# Measurements and Modeling of Thermal Conductivity of Recycled Aggregates from Concrete, Clay Brick, and Their Mixtures with Autoclaved Aerated Concrete Grains

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^{*}

## Abstract

**:**

_{e}) and effective saturation (S

_{e}). The new λ

_{e}(S

_{e}) models performed well for the measured data compared to previously proposed models and would be useful to evaluate λ of recycled aggregates for roadbed materials.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Materials

#### 2.2. Testing Methods

#### 2.2.1. Preparation of the Samples

^{−3}). The dry density and total porosity of the tested samples are shown in Table 2.

#### 2.2.2. Water Retention Curve Measurement

^{3}m

^{−3}), θ

_{r}(m

^{3}m

^{−3}), and θ

_{s}(m

^{3}m

^{−3}) are the volumetric water content, residual volumetric water content, and the saturated volumetric water content, respectively; k is the number of the peak pore size density or subsystems which form the total pore size distribution; w

_{i}represents the sub-curve weighting factors; and α

_{i}, n

_{i}, and m

_{i}are the parameters of the sub-curves (m

_{i}= 1 − 1/n

_{i}) that indicate fitted parameters. The equivalent pore size distributions of the tested samples were calculated by using the equation which was defined by Durner [20]:

^{*}is the specific moisture capacity (C

^{*}= dθ/dψ).

_{s}) increased with the increased proportion of AAC grains for both blended RC and RCB samples, i.e., the θ

_{s}values of samples with 20% and 40% blended AAC were 1.8 to 2.2 times higher than those of the single material for RC mixtures and 1.2 to 1.4 times higher than those of the single material for RCB mixtures. This indicated that the water retention capacities of RC and RCB were improved by blending in AAC grains as expected above. In addition, in air-dried conditions, tested samples retained a high residual volumetric water content (θ

_{r}). This may be because tested materials, which were compacted before measurement, had high water absorption capacity. Compacted samples are usually very dense and contain many micropores (see Figure 3b,d), which prevent the movement of water out of the samples under air-dried conditions.

#### 2.2.3. Measurement of Thermal Properties

#### 2.3. Statistical Evaluation of the Model Prediction

_{i}(W m

^{−1}K

^{−1}) is the difference between the ith predicted and measured thermal conductivities, n is the number of measurements, and k is the number of model parameters.

## 3. Model Development for Estimating Thermal Conductivity

#### 3.1. Existing Models for Estimating Thermal Conductivity

_{f}, λ

_{s}(W m

^{−1}K

^{−1}) are thermal conductivities of pore fluid and solid phases; ϕ (m

^{3}m

^{−3}) is the total porosity of the soil. According to Beziat [36] and Zhang et al. [37], when soil consists of three phases (solid, water, and gas), thermal conductivity can be calculated as below:

_{s}and λ

_{w}, and λ

_{a}(W m

^{−1}K

^{−1}) are the thermal conductivities of the solid phase, water, and air, respectively; S

_{r}(%) is the degree of saturation:

_{s}(m

^{3}m

^{−3}) are the volumetric water content and the saturated volumetric water content. Note here that water is more thermally conductive than air by a factor of >20 (λ

_{w}= 0.57 W m

^{−1}K

^{−1}and λ

_{a}= 0.025 W m

^{−1}K

^{−1}[38]). Thus, the higher the volumetric water content, the higher the apparent thermal conductivity.

_{w}and λ

_{s}, (W m

^{−1}K

^{−1}) are the thermal conductivity of water and the solid phase, respectively; λ

_{app}(W m

^{−1}K

^{−1}) is the apparent thermal conductivity of the air-filled pore space, made up partly of normal heat conduction (λ

_{a}) and partly of vapor movement (λ

_{v}); θ, σ, and ε (m

^{3}m

^{−3}) are the volumetric fractions of water, solids, and air, respectively; k

_{s}and k

_{a}are the weighting factors for the solid and air phases determined by λ for each phase and geometric shape of the soil particles; and g

_{a}, g

_{b}, and g

_{c}represent the depolarization factor of the ellipsoid in the different directions, satisfying g

_{a}+ g

_{b}+ g

_{c}= 1. Farouki [7] suggested that ${g}_{a}=0.333-\frac{\epsilon}{\varphi}(0.333\text{}-\text{}0.035)$ and g

_{c}= 1 − 2g

_{a}.

_{e}):

_{e}is normalized thermal conductivity, and λ

_{sat}and λ

_{dry}(W m

^{−1}K

^{−1}) are thermal conductivities under full saturation and dry conditions, respectively. Johansen [39] proposed λ

_{e}as a function of S

_{r}:

_{w}and λ

_{s}(W m

^{−1}K

^{−1}) are the thermal conductivity of water and the solid phase, respectively; ϕ (m

^{3}m

^{−3}) is the total porosity of the soil. Note that Equation (16) is equivalent to Equation (7). Thermal conductivity in dry conditions has an empirical form:

_{d}/1000) + 0.60(ρ

_{d}/1000)

^{2}, B = 1.06(ρ

_{d}/1000)θ, C = 1 + 2.6(1000m

_{c})

^{−0.5}, D = 0.03 + 0.10(ρ

_{d}/1000)

^{2}, and E = 4, where ρ

_{d}(kg m

^{−3}) is the dry bulk density and m

_{c}(kg) is the clay mass fraction of the soil; θ (m

^{3}m

^{−3}) is the volumetric water content.

_{r}):

_{e}–S

_{r}relationship. They suggested that κ is 4.6 for gravel and coarse sand; 3.55 for median and fine sand; and 1.9 for silt and clay. Thermal conductivity of saturated soil was calculated by Equation (16). Thermal conductivity of dry soils was predicted:

_{r}(%) is the degree of saturation; F is a soil texture-dependent parameter, F is suggested to be 0.96 and 0.27 for coarse and fine soils, respectively. Thermal conductivity of saturated soil was calculated by Equation (16). Lu et al. [42] presented a simple linear model for predicting the thermal conductivity of dry soils from the total porosity of the soil:

#### 3.2. The New Models for Estimating Thermal Conductivity

#### 3.2.1. Linear Model

_{r}) in air-dried conditions. Therefore, to develop the model for estimating λ of tested samples with the change of θ, we suggest that λ should be a function of (θ − θ

_{r}). A linear model was expected for estimating λ as:

_{dry}can be calculated from σ by a simple linear equation:

_{1}and b

_{1}are empirical parameters. We assumed b

_{1}is equal to the thermal conductivity of air, b

_{1}= λ

_{a}= 0.025 W m

^{−1}K

^{−1}[38]). Therefore, the linear model for thermal conductivity can be written as the equation:

#### 3.2.2. Simple Closed-Form Model

_{e}[39]. Previous studies proposed normalized thermal conductivity, λ

_{e}, as a function of S

_{r}[39,41,42]. In these, the λ

_{e}models of Johansen [39] and Lu et al. [42] are convex curves, while Cote and Konrad’s λ

_{e}model [41] is a convex curve when κ > 1 and a concave curve when κ < 1. Cote and Konrad’s λ

_{e}model seems to be more general than the other two models. Therefore, the simple closed-form model for λ

_{e}in this study was developed based on Cote and Konrad’s λ

_{e}model (Equation (19)) [41]. As mentioned above, in air-dried conditions, the tested samples contained a significant amount of water to make the S

_{r}values of tested samples large, which may not be expressed well by the previous λ

_{e}models. Thus, we incorporated the effective saturation, S

_{e}[S

_{e}= (θ − θ

_{r})/(θ

_{s}− θ

_{r}); Equation (1)], into the model development. The resulting new model for estimating λ

_{e}is shown by the equation below:

_{sat}and λ

_{dry}, we applied two models: the geometric mean (GM) model [31] and the linear model. By applying the GM model for estimating λ

_{sat}and λ

_{dry}, modified geometric mean equations were proposed to calculate λ

_{sat}and λ

_{dry}of tested samples for not only single materials (i.e., RC100%, RCB100%, AAC100%) but also mixtures (i.e., RC80% + AAC20%, RC60% + AAC40%, RCB80% + AAC20%, and RCB60% + AAC40%) with the following equations:

_{s1}and λ

_{s2}(W m

^{−1}K

^{−1}) are the thermal conductivity of the solid phase of aggregates 1 and 2 in the mixtures, respectively; f is the proportion of aggregate 1 in the mixtures; and σ (m

^{3}m

^{−3}) is volumetric solid content. We assumed that λ

_{s}values of tested materials (RC, RCB, and AAC) could be estimated from λ

_{sat}based on Equation (16) and are shown in Table 4.

_{sat}and λ

_{dry}, λ

_{dry}can be calculated by Equation (24), while λ

_{sat}can be calculated by substituting θ = θ

_{s}in Equation (25) to obtain:

## 4. Results and Discussion

#### 4.1. Measured Thermal Conductivity and Heat Capacity

_{abs,c/f}= [m

_{c/f}× (w

_{abs,c/f}/100)]/(1000V), where m

_{c/f}is the mass of coarse or fine aggregates in the tested sample (kg), w

_{abs,c/f}is the water absorption of coarse or fine aggregates (kg/kg in %), and V is the volume of the sample (m

^{3}). The maximum volumetric absorbed water by coarse and fine aggregates for the tested materials is shown in Table 1.

_{w}= 0.57 W m

^{−1}K

^{−1}and λ

_{a}= 0.025 W m

^{−1}K

^{−1}[38]); when θ increases, water displaces air, leading to an increase in the thermal conductivity of the sample. Turning to the comparison of tested materials, at the same θ value, RC had the highest λ value followed by RCB and AAC. This may be due to the difference in the water absorption capacity of materials. AAC and RCB had a higher water absorption capacity than RC (see Table 1), which resulted in the water films surrounding the AAC and RCB particles being thinner than the water films surrounding the RC particles at the same θ value. Hence, AAC and RCB had lower thermal conductivity than RC.

^{−3}K

^{−1}[38]).

_{dry}and HC

_{dry}) of the tested samples are shown in Figure 7. The predictive models from previous studies for λ

_{dry}and HC

_{dry}were also plotted. λ

_{dry}and HC

_{dry}depend on a variety of factors such as particle shape, dry density, total porosity, and volumetric solid content [38,39,41,42,47]. Both λ

_{dry}and HC

_{dry}tended to increase as the volumetric solid content (σ) increased, similar to the trend of previous studies. However, the measured data were not expressed well by the previous predictive models. The measured data were fitted by using Equation (24) with a

_{1}= 0.35 and b

_{1}= 0.025. The regression line for HC

_{dry}–σ relationship is shown in Figure 7b.

_{r}). The proposed linear model for λ based on Equation (25) gave good regressions (R

^{2}> 0.81) for the tested samples, except for the RC80% + AAC20% sample (R

^{2}= 0.71). The regression lines for HC values correlated with (θ − θ

_{r}) of the tested samples are shown in the figure with relatively high values of R

^{2}(≥0.71).

_{e}and S

_{e}for the tested samples, sands [48], silt loam, and silty clay loam [42]. It is clearly seen that the new λ

_{e}model performed very well not only in data measured in this study but also in data sets from previous studies. The R

^{2}values for the λ

_{e}–S

_{e}relationship based on Equation (26) for each data set ranged from 0.91 to 0.97.

_{50}). The κ values of the tested samples using recycled materials in this study were less than 1, while those of natural materials (i.e., sands, silty loam, and silty clay loam) were more than 1. This means that λ

_{e}values of recycled materials increase less rapidly than those of natural materials with increasing S

_{e}. This may be because recycled materials had higher water absorption capacity than natural materials, which caused water films between particles of recycled materials to form at a higher water content than natural materials. κ values tended to increase with increasing D

_{50}for natural materials, while κ values of recycled materials fluctuated with the increasing D

_{50}. Turning to the effect of gradation, the relationship between κ values and the coefficient of uniformity (C

_{u}; C

_{u}= D

_{60}/D

_{10}; where D

_{60}and D

_{10}(mm) are particle diameters at which 60% and 10% of particles, respectively are smaller) are shown in Figure 10b. The κ values of samples decreased with increasing C

_{u,}and the regression line was fitted: κ = 0.73[log(C

_{u})]

^{−0.67}(R

^{2}= 0.75).

#### 4.2. Performance of the Predictive Models for Thermal Conductivity

## 5. Conclusions

_{e}) and effective saturation (S

_{e}). The refined models estimated sufficiently well the measured data of tested samples, and the J-CK-L model especially had the best performance among tested thermal conductivity models. Because the J-CK-L model with a single variable of κ adopted measured data from different mixed proportions of recycled concrete and clay brick aggregate and their mixtures with AAC grains, the model would be useful for quick assessment of the thermal conductivity of roadbed materials and for evaluating the heat balance to mitigate urban heat islands.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations and Symbols

AAC | Autoclaved aerated concrete |

AIC | Akaike’s information criterion |

CDW | Construction and demolition waste |

LA | Los Angeles abrasion |

RC | Recycled concrete |

RCB | Recycled clay brick |

RMSE | Root mean square error |

WRCs | Water retention curves |

A, B, C, D, E | Parameters dependent on physical properties of the soil (Equation (18)) (-) |

a_{1}, b_{1} | Empirical parameters in the linear model used for estimating λ_{dry} (Equation (24)) (-) |

a_{2} | Parameter in linear model (Equation (25)) (-) |

C* | Specific moisture capacity (-) |

C_{u} | Coefficient of uniformity (-) |

D_{50} | Mean particle size (mm) |

F | Soil texture dependent parameter in Equation (21) (-) |

f | Proportion of aggregate in the mixtures |

G, H | Coefficients in Equation (22) (-) |

g_{a}, g_{b}, g_{c} | Depolarization factor of the ellipsoid in different directions (-) |

HC | Heat capacity (MJ m^{−3} K^{−1}) |

HC_{dry} | Heat capacity at air dried (MJ m^{−3} K^{−1}) |

k | Number of model parameters (-) |

k_{a} | Weighting factors for the air phase (-) |

k_{s} | Weighting factors for the solid phase (-) |

m_{c} | Clay mass fraction of the soil (kg) |

r | Equivalent pore radius (μm) |

S_{e} | Effective saturation (-) |

S_{r} | Degree of saturation (%) |

w_{abs} | Water absorption capacity (%) |

w_{AD} | Water content in air-dried condition (%) |

α, n, m | Parameters of van Genuchten WRC (Equation (1)) |

α_{i}, w_{i}, n_{i}, m_{i} | Parameters of Durner WRC (Equation (2)) (-) |

χ | Coefficient accounting for soil type (-) |

ε | Air-filled porosity (m^{3} m^{−3}) |

ϕ | Total porosity (m^{3} m^{−3}) |

η | Coefficient accounting for grain shape (-) |

κ | Material dependent parameter (-) |

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

λ_{a} | Thermal conductivity of air (W m^{−1} K^{−1}) |

λ_{app} | Apparent thermal conductivity of the air-filled pore space (W m^{−1} K^{−1}) |

λ_{dry} | Thermal conductivities at air dry (W m^{−1} K^{−1}) |

λ_{e} | Normalized thermal conductivity (-) |

λ_{s} | Thermal conductivity of solid phase (W m^{−1} K^{−1}) |

λ_{sat} | Thermal conductivity at water saturation (W m^{−1} K^{−1}) |

λ_{w} | Thermal conductivity of water (W m^{−1} K^{−1}) |

λ_{v} | Apparent thermal conductivity of vapor movement (W m^{−1} K^{−1}) |

θ | Volumetric water content (m^{3} m^{−3}) |

θ_{r} | Residual volumetric water content (m^{3} m^{−3}) |

θ_{s} | Saturated volumetric water content (m^{3} m^{−3}) |

ρ_{d} | Dry density (kg m^{−3}) |

ρ_{s} | Density of solid phase (kg m^{−3}) |

σ | Volumetric solid content (m^{3} m^{−3}) |

|ψ| | Water potential (kPa) |

## Appendix A

_{dry}values of most tested samples were lower than those of concrete and lightweight concrete and higher than those of pervious concrete at the same dry density value except for λ

_{dry}value of RC100% sample. In the saturated conditions, λ

_{sat}values of tested samples were higher than those of pervious concrete.

**Figure A2.**Comparison of measured thermal conductivities of tested samples in this study with those for (

**a**) lightweight and normal concrete in air dried condition and (

**b**) pervious concrete in saturated condition.

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**Figure 1.**Particle size distributions (PSDs) of tested samples in this study. The upper and lower boundaries of PSD for road base materials prescribed in TCVN 8859:2011 and TCVN 8857:2011, respectively, are indicated by dotted lines.

**Figure 3.**Water retention curves (WRCs) (Equations (1) and (2)) and pore size density of tested samples (Equation (3)). (

**a**,

**b**) RC 100%, RC–AAC mixtures, and AAC100% and (

**c**,

**d**) RCB100%, RCB–AAC mixtures, and AAC100%. Measured values of saturated volumetric water content (θ

_{s}) were plotted at |ψ| = 10

^{−2}kPa.

**Figure 4.**Comparison of measured λ values by two different methods. The 95% confidence curves are indicated by broken lines.

**Figure 5.**Thermal conductivity as a function of volumetric water content and water retention curves of tested samples. The filled points present the data of thermal conductivity, and the unfilled points present the data of water retention curves.

**Figure 6.**Heat capacity as a function of volumetric water content and water retention curves of tested samples. The filled points present the data of heat capacity, and the unfilled points present the data of water retention curves.

**Figure 7.**Measured λ

_{dry}and HC

_{dry}of tested samples as a function of volumetric solid content, σ.

Tested Materials | ρ_{s} | w_{AD} | w_{abs} (%) | θ_{abs} (m^{3} m^{−3}) | LA | ||
---|---|---|---|---|---|---|---|

kg m^{−3} | % | Fine Aggregate (<4.75 mm) | Coarse Aggregate (≥4.75 mm) | Fine Aggregate (<4.75 mm) | Coarse Aggregate (≥4.75 mm) | % | |

RC | 2630 | 0.85 | 8.5 | 5.2 | 0.06 | 0.06 | 38.0 |

RCB | 2640 | 0.34 | 14 | 13 | 0.09 | 0.13 | 45.6 |

AAC | 2510 | 2.07 | 61 | - | 0.50 | - | 55.6 |

Tested Samples | Percentage in Mixture (%) | ρ_{d}(kg m ^{−3}) | ϕ (m ^{3} m^{−3}) | ||
---|---|---|---|---|---|

RC | RCB | AAC | |||

RC100% | 100 | 0 | 0 | 1980 | 0.24 |

RC80% + AAC20% | 80 | 0 | 20 | 1560 | 0.42 |

RC60% + AAC40% | 60 | 0 | 40 | 1260 | 0.52 |

RCB100% | 0 | 100 | 0 | 1650 | 0.38 |

RCB80% + AAC20% | 0 | 80 | 20 | 1410 | 0.47 |

RCB60% + AAC40% | 0 | 60 | 40 | 1150 | 0.54 |

AAC100% | 0 | 0 | 100 | 820 | 0.70 |

Tested Samples | θ_{s}(m ^{3} m^{−3}) | θ_{r}^{(a)}(m ^{3} m^{−3}) | α_{1} | n_{1} | m_{1} | w_{1} | α_{2} | n_{2} | m_{2} | w_{2} |
---|---|---|---|---|---|---|---|---|---|---|

RC100% | 0.24 | 0.09 | 0.06 | 1.2 | 0.17 | - | - | - | - | - |

RC80% + AAC20% | 0.42 | 0.11 | 0.20 | 1.4 | 0.29 | 0.50 | 1.7 × 10^{−04} | 3.1 | 0.68 | 0.50 |

RC60% + AAC40% | 0.52 | 0.14 | 0.11 | 1.4 | 0.28 | 0.61 | 1.1 × 10^{−04} | 3.8 | 0.73 | 0.39 |

RCB100% | 0.38 | 0.06 | 0.07 | 1.2 | 0.15 | - | - | - | - | - |

RCB80% + AAC20% | 0.47 | 0.09 | 0.09 | 1.6 | 0.35 | 0.45 | 9.0 × 10^{−05} | 2.7 | 0.63 | 0.55 |

RCB60% + AAC40% | 0.54 | 0.10 | 0.07 | 1.9 | 0.48 | 0.42 | 8.0 × 10^{−05} | 2.8 | 0.64 | 0.58 |

AAC100% | 0.70 | 0.16 | 0.08 | 1.3 | 0.22 | 0.56 | 9.0 × 10^{−05} | 3.8 | 0.74 | 0.44 |

^{(a)}θ

_{r}was assumed to be equal to θ

_{AD}.

**Table 4.**Estimated thermal conductivity of solid (λ

_{s}) from saturation condition (λ

_{sat}) based on geometric mean model.

Samples | ϕ (m ^{3} m^{−3}) | 1 − ϕ (m ^{3} m^{−3}) | λ_{sat}(W m ^{−1} K^{−1}) | λ_{w}^{(a)}(W m ^{−1} K^{−1}) | λ_{s}^{(b)}(W m ^{−1} K^{−1}) |
---|---|---|---|---|---|

AAC100% | 0.70 | 0.30 | 0.935 | 0.57 | 2.933 |

RC100% | 0.24 | 0.76 | 1.974 | 0.57 | 2.898 |

RCB100% | 0.38 | 0.62 | 1.197 | 0.57 | 1.884 |

^{(a)}Described by de Vries (1963),

^{(b)}Calculated by Equation (16).

Models | All Tested Samples | Mixtures | ||||
---|---|---|---|---|---|---|

RMSE | Bias | AIC | RMSE | Bias | AIC | |

Woodside and Messmer (1961; Equation (8)) | 0.34 | −0.18 | 42.9 | 0.25 | −0.15 | 11.8 |

de Vries (1963; Equation (10)) | 0.45 | −0.32 | 78.6 | 0.37 | −0.29 | 40.5 |

Johansen (1975; Equations (13), (15)–(17)) | 0.47 | −0.34 | 75.1 | 0.37 | −0.30 | 33.7 |

Campbell (1985; Equation (18)) | 0.46 | 0.34 | 68.0 | 0.34 | 0.27 | 24.7 |

Conte and Konrad (2005; Equations (13), (16), (19) and (20)) | 0.49 | −0.36 | 80.4 | 0.39 | −0.32 | 36.6 |

Lu et al. (2007; Equations (13), (16), (21) and (22)) | 0.45 | −0.33 | 70.9 | 0.35 | −0.29 | 31. 7 |

Linear model (this study, Equation (25)) | 0.14 | −0.02 | −46.7 | 0.12 | −0.01 | −33.1 |

J-CK-GM model (this study, Equation (30)) | 0.25 | −0.13 | 17.9 | 0.21 | −0.15 | 7.1 |

J-CK-L model (this study, Equation (32)) | 0.14 | 0.06 | −48.9 | 0.12 | 0.06 | −32.7 |

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## Share and Cite

**MDPI and ACS Style**

Thai, H.N.; Kawamoto, K.; Nguyen, H.G.; Sakaki, T.; Komatsu, T.; Moldrup, P.
Measurements and Modeling of Thermal Conductivity of Recycled Aggregates from Concrete, Clay Brick, and Their Mixtures with Autoclaved Aerated Concrete Grains. *Sustainability* **2022**, *14*, 2417.
https://doi.org/10.3390/su14042417

**AMA Style**

Thai HN, Kawamoto K, Nguyen HG, Sakaki T, Komatsu T, Moldrup P.
Measurements and Modeling of Thermal Conductivity of Recycled Aggregates from Concrete, Clay Brick, and Their Mixtures with Autoclaved Aerated Concrete Grains. *Sustainability*. 2022; 14(4):2417.
https://doi.org/10.3390/su14042417

**Chicago/Turabian Style**

Thai, Hong Nam, Ken Kawamoto, Hoang Giang Nguyen, Toshihiro Sakaki, Toshiko Komatsu, and Per Moldrup.
2022. "Measurements and Modeling of Thermal Conductivity of Recycled Aggregates from Concrete, Clay Brick, and Their Mixtures with Autoclaved Aerated Concrete Grains" *Sustainability* 14, no. 4: 2417.
https://doi.org/10.3390/su14042417