$\left(\begin{array}{c}{\theta}_{c}\\ {\Phi}_{hpo}\end{array}\right)=\left(\begin{array}{cc}1& S{R}_{hpo}\\ 0& 1\end{array}\right)\left(\begin{array}{cc}{A}_{po}& {B}_{po}\\ {C}_{po}& {D}_{po}\end{array}\right)\left(\begin{array}{c}0\\ {\Phi}_{2}\end{array}\right)$ (19)

The total heat flux density in the Laplace field is calculated from relation:

${\Phi}_{0}={\Phi}_{1}+{\Phi}_{2}$ (20)

Combining relations (18); (19) and (20), the system leads to:

${\theta}_{c}\left(z,p\right)=\frac{{B}_{po}{B}_{s}}{{D}_{s}{B}_{po}+{D}_{po}{B}_{s}}\Phi \left(0,p\right)$ (21)

The theoretical curve
${T}_{asym}\left(t\right)={L}^{-1}\left({\theta}_{c}\left(z,p\right)\right)$ was obtained by numerical inversion of the Equation (21) using the De Hoog algorithm [18] . The Levenberg Marquert algorithm [19] [20] used with Matlab helps to calculate the E and ρC_{p} parametres with the complete model. This assessment can only drop when converged (reducing the sum of the quadratic difference: Equation (22)) between the experimental curve
$\Delta {T}_{exp}\left(t\right)=T\left(0,t\right)-{T}_{a}$ and theoretical curve, taking as initial values the pre-estimated values (Equations (15) and (16)).

$\Sigma ={\displaystyle \underset{i=1}{\overset{n}{\sum}}{\left[\Delta {T}_{exp}\left({t}_{i}\right)-{T}_{asym}\left({t}_{i}\right)\right]}^{2}}$ (22)

The calculated values of apparent thermal conductivity of composite material can also be inferred from the Equation (23):

${\lambda}_{est}=\frac{{E}^{2}}{\rho {C}_{p}}$ (23)

4. Research Procedure for Measuring Thermal Capacity C_{p}_{ }

Establishing thermal capacity of composite material with different fibre proportions that was achieved had as objective to confirm the results of the pilot study carried out with Equation (23). This test achievement was made known through its water content, colometry sizing which helps to validate the results. To establish the water content, several methods do exist among which the dehydration method (DM) uses in this research. Three research instruments (Figure 9) were used to get the results of the pilot study: drying oven regulated at 105˚C for 24 h to dry the sample; Omega Engineering probe; Dewar Vapour calorimetre 12V_{max}.

4.1. Method of Determining C_{P}_{ }

This method was inspired by the manner in which objects transfer heat. The C_{p} was measured in cold water as well as normal temperature. The thermal heat emitted by the object is as follows:

$Q=m{C}_{m}\Delta \theta $ (24)

In cases where heat loss is neglected within the exchange system, we can therefore conclude that the energy produced by the object is exactly equivalent to that absorbed by water:

${Q}_{hw}+{Q}_{cw}+{Q}_{ca}=0$ (25)

where Q_{hw} equals energy emitted by the object, Q_{cw} energy absorbed by water and Q_{ca} energy absorbed by calorimeter.

For calorimeter sizing, the following equation was used:

${m}_{2}{C}_{e}\left({\theta}_{eq}-{\theta}_{i}\right)+C\left({\theta}_{eq}-{\theta}_{i}\right)+{m}_{2}{C}_{e}\left({\theta}_{eq}-{\theta}_{c}\right)=0$ (26)

If the mass of hot water is equal to that of cold water, we’ll then say that:

$C=\frac{{m}_{2}{C}_{e}\left(2{\theta}_{eq}-{\theta}_{i}-{\theta}_{c}\right)}{{\theta}_{i}-{\theta}_{eq}}$ (27)

To determine the C_{p} sample; we have:

${m}_{1}{C}_{e}\left({\theta}_{eq}-{\theta}_{i}\right)+C\left({\theta}_{eq}-{\theta}_{i}\right)+{m}_{2}{C}_{p}\left({\theta}_{eq}-{\theta}_{wo}\right)=0$ (28)

Leading to:

${C}_{p}=\frac{{m}_{1}\left({C}_{e}+C\right)\left({\theta}_{eq}-{\theta}_{i}\right)}{{m}_{2}\left({\theta}_{wo}-{\theta}_{eq}\right)}$ (29)

where: m_{1} mass of hot water, m_{2} mass of cold water, θ_{c} hot water temperature, θ_{wo} Cold water temperature, θ_{eq} balance point, Calorie capacity of calorimeter

4.2. Experimental Result of C_{p}_{ }

To validate the research method used, we tested the value of the thermal capacity of the calorimeter used. The specification sheet provided shows C_{P} = 30 J・K^{−1}. (Table 5).

We notice a correlation between the average value of the thermal capacity of calorimeter calculated with dehydration method and that shown on the researcher’s specification sheet. The relative difference obtained between the two values is less than 3%. This result enables us to validate the results of the thermal capacities of the composite materials studied. Table 6 presents the set of experimental results obtained for samples E0, E1, E2, E3 and E4.

5. Discussion of Findings

Measurements were carried out on five different samples E0, E1, E2, E3 and E4.

(a) (b) (c)

Figure 9. Testing material for establishing thermal heat capacity; (a) Adjustable oven; (b) Omega engineering probe; (c) Dewar vapor calorimetre.

Table 5. Equilibrium temperatures and thermal heat capacity C_{p} of the calorimeter obtained.

M_{ce}: Mass of cold water; M_{he}: Mass of hot water; T_{ce}: cold water temperature; T: hot water temperature; D: deviation.

Table 6. Experimental thermal capacity of samples.

Three were performed on each sample as well as the average value of each of the samples. For the mechanical tests, Table 7 summarizes all the experimental results.

We notice that the apparent density ρ_{app} and the porosity ε were subsequently obtained from the Equation (30):

${\rho}_{app}=\frac{{M}_{s}}{{V}_{app}}$ (30)

$\epsilon =\frac{{\rho}_{apps}}{{\rho}_{e}}\frac{{M}_{h}-{M}_{s}}{{M}_{s}}$ (31)

where: V_{app} is the apparent density of the composite material, M_{h} wet density of composite material, M_{s} is dry density of composite material and ρ_{e} is the mass density of water.

It should be noted that control protocol on building site recommended by MIPROMALO (Mission de Promotion des Matériaux Locaux) [12] BTC dry compressive strength and with a 14 day greater than 2 MPa. Generally, cases with filling material of 0.6 ≤ R_{C} ≤ 1 MPa blocks should be used to construct houses with first floor and large borders. Figure 10 represents the compressive strength of bricks made according to the proportion of fibres. We noticed that as quantity of fibres increases, so is the solidity of the compressive strength of the composite material. The reverse was supposed to be true, but since the lateritic soil used is very rich in clay, these thatch fibres from the Adamawa region, known for its firm stems (especially when dried) would however, naturally increase the strength of these bricks. At the same time adding fibres increase the porosity (Figure 11) of the material and being too clayish, it therefore creates micropores that help increase the insulation power as well as its flexural strength. (Figure 12) Looking at the most general case, that is: standards of construction on the compressive samples R_{C} ≥ 2 MPa of the observation results, we found out that the brick can be used even at a grade of 4% in fibres content.

Figure 10. Compressive strength of 28 days according to the fibrous content.

Figure 11. Compressive strength of 28 days according porosity of composites materials.

Figure 12. Flexural strength to 28 days according to the fibrous content.

Table 7. Result of mechanical experiment.

As concerns thermophysical characterisation, we initially verified if the new model could react to thermophysical properties when calculated with transient hot plate method. Since these thermophysical parametre are not yet known, we noticed a reduced sensitivity of the thermophysical parametres
$\beta \frac{\partial T}{\partial \beta}$ by using the result obtained with the simplified model (Equations (15) and (16)) Values E_{pres} and (ρC_{p})_{pres} were calculated within a period of time [t_{1}, t_{2}] as much as the thermogram T_{asym}(t) and T_{sinf}(t) superimposed (Figure 13(a)). For example, with the case of sample E3, the numerical calculations of slope δ and η (Figure 13(b)) between limits 100 - 150 s gave us: E_{pres} = 878.34 J・m^{−2} K^{−1}・s^{−1/2} and (ρC_{p})_{pres} = 1.531 × 10^{6} J・m^{−3}・K^{−1}.

The reduced sensitivities of the temperature of parameter E, ρC_{p} and SR_{chs} were calculated as presented in Figure 14.

(a)(b)

Figure 13. Temperature curve for sample E3 (a) T = f(t); (b) T = f(rac(t)) with residues curves.

(a)(b)

Figure 14. Reduced sensitivities curves: (a) for samples E0; (b) for samples E3.

- We noticed that the reduced sensitivities of T to the heating sensor becomes constant after 10 s, this shows that the inertia of the heating sensor could be neglected only 10 s after having measured T. This shows that, although the effect of the inertia becomes important as the proportion of fibres in the material increases, the developed model cannot provide a reliable estimate of the thermal contact resistance at the interface of the heat sensor/material.

- Sensitivities of T to ρC_{p} and to E are decorrelated, but T is sensible to E only between 0 and 500 s and sensible to ρC_{p} practically after 300 s. The link observed between ρC_{p} and R_{i} is as a result of strong thermal initial observed 100 s after having started registering the temperature. This could be fully seen on the residual graph that shows the plotting in Figure 14(b). After having well focused the residual at the centre between 100 s and 220 s and the temperature being very sensitive to E within the gap limit (Figure 14(a) and Figure 14(b), a pre-estimate of E was carried out on all the samples within the set interval of 100 - 220 s. On the contrary ρC_{p} pre-estimate was done from the moment there was no correlation with R_{i}, i.e. between 400 s and 450 s.

Contrary to the simplified model which does not reduce the sum of quadratic differences (Equation (22)) between the model and the experimental temperature, the completed model instead reduces inversely. We therefore observed a convergence between pre-estimated and experimental results as seen in Figure 15, explaining why the completed model portrays a reliable estimation of the thermophysical parameters ρC_{p} and E.

In a bid to validate and confirm the experimental results with the asymmetric hot plate method, we compared the estimated value of (ρC_{p})_{est} with that of ρ_{app} × (C_{p})_{DM} where the value of (C_{p})_{DM} and p_{app} are respectively shown in Table 6 and Table 7. We equally compared the value of λ_{est} (Equation (23)) with λ_{exp} as shown by the Equation (32):

${\lambda}_{exp}=\frac{{E}^{2}}{{\rho}_{app}\ast {\left({C}_{p}\right)}_{DM}}$ (32)

All the results obtained are shown in Table 8.

When comparing results from both methods which aim at determining the volumetric heat capacity, we can clearly see the mean value of apparent thermal conductivity obtained by the asymmetric hot plate method on one hand and that obtained by the Equation (32), perfectly match (Figure 16). As the use of fibre increases the porosity of composite material, so too decreases the thermal conductivity with the addition of fibres. The results of the experiment showed that blocks E4 have a much lower thermal conductivity than the blocks E0. We came to the conclusion that the use of plant fibres increases the weightlessness of the material without reducing its resistance to compression. This can also mean a reduced thermal conductivity.

Table 8. Results of the thermophysical properties.

Simpl: simplified model; compl: completed model; est: estimated; exp: experimental; D: standard deviation.

Figure 15. Results mixture with the complete model for the E4 sample.

Figure 16. Thermal conductivity of materials according to the fibrous content (ASYM: obtained from Equation (23); DM: obtained with Equation (32)).

6. Conclusion

This research work sets out to investigate the energetic and mechanical reinforcement of sun-dried bricks made with laterite and different proportions of thatch fibres from the Meiganga locality of the Adamawa region in Cameroon. Mechanical characterisation studies have shown that with up to 4% fibre incorporated, the composite materials produced have a very good compressive and flexural strength that meets civil engineering standards. The asymmetric hot plate method used to calculate the thermophysical parametres of these different materials was validated with another method, i.e. the dehydration method, used for determining the heat capacity. With the use of apparent thermal conductivity study, we came to the conclusion that with fibres used, a better thermal insulating composite material was the bi-product. These materials can then contribute to ensuring a better thermal comfort of the interior of a building. The extremely hot climate of this region gives credence to the treatment and use of plant fibres to make sun-dried bricks. These could be used to reduce energy consumption thus, limiting the emission of greenhouse gas (especially CO_{2}) when air-conditioners are used.

Nomenclature

T. Temperature (˚C)

E. Thermal effusivity (J・m^{−2}・C^{−1}・s^{−1⁄2})

λ. Thermal conductivity (W・m^{−1}・K^{−1})

h. Convective heat loss coefficient (W・m^{−2}・˚C^{−1})

R. Thermal contact resistance (˚C^{−1}・W^{−1})

C_{p}. Heat capacity (J・kg^{−1}・K^{−1})

ρ. Density (kg・m^{−3})

e. Thickness (m)

θ. Laplace transform of temperature

P. Laplace parameter

Φ. Laplace transform of heat flux

φ_{0}. Heat flux density (W・m^{−2})

ρC_{p}. Volumetric heat capacity (J・m^{−3}・K^{−1})

Subscripts

P_{0}. Insulating blocks

s. Sample

comp. Completed model

simpl. Simplified model

h. Heation element

exp. Experimental

Pre. Pre-estimated

DM. Dehydration Method

Sinf. Semi-infinite medium

c. Center

Cite this paper

Nitcheu, M. , Meukam, P. , Damfeu, J. and Njomo, D. (2018) Thermomechanical Characterisation of Compressed Clay Bricks Reinforced by Thatch Fibres for the Optimal Use in Building.*Materials Sciences and Applications*, **9**, 913-935. doi: 10.4236/msa.2019.912066.

Nitcheu, M. , Meukam, P. , Damfeu, J. and Njomo, D. (2018) Thermomechanical Characterisation of Compressed Clay Bricks Reinforced by Thatch Fibres for the Optimal Use in Building.

References

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[2] Meggers, F., Leibundgut, H., Kennedy, S., Qind, M., Schlaiche, M., Sobekf, W. and Shukuyag, M. (2012) Reduce CO2 from Buildings With Technology to Zero Emissions. Sustainable Cities and Society, 2, 29-36.

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[3] Osseni, S.O.G, Apovo, B.D., Ahouannou, C., Sany E.A. and Jannot, Y. (2016) Caractérisation thermique des mortiers de ciment dopés en fibres de coco par la méthode du plan chaud asymétrique à une mesure de température. Afrique Science, 12, 119-129.

[4] Meukam, P., Noumowe, A., Jannot Y. and Duval, R. (2003) Caractérisation thermophysique et mécanique des briques de terre stabilisées en vue de l’isolation thermique de batiment. Materials and Structures, 36, 453-446.

https://doi.org/10.1007/BF02481525

[5] Lamkharouet, N., Boussaid, S., Ezbakhe, H., EL Bakkouri, A., Ajzoul, T. and El Bouard, A. (2001) Etude thermique de la terre de Larache stabilisée au Ciment. Laboratoire de Thermique, Energie solaire & Environnement, Faculté des Sciences, Université Abdelmalek Essaadi, Tétouan, 541-546.

[6] Bal, H., Jannot, Y., Gaye, S. and Demeurie, F. (2013) Measurement and Modelisation of the Thermal Conductivity of a Wet Composite Porous Medium: Laterite Based Bricks with Millet Waste Additive. Construction and Building Materials, 41, 586-593.

https://doi.org/10.1016/j.conbuildmat.2012.12.032

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https://doi.org/10.1016/j.conbuildmat.2004.03.010

[8] Damfeu, J.C., Meukam, P. and Jannot, Y. (2016) Modeling and Measuring of the Thermal Properties of Insulating Vegetable Fibers by the Asymmetrical Hot Plate Method and the Radial Flux Method: Kapok, Coconut, Groundnut Shell Fiber and Rattan. Thermochimica Acta, 630, 64-77.

https://doi.org/10.1016/j.tca.2016.02.007

[9] Damfeu, J.C., Meukam, P., Jannot, Y. and Wati, E.(2017) Modelling and Experimental Determination of Thermal Properties of Local Wet Building Materials. Energy and Buildings, 135, 109-118.

https://doi.org/10.1016/j.enbuild.2016.11.022

[10] Elhamdouni, Y., Khabbazi, A., Benayad, C., Mounir, S. and Dadi, A. (2017) Thermophysical and Mechanical Characterization of Clay Bricks Reinforced by Alfa or Straw Fibers. IOP Conference Series: Materials Science and Engineering, Vol. 186, Beijing, 12-35.

[11] Houben, H. (2009) Blocs de Terre Comprimé: Normes. CDI, Philadelphia.

[12] Elimbi, A. (2005) MIPROMALO: Protocoles d’analyses des matières premières et des produits finis au laboratoire.

[13] MINTP (2009) Séminaire de renforcement des capacités des controleurs du Ministère des Travaux Publiques aux techniques de controle de qualité des constructions en BTC. Yaounde.

[14] Petkova, R. and Zlatev, P. (2017) Thermal Insulating Properties of Straw Filled Environmentally Friendly Building Materials. Vol. 13, 52-57.

[15] Azhary El, K., Chihab, Y., Mansour, M., Laaroussi, N. and Garoum, M. (2017) Energy Efficiency and Thermal Properties of the Composite Material Clay-Straw. Energy Procedia, 141, 160-164.

https://doi.org/10.1016/j.egypro.2017.11.030

[16] Kossi, I.B., Florent, P.K. and Ouedraogo, E. (2014) Thermal and Mechanical Study of the Adobe Stabilized with Straws and/or Cement at Different Rate. International Journal of Engineering and Innovative Technology, 4.

[17] Maillet, D., Andre, S., Batsale, J.C., Degiovanni, A. and Moyne, C. (2000) Thermal Quadrupoles. Wiley, New York.

[18] De Hoog, F.R. and Stokes, A.N. (1982) A Improved Method for Numerical Inversion of Laplace Transforms. SIAM Journal on Scientific and Statistical Computing, 3, 357-366.

https://doi.org/10.1137/0903022

[19] Gill, P.E. and Murray, W. (1978) Algorithms for the Solution of the Nonlinear Least-Squares Problems. Journal of Numerical Analysis, 15, 977-992.

[20] Bories, S. and Prantt, M. (1995) Transferts de chaleur dans les milieux poreux. Tech. Ing., B8 (250).

[1] Olivier, J.G.J., Schure, K. and Peters, J. (2017) Trends in Global CO2 and Total Greenhouse Gas Emissions. Summary of 2017 Report. PBL Netherlands Environmental Assessment Agency.

[2] Meggers, F., Leibundgut, H., Kennedy, S., Qind, M., Schlaiche, M., Sobekf, W. and Shukuyag, M. (2012) Reduce CO2 from Buildings With Technology to Zero Emissions. Sustainable Cities and Society, 2, 29-36.

https://doi.org/10.1016/j.scs.2011.10.001

[3] Osseni, S.O.G, Apovo, B.D., Ahouannou, C., Sany E.A. and Jannot, Y. (2016) Caractérisation thermique des mortiers de ciment dopés en fibres de coco par la méthode du plan chaud asymétrique à une mesure de température. Afrique Science, 12, 119-129.

[4] Meukam, P., Noumowe, A., Jannot Y. and Duval, R. (2003) Caractérisation thermophysique et mécanique des briques de terre stabilisées en vue de l’isolation thermique de batiment. Materials and Structures, 36, 453-446.

https://doi.org/10.1007/BF02481525

[5] Lamkharouet, N., Boussaid, S., Ezbakhe, H., EL Bakkouri, A., Ajzoul, T. and El Bouard, A. (2001) Etude thermique de la terre de Larache stabilisée au Ciment. Laboratoire de Thermique, Energie solaire & Environnement, Faculté des Sciences, Université Abdelmalek Essaadi, Tétouan, 541-546.

[6] Bal, H., Jannot, Y., Gaye, S. and Demeurie, F. (2013) Measurement and Modelisation of the Thermal Conductivity of a Wet Composite Porous Medium: Laterite Based Bricks with Millet Waste Additive. Construction and Building Materials, 41, 586-593.

https://doi.org/10.1016/j.conbuildmat.2012.12.032

[7] Meukam, P., Jannot, Y., Noumowe, A. and Kofane, T.C. (2004) Thermophysical Characteristics of Economical Building Materials. Construction and Building Materials, 18, 437-443.

https://doi.org/10.1016/j.conbuildmat.2004.03.010

[8] Damfeu, J.C., Meukam, P. and Jannot, Y. (2016) Modeling and Measuring of the Thermal Properties of Insulating Vegetable Fibers by the Asymmetrical Hot Plate Method and the Radial Flux Method: Kapok, Coconut, Groundnut Shell Fiber and Rattan. Thermochimica Acta, 630, 64-77.

https://doi.org/10.1016/j.tca.2016.02.007

[9] Damfeu, J.C., Meukam, P., Jannot, Y. and Wati, E.(2017) Modelling and Experimental Determination of Thermal Properties of Local Wet Building Materials. Energy and Buildings, 135, 109-118.

https://doi.org/10.1016/j.enbuild.2016.11.022

[10] Elhamdouni, Y., Khabbazi, A., Benayad, C., Mounir, S. and Dadi, A. (2017) Thermophysical and Mechanical Characterization of Clay Bricks Reinforced by Alfa or Straw Fibers. IOP Conference Series: Materials Science and Engineering, Vol. 186, Beijing, 12-35.

[11] Houben, H. (2009) Blocs de Terre Comprimé: Normes. CDI, Philadelphia.

[12] Elimbi, A. (2005) MIPROMALO: Protocoles d’analyses des matières premières et des produits finis au laboratoire.

[13] MINTP (2009) Séminaire de renforcement des capacités des controleurs du Ministère des Travaux Publiques aux techniques de controle de qualité des constructions en BTC. Yaounde.

[14] Petkova, R. and Zlatev, P. (2017) Thermal Insulating Properties of Straw Filled Environmentally Friendly Building Materials. Vol. 13, 52-57.

[15] Azhary El, K., Chihab, Y., Mansour, M., Laaroussi, N. and Garoum, M. (2017) Energy Efficiency and Thermal Properties of the Composite Material Clay-Straw. Energy Procedia, 141, 160-164.

https://doi.org/10.1016/j.egypro.2017.11.030

[16] Kossi, I.B., Florent, P.K. and Ouedraogo, E. (2014) Thermal and Mechanical Study of the Adobe Stabilized with Straws and/or Cement at Different Rate. International Journal of Engineering and Innovative Technology, 4.

[17] Maillet, D., Andre, S., Batsale, J.C., Degiovanni, A. and Moyne, C. (2000) Thermal Quadrupoles. Wiley, New York.

[18] De Hoog, F.R. and Stokes, A.N. (1982) A Improved Method for Numerical Inversion of Laplace Transforms. SIAM Journal on Scientific and Statistical Computing, 3, 357-366.

https://doi.org/10.1137/0903022

[19] Gill, P.E. and Murray, W. (1978) Algorithms for the Solution of the Nonlinear Least-Squares Problems. Journal of Numerical Analysis, 15, 977-992.

[20] Bories, S. and Prantt, M. (1995) Transferts de chaleur dans les milieux poreux. Tech. Ing., B8 (250).