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Estimation Model for Effective Thermal Conductivity of Reinforced Concrete Containing Multiple Round Rebars
International Journal of Concrete Structures and Materials volume 12, Article number: 65 (2018)
Abstract
The objective of this research is to estimate the effective thermal conductivity model for the reinforced concrete containing round rebars. The thermal network concept and Fourier’s law are used to develop a mathematical model to calculate the effective thermal conductivity (k_{eff}) of reinforced concrete with different numbers of round rebars oriented either normal or parallel to the heattransfer direction, and different volume fractions of steel. Model predictions generally agree well with results of numerical simulations using the finitevolume method (FVM). FVM results suggest that at fixed volume fractions of steel, the effective thermal conductivity decreases as the number of round rebars increases if the rebars are in the normal orientation but increases as the number of round rebars increases if they are in the parallel orientation. This research can be used in practical conditions to predict the effective thermal conductivity of reinforced concrete containing round rebars accurately.
Background
Thermal analysis of concrete structures can be used to evaluate the insulation of buildings and to estimate the extent of concrete cracks (Kim et al. 2003). Demand for precise analysis in concrete structures is increasing, for example for analysis of heat sinks in nuclear power plants after Fukushima accident (Noh et al. 2017). Thermal conductivity of concrete is an important property for thermal analysis of concrete structures such as buildings and nuclear power plants, and in general civil engineering.
Many researchers have investigated key factors determining thermal conductivity of concrete (Kim et al. 2003; Uysal et al. 2004; Davraz et al. 2015). However, reinforced concrete that includes steel rebars is used in most concrete structures. Therefore, predicting effective thermal conductivity k_{eff} of reinforced concrete is very important in many industries. Generally, reinforced concrete is composed of concrete and steel as composite material. Models of k_{eff} have been developed for evaluation of composite materials.
Maxwell (1904) proposed the first effective model of thermal conductivity in composite materials that included a dilute suspension of spheres of diverse size. The model is only applicable for volume fractions < 25% of spheres. The Rayleigh sphere model (Strutt 1892) assumed that spherical particles are regularly arranged in a continuous matrix. The model additionally considered thermal interaction between particles and Maxwell’s model. The Rayleigh cylinder model (Pietrak and Wisniewski 2015) considered cylindrical particles placed uniformly. Noh et al. (2017) evaluated the applicability of these models to the containment wall of OPR1000, and presented a modified Rayleigh model that considers the orientations of rebar and tendons.
However, these models do not consider the number of round rebars N_{re} and their arrangement θ; development of such a model is the main purpose of this paper. This paper presents a mathematical model that considers one round rebar or multiple round rebars; the model uses the thermal network concept and Fourier’s law. The model considers N_{re} and their orientations (normal to (θ_{⊥}) or parallel to (θ_{∥}) the heatflow direction) at the same volume fraction Φ_{S} of steel. This model can predict k_{eff} at various geometries in practical use.
Numerical simulations were performed to validate the mathematical model with ~ 1% ≤ Φ_{S} ≤ 14% of rebars based on KS R 3504. Numerical simulations also consider N_{re} and θ. Results suggest that at a given Φ_{S}, k_{eff} decreases as N_{re} increases if the rebars are in the θ_{⊥} orientation, but increases as N_{re} increases if they are in the θ_{∥} orientation. The proposed model could increase the accuracy of thermal analysis in structures that use reinforced concrete.
Modeling Method
A thermal resistance network is a useful way to model k_{eff} theoretically in composite materials (Agrawal and Satapathy 2015). The method assumes that the heat transfer is adiabatic in any plane parallel to the heatflow direction. Two models are developed; one considers reinforced concrete containing one round rebar (Fig. 1) and one considers multiple round rebars (Fig. 2). k_{eff} model at the reinforced concrete can be derived by solving complex thermal network and using Fourier’s law (Figs. 3, 4). The equations are developed as follows.
Reinforced Concrete Containing One Round Rebar
The concrete is modelled as a cube with side length L. The block can be divided into concrete and steel components (Fig. 3, left). The corresponding thermal network (Fig. 3, right) to model reinforced concrete with N_{re} = 1 is composed of five thermal resistances. The round rebar is oriented along the z axis in the concrete, and heat is applied along the x axis. The k_{eff} model can be derived by developing the following equations as function of Φ_{S}, and the thermal conductivities of concrete and of steel.
To calculate R_{2}, formulas are developed in sequence
Twodimensional integration over 2r is performed to obtain the value of in Eq. (6) because A_{s} and A_{c} are expressed as functions of x. For convenience of calculation, the coordinate at the center of the round rebar is set to (0,0).
A_{s}(x) and A_{c}(x) are inserted into Eq. (6), then k_{2,2} can be derived as
To enable simple solution of the formula, rsin(u) is substituted for x
Thus, R_{2} can be expressed as
Moreover, total thermal network can be expressed as
Therefore, k_{eff} can be derived as (15). Detailed calculation is omitted.
Due to mathematical conditions, 2r must be < L (ϕ_{s} < 0.7854), so Φ_{S} should not exceed 0.7854.
Reinforced Concrete Containing Multiple Round Rebars
This model also considers a concrete cube of side L. All round rebars are oriented along the z axis (Fig. 2). In this model, m rows of n round rebars are embedded in reinforced concrete. k_{eff} models consider rebars in the θ_{⊥} and θ_{∥} orientations. The k_{eff} model of reinforced concrete containing multiple round rebars can be acquired by solving complex thermal network (Fig. 4). The thermal network including many round rebars can be generally expressed in two parts: concrete layer that is modelled by using the thermal resistance of concrete; and a mixed layer composed of alternating thermal resistances of concrete and steel. The two layers are stacked alternately beginning and ending with concrete layers (Fig. 4). The theoretical model of k_{eff} is derived as follows.
Total thermal network can be represented as the sum of the two major parts:
The concrete zone is described as
and mixed zone is also given as follow:
To obtain R_{2}, the formula is derived sequentially:
Integration is conducted over 2r to obtain R_{2,2} on k_{2,2}; the process is the same as in Sect. 2.1, so details are omitted. Thus, R_{2} can be expressed as
Therefore, k_{eff} can be expressed by deriving equation of R_{total} (23). Calculation details are omitted.
Because 2mr and 2nr must both be < L, Φ_{S} must simultaneously be < nπ/(4 m) and < mπ/(4n).
Numerical Simulations
The numerical simulations to analyze the k_{eff} of reinforced concrete as Φ_{S} and N_{re} were performed using ANSYS CFX16.2 in the steadystate condition. The Finite Volume Method (FVM) adopted by ANSYS CFX16.2 is used for numerical simulations because it conserves mass, momentum, and energy better than the Finite Difference Method (FDM) does. Thermal properties (Table 1) for the numerical simulations are based on the Korean design standard of buildings (Korean Ministry of Land, Infrastructure and Transport (KMOLIT) 2015).
The numerical domain is a cube with side length 0.1 m. Thirty test cases are conducted with various Φ_{S} and N_{re}. The rebars have diameters of 6–42 mm, following KS R 3504 for practical and actual uses of reinforced concrete (Fig. 5), with ~ 1% ≤ Φ_{S} ≤ ~ 14% (Table 2).
Conditions of the simulations are as follows. The contact thermal resistance between concrete and steel is neglected. For the boundary conditions, constant heat flux is applied to the inside surface of the concrete wall and constant convective heat transfer coefficient and room temperature are adopted at the outside air of the concrete surface (Fig. 6). Adiabatic conditions in the heat transfer direction are assumed at the four surfaces.
Convergence is assumed when residuals are < 10^{−10} and energy balance was > 99.9%. Tetrahedron grid is used as mesh type and the number of grids is about 0.92 million. For verification, numerical simulation was conducted using concrete with no rebar. To confirm that the boundary conditions did not effect k_{eff}, additional numerical analysis was conducted that considered heat flux inside the concrete, and heat transfer coefficient and temperature of the outside air.
Results and Discussion
k_{eff} of reinforced concrete containing multiple round rebars was numerically determined versus Φ_{S}, and N_{re} and θ. The average temperature gradually decreases as Φ_{S} increases at the left side of concrete wall (Figs. 7, 8); this trend means that k_{eff} of reinforced concrete increases as Φ_{S} increases.
As the number m of rows of rebars increases, temperature field becomes uniform. Increase in m affects the reduction of k_{eff} at the same Φ_{S} (Fig. 7). On the contrary, as the number n of rebars in a row increases, the temperature field becomes distorted; increase in n has a significant effect on the increase in k_{eff} at the same Φ_{S} (Fig. 8).
To evaluate how Φ_{S} and N_{re} affected the k_{eff} of reinforced concrete, the results of FVM were compared with the mathematical model. Considering m, the model results for reinforced concrete with rebars in the θ_{⊥} orientation matches the results of FVM (Fig. 9) with a standard deviation of 0.41–0.43% and a maximum error of ~ 1.5%. The large discrepancy is shown between sphere models and the numerical simulation due to difference of shape. Rayleigh cylinder model can’t estimate reduction of k_{eff} as increasing m.
Considering n, the model also well predict the results of FVM when the rebars in the θ_{∥} orientation (Fig. 10), with a standard deviation of 0.31–1.05% and maximum error of ~ 2.8%. The highest discrepancy between model value and results of FVM occurs with n = 3 at the largest Φ_{S}. This tendency may be due to temperature distortion at both concrete walls (Fig. 8). Sphere models and cylinder model are hard to explain this tendency of k_{eff} depending on n due to different shape and assumption of regular arrangement.
k_{eff} of reinforced concrete containing rebars is strongly affected by Φ_{S} at the same N_{re}. k_{eff} decreases as N_{re} increases if the rebars are in the θ_{⊥}orientation, but increases as N_{re} increases if they are in the θ_{∥} orientation. For reinforced concrete with N_{re} = 1, a correlation from FVM is expressed as a function of Φ_{S} as
Experimental data, FVM results, and k_{eff} models predictions are represented over Φ_{S} (Fig. 11). Thermal conductivity of concrete varies with conditions such as density, w/c ratio and its types; therefore, nondimensional k_{eff} is used for quantitative comparison at identical conditions. As supplementary information, we consider the results of Zhao et al. who manufactured reinforced concrete with rebars in the θ_{⊥} orientation (Zhao et al. 2013); the rebars are inserted horizontally at the center of the concrete specimen; Φ_{S} is 1.7 and 2.7%. In Zhao’s experiment, k_{eff} is 1.4313 W/(m K) in the bare concrete, 1.5145 W/(m K) in the reinforced concrete with Φ_{S} = 1.7%, and 1.5524 W/(m K) in the reinforced concrete with Φ_{S} = 2.7%. The mathematical model predicts Zhao’s experimental data well, with error of 3.38% in the reinforced concrete with Φ_{S} = 1.7 and 4.53% in the reinforced concrete with Φ_{s} = 2.7%. Overall, the mathematical model matches the results of FVM well, with average standard deviation of 0.52%.
Conclusion
This paper presents a model based on the thermal network concept and Fourier’s law, a model to predict the effective thermal conductivity(k_{eff}) of reinforced concrete containing round rebars. To validate the mathematical models for the effective thermal of reinforced concrete containing round rebars oriented normal (θ_{⊥}) or parallel (θ_{∥}) to the heatflow direction was investigated numerically using ANSYS CFX16.2. Mathematical models were compared with FVM results at various volume fraction of steel, number of rebars and orientations of rebars. At the same number of rebars, the effective thermal conductivity of reinforced concrete is affected by volume fraction of steel. Furthermore, at the same volume fraction of steel, the effective thermal conductivity decreases as the number of rebars increases if the rebars are in the θ_{⊥} orientation, but increases as the number of rebars increases if they are in the θ_{∥} orientation. The mathematical model generally matches the results of FVM well. The average standard deviation between models and FVM results is 0.52%. The small magnitude of this disagreement indicates that these mathematical models can be used to estimate k_{eff} of reinforced concrete containing multiple round rebars at volume fraction of steel less than 15%.
This research has developed a reliable mathematical model to predict the effective thermal conductivity of reinforced concrete containing round rebars. However, to increase the adhesion between rebar and concrete, many industries generally use deformed rebars that have ribs and nodes. Nevertheless, ribs and nodes increase the volume fraction of steel contribution of the bar only slightly. Therefore, this model could be also applicable to reinforced concrete that uses rebars that have ribs and nodes, because their effect on volume fraction of steel is probably negligible.
Abbreviations
 A _{ c } :

area of concrete in mixed part (m^{2})
 A _{ s } :

area of steel in mixed part (m^{2})
 A _{ i } :

area of ith section (m^{2})
 k _{ eff } :

effective thermal conductivity in reinforced concrete (W/(m K))
 k _{ c } :

concrete thermal conductivity (W/(m K))
 k _{ s } :

steel thermal conductivity (W/(m K))
 l _{ i, j } :

jth length of i direction (m)
 L:

one side length of concrete wall (m)
 m :

the number of round rebars at normal to heat transfer direction
 n :

the number of round rebars at parallel to heat transfer direction
 N _{ re } :

number of rebars in the unit cell
 R _{ total } :

total thermal resistance (K/W)
 R _{ i } :

ith thermal resistance (K/W)
 Q _{ total } :

total heat quantity (W)
 Q _{ c } :

heat quantity to concrete (W)
 Q _{ s } :

heat quantity to steel (W)
 S _{ o } :

sum of one over odd number thermal resistance (W/K)
 S _{ e } :

sum of one over even number thermal resistance (W/K)
 r:

radius of round rebar (m)
 ϕ _{ s } :

volume fraction of steel
 θ :

orientation of rebar
 θ _{⊥} :

orientation of rebar normal to heattransfer direction
 θ _{∥} :

orientation of rebar parallel to heattransfer direction
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Authors’ contributions
HGN carried out the effective thermal conductivity studies of concrete, participated in the sequence alignment and drafted the manuscript. HCK participated in discussing modeling part of this research, specifically. MHK discussed about the overall research such as introduction, result and conclusion parts. HSP conducted optimization of this study. Moreover, He mainly participated in result graphs. All authors read and approved the final manuscript.
Acknowledgements
This work was supported by the Nuclear Safety Research Program through the Korea Foundation of Nuclear Safety (KOFONS), granted financial resource from the Nuclear Safety and Security Commission (NSSC), Republic of Korea (No. 1305008).
Competing interests
The authors declare that they have no competing interests.
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Noh, H.G., Kang, H.C., Kim, M.H. et al. Estimation Model for Effective Thermal Conductivity of Reinforced Concrete Containing Multiple Round Rebars. Int J Concr Struct Mater 12, 65 (2018). https://doi.org/10.1186/s4006901802912
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DOI: https://doi.org/10.1186/s4006901802912
Keywords
 reinforced concrete
 effective thermal conductivity
 rebar orientation
 volume fraction
 composite materials