Clay samples should be taken at depths of −30 cm and −50 cm to avoid sampling the organic part of the soil (Figure 4). For our study, we needed samples from Bangangté town, specifically from the entrance to the town in a mud brick production area (Figure 5).

After removing the soil, leave it in the sun for a few days to dry at room temperature. 24 hours before making the test pieces, place them in an oven set at room temperature to maximise drying.

The robust cement used, from the Cameroon Cement Works (CIMENCAM), is of the Portland grey type with additives (pozzolan or limestone), with the characteristics prescribed by the Cameroonian standard for hydraulic binders. It corresponds to the cement type CEM II B-P 42.5R. Cement, a hydraulic binder, is suitable for soils with low clay content or low plasticity and for soils with low-activity clays. During its hydration with the water contained in the soil, cement is transformed over time into stable, water-resistant crystals that form bonds between the grains of the soil. This reaction requires a sufficient amount of water. Furthermore, as the cement reaches its hardness and maximum strength in 28 days (the time required for the cement to set well in the material), the blocks must remain wet during this period. They are covered with a plastic sheet, or if this is not possible, they are watered. In their work, the authors of [8] show an increase in compressive strength with increasing cement content in the material.

The tree known locally as padauk (Figure 6) belongs to the Papilionaceae family (Leguminosae, Papilionoideae, Fabaceae) and has the scientific name Pterocarpus soyauxii. It is located in the equatorial forests of Nigeria up to the Central African Republic, as well as the Democratic Republic of Congo and Angola. Pterocarpus soyauxii is distributed from Cameroon to the Democratic Republic of Congo. This tree is a substantial species found in the tropical forest, with a straight, cylindrical trunk that can grow up to 15 - 20 metres in height and regularly exceeds 1.50 metres in diameter. Its sapwood, which is 10 cm thick, is white in colour, and it is clearly distinct from the heartwood, which has a blood-red colour. The colour changes to purplish-brown in the presence of air. A perfect

wood demonstrates good resistance against weathering agents and is thus classified as durable. As with all sapwood, padauk sapwood is less resistant to attack by wood-wetting agents than heartwood. Padauk is a semi-heavy wood with an average density of 770 kg/m³. It is highly stable and has high hygroscopicity. Padauk exhibits low mechanical strength. The analysis of the chemical composition by the Center Technique Forest-ier Tropical (C. T. F. T.) revealed the following: alcohol-benzene extractables at a rate of 12.00%, water at 1.37%, pentosans at approximately 10.95%, cellulose at 41.85%, and lignin at 30.55%. Due to its durability, padauk is suitable for interior carpentry such as stairs, floors, and notably truck-bed floors. It can also be used for building structures, frames, canoes, and exterior carpentry such as windows and doors. The wood’s colour is valued for carving, furniture, cabinets, knife and tool handles, combs, walking sticks, and musical instruments (C. T. F. T., 1978). The reddish colour of the heartwood makes it a source of dye for the textile industry. This dye is used to give a red colour to fabrics and clothes. The wood contains high levels of tannins, which facilitate mordanting during fibre dyeing. In 1980, the C. T. F. T. conducted a study on the ethereal extract and identified the following compounds: homopterocarpine, pterostilbene, prunetine (a dihydroxy-4’,5,methoxy-7 isoflavone), and sandalwood (a trihydroxy4’,5,5,7-methoxy isoflavone).

After collecting the sawdust without a sawmill, allow it to dry at room temperature for a week before using.

The asymmetrical hot-plane method (Figure 7) is advantageous due to its simplicity, precision, speed, and ease of implementation. This method enables the

estimation of both thermal conductivity and effusivity [24] . In comparison to other methods, such as the hot-disk method or the hot-wire method (which is not suitable for lightweight materials), it requires little financial investment. This method will be used for our studies, as it provides both technological and financial benefits.

• The method’s principle involves applying a constant heat flow step (φ = 0 for t < t₀ and φ = φ₀ for t > t₀) to the heating resistor. The temperature evolution of Ts(t) is recorded in the centre of this resistor, where a thermocouple is in place. While the disturbance has not affected the other faces, and assuming the semi-infinite medium hypothesis is valid (i.e., while T_e(t) remains constant), we assume the transfer in the centre of the sample to be unidirectional. Modelling this heat transfer enables the calculation of the temperature evolution in the centre of the sample (Figure 8). The values of the following are calculated by applying the following parameter-estimation method.

• The thermal effusivity, E, can be calculated using the formula E = λ ρ c , where λ represents the thermal conductivity of the material, ρ represents the density of the material, and c represents the specific heat capacity of the material.

• The thermal capacitance (mc)_s of the probe and assembly of the heating resistor.

• The contact resistance R_c at the interface between the probe and the sample should be minimised to reduce the difference between the theoretical and experimental T_s (t) curves.

• Modelling of the semi-infinite heated plane (Figure 9).

In a hot-plane device, an electric resistor with small thickness, mass (m_s), and heat capacity (C_s) is placed between two samples of the material to be characterised. The resistor is subjected to a density of heat flow Φ and has a supposedly uniform temperature T_s.

The sample demonstrates the heat equation:

∂ 2 T ∂ x 2 = 1 a ∂ T ∂ t (1)

Consider the following boundary conditions:

{ T ( x , 0 ) = T s ( 0 ) = T i                                           ( 2 ) T ( ∞ , t ) = T i                                           ( 3 ) h [ T s ( t ) − T ( 0 , t ) = − λ ∂ T ( 0 , t ) ∂ x ]                                           ( 4 ) ∅ 0 S 2 = m s c s d T s d t + h S [ T s ( t ) − T ( 0 , t ) ]                                           ( 5 )

Relation (1) expresses the conservation of heat flux at the surface of the semi-infinite medium, while relation (2) expresses the conservation of heat flux at the level of electrical resistance.

The variable is changed as follows: T ¯ = T − T i from ∂ T ¯ ∂ X = ∂ T ∂ X , ∂ 2 T ¯ ∂ X = ∂ 2 T ∂ X et ∂ T ¯ ∂ t = ∂ T ∂ t .

We can express Equation (1) as follows:

∂ 2 T ¯ ∂ X 2 = 1 α ∂ T ¯ ∂ t (6)

{ T ¯ ( x , 0 ) = 0                                             ( 7 ) T ¯ ( ∞ , t ) = 0                                             ( 8 ) h [ T ¯ s ( t ) − T ¯ ( 0 , t ) = − λ ∂ T ¯ ( 0 , t ) ∂ x ]                                             ( 9 ) ∅ 0 S 2 = m s c s d T ¯ s 2 d t + h S [ T ¯ s ( t ) − T ¯ ( 0 , t ) ]                                           ( 10 )

θ ( X , p ) = A e − q x ; A = θ ( 0 , p ) .

θ s ( p ) = ∅ 0 S 2 P 1 + R C E S P m s c s 2 p + [ R C m s c s 2 P + 1 ] E S P

Equations (4) and (5) are transformed into Laplace space as follows:

{ h [ θ s ( p ) − θ ( 0 , p ) ] = − λ ∂ θ ( 0 , p ) ∂ x = P a λ θ ( 0 , p ) = E P θ           ( 11 ) θ 0 S 2 P = m s c s 2 p θ ( p ) + h s [ θ s ( p ) − θ ( 0 , p ) ]           ( 12 )

θ s ( p ) = ∅ 0 S 2 P 1 + R C E S P m s c s 2 p + [ R C m s c s 2 P + 1 ] E S P (13)

∅ s ( p ) = ∅ 0 S 2 1 + R c E S p ( m c 2 ) s p + E S p = ∅ 0 2 E p 3 / 2 ( 1 + R C E S p ) [ 1 − ( m c ) s 2 E S p ] (14)

θ s ( p ) = ∅ 0 2 E p 3 / 2 [ 1 + ( R C E S p − ( m c ) s 2 E S ) p ] = ∅ 0 2 E p 3 / 2 + ∅ 0 2 ( R C − ( m c ) s 2 E S ) (15)

T s ( t ) − T s ( 0 ) = ∅ 0 S ( R c − ( m c ) s 2 E S ) + ∅ 0 E π t (16)

Plotting T s ( t ) − T s ( 0 ) against t results in a straight line with a slope of ∅ 0 E π , which can be used to determine the thermal effusivity E. The temperature is not affected by the probe’s inertia or the contact resistance over a significant duration. The assumption of a semi-infinite medium must be valid for the chosen estimation interval to apply this estimation method.

Complete 1D model

Heat transfer in a solid that is opaque, homogeneous, and isotropic can be mathematically represented using the heat equation

1 a ∂ T ∂ t = ∇ 2 T (17)

The parameter “a” represents the thermal diffusivity and is a function of λ and ρcP.

The system is modelled with the assumption that the heat transfer remains unidirectional towards the centre of the device throughout the measurement period. The hypothesis is verified through a 1D simulation on COMSOL and an analysis of the estimation residuals, which are the differences between the temperature obtained through the 1D model T_mod(t) and the experimental temperature T_exp(t).

Based on these assumptions, we can express the following.

• By considering the heat flux density ∅ 01 leaving the probe towards the sample.

[ θ ∅ 01 ] = [ 1 0 C h p 1 ] [ 1 R c 0 1 ] [ A B C D ] [ A i B i C i D i ] [ 0 ∅ 1 ] = [ A 1 B 1 C 1 D 1 ] [ 0 ∅ 1 ] (18)

∅ 01 = θ 1 D 1 B 1 (19)

[ θ ∅ 02 ] = [ A i B i C i D i ] [ 0 ∅ 2 ] (20)

∅ 02 = θ D i B i (21)

∅ 0 = ∅ 01 + ∅ 02 = ∅ 0 S (22)

A = D = cosh ( q e ) ; B = sinh ( q e ) λ q ; C = λ q sin ( q e ) ; avec q = p a (23)

A i = D i = coshi ( q i e i ) ; B i = sinh ( q i e i ) λ i q i ; C i = λ i q i sinh ( q i e i )

q i = p a i (24)

λ: The thermal conductivity of the sample.

a: The thermal diffusivity of the sample.

e: The thickness of the sample.

λ_i: The thermal conductivity of polystyrene.

a_i: The thermal diffusivity of polystyrene.

e_i: The thickness of the polystyrene.

θ ( p ) = ∅ 0 ( p ) D 1 B 1 + D i B i (25)

θ: The Laplace transform of T(t).

∅ 01 : The Laplace transform of the flux dissipated towards the sample.

∅ 02 : The Laplace transform of the heat flux dissipated towards the polystyrene.

∅ 0 : The Laplace transform of the total heat flux produced by the heating element.

φ₀: The total heat flux produced by the heating element.

C_h: The heat capacity of the heating element per unit area: C_h = ρ_hc_he_h.

R_c: The contact resistance between the heating element and the sample.

∅ 1 : The Laplace transform of the flux density transmitted to the upper aluminium block.

∅ 2 : The Laplace transform of the flux density transmitted to the lower aluminium block Simplified model.

Simplified model

At the beginning of the heating process, the sample behaves as a semi-infinite medium. The temperature at the centre of the heated surface confirms this statement.

θ ( p ) = ∅ 0 ( p ) E i p + C h p + ( 1 + R c C h p ) E p 1 + R c E p (26)

For sufficiently long times (p → 0) and considering a flux step, ∅ 0 ( p ) = φ 0 p limited development, which leads to:

θ ( p ) ≈ φ 0 p 3 / 2 ( E + E i ) + φ 0 R c E 2 − C h p ( E + E i ) 2 (27)

The thermal effusivity can be estimated from the slope α of the linear segment of the curve using inverse Laplace transformation.

(t) = ( t ) with:

E = φ 0 α π − E i (28)

Furthermore, assuming a semi-stationary regime at longer times enables us to express

∅ = [ ρ c e + ( ρ c e ) i − ( ρ c e ) s ] d T d t (29)

The heat capacity ρc can be estimated from the slope β of the curve T(t) using the following relationship:

ρ c = ∅ β − ( ρ c e ) i − ( ρ c e ) s (30)

The experimental equipment (Figure 10) for producing test specimens comprises the metal mould, along with its accessories and the hydraulic press. The mould is constructed using an 8 mm thick steel sheet designed specifically to resist deformation under compaction pressure. While the openings on the lid and the bottom of the mould enable the filling of the mould, they also permit the unmoulding of the specimen after compaction. The sliding action of two metal plates inside the mould enables the flattening of the specimen surfaces.

• Sieving

The soil has been double-sifted in order to retain only particles with a grain size with a diameter between 4 mm and 12.5 mm. This grain size makes it possible to eliminate the fine particles that are not sufficiently porous. In fact, the presence of pores (filled with air) in the soil aggregates helps to reduce the thermal conductivity of the material and makes it a better thermal insulator [25] . Furthermore, the mechanical resistance increases with the fineness of the soil due to the reduction in the volume of air. A granulometry of diameters between 4 mm and 12.5 mm was chosen to find a compromise between good thermal insulation and acceptable mechanical resistance [25] .

The grain size of the soil used is less than 12.5 mm.

• Preparation of the mix to be compacted (Figure 11)

• After the materials have dried, and to carry out the compaction, water is added to the mixture of cement and laterite until the desired consistency is reached. The amount of water needed to obtain this consistency is 13% of the total mass (Table 1).

Evolution of the parameters with the stabilizer (Conductivity, Table 2 and Figure 12).

Evolution of the parameters with the stabilizer (Effusively, Figure 13).

Evolution of the parameters with the stabilizer (Diffusivity, Figure 14).

Evolution of the parameters with the stabilizer (Heat capacity, Figure 15).

In the literature, many authors have evaluated the thermal properties of compressed and stabilised earth blocks (Table 3):

For bricks stabilised with sawdust, the thermal conductivities range from 1.04 W/m/K to 0.64 W/m/K. These values are close to those obtained for the thermal conductivities of CCR-stabilized earth blocks and fibres (Table 4) as reported in [37] , which range from 1.02 W/m/K to 0.84 W/m/K. Bricks stabilised with sawdust are classified as insulating materials due to their relatively low conductivity values. This characteristic results in a slowdown of heat transfer and heat flow within the materials. These low values can be attributed to the high porosity levels of the blocks, as the pores contain air, which is a good insulator. The specific heat capacity of the cement-stabilised brick is higher than that of the sawdust-stabilised brick. Therefore, cement-stabilised brick will have a greater heat storage capacity than sawdust-stabilised brick.

Thermal diffusivity values are 4.32 × 10⁻⁷ m²/s for unstabilised bricks, ranging from 4.32 × 10⁻⁷ m²/s to 5.90 × 10⁻⁷ m²/s for bricks stabilised with sawdust and from 4.32 × 10⁻⁷ m²/s to 9.82 × 10⁻⁷ m²/s for bricks stabilised with cement. These values are consistent with the findings of study [37] . The low values are attributable to the insulation properties of these materials. Heat flow through walls constructed with these bricks, particularly those stabilised with sawdust, will take a considerable amount of time. Diffusivity is a measure of the material’s ability to transfer heat.

CCR: Calcium Carbide Residue.

The thermal effusivity value for the unstabilised brick is 1404 (J/m²∙K∙s^−1/2). The values for the cement-stabilised bricks are, respectively, 1182 (J/m²∙K∙s^−1/2), 1100 (J/m²∙K∙s^−1/2), and 1096 (J/m²∙K∙s^−1/2) for cement percentages of 4%, 6%, and 8%. The effusivity values for the sawdust stabilised bricks are, respectively, 1150 (J/m²∙K∙s^−1/2), 1050 (J/m²∙K∙s^−1/2), and 906 (J/m²∙K∙s^−1/2) for sawdust percentages of 4%, 6%, and 8%. The high effusivity values result in bricks quickly absorbing a significant amount of energy without noticeable surface heating.

The estimation of a parameter is only possible if its reduced sensitivity is non-zero and if all other examined parameters have a zero or constant reduced sensitivity.

The principle of reduced sensitivity analysis has been described by several authors [25] . The reduced temperature sensitivity with respect to a parameter is calculated by

k i ∂ T ∂ k i (31)

In practice, the sensitivity of T to the parameter x is calculated as follows:

x ∂ T ∂ x = x T ( x ) − T ( 1.001 x ) 0.001 x = 1000 [ T ( x ) − T ( 1.001 x ) ] (33)

Figure 16 shows the reduced sensitivities of the temperature at the centre of the probe to the parameters E, R_c, and ρc. It can be seen that the temperature at the centre of the probe is at all times significantly sensitive to the effusivity E of the material. Moreover, this sensitivity is not correlated with that of the other parameters. We also note that the temperature is not sensitive to the volumetric heat capacity of the sample at the beginning (up to a time t₁ = 150 s in this case). A simultaneous estimation of the parameters E and ρc from the full model is possible if the estimation time is greater than t1 but within the limits of the unidirectional transfer hypothesis.

Residual curves show the difference between experimental and theoretical data. When residuals are zero (curves at the centre value of zero) within a specified interval, the theoretical and experimental findings are in agreement. Residual curve analysis supports the choice of estimation duration. They can also verify the validity of the hypotheses of unidirectional transfer and the semi-infinite medium.

The residues curve in Figure 17 indicates that the experimental and simulated curves perfectly overlap, indicating that the complete model minimises the sum of quadratic deviations between the two curves. The residues are not signed. The residues are entirely flat throughout the estimated duration. This suggests that accurate thermal parameter estimation can be achieved over the 0 to 500 s time interval.

The Levenberg-Marquart algorithm minimises errors between experimental and simulated temperatures. Figure 18 indicates that the results have converged. The perfect overlap of the two curves demonstrates the successful minimisation of differences between the model and the experiment.

In Figure 19, a manifestation of three-dimensional effects during the sample test can be observed. The residues remain perfectly flat until 400 s; a detachment is then observed. This change indicates that the heat transfer is no longer centred and that the convection phenomenon has started. An accurate estimate of the

sample’s thermophysical parameters should be made between 100 and 300 s. This observation is generalisable to much longer periods. It has been observed that the densest materials demonstrate less sensitivity to heat flow.

During data collection, the experimental curve and the theoretical curve may converge relatively, but the residues may not be centred at the end. Possible causes of this phenomenon are the low quality of the energy source, a defect in the specimen’s dimensioning, or an unevenness in the specimen’s surface.

This work, which is part of the theme of sustainable development, is applied in various sectors. In the field of recycling, it shows a way of valorising sawdust, which is very often used only as firewood. It also contributes to the fight against global warming, as we show that the stabilisation of blocks of earth with agricultural residues can be used as a substitute for cement, thus reducing the emission of greenhouse gases. Understanding the properties of materials helps engineers to better design homes by facilitating the assessment of thermal comfort.