,𝑝 𝑐 ,𝑛 𝑐 )

⟨𝑑⟩,𝑝 𝑖𝑛𝑡𝑟𝑎 ) (12) 𝜆 𝑚𝑜𝑑 𝑒 (𝑇 , ⟨𝑑⟩,

Table 1

 1 Properties and parameters of the powder bed samples used in different studies (our measurements made by X-ray tomography are marked by *)

	Studies	⟨𝑑⟩ (µm) 𝑑 𝑝 (µm)	𝑝 𝑒	𝑝 𝑖𝑛𝑡𝑒𝑟	𝑇 𝑚𝑒𝑎𝑠 (K) 𝜆 𝑚𝑒𝑎𝑠 𝑒	(W(mK) -1 ) 𝑛 •
	This study
	spray-dried 𝛼-alumina (SD)		1.35	0.83		298	0.089	1
			4.93	0.645		298	0.083	2
		0.1	23.3	0.66		298	0.122	3
		0.1	44.8	0.66		298	0.127	4
		0.1	56.4	0.70 0.405 ± 0.005* 298	0.106	5
		0.1	65.8	0.66 0.37 ± 0.004* 298	0.202	6
	This study
	𝛼-alumina produced by Bayer process (BP)		1.63	0.72		298	0.092	7
			7.23	0.71		298	0.109	8
		5	49	0.73		298	0.178	9
		5	82.25	0.73		298	0.179	10
	Laubitz (1959)		80	0.75		473	0.22	11
			400 ± 100 0.525 0.505	473	0.376	12
	Luikov et al. (1968)		60	0.75		300	0.171	13
			263	0.42		373	0.42	14
	Shapiro et al. (2004)		60	0.46		333	0.21	15
			166	0.51		333	0.286	16
			211	0.42		333	0.381	17
	Hu et al. (2007)		0.011	∼ 0.8		300	0.035	18
			0.3	∼ 0.8		300	0.085	19
			0.5	∼ 0.8		300	0.105	20
	Van Norman et al. (2015)		90	0.83 0.33 -0.37	320	0.48	21
	Pujula et al. (2016)		125	0.745		293	0.11	22
	𝑛

𝑐

Optimized value of 𝑛 𝑐 for the different studies with 𝑝 𝑐 = 0.168, and 𝑝 𝑖𝑛𝑡𝑒𝑟

∈ [0.33,0.42]

  table 1). (b) Plot of the critical porosity 𝑝 𝑐 for different studies with 𝑛 𝑐 fixed at 5.02 and 𝑝 𝑖𝑛𝑡𝑒𝑟 varying from 0.33 to 0.42 except for values where 𝑝 𝑖𝑛𝑡𝑒𝑟 is known (see table 1)

  𝜆 𝑚𝑜𝑑 𝑒 ∕𝜆 𝑎𝑖𝑟 = 1 -𝛾 2 𝑎 -2𝛾 𝑐 𝛾 𝑎 + 2𝛾 𝑐 𝛾 2 𝑎 + (𝛾 𝑐 𝛾 𝑎 ) 2 𝜅 + 𝛾 2 𝑐 -(𝛾 𝑐 𝛾 𝑎 ) 2 1 -𝛾 𝑎 + 𝛾 𝑎 ∕𝜅 + 2(𝛾 𝑐 𝛾 𝑎 -𝛾 𝑐 𝛾 2 𝑎 ) 1 -𝛾 𝑐 𝛾 𝑎 + 𝛾 𝑐 𝛾 𝑎 ∕𝜅 With 1 -𝑝 𝑒 = (1 -3𝛾 2𝑐 )𝛾 3 𝑎 + 3(𝛾 𝑐 𝛾 𝑎 ) 2 and 𝛾 𝑐 = 0.13Hsu et al. (1995) (non-touching cubes)

	Hsu et al. (1995)
	(touching cubes)
	𝜆 𝑚𝑜𝑑 𝑒 ∕𝜆 𝑎𝑖𝑟 = (1 -(1 -𝑝 𝑒 ) 2∕3 ) +	(1 -𝑝 𝑒 ) 2∕3 1 + (1∕𝜅 -1)(1 -𝑝 𝑒 ) 1∕3

Acknowledgements

The authors are grateful for the financial support provided by the French Alternative Energies and Atomic Energy Commission (CEA), Électricité de France (EDF) and FRAMATOME. The project leading to this publication has received funding from the Excellence Initiative of Aix-Marseille University -A*Midex, a French Investissements

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.