# Equations for Calculating the Thermal Conductivity of Capillary-Porous Materials with over Sorption Moisture Content

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Equations for Determining the Effective Thermal Conductivity of Moist Porous Materials with Closed Gas Inclusions

^{2}.

## 3. Thermal Conductivity of a Three-Component System with a Specified Structure of the Gas Component

_{m}at a known volumetric gas concentration ${m}_{2}\ge {c}_{g}^{3}$. Figure 2 shows that the volume of the pore channels is ${V}_{m}=3{\Delta}_{m}^{2}\left(L-{\mathsf{\Delta}}_{g}\right)$. On the other hand, the same volume is equal to ${V}_{2}-{\Delta}_{g}^{3}$, where ${V}_{2}$ is the total volume of gas in the cell. After dividing these volumes by the cell volume $V={L}^{3}$, we come to the equation

## 4. Thermal Conductivity of a Three-Component System with a Specified Structure of the Liquid Component

## 5. Limitations of the Application of the Formulas

_{3}= 0.4288 and ${m}_{2}=P-{m}_{3}=0.0712$ were found for this ${K}_{m}$. Further, using formulas (20) and (21), the thermal conductivities ${\lambda}_{a}=0.6149$λ W/(m·K) and ${\lambda}_{iz}=0.7324$W/(m·K) were determined, and the estimate of the effective thermal conductivity $\lambda =0.5\left({\lambda}_{a}+{\lambda}_{iz}\right)=0.6736$ W/(m·K) of the three-component system was calculated.

## 6. Preparation of Initial Data for the Calculation

^{2}/s;

## 7. Calculation Examples

^{3}

^{3}[16], the coefficient of variation ${V}_{\lambda}$ (the ratio of s to the sample mean $\stackrel{\u0304}{\lambda}$) was 0.16 (16%). According to our data obtained for nine dry samples with apparent density ρ from 1510 to 1900kg/m

^{3}, the coefficient of variation was 0.073.

**Example**

**1.**

^{3}, the density of the solid skeleton ${\rho}_{s}=2650$ kg/m

^{3}according to pycnometry, and the thermal conductivity of material in the dry state, ${\lambda}_{0}=0.74$ W/(m∙K), and in the water saturated state, ( $\psi =0.291$) ${\lambda}_{w}=1.107$ W/(m∙K). The thermal conductivity was measured by a stationary method at a temperature of 20 °C. The contact angle $\theta $ was taken as equal to 45°. The same temperature and angle θ values were also used in the remaining examples.

**Example**

**2.**

^{3}had the thermal conductivity ${\lambda}_{0}=0.63$ W/(m∙K) in a dry state and ${\lambda}_{w}=1.02$ W/(m∙K) in a water-saturated state ( $\psi =0.3$ ). The density of the skeleton of the material ${\rho}_{s}=2647$ kg/m

^{3}was determined by Formula (36).

**Example**

**3.**

^{3}, ${\lambda}_{0}=0.69$ W/(m∙K), and ${\lambda}_{w}=1.0$ W/(m∙K) at $\psi =0.23$ (water saturation). By Formula (36), ${\rho}_{s}=2683$ kg/m

^{3}. On the basis of this information, as before, the initial data were prepared by calculations using formulas (20) and (21): ${\lambda}_{a}=0.804$ W/(m∙K), and ${\lambda}_{iz}=1.031$ W/(m∙K). The average value $\lambda =0.5\left({\lambda}_{a}+{\lambda}_{iz}\right)=0.9175$ W/(m∙K), taken as an estimate of the effective thermal conductivity of the material, turned out to be 8.25% lower than the experimental value ${\lambda}_{w}=1.0$ W/(m∙K). As is shown in Table 1, the deviations of the calculated values of λ from the experimental ones observed in these examples are practically two times less than the mentioned coefficient of variation ${V}_{\lambda}=16\%$ (relative standard deviation). Therefore, there is reason to believe that the proposed formulas with reliable experimental data are capable of predicting the thermal conductivity of moist capillary-porous wall materials with sufficient accuracy.

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The eighth part of a unit cell with an isolated inclusion of gas and components of the total heat flux ${G}_{1}\dots {G}_{4}$ (

**a**); connection diagram of the resistances of individual elements during adiabatic (

**b**) and isothermal (

**c**) division of the unit cell; 1, 2, 3—solid, gas, and liquid components, respectively.

**Figure 2.**The eighth part of a unit cell with a specified structure of a continuous gas component; 1, 2, 3—solid, gas, and liquid components, respectively.

**Figure 3.**The eighth part of a unit cell with a specified structure of a continuous liquid component; 1, 2, 3—solid, gas, and liquid components, respectively.

**Table 1.**Experimental data for clay bricks obtained by us and published in [15] (ρ, ${\rho}_{s}$, ${\lambda}_{0}$,$\psi $, ${\lambda}_{w})$, results of calculations (${\lambda}_{a,}$ ${\lambda}_{iz}$, $\lambda ),$ and deviation between experimental (${\lambda}_{w})$ and computational ($\lambda $) values of thermal conductivity.

Calculation Example | Experimental Data | Results of Calculations | $\mathbf{Deviation}\mathbf{between}{\mathit{\lambda}}_{\mathit{w}}$$\mathbf{and}\mathit{\lambda}$ | ||||||
---|---|---|---|---|---|---|---|---|---|

Ρkg/m^{3} | ${\mathit{\rho}}_{\mathit{s}}\mathbf{kg}/{\mathbf{m}}^{3}$ | ${\mathit{\lambda}}_{0}\mathbf{W}/(\mathbf{m}\xb7\mathbf{K})$ | $\mathit{\psi}$ | ${\mathit{\lambda}}_{\mathit{w}}\mathbf{W}/(\mathbf{m}\xb7\mathbf{K})$ | ${\mathit{\lambda}}_{\mathit{a}}\mathbf{W}/(\mathbf{m}\xb7\mathbf{Kne})$ | ${\mathit{\lambda}}_{\mathit{i}\mathit{z}}\mathbf{W}/(\mathbf{m}\xb7\mathbf{K})$ | $\mathit{\lambda}$ | ||

1 | 1640 | 2650 | 0.74 | 0.291 | 1.107 | 0.9239 | 1.1677 | 1.0458 | 5.53 |

2 | 1600 | 2647 | 0.63 | 0.3 | 1.02 | 0.8148 | 1.0374 | 0.9261 | 9.2 |

3 | 1820 | 2683 | 0.69 | 0.23 | 1.0 | 0.804 | 1.031 | 0.9175 | 8.25 |

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**MDPI and ACS Style**

Alsabry, A.; Backiel-Brzozowska, B.; Nikitsin, V.I.; Nikitsin, S.K.
Equations for Calculating the Thermal Conductivity of Capillary-Porous Materials with over Sorption Moisture Content. *Sustainability* **2022**, *14*, 5796.
https://doi.org/10.3390/su14105796

**AMA Style**

Alsabry A, Backiel-Brzozowska B, Nikitsin VI, Nikitsin SK.
Equations for Calculating the Thermal Conductivity of Capillary-Porous Materials with over Sorption Moisture Content. *Sustainability*. 2022; 14(10):5796.
https://doi.org/10.3390/su14105796

**Chicago/Turabian Style**

Alsabry, Abdrahman, Beata Backiel-Brzozowska, Vadzim I. Nikitsin, and Serafim K. Nikitsin.
2022. "Equations for Calculating the Thermal Conductivity of Capillary-Porous Materials with over Sorption Moisture Content" *Sustainability* 14, no. 10: 5796.
https://doi.org/10.3390/su14105796