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# Enhancing the accuracy of thermal conductivity measurements with insulating construction materials: Addressing edge effects and methodological improvements

K. Mourda, Y. Zou, V.T. Nguyen, P. Coussot*

Laboratoire Navier (Ecole Nationale des Ponts et Chaussées, Univ. Gustave Eiffel, CNRS), Champs sur Marne, France

## ABSTRACT

Accurate and reliable thermal conductivity measurements are essential for evaluating the insulation properties of materials used in construction. Traditional insulating materials like glass or rock wool, or more recently developed biobased materials such as hemp wool, cellulose, flax or wood fiber panels, have very low thermal conductivity influenced by density, porosity, and temperature. However, edge effects often compromise the reliability of measurements, especially for low-conductivity materials. This study revisits the standard measurement technique with a heat flow meter proposing a refined methodology to minimize edge effects. We first show that the standard procedure based on a preliminary calibration of the heat flow meter apparatus on a reference sample to mitigate edge effects must be used with great care as it can afterwards lead to significant error on tested samples. With a model (cellulosic) material we then study systematically how the sample thickness and the temperature difference between sample and ambient air can impact measurements. In particular we show that the material anisotropy can significantly enhance edge effects, and in some cases edge effects are so large that the heat flux along one plate is negative. The analysis is completed by numerical simulations and measurements of the temperature distribution inside the sample. This work finally allows to establish the optimal thermal conductivity testing conditions for insulating materials.

## 1. Introduction

Accurate measurements of thermal conductivity are crucial for the evaluation and application of insulating materials in construction. Insulating materials are integral to building design due to their ability to reduce heat transfer, thereby enhancing energy efficiency. Traditional insulating materials, such as glass or rock wool, have been extensively studied. Glass wool, for instance, is known for its low thermal conductivity, typically in the range of 0.03–0.04 W m⁻¹ K⁻¹ at standard temperature (10 °C) [1,2]. The thermal performance of glass wool and similar materials can be influenced by factors such as density, porosity and temperature. Thermal conductivity generally decreases as density increases, up to an optimal point [1]. Higher porosity often leads to lower thermal conductivity, as the material contains more air, which is a poor conductor of heat. Additionally, thermal conductivity of glass wool increases with temperature [2].

In recent years, there has been a growing interest in biobased insulating materials. These materials, derived from renewable resources, have a lower carbon footprint compared to traditional construction materials, as they often sequester carbon during growth and require less energy to produce [3,4]. Common biobased materials used in construction include cellulose, hemp, flax, and cork. Wood fiber insulation boards as an example have a thermal conductivity of 0.040 W m⁻¹ K⁻¹ [5]. The increasing adoption of biobased materials is driven by their ecological advantages and their additional potential to improve indoor air quality by regulating humidity [6–8].

<footnote>* Corresponding author.
E-mail address: philippe.coussot@univ-eiffel.fr (P. Coussot).</footnote>
Thermal conductivity measurements, particularly for low-conductivity materials [9] such as biobased insulators, can be critically influenced by edge effects. This is due to the fact that heat losses along the sample edges can easily be in the same order as the measured heat flux, which is naturally low for such low conductivity materials. Edge effects can lead to thermal performance degradation and inaccurate measurements due to thermal bridging at the edges of insulation panels [10]. These effects can significantly distort the results, leading to inaccurate measurements if not properly addressed. As a result, ensuring the reliability of these measurements requires careful consideration and correction of edge effects [11], a challenge that becomes especially pronounced when comparing different materials and testing methods across the construction industry. The width of the sample is crucial. Plates should be wide enough so that edge effects in certain directions do not influence the measurements [12]. The thermal contrast between the sample and surrounding materials can affect the extent and intensity of edge effects. However, the decay rate of edge effects, which refers to how quickly their influence diminishes as one moves away from the boundary, appears to be independent of the thermal properties of the materials involved [13]. The complexity of this issue is compounded by the varying physical properties of insulating materials, making it difficult to standardize measurement protocols universally [14]. To mitigate these challenges, researchers have explored a variety of approaches, including the use of guard rings, advanced mathematical correction techniques, and specialized testing setups designed to minimize lateral heat transfer [11]. These methods, while effective to varying degrees, underscore the ongoing need for a deeper understanding and more robust solutions to ensure the accuracy of thermal conductivity measurements in both traditional and modern insulating materials.

As for standard approaches, such as those recommended by ASTM C518 [15] (for thermal conductivity measurement using the Heat Flow Meter (HFM) method) and ISO norms (ISO 8301 [16] for steady-state measurements using the heat flow meter method and ISO 8302 using the guarded hot plate method). In the HFM method, a known temperature gradient is established across a sample positioned between two plates, with heat flux sensors measuring the rate of energy flow. This approach involves a preliminary calibration stage with standard samples to mitigate these effects [17]. The instruments should be calibrated using certified reference materials traceable to national or international standards [18]. Despite its widespread adoption due to relative simplicity and efficiency, the accuracy of HFM measurements can be adversely affected by edge losses and imperfect thermal contact, especially for samples exhibiting low thermal conductivity and complex geometries. On the other hand, the guarded hot plate method, often considered the “primary” standard technique for measuring thermal conductivity of insulating materials, employs a central heater plate surrounded by a guard zone [19]. This configuration ensures a unidirectional heat flow through the sample by minimizing lateral heat losses. Although more robust in its theoretical foundation, the guarded hot plate setup also demands meticulous assembly and calibration to limit systematic errors, such as heat leakage around sample edges [20].

To address these challenges, researchers have developed alternative techniques for small sample measurements. Methods such as the 3-Ω technique [21], which measures thermal conductivity in both cross-plane and in-plane directions [22], and transient methods like laser flash analysis [23], have been employed. These techniques are capable of characterizing samples with dimensions as small as 50 × 50 mm while accounting for certain edge effects. However, their application often requires complex setups, higher costs, or assumptions about material homogeneity that may not always be valid. Furthermore, their ability to accurately quantify anisotropy in fibrous or porous materials remains limited.

In this work, we critically revisit the standard technique for measuring thermal conductivity with heat flow meter, analyzing its limitations and proposing improved methodologies. We start by recalling the basic theoretical approach for steady-state heat transfer under ideal conditions (infinitely large sample and plates) and, in the finite sample case, we discuss the parameters influencing heat losses through rough estimations (Section 2: Theoretical approach). Next, we show that the standard procedure based on a preliminary calibration of the heat flow meter apparatus on a reference sample to mitigate edge effects must be used with great care as it can afterwards lead to significant error on tested samples (Section 4.1. Initial observations and calibration process). We then study systematically, with a model (cellulosic) material, how the sample thickness and the temperature difference between sample and ambient air can impact measurements (Sections 4.2 and 4.3: Flux analysis and advanced flux analysis). Particular attention is given to the impact of material anisotropy, which can significantly enhance edge effects, and in some cases edge effects are so large that the heat flux along one plate is negative. The analysis is supported by experimental measurements of the temperature distribution inside the samples (Section 4.4. Temperature fields inside the samples at steady states) and numerical simulations (last part of the results). Finally, this work allows to establish optimal thermal conductivity testing conditions for insulating materials, offering insights into minimizing edge effects and accounting for anisotropy. This study provides a comprehensive framework for improving the reliability of thermal conductivity measurements for insulating materials.

## 2. Steady state heat transfer between parallel plates: theoretical approach

Understanding thermal conductivity measurement begins with foundational thermal physics [24]. This study analyzes the heat transfer between two isothermal plates separated by a sample. Fourier's law states that the heat flux through a material, $q$ (W m⁻²), is proportional to the temperature gradient and is quantified by the material's thermal conductivity [25]:

$$q = -k\nabla T \qquad (1)$$

where $T(z)$ is the temperature distribution in space, and $k$ the thermal conductivity (W m⁻¹ K⁻¹) of the material.

Let us consider the case of an infinitely large sample between infinitely large parallel plates with surfaces situated in $z = 0$ and $z = L$ with the temperatures imposed as $T(z = 0) = T_1$ and $T(z = L) = T_2$, where $z$ is the spatial coordinate in the direction of heat flow (m), and $L$ the length of the material along the $z$-direction. In steady state there is no heat loss or source in the material so that for a ho-
mogeneous material the heat flux (q) is constant throughout the material. Moreover, by symmetry the temperature varies only as a function of z. From the Fourier's law Eqn (1) it follows that $\partial T/\partial z = c_{st}$, and using the boundary conditions we deduce the following distribution of temperature along the direction of heat flow:

$$T(z) = \frac{T_2 - T_1}{L} z + T_1 \qquad (2)$$

Let us now consider a cylindrical sample (with radius R and height L) and plates of same finite diameter (see Fig. 1). In that case, heat is not only transferred vertically through the sample but it can also be transferred laterally to the surroundings, making the process inherently multidimensional, as also seen in tunnel simulations where stress, geometry, and heat flux interactions complicate analysis [26]. The typical boundary condition along the edges consists in some external sample surface exposed to air, leading to a heat exchange by convection. To determine the convective heat transfer, we must consider the velocity field resulting from the air flux imposed over the surface [27], and resulting in the formation of a velocity boundary layer near the surface. From a similarity analysis of the momentum equation and energy conservation equation, it may be shown [27] that the total heat flux across the interface between the material and the air can be expressed in the following form:

$$q_{convection} = -k \left. \frac{\partial T}{\partial x} \right|_{\text{lateral surface}} = h(T_s - T_{\infty}) \qquad (2a)$$

where $T_s(z)$ is the surface temperature (along the edge), $T_\infty$ the temperature in the air far from the surface, and the coefficient h expresses as

$$h = f(\text{Re}, \text{Pr}) \frac{k}{L} \qquad (3)$$

in which the Reynolds (Re) and Prandtl (Pr) numbers are defined as: Re = $\rho_a V L / \mu$ and Pr = $\mu C_{p,a} / k_a$, with $\mu$ the air viscosity, L a characteristic length of the system (such as the sample's diameter), V a characteristic velocity of the air along the sample surface, and $\rho_a$, $C_{p,a}$ and $k_a$ the density, specific heat capacity, and thermal conductivity of the air, respectively. In this manner, h represents a coefficient averaged over the surface area of the sample, accounting for both the fluid dynamics around the sample and its thermal properties. It appears that the value of h increases with higher fluid velocities (captured by the Reynolds number) and varies with changes in the thermal properties of the air such as its thermal conductivity, specific heat capacity, and dynamic viscosity (captured by the Prandtl number), leading to more or less pronounced edge effects. However, it is worth emphasizing that even if one can take some reasonable value for h (see below: Simulation) from literature, the heat losses effectively associated with this exchange with ambient air cannot be evaluated a priori, as they also depend on the value of $T_s$ (see equation (2a)), which depends on the temperature gradient established inside the material. Thus, the heat losses depend on the whole temperature distribution, which in turn depends on the equilibrium resulting from the coupling between the different heat fluxes in the system.

Let us roughly evaluate the role of the main parameters for a typical experiment with a heat transfer imposed by horizontal plates, and with the sample offering free surfaces in contact with air (see Fig. 1). Here, for simplicity, it is assumed that heat transfers occur in both the z and r directions with the same thermal conductivity (isotropic sample). Due to edge effects (i.e., heat losses) the temperature distribution differs from that given by equation (2). We nevertheless can estimate the order of magnitude of the axial heat flux by assuming a linear variation of the temperature: $q_z = -k(T_2 - T_1)/L$, from which we deduce the total heat transfer rate, $Q_z$, through the cross-sectional area of the cylinder:

$$Q_z = k(T_1 - T_2)\pi R^2 / L \qquad (4)$$

On the other side, we estimate the radial heat flux by considering that the temperature decreases from the central axis to the periphery, which roughly leads to a heat flux $q_r = -k(T_\infty - T_0)/R$, so that the corresponding total heat transfer rate along the exposed lateral surface of the cylinder is

$$Q_r = 2\pi L k (T_0 - T_\infty) \qquad (6)$$

<figure><img src="image_2.png" /><figcaption>Fig. 1. Heat fluxes in a cylindrical sample between two plates at different temperatures.</figcaption></figure>
Here the temperature in the central region is estimated as $T_0 = (T_1 + T_2)/2$. We can finally estimate the order of magnitude of the ratio of radial heat losses to the main measured heat transfer:

$$ \frac{Q_r}{Q_z} = \frac{2(T_0 - T_\infty)}{(T_1 - T_2)} \left(\frac{L}{R}\right)^2 \qquad (7) $$

This ratio increases with the aspect ratio of the sample, which is the thickness relative to the radius ($L/R$). This relationship Eqn (7) thus emphasizes that, as usual when one considers edge effects for any physical process, it is not the absolute thickness alone that matters, but how the thickness compares to the sample's radius. The graph in Fig. 2 shows that as the aspect ratio increases, $Q_r/Q_z$ becomes more pronounced, particularly at higher mean temperatures (as compared to the ambient one). Additionally, negative values of this ratio indicate scenarios where the mean temperature within the sample is lower than the ambient temperature, resulting in heat flux (losses) occurring from the outer surface to the inner part of the sample. This analysis suggests that reducing the sample thickness can help mitigate the influence of edge effects, making it easier to achieve accurate thermal conductivity measurements. This is a common approach for minimizing such errors in various thermal studies. Additionally, keeping the sample's mean temperature close to the ambient temperature is also crucial for minimizing these edge effects.

In this study, we use the term “edge effects” to describe the deviation from 1D heat flow caused by lateral heat exchanges between the sample and its surroundings. “Lateral flux” (or radial flux) refers to the heat flow component perpendicular to the primary axial direction (through the sample thickness). “Heat losses” denote the net energy leaving the system due to these lateral fluxes, particularly via convection at the sample edges. While interrelated, we use these different terms to reflect their distinct physical meanings.

# 3. Methodology

## 3.1. Materials and sample preparation

The experiments were conducted using Arbocel® BC 200 cellulose fiber, sourced from Kremer Pigmente. These cellulose fibers are strip-like, with an average length of 300 µm, a width of 10 µm and a thickness about 1 µm. The density of the dry cellulose fiber is $ρ_s \approx 1500$ kg m⁻³. Cellulose samples were prepared in cylindrical shape with a diameter of 7 cm. The preparation of these cylindrical samples involved a carefully controlled process to maintain consistent properties across different samples, even with varying dimensions. This consistency was achieved by calculating the mass of cellulose required to be compressed into a given volume, ensuring that the orientation of the cellulose fibers and the final porosity were consistent across samples. The porosity of each cylindrical sample was controlled according to the following equation:

$$ \varepsilon = \frac{V_{\text{voids}}}{V_{\text{total}}} = 1 - \frac{V_s}{V_{\text{total}}} = 1 - \frac{m_s/\rho_s}{V_{\text{total}}} \qquad (8) $$

where $\varepsilon$ is the porosity, $V_{\text{voids}}$ is the volume of voids, $V_{\text{total}}$ is the total volume of the sample, $V_s$ and $m_s$ are the volume and mass of cellulose respectively.

Cylindrical samples were prepared by compressing cellulose to achieve a target porosity of 0.5, with thickness ranging from 0.5 cm to 3.5 cm. The compression was performed in a controlled manner to reach a precisely defined final thickness, which was measured and maintained throughout the experiments. This thickness corresponds to the post-compression state and is the reference dimension used in all analyses. During thermal conductivity measurements, the samples remained mechanically stable under the applied thermal gradients, ensuring that both thickness and porosity were constant, which is critical for reproducibility and for properly characterizing anisotropy due to fiber alignment. Due to the compression process, the cellulose fibers naturally tend to align in planes perpendicular

<figure><img src="image_3.png" /><figcaption>Fig. 2. Comparison of the orders of magnitude of axial to radial heat flux as a function of the aspect ratio for a heat flow meter test with a cylindrical sample, for different mean temperatures in the gap under an ambient temperature of 22 °C. It is worth noting that the temperature difference $T_1 - T_2$ is maintained at 5 °C in this analysis.</figcaption></figure>
to the compression direction, resulting in significant anisotropy (see Ref. [28]). With these samples, the main heat flow direction during thermal conductivity measurements was parallel to the direction of compression. Under these conditions, the thermal conductivity in the radial direction of cylindrical samples may differ from that in the axial direction. Thus, to estimate the impact of fiber orientation on thermal conductivity, a cuboid-shaped sample (75 mm × 75 mm × 10 mm) was also prepared by compression. The sample was then rotated by 90°, ensuring that when placed between the plates of the heat flow meter, the primary heat flux direction was now perpendicular to both the compression direction and the fiber alignment.

Dried samples were prepared by compressing cellulose, initially at ambient conditions, into the desired porosity, then placing them in an oven until a constant (dry) mass was achieved, which typically took approximately 2–3 days. After each thermal conductivity measurement with the Heat Flow Meter (HFM), the sample was returned to the oven to maintain the dried state before another measurement.

The Standard Reference Material (SRM) 1450E, provided by the National Institute of Standards and Technology (NIST), is a high-density fibrous glass board, fabricated using fibrous glass combined with a phenolic binder. Each standard sample unit is formed as a semi-rigid board with nominal dimensions of 610 mm × 610 mm × 25 mm and a certified bulk density ranging from 110 to 154 kg m⁻³ [29]. The certified thermal conductivity values, determined under steady-state conditions, are traceable to the International System of Units (SI) and calibrated using the guarded hot plate method in accordance with ASTM standards [29]. To investigate the possible anisotropy of thermal properties of this material, we studied the sample in two configurations. For the **axial direction**, the standard fibrous glass board was used in its original form without cutting, preserving the manufacturer's layout and fiber alignment. The fibers are naturally aligned along the plane perpendicular to the compression axis, allowing us to measure the thermal conductivity parallel to the compression direction. To explore the **radial conductivity**, the board was sliced into several identical pieces, which were then reoriented by rotating them 90° such that the primary heat flux direction was now perpendicular to the original fiber alignment. The reconfigured stack retained the same overall dimensions as the original sample, with a total thickness of 2.5 cm. Although the exact manufacturing process is not disclosed by the supplier, our visual observations and results suggest that the material is likely formed via compression and directional fiber deposition, which can lead to in-plane/out-of-plane anisotropy (e.g., $k_x \neq k_y$). However, as we will see in the analysis section, the best agreement between experimental data and simulations is obtained when considering three distinct thermal conductivities ($k_x \neq k_y \neq k_z$), indicating the presence of 3D anisotropy. This behavior may result from layer-by-layer deposition, imperfect fiber orientation, or uneven dispersion in the binder matrix during manufacturing.

In this study, rigid polyurethane foam (PUR/PIR) insulation material was also employed to investigate the influence of edge effects on thermal conductivity measurements. Rigid polyurethane foam is a closed-cell plastic known for its exceptionally low thermal conductivity, mechanical strength, and durability. The foam used had a density ranging between 30 and 45 kg m⁻³, and exhibited thermal conductivity values influenced by factors such as cell gas composition, temperature, and moisture absorption. This material's thermal conductivity is about 0.02 W m⁻¹ K⁻¹ as measured by our heat flow meter. For the tests, the insulation was wrapped around the cellulose and standard NIST1450E samples to investigate the impact of edge effects and heat losses. The setup allowed for a controlled comparison between insulated and non-insulated conditions, isolating the influence of direct air contact on heat flux measurements.

## 3.2. Thermal conductivity measurements

Thermal conductivity measurements were performed using a Thermtest HFM-25 Heat Flow Meter (Fig. 3). The HFM-25 device utilizes two parallel plates held at different, but constant, temperatures $T_1$ (hot plate) and $T_2$ (cold plate). Each plate is instrumented with an embedded temperature sensor—positioned near the plate surfaces to accurately monitor their respective setpoint temperatures—and a 25 × 25 mm² heat flow transducer that converts the measured heat flux into an electrical signal. In accordance with ISO 8301 recommendations, the sample width was chosen to be significantly larger than the 25 × 25 mm² heat flow sensors.

To begin each measurement, the system is calibrated based on National Institute of Standards and Technology (NIST) [30] using traceable reference material (e.g., NIST 1450E) with a well-characterized thermal conductivity. This involves placing the reference sample between the plates, applying a known temperature difference, and allowing the system to reach thermal equilibrium. The

<figure><img src="image_4.png" /><figcaption>**Fig. 3.** Principle of a test with the Thermtest HFM-25 Heat Flow Meter Setup. The red lines correspond to the top and bottom plates. The thick black lines correspond to the sensors. The heat fluxes are considered as positive in the direction of the arrow.</figcaption></figure>
measured thermal conductivity is then compared to the reference value, and a calibration factor is computed to correct any systematic offsets in the instrument.

After this calibration, the cellulose sample is positioned between the plates of the HFM-25. Care is taken to ensure that the sample is flat and that their surfaces are well in contact with the plates, a procedure consistent with approaches used in thermal testing of low-conductivity backfill materials under cyclic heating [31]. For all our measurements, we maintain a constant temperature difference of 5 °C between the top and bottom plates (see Fig. 3). Measurements were conducted within the temperature range of 10 °C–45 °C, while the ambient temperature was 22 °C. This range is dictated by the operational limits of the machine. In the following, we use the term "Mean temperature" to refer to the average of the top and bottom temperatures i.e., ((T₁ + T₂)/2, imposed on the sample.

Once the sample is in place and the desired setpoint temperatures are reached, the device is allowed to reach thermal equilibrium. In our experiments, it typically takes around 0.5–2 h to reach thermal equilibrium, depending on the samples’ thermal properties and thickness *L*. During this period, the heat flow meter continuously monitors the heat flux until it becomes constant, indicating that the system has reached thermal equilibrium. After stabilization (at equilibrium), the system records the temperatures (*T*₁ and *T*₂) and the corresponding heat fluxes, *Q*<sub>top</sub> and *Q*<sub>bottom</sub>, at the top and bottom plates. We define the mean flux passing through the sample as *Q*<sub>avg</sub>, and adopt a convention where positive flux indicates heat flow from the hot plate to the cold plate as illustrated in Fig. 3.

Finally, the software applies the previously determined calibration factor to compute the sample's thermal conductivity (k). The thermal conductivity (k) is represented as a scalar for isotropic cases or as a tensor for anisotropic cases. The heat flux, however, is treated as a scalar with positive or negative values depending on the direction of heat transfer, as it is assumed to flow along a fixed axis in the experimental setup. According to ISO 8301, the sample dimensions should ensure that lateral dimensions (width and length) are sufficiently large compared to the sensor size to minimize edge effects and ensure accurate heat flux measurements. While no strict ratio is specified, common practices indicate that a sample width at least three times larger than the sensor size is typically sufficient for low-conductivity materials. In our study, we used samples with a width of 7.5 cm, which is three times larger than the 2.5 × 2.5 cm² heat flow sensors in the HFM-25 device. This choice aligns with ISO 8301 recommendations for minimizing edge effects. However, as we will see below, edge effects and lateral heat losses were still observed in our results, particularly for thicker samples and extreme temperatures. These findings suggest that, while the dimensions used in this study are generally adequate for low-conductivity materials, additional measures, such as insulating the sample edges, may be required to further mitigate edge effects in certain cases, and allow us to study and understand better the heat transfer process inside the material. This highlights the importance of validating experimental setups through numerical simulations and reference material testing to ensure compliance with ISO standards and achieve accurate thermal conductivity measurements.

### 3.3. Measurement techniques

For some experiments with cellulose, in order to get a detailed view of the temperature distribution inside the sample and compare with numerical predictions, the temperature distribution within the sample was monitored with the help of thermocouples. The temperature measurements within the cellulose samples were performed using *type T* thermocouples. *Type T* thermocouples are composed of copper and constantan and are known for their accuracy in the temperature range of -200 °C to 350 °C, making them suitable for our experimental conditions. The thermocouples used in our experiments had a diameter of 0.45 mm, ensuring minimal disturbance to the sample structure and minimal heat loss by conduction along the radial direction, while providing precise temperature readings at different points within the material. The thermocouples were placed at different heights and radial positions within the sample by creating holes using a specialized drill, as represented in Fig. 4. This method ensures precise placement of the holes and accurate positioning of the thermocouples. Furthermore, experimental validation confirmed that this approach yields results comparable to directly compressing the thermocouples against the sample. Thermocouples were positioned at different radial distances (0 cm, 0.5 cm, 1 cm, 2 cm, 3 cm and 3.4 cm from the center) and at three different depths: middle plane, near the top surface, and near the bottom surface. The experimental conditions were carefully maintained under controlled room settings. Humidity levels were monitored using a Testo device, while room temperature was measured with a Pt100 thermal resistance sensor. The recorded relative humidity was consistently 30 %, and the temperature remained stable with no significant fluctuations throughout the entire experiment.

<figure><img src="image_5.png" /><figcaption>**Fig. 4.** Schematic of thermocouple placement within the sample to obtain detailed temperature distribution during a test with the HFM-25. (a) Side view of the cylindrical sample, where the orange lines correspond to the planes in which thermocouples are placed. These planes include one at the mid-height of the sample, one at the center of the upper half, and one at the center of the lower half. (b) Top view illustrating the six radial positions (R, in cm) of the thermocouples within each plane.</figcaption></figure>
## 3.4. Numerical simulation

To investigate the thermal behavior of cellulose samples under steady-state conditions, a comprehensive heat transfer model was developed using COMSOL Multiphysics. The governing equation for the heat transfer in the isotropic sample is used:

$$ (\rho C_p)_{eff} \frac{\partial T}{\partial t} = k \nabla^2 T \qquad (9) $$

where $(\rho C_p)_{eff} = (1 - \varepsilon)\rho_s C_{p,s} + \varepsilon\rho_a C_{p,a}$ represents the effective volumetric heat capacity accounting for both solid cellulose and air voids, with $C_{p,s} = 1250 \text{ J kg}^{-1} \text{ K}^{-1}$ and $C_{p,a} = 1000 \text{ J kg}^{-1} \text{ K}^{-1}$ the specific heat capacities of solid cellulose and air, and $\rho_s = 1500 \text{ kg m}^{-3}$ is the density of dry cellulose and $\rho_a = 1.2 \text{ kg m}^{-3}$ is the density of air. Within the framework of these simulations we used the values of the different parameters at ambient temperature, whatever the range of imposed temperature. This is a good approximation considering the weak variations of these parameters in the range of temperature tested, even for the thermal conductivity of our cellulose samples (see below). Boundary conditions were defined as follows: the top and bottom plate temperatures were assumed to be respectively equal to $T_{top}$ and $T_{bottom}$. Heat flux at the lateral surfaces accounted for convective losses to ambient air at $T_{\infty} = 22^{\circ}\text{C}$, was expressed by $q = h(T - T_{\infty})$. In order to get a quantitative solution, the governing equations are discretized in space by Finite Element Method using a Galerkin procedure and in time by backward Euler method. The mesh was carefully designed to ensure that mesh size is big enough to optimize the computing time and fine enough to well follow the fields. Moreover, the time step is adaptive according to the convergence rate to obtain quantitative solutions of the governing equations efficiently. Boundary layer refinement was applied near the lateral surfaces to capture localized heat flux variations accurately. The simulation aimed to replicate the steady-state temperature distribution and heat fluxes within the sample, validating the model against experimental measurements.

In our study, we employ a heat transfer coefficient ($h$) derived from kirigami-processed cellulose nanofiber (CNF) films, which demonstrated significant heat dissipation capabilities under both natural and forced convection conditions. The coefficient value, such as $30 \text{ W m}^{-2} \text{ K}^{-1}$ under natural convection (air velocity of $0.12 \text{ m s}^{-1}$), the value that will be adopted in our simulations, was determined in a specific experimental setup that involved controlled ambient environment [32]. In addition, experiments conducted in our lab to estimate $h$ under similar natural convection conditions also yielded a value of approximately $30 \text{ W m}^{-2} \text{ K}^{-1}$. This agreement reinforces the suitability of the chosen value for our study. It is essential to note, however, that the conditions in the CNF study differ from those in our experiment, where the cellulose sample is confined between two plates at imposed temperatures, with lateral surfaces exposed to ambient conditions. Consequently, while adopting the $h$ value from the kirigami CNF study and our lab experiment provides a useful baseline, adjustments might be required to represent more precisely the heat transfer dynamics in our setup. For comparison, simulations were also conducted using $h = 10 \text{ W m}^{-2} \text{ K}^{-1}$, a value commonly reported in the literature for natural convection under standard conditions [27,33]. The motivation for studying this value was to evaluate whether it could adequately represent the heat transfer behavior in our experimental setup. As detailed in Appendix B, simulations with $h = 10 \text{ W m}^{-2} \text{ K}^{-1}$ did not align well with the experimental data, further confirming the appropriateness of adopting $h = 30 \text{ W m}^{-2} \text{ K}^{-1}$. This discrepancy highlights the importance of tailoring the heat transfer coefficient to the specific experimental conditions.

<figure><img src="image_6.png" /><figcaption>Fig. 5. Thermal conductivity of the standard sample NIST 1450E (7.5 cm × 7.5 cm) and air as a function of mean temperature. Measurements were conducted under various experimental conditions, including different sample thicknesses (0.5 cm, 1 cm, and 2.5 cm), with and without insulation around the sample, and under normal ambient air or forced airflow (fan) for the 2.5 cm thick sample. Theoretical values are shown for comparison, with dashed lines representing air conductivity and solid lines representing the theoretical thermal conductivity of NIST 1450E. Error bars represent the variability across repeated measurements.</figcaption></figure>
# 4. Results

## 4.1. Initial observations and calibration process

In the initial phase of our investigation, we analyze the thermal conductivity measurements obtained with the standard material, NIST 1450E. The standard calibration process involves measuring the heat flux (i.e., $Q_{avg} = (|Q_{top}| + |Q_{bottom}|)/2$) across the sample, calculating the sample's thermal conductivity using Fourier's law, i.e., $k_{exp} = Q_{avg}L/(T_2 - T_1)$, and deducing the calibration factor as $f_{cal} = k_{theo}/k_{exp}$, in which $k_{theo}$ is the theoretical value of the conductivity of the standard material, known from precise independent measurements carried out by NIST. The value of $k_{theo}$ as a function of the temperature is shown in Fig. 5. This calibration factor is then used to adjust subsequent measurements with the sample to be specifically studied, by multiplying the apparent conductivity, measured in the same as above, by the factor $f_{cal}$.

First of all, with the standard NIST 1450E sample, we notice variations in the measured heat flux for repeated tests under a priori similar conditions. The resulting uncertainty is indicated by the error bars in Fig. 5. It likely finds its origin in some fluctuations of the ambient conditions (air flux, temperature) or some variations in the contact between sample and plates. This appears as a first source of uncertainty on the subsequent measurements, since an error on the calibration factor value will not necessarily be compensated by the same error in these measurements.

Let us first focus on the calibration factor value obtained for different temperatures under basic experimental conditions, i.e., with the sample thickness of 2.5 cm as provided by the supplier, and with the sample periphery in contact with ambient air. By comparing the values (red circles) for $k_{exp}$ and $k_{theo}$ as shown in Fig. 5, it is clear that $f_{cal}$ widely varies, i.e., typically between 1.3 and 3, for temperature in the range 10–40 °C, with larger values observed at the boundaries of this range. This means that in this context this calibration may not be a slight correction of the data, it can introduce a very large change of the apparent value to obtain the supposedly relevant value. This observation highlights the importance of carefully accounting for edge effects in thermal conductivity measurements to ensure accurate data interpretation. However, it is worth noting that this value does not depend too much on the exact air flow conditions around the sample, since rather close data were obtained with quiet ambient air or strong ventilation (see Fig. 5). Remarkably, the calibration factor is smaller when some insulation is used and placed around the sample: the apparent conductivity is closer to the theoretical value (see Fig. 5). This result, along with our above theoretical analysis of edge effects, suggests that the calibration factor may be sensitive to the sample thickness. To verify this point, we tested samples of different thicknesses. Remarkably, for sample thickness as low as 0.5 or 1 cm, the values seem to align with expected results (from theory) in the full temperature range, with or without isolation at the periphery (see Fig. 5). This suggests that under these conditions, edge effects or heat losses at the edges are now negligible as compared to the main (axial) heat flux. However, a consistent slight discrepancy remains: in such a case measured values are approximately still about 15 % higher than the theoretical values whatever the sample thickness and ambient temperature. This deviation aligns with findings from air conductivity measurements (see Fig. 5 and details in Appendix A), suggesting a correction factor about 1.15. Therefore, all calculated thermal conductivities based on raw heat flux measurements were systematically divided by this factor for consistency. Note that this preliminary correction a priori concerns the measurement system itself, i.e., the sensor and/or data analysis by the apparatus, not the measurement procedure that we are here studying. We believe this effect finds its origin in an error made on the temperatures imposed at the top and bottom surfaces: the heat fluxes are measured, but the temperature is supposedly imposed by the system; there is likely a slight discrepancy between the temperature effectively imposed along the interfaces with the sample and the assumed temperature; this leads to an underestimation of the temperature difference ($T_2 - T_1$) which induces an overestimation of the conductivity; an error of 7 % on each temperature is sufficient to explain the error on the final values.

<figure><img src="image_7.png" /><figcaption>**Fig. 6.** Thermal conductivity, obtained using the standard calibration process based on the reference material (NIST1450E), of cellulose samples with a porosity of 0.5, measured by the Thermtest HFM-25, at different thicknesses ($L = 0.5$ cm–$3.5$ cm) as a function of mean temperature. The secondary axis (red curve) represents the calibration factor variation with temperature.</figcaption></figure>
These findings underline the critical role of sample thickness (or aspect ratio, L/R) in controlling edge effects. The curvature observed with thicker samples, indicative of higher thermal conductivity or enhanced heat flux at extreme temperatures, necessitates further investigation to fully understand and mitigate this behavior for accurate measurements.

Let us now look at the data obtained for our cellulose sample following the standard procedure, including the calibration stage (using the calibration factor determined in a preliminary stage at each temperature). The data with a sample of same thickness as the standard sample (2.5 cm) appear to vary significantly with the tested temperature around a value for the conductivity of the order of 0.08 W m⁻¹ K⁻¹. This is unexpected as bio-based materials generally show less pronounced variations in thermal conductivity compared to synthetic or mineral materials [21]. The results obtained for samples of different thicknesses reveal further unexpected variations in thermal conductivity within this relatively small temperature range (see Fig. 6). For larger thickness (say, up to 3.5 cm), the conductivity at low and high temperature is now about twice that observed for a temperature around 25 °C. On the contrary, for smaller thickness (say, down to 0.5 cm), the conductivity vs temperature has a completely different shape, exhibiting smaller values for extreme temperatures than for 25 °C. It is worth noting that the latter variation with temperature has the same shape as for the calibration factor. This suggests that this calibration has a strong impact on the apparent variations of the conductivity of cellulose with temperature. These inconsistencies are in fact not so surprising considering the problems observed for different thicknesses with the standard material. This justifies that we will study in more details the mechanisms of heat transfers within the material. In this context, we will obviously give up the preliminary calibration process and only use the raw data concerning the heat fluxes, obtained from the sensors of the apparatus.

## 4.2. Flux analysis

To resolve the observed anomalies in direct thermal conductivity measurements, we now focus on the apparent thermal conductivity directly deduced from the measured value for $Q_{\text{avg}}$: $k_{\text{app}} = QL/\Delta T$, i.e., without going through the calibration stage (see above). The variations of this value with the temperature and thickness are now quite clear (see Fig. 7):

i) $k_{\text{app}}$ significantly varies with the temperature when the thickness is larger than 2 cm, and it takes larger values for temperature farther from the ambient one;

ii) $k_{\text{app}}$ decreases when the thickness decreases

iii) When the thickness tends to zero, $k_{\text{app}}$ reaches an asymptotic, finite value, approximately independent of the temperature.

The significant variations of the conductivity observed for large thickness and their disappearance at small thickness suggest that the impact of edge effects is significant for large thickness and progressively disappears when decreasing the thickness. In the following we study this phenomenon in more details.

## 4.3. Advanced flux analysis

Considering the impact of edge effects detected above at large thickness we can suspect that under some conditions the temperature distribution significantly differs from the ideal distribution given by (2), even around the central region. The first way to get some information on this point consists to look at the values of the top and bottom heat fluxes through each of the plates, as measured by the apparatus. Indeed, for the ideal distribution the two fluxes are identical. Here, the measurements reveal significant differences between the top and bottom heat flux values within sufficiently thick samples and at temperatures far from ambient one (see Fig. 8). More precisely, the top heat flux is larger than the bottom one, which indicates asymmetric heat transfer likely caused by edge effects—an

<figure><img src="image_8.png" /><figcaption>Fig. 7. Thermal conductivity of cellulose samples, calculated from the average absolute heat flux measured by the Heat Flow Meter (HFM-25), as a function of mean temperature. The measurements were conducted for samples of varying thicknesses ($L$ = 0.5 cm–3.5 cm).</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>**Fig. 8.** Apparent thermal conductivity ($k_{app}$) derived either from the top or the bottom heat flux measured during Heat Flow Meter (HFM-25) tests on cellulose samples, as a function of mean temperature for thicknesses ranging from 0.5 cm to 3.5 cm.</figcaption></figure>

issue also highlighted in computational studies where boundary-induced thermal losses disrupt expected flow profiles [34]. This effect is even more critical for thick sample and extreme temperatures, since in that case the bottom heat flux is negative, indicating that edge effects are so large that they induce a heat flux from the bottom plate fully counteracting the expected heat flux. In that case we obtain unrealistic negative apparent conductivities. These differences tend to disappear at sufficiently small thickness, which means that edge effects will tend to have a negligible impact on the measurements.

As a corollary of these observations we deduce that the way the average flux is computed by the apparatus, i.e., from the absolute value of the top and bottom fluxes, is not appropriate. An alternative way consists to compute the average flux from the arithmetic mean of real values (negative fluxes retained as negative and positive fluxes retained as positive), leading to $Q = (Q_{top} + Q_{bottom})/2$. Note that this approach has no strict scientific justification, it should just provide an estimation of the conductivity closer to the real one as $Q$ is closer to some mean value of the flux than $Q_{avg}$. Fig. 9 shows how, by comparison with Fig. 7, the correct averaging process leads to conductivity values at large thickness closer to the constant one obtained at low thickness. Thus, despite the existence of significant edge effects, possibly leading to negative bottom flux, the averaging operation, based on heat fluxes measured in the central region of the sample, allows to partly counterbalance the impact of edge effects and finally lead to obtain conductivity values not too far from the real one.

Based on these findings, the conductivity values obtained for 0.5 cm and 1 cm thicknesses can be considered reliable. These thin samples minimize edge effects, and the resulting measurements provide accurate representations of the material's thermal conductivity. This conclusion emphasizes the importance of selecting appropriate sample thicknesses to ensure accurate and meaningful thermal conductivity measurements in the presence of edge effects.

<figure><img src="image_10.png" /><figcaption>**Fig. 9.** Thermal conductivity of cellulose samples, calculated from the arithmetic mean of the top and bottom heat fluxes measured during Heat Flow Meter (HFM-25) tests, as a function of mean temperature. Measurements were conducted for samples of varying thicknesses ($L = 0.5$ cm–3.5 cm).</figcaption></figure>
## 4.4. Simulation results

Numerical simulations using Finite Element Method (FEM) models in Comsol were employed to better understand the heat transfer processes within the samples. In that aim we used the value for the material conductivity as determined for the smallest thickness (around 0.07 Wm⁻¹K⁻¹) without insulation material around to see the heat losses at the edges. Note that for these simulations, the heat transfer coefficient (h) was set to 30 Wm⁻²K⁻¹, as discussed earlier (see Appendix B), a value consistent with previous modeling of anisotropic and layered soil-thermal systems under operational conditions [35]. To illustrate the way the edge effects develop, two cases were selected to emphasize the typical possible characteristics (Fig. 10):

1. **Case a):** Large thickness (2 cm) and temperatures (45 °C at the top and 40 °C at the bottom) far from ambient conditions. This scenario highlights pronounced edge effects, clearly visible in the contour lines of temperature (isothermal lines) with a significant curvature extending up the central region and leading to a non-linear increase of the temperature along the central axis (see Fig. 10-a). The edge effects will obviously be more intense for larger thickness.

2. **Case b):** Smaller thickness (1.5 cm) and moderate temperatures (35 °C at the top and 30 °C at the bottom). In that case edge effects still exist, but they do not extend up to the central axis, and the temperature in the central region increases linearly along the vertical axis (see Fig. 10-b), which agrees with field observations of lateral temperature field evolution around buried thermally stressed systems [36]. The edge effects will be less important for smaller thickness and temperature closer to the ambient one.

These simulated temperature fields can be compared with our measurements of the temperature distribution inside the sample. Assuming that the temperature field is symmetrical by rotation around the central axis, we can consider that the data obtained at different distances even along different angles provides the temperature as a function of the radial distance. We can thus compare these profiles obtained at different depths with the results from simulation (see Fig. 11).

The variations of the temperature along the radial direction are qualitatively consistent with the numerical predictions: the temperature decreases from the central axis towards the exterior for the thicker sample and highest temperature, typically by 3 or 4 °C over 2 cm, while the decrease is somewhat slower (typically by 1 or 2 °C over 2 cm) in the central region for the thinner sample and lowest temperature. Also, in both cases the slope is initially small and increases a lot at the approach of the sample periphery. However, there remains some apparent systematic discrepancy between the experiments and the modelling: in the central region the decrease of the measured temperature with the distance is generally faster than the simulated value (see Fig. 11). This suggests that some important aspect is not taken into account in the simulations.

Actually, remarking that the material has been prepared by compression, tending to align the fibers along the plane perpendicular to the axis of compression we can fear some difference of conductivity in the radial and axial directions. To clarify this point, we measured the thermal conductivity in the radial direction. The sample preparation process is detailed in the *Materials and Sample Preparation* section, where for radial conductivity measurement cylindrical set-ups to compression were replaced by square ones to be able to rotate the sample compressed by 90° before placing it between the plates of the Heat flow meter, ensuring that the primary heat flux direction was now perpendicular to both the compression axis and the fiber alignment. The sample had a final thickness of 1 cm, which was selected based on prior analyses of axial conductivity demonstrating that a thickness of 1 cm minimizes the impact of edge effects. The thermal conductivity measurement was carried out at a mean temperature of 22.5 °C, achieved by setting the top and bottom plate temperatures to 25 °C and 20 °C, respectively. These conditions, previously validated during the axial conductivity experiments, were found to ensure homogeneous heat flux inside the sample, effectively eliminating edge losses. Under these optimized conditions, the radial thermal conductivity was determined to be 0.14 Wm⁻¹K⁻¹. Given the preparation method of cylindrical

<figure><img src="image_11.png" /><figcaption>**Fig. 10.** Temperature contour plots in a half-radial cross-section (central axis on left, interface with air on the right) of the sample for simulated heat transfer in cellulose, considering anisotropic thermal properties ($k_r = 2k_s$), in two cases: (a) $L = 2$ cm and temperature between 40 and 45 °C, (b) $L = 1.5$ cm and temperature between 30 and 35 °C. Simulations represent heat flow meter test conditions with imposed top ($T_1$) and bottom ($T_2$) temperatures, accounting for the anisotropic material properties.</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>**Fig. 11.** Simulated and experimental temperature distributions in cellulose samples of varying thicknesses and radial positions. Solid lines represent simulations with $k_r = 2k_z$, while dashed lines represent simulations with $k_r = k_z$. (a) Temperature profiles for a 2 cm thick sample, measured at different heights, with top and bottom temperatures set to 45 °C and 40 °C, respectively. (b) Temperature profiles for a 1.5 cm thick sample, measured at different heights, with top and bottom temperatures set to 35 °C and 30 °C, respectively. Note that the uncertainty on the measurements is typically of 1 mm concerning the vertical position of the sensors. It can hardly be represented in such graphs.</figcaption></figure>

samples, the material inherently exhibits anisotropy, characterized by two primary conductivities: axial ($k_x$) and radial ($k_r$).

The simulations now carried out with a conductivity equal to 0.14 W m⁻¹ K⁻¹ in the radial direction appear in better agreement with the data (see Fig. 11): for both conditions (thick and thin samples) the temperature profiles exhibit a steeper slope from the central axis, which is closer to the experimental data.

These differences observed by taking into account the material anisotropy can be studied in a more systematic way, now focusing on the corresponding heat losses, and thus, indirectly, the impact on the measured conductivity. In that aim, we compare the top and bottom heat fluxes obtained with the above isotropic and anisotropic conditions.

The introduction of anisotropy ($k_r = 2k_z$) significantly improves the alignment of simulated results with experimental data, particularly for thicker samples. The experimental trend of increasing divergence between the top and bottom fluxes is closely captured by the anisotropic simulations, highlighting the critical role of anisotropy in modeling heat transfer. For thinner samples (e.g., 0.5 cm and 1 cm), the behavior remains similar in both the isotropic and anisotropic cases, with minimal divergence between the top and bottom fluxes (Fig. 12).

The impact of mean temperature becomes more pronounced for thicker samples, amplifying the divergence between the top and bottom fluxes. In both isotropic and anisotropic cases, this divergence increases with sample thickness. However, the isotropic case consistently underestimates this difference (Fig. 12-b), failing to match the experimental data, whereas the anisotropic model successfully captures the observed trends (Fig. 12-a). The effect of anisotropy becomes evident starting from samples with a thickness of 1.5 cm. In this range, the isotropic simulations still show top and bottom fluxes that are very close to each other, while the anisotropic simulations align with the experimental data by reproducing the same difference between the fluxes.

For thicker samples, the anisotropic model not only captures the larger difference between the top and bottom fluxes but also aligns
<figure><img src="image_13.png" /><figcaption>**Fig. 12.** Experimental and simulated values of the product $Q^*L$ representing the apparent thermal conductivity ($k_{app}$) of cellulose samples with porosity 0.5, at the top and bottom during the Heat Flow Meter test, as a function of mean temperature and varying thicknesses. (a) Simulation results under the assumption of anisotropy ($k_r = 2k_z$). (b) Simulation results under the assumption of isotropy ($k_r = k_z$).</figcaption></figure>

more closely with experimental data than the isotropic model. In the isotropic case, the difference between fluxes is smaller than what is observed experimentally, further reinforcing the inadequacy of isotropic assumptions for modeling real cases.

The effect of anisotropy became evident in the cellulose samples, where the deliberate preparation process allowed us to characterize directional thermal conductivities. This observation motivated further investigation of the standard sample, NIST 1450E, provided by the manufacturer.

As previously discussed, initial thermal conductivity measurements for the standard sample across different thicknesses revealed important trends. While thin samples (e.g., 0.5 cm and 1 cm) exhibited a linear relationship with temperature consistent with theoretical predictions, the 2.5 cm sample displayed an unusual parabolic curvature. This deviation, initially unexplained, would be attributed to heat losses at the sample's edges and radial heat transfer effects. Moreover, comparisons between insulated and non-insulated setups showed no difference for thinner samples, while thicker samples (e.g., 2.5 cm) exhibited significant deviations, reinforcing the role of edge effects. These observations prompted a detailed study of the standard sample's anisotropic properties, similar to the approach used for cellulose samples.

To investigate this anisotropic behavior, the thermal conductivity of NIST 1450E was measured in the radial direction, and the resulting data guided the application of anisotropic lattice modeling strategies previously validated in geomaterials [37], sample was prepared as described in paragraph 3.1 (Materials and sample preparation). The experimental data indicate a clear distinction between axial and lateral thermal conductivities (Fig. 13). Simulations were then conducted to further analyze the experimental results. Initially, the study considered a simplified anisotropic assumption with only two distinct conductivities, $k_r$ and $k_z$, which is natural for cylindrical samples where compression induces alignment of fibers in a specific direction. For cellulose samples, this assumption is
<figure><img src="image_14.png" /><figcaption>Fig. 13. Thermal conductivity measurements of NIST1450E in two directions and analogy with the simulation.</figcaption></figure>

appropriate due to the controlled preparation and known (average) alignment of fibers resulting from the compression starting from isotropic orientation. However, when analyzing the standard NIST 1450E sample, additional complexity may be assumed due to the uncertainty in its preparation and internal structure. To account for this, simulations incorporated a more general anisotropy by assigning distinct thermal conductivities to all three principal directions ($k_x, k_y, k_z$), assuming the axes of conductivity correspond to the main geometrical axes of the parallelepiped sample. Although this approach simplifies the anisotropic tensor (e.g., neglecting possible misalignments between geometrical and conductivity axes), it provided an adequate first approach for further exploring the thermal behavior of the sample. These simulations utilized the theoretical temperature-dependent thermal conductivity of the standard material as input, referred to as *k* in this context. Under isotropic assumptions ($k_x = k_y = k_z = k$), the simulations successfully reproduced the theoretical linear variation of thermal conductivity with temperature, as expected for calibration. However, these results could not account for the experimentally observed parabolic trend, emphasizing that isotropy alone was insufficient to describe the behavior of the standard sample.

The best agreement with experimental data was achieved by introducing anisotropy into the simulation model, assigning distinct thermal conductivities to each principal direction ($k_x, k_y, k_z$). Specifically, the configuration 1.5k, 3k, representing relative anisotropic conductivities, provided the best match to the experimental data. This setup effectively captured the variations observed in the measurements, emphasizing the importance of anisotropy in accurately modeling heat transfer.

To explore the effect of anisotropy, simulations were performed by varying the conductivity values along the principal axes, with the Z-axis representing the primary heat flux direction. Different configurations of anisotropic conductivity factors were tested to determine their impact on the effective thermal conductivity, building upon earlier work in computational estimation of anisotropic heat flow in porous media [38].

* *k*, 1.5* *k*, 3*k* (Z Y X): Here, the Z-direction conductivity is the lowest (*k*), with intermediate and higher values for the Y and X directions, respectively. This configuration aligns with the measured axial conductivity of the compressed fiber arrangement, where the heat flux flows along the compression axis, and echoes earlier lattice-based modeling of heat transfer in engineered composite systems [39].

* *k*, 1.5* *k*, 3* *k* (X Z Y): Here, the anisotropic conductivities are arranged such that the Z-direction, still representing the primary heat flux direction, has an intermediate conductivity (1.5* *k*), while the X and Y directions have values of *k* and 3* *k*, respectively. This setup represents another fiber arrangement experimentally measured, where the heat flux is lateral relative to the compression direction.

* *k*, 1.5* *k*, 3* *k* (X Y Z): This assumes the primary heat flux direction (Z) has the highest conductivity (3* *k*), with lower conductivities in the X and Y directions, it could be related to the third arrangement or alignment of fibers. Despite this success, experimental limitations prevented full characterization along the third direction, as the fibrous structure within the xy-plane could not be fully separated or analyzed.

Ultimately, the practical measurements of thermal conductivity, which deviated significantly from theoretical expectations, could only be explained by incorporating anisotropy into the model. This approach successfully reconciled the experimental results with the observed behavior, shedding light on the complexities introduced by edge effects and internal material structure.

While anisotropy and edge effects are fundamentally different phenomena, our study shows that they can strongly interact. Anisotropy does not cause edge effects, but it enhances them by increasing lateral heat redistribution, especially in samples where $k_r > k_z$. This makes edge losses more significant and complicates interpretation of measured fluxes, particularly in thicker samples or when the temperature gradient with ambient air is large. This interaction must be accounted for in both experimental protocols and modeling approaches.
## 5. Conclusion

This study provides a comprehensive analysis of thermal conductivity measurements in insulating materials, with a focus on cellulose samples and the standard material NIST 1450E. Key findings emphasize the importance of accounting for edge effects, anisotropy, and experimental conditions to achieve accurate results.

The results confirm that anisotropy, particularly in fibrous materials like cellulose, plays a critical role in thermal behavior. Axial and radial thermal conductivities exhibit significant differences due to fiber alignment during compression, a feature effectively captured by modeling with $k_r = 2k_z$. This highlights the need for directional measurements to fully characterize thermal properties in anisotropic materials.

For the standard material NIST 1450E, the isotropic assumption was inadequate to explain the experimental data, which displayed a parabolic trend with temperature. Incorporating anisotropic properties into simulations provided a better fit, demonstrating that even standard calibration methods need refinement when applied to anisotropic materials. This underscores the broader applicability of the insights gained in this study, extending to both standard and bio-based insulating materials.

These findings show that the measurements require careful optimization of sample geometry and boundary conditions. For instance, the inclusion of insulation around the sample effectively reduces lateral heat losses for thicker samples, highlighting the importance of considering surrounding conditions during experiments. Additionally, maintaining a uniform temperature distribution along the heating and cooling plates remains a challenge. Simplified assumptions, such as high heat transfer velocity, can introduce uncertainties, emphasizing the need for refined setups to minimize errors.

All in all we find that, in our specific context of 7 cm wide samples, in order to get reliable data for the thermal conductivity measurement in the range of temperatures [10–45 °C], it is necessary to use a sample thickness of about 1 cm, a conclusion valid with or without insulation around the sample. Also, an error of about 10 % can be made when using a larger thickness, i.e., 1.5 cm.

The appropriate thickness found in our study for negligible edge effects is obviously dependent on the specific conditions of our experiments (material properties, sample aspect ratio, boundary conditions). This result can nevertheless be used and extrapolated to other conditions to get an approximate value of the critical thickness. In that aim, we first note that the heat equation (9) becomes independent of the sample size if all distances are rescaled by a characteristic length of the sample, and the time appropriately rescaled. Thanks to the expression Eqn (3) for $h$, a similar conclusion is reached for the boundary condition Eqn (2a) if, in addition, we assume that $f$ slowly varies with $Re$, which is generally verified (it typically varies as the cubic root of $Re$[20]), which allows to neglect this variation as a first approximate. Under these conditions, we deduce that, as a first approximate, similar results for the heat flux distribution will be obtained for two systems of all geometrical characteristics similar by a given factor, and thus similar thermal conductivity will be deduced from these measurement. Thus, as a first approximate, our conclusions can be extrapolated to samples of similar aspect ratios (thickness to diameter ratio). We thus conclude that rather precise data for the thermal conductivity measurement in the range of temperatures [10–45 °C] will in general be obtained with samples of aspect ratio (diameter to thickness ratio) larger than 7, while an error of about 10 % may be done for aspect ratio smaller than 5. This in particular that for thermal conductivity measurement with a heat flow meter the thickness of a typical biobased construction material sample of diameter 50 cm should ideally be smaller than 7 cm.

Finally, the proposed workflow for accurate thermal conductivity measurement can be resumed as follow.

* Avoid standard calibration for low-conductivity or anisotropic samples; instead use raw measured fluxes and apply a known correction factor if needed (e.g. 1.15 in our machine after comparing values of thermal conductivity of air measured with the theoretical ones).

* Use thin samples (i.e., aspect ratio lower than 1/7) to reduce edge losses, especially for bio-based or fibrous materials.

* If thick samples (i.e., larger aspect ratio) are to be used:

* Wrap lateral surfaces in insulating material to limit edge effects.

* Measure both top and bottom fluxes; use the arithmetic mean of real values to estimate effective flux.

* Monitor temperature field inside the sample to detect asymmetries or lateral gradients (use thermocouples).

* Model anisotropic behavior numerically when interpreting non-uniform heat fluxes or comparing with simulations.

* When using standard samples like NIST 1450E, test for anisotropy, even if the material appears homogeneous.

Looking forward, this work offers valuable guidelines for optimizing experimental procedures. For practical applications, researchers should prioritize experimental conditions that minimize edge effects, such as using thinner samples and adding insulation when appropriate. In terms of future research, characterizing thermal conductivity along all three principal axes in fibrous materials remains a key objective. Moreover, further development of boundary conditions and testing configurations could enhance the accuracy and generalization of the results, especially for more complex anisotropic or bio-based materials.

In conclusion, this study lays the groundwork for improving the reliability of thermal conductivity measurements. The integration of advanced models and experimental techniques that account for anisotropy and edge effects represents a promising avenue for future studies, contributing to a more precise characterization of insulating materials in both research and practical applications.

### CRediT authorship contribution statement

**K. Mourda:** Writing – original draft, Validation, Methodology, Investigation, Conceptualization. **Y. Zou:** Writing – review & editing, Validation, Software, Formal analysis. **V.T. Nguyen:** Writing – original draft, Visualization, Investigation, Data curation. **P.**
**Cousso:** Writing – review & editing, Writing – original draft, Validation, Methodology, Investigation, Conceptualization.

**Declaration of competing interest**

The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: COUSSOT reports financial support was provided by ERC. If there are other authors, they declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

**Acknowledgements**

Funded by the European Union (ERC PHYSBIOMAT - 101095764). Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them.

**Appendix A. Measurement of Thermal conductivity of air to ensure an adequate calibration**

To verify the measurement process, thermal conductivity measurements of air were conducted using the Heat Flow Meter (HFM) with hollow sample setups surrounded by insulation material. Two configurations were designed: a hollow cylinder and a hollow cube. The hollow cube setup was specifically used for the smallest thickness (0.5 cm) due to challenges in cutting such thin cylindrical samples, while hollow cylinders were used for thicker samples. Both configurations had the same diameter as the cellulose sample, 7 cm. The tested sample was air, enclosed within the insulating material, as illustrated in Fig. 14.

<figure><img src="image_15.png" /><figcaption>**Fig. 14.** Rigid polyurethane foam samples prepared and cut to measure thermal conductivity of air.</figcaption></figure>

The known theoretical thermal conductivity of air as a function of temperature [40] provided a basis for comparison. For thinner samples, the measured thermal conductivity was approximately 15% higher than the theoretical values. This discrepancy validates the correction factor identified earlier during the measurements of the NIST1450E sample (with its theoretical conductivity already known) and subsequently applied to cellulose samples.

For the thicker sample (2 cm), the measured values were significantly higher than the theoretical ones, emphasizing the impact of edge effects and heat losses. Notably, unlike the cellulose and NIST1450E samples, the parabolic trend in thermal conductivity was absent in the air measurements, as air is isotropic. Consequently, the linear relationship between thermal conductivity and temperature was maintained, albeit with consistently higher values due to the edge effects.

**Appendix B. Impact of heat transfer coefficient on heat flux**

In addition to the simulations presented in the main text, which utilized a heat transfer coefficient ($h$) of $30 \ W m^{-2} K^{-1}$ (Fig. 12a and b), we conducted additional simulations using a reduced $h$ value of $10 \ W m^{-2} K^{-1}$. This value corresponds to typical conditions for natural convection in ambient air with minimal airflow disturbances.

The results of these simulations, illustrated in Fig. 15a and b, show a less accurate fit with the experimental data compared to the results obtained using $h = 30 \ W m^{-2} K^{-1}$. Specifically, the variation in heat fluxes between the top and bottom plates of the sample, which is a key indicator of edge effects and heat losses, is significantly underestimated with $h = 10 \ W m^{-2} K^{-1}$. The simulated apparent thermal conductivities deviate from the experimental values, particularly at higher mean temperatures and for thicker samples.

These observations highlight the sensitivity of the simulations to the choice of the heat transfer coefficient. Including these results in the Appendix allows for a comprehensive understanding of the influence of the heat transfer coefficient on the simulation outcomes and validates the choice of in the main analysis.
<figure><img src="image_16.png" /><figcaption>Fig. 15. Experimental and simulated values with $h = 10 \ W \ m^{-2} \ K^{-1}$ of the product Q\*L representing the apparent thermal conductivity ($k_{app}$) of cellulose samples with porosity 0.5, at the top and bottom during the Heat Flow Meter test, as a function of mean temperature and varying thicknesses. (a) Simulation results under the assumption of anisotropy ($k_r = 2k_z$). (b) Simulation results under the assumption of isotropy ($k_r = k_z$).</figcaption></figure>

## Data availability

Data will be made available on request.

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