1 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 keff of reinforced concrete is very important in many industries. Generally, reinforced concrete is composed of concrete and steel as composite material. Models of keff 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 Nre 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 Nre and their orientations (normal to (θ⊥) or parallel to (θ∥) the heat-flow direction) at the same volume fraction ΦS of steel. This model can predict keff at various geometries in practical use.

Numerical simulations were performed to validate the mathematical model with ~ 1% ≤ ΦS ≤ 14% of rebars based on KS R 3504. Numerical simulations also consider Nre and θ. Results suggest that at a given ΦS, keff decreases as Nre increases if the rebars are in the θ⊥ orientation, but increases as Nre increases if they are in the θ∥ orientation. The proposed model could increase the accuracy of thermal analysis in structures that use reinforced concrete.

2 Modeling Method

A thermal resistance network is a useful way to model keff theoretically in composite materials (Agrawal and Satapathy 2015). The method assumes that the heat transfer is adiabatic in any plane parallel to the heat-flow direction. Two models are developed; one considers reinforced concrete containing one round rebar (Fig. 1) and one considers multiple round rebars (Fig. 2). keff 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.

Fig. 1
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Schematic of reinforced concrete containing one round rebar.

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Fig. 2
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Schematic of reinforced concrete containing multiple round rebars.

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Fig. 3
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2-dimensional view of reinforced concrete containing one round rebar and its thermal network.

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Fig. 4
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2-dimensional view of reinforced concrete containing multiple round rebars and its thermal network.

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2.1 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 Nre = 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 keff model can be derived by developing the following equations as function of ΦS, and the thermal conductivities of concrete and of steel.

$$\frac{1}{{R_{total} }} = \frac{1}{{R_{1} }} + \frac{1}{{R_{2} }} + \frac{1}{{R_{3} }}$$
(1)
$$\frac{1}{{R_{1} }} + \frac{1}{{R_{3} }} = \frac{{k_{c} }}{\text{L}}\left( {A_{1} + A_{3} } \right)\left( {\because k_{1} = k_{3} = k_{c} , {\text{R}} = \frac{L}{kA}} \right).$$
(2)

To calculate R2, formulas are developed in sequence

$${\text{R}}_{2} = {\text{R}}_{2,1} + {\text{R}}_{2,2} + {\text{R}}_{2,3} .$$
(3)
$${\text{R}}_{2,1} + {\text{R}}_{2,3} = \frac{L - 2r}{{k_{c} A_{2} }}\quad \left( {\because l_{x,1} + l_{x,3} = L - 2r} \right).$$
(4)
$$Q_{\text{total}} = Q_{\text{s}} + Q_{\text{c}} {\text{at R}}_{2,2} .$$
(5)
$$k_{2,2} = \frac{{Q_{\text{s}} + Q_{\text{c}} }}{{\frac{dT}{\text{dx}} \cdot A_{2} }} = \frac{{k_{\text{s}} A_{s} }}{{A_{2} }} + \frac{{k_{\text{c}} A_{c} }}{{A_{2} }} \quad \left( {\because {\text{Q}} = {\text{k}}A\frac{dT}{\text{dx}}} \right).$$
(6)

Two-dimensional integration over 2r is performed to obtain the value of in Eq. (6) because As and Ac are expressed as functions of x. For convenience of calculation, the coordinate at the center of the round rebar is set to (0,0).

$$A_{\text{s}} \left( {\text{x}} \right) = 2{\text{yL}} = 2(r^{2} - x^{2} )^{{\frac{1}{2}}} {\text{L }}\quad \left( {\because x^{2} + y^{2} = r^{2} } \right),$$
(7)
$$A_{\text{c}} \left( {\text{x}} \right) = 2rL - 2{\text{yL}} = 2rL - 2(r^{2} - x^{2} )^{{\frac{1}{2}}} {\text{L}}.$$
(8)

As(x) and Ac(x) are inserted into Eq. (6), then k2,2 can be derived as

$$k_{2,2} = \frac{1}{{2rA_{2} }}\mathop \smallint \limits_{ - r}^{r} (k_{s} A_{s} { + }k_{c} A_{c} )dx = \frac{1}{{2rA_{2} }}\mathop \smallint \limits_{ - r}^{r} (2k_{s} (r^{2} - x^{2} )^{{\frac{1}{2}}} L { + }k_{c} (2rL - 2 (r^{2} - x^{2} )^{{\frac{1}{2}}} L ) )dx$$
(9)

To enable simple solution of the formula, rsin(u) is substituted for x

$$\begin{aligned} k_{2,2} & = \frac{{k_{s} \cdot 2L}}{{2rA_{2} }}\left[ {\frac{{x\sqrt {r^{2} - x^{2} } }}{2} - \frac{{r^{2} }}{2}arcsin\frac{x}{r}} \right]_{ - r}^{r} + \frac{{k_{c} }}{{2rA_{2} }}\left[ {2rLx - 2L\left( {\frac{{x\sqrt {r^{2} - x^{2} } }}{2} - \frac{{r^{2} }}{2}arcsin\frac{x}{r}} \right)} \right]_{ - r}^{r} \\ & = \frac{1}{{2rA_{2} }}k_{s} \left( {\pi r^{2} L} \right) + \frac{1}{{2rA_{2} }}k_{c} \left( {4r^{2} {\text{L}} - \pi r^{2} L} \right) \\ \end{aligned}$$
(10)
$${\text{R}}_{2,2} = \frac{{4r^{2} }}{{k_{s} \left( {\pi r^{2} L} \right) + k_{c} \left( {4r^{2} {\text{L }} - \pi r^{2} L} \right)}}$$
(11)

Thus, R2 can be expressed as

$${\text{R}}_{2} = \frac{L - 2r}{{k_{c} A_{2} }} + \frac{{4r^{2} }}{{k_{s} \left( {\pi r^{2} L} \right) + k_{c} \left( {4r^{2} {\text{L }} - \pi r^{2} L} \right)}}$$
(12)

Moreover, total thermal network can be expressed as

$$R_{\text{total}} = \frac{1}{{\frac{1}{{R_{1} }} + \frac{1}{{R_{2} }} + \frac{1}{{R_{3} }}}},$$
(13)
$$k_{eff} = \frac{L}{{R_{total} \cdot A}}$$
(14)

Therefore, keff can be derived as (15). Detailed calculation is omitted.

$$\therefore k_{eff} = k_{c} \left[ {1 - \left( {\frac{{4\phi_{s} }}{\pi }} \right)^{1/2} } \right] + \frac{1}{{ \frac{1}{{k_{c} }}\left[ {\left( {\frac{\pi }{{4\phi_{s} }}} \right)^{1/2} - 1} \right] { + }\frac{4}{{\pi \left( {k_{s} - k_{c} } \right) + 4k_{c} }}}},\quad \left( {\because \phi_{s} = \frac{{\pi r^{2} }}{{L^{2} }}} \right).$$
(15)

Due to mathematical conditions, 2r must be < L (ϕs < 0.7854), so ΦS should not exceed 0.7854.

2.2 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. keff models consider rebars in the θ⊥ and θ∥ orientations. The keff 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 keff is derived as follows.

Total thermal network can be represented as the sum of the two major parts:

$$\frac{1}{{R_{total} }} = \frac{1}{{R_{1} }} + \frac{1}{{R_{2} }} + \frac{1}{{R_{3} }} + \cdots + \frac{1}{{R_{2m + 1} }}.$$
(16)

The concrete zone is described as

$$S_{\text{o}} = \frac{1}{{R_{1} }} + \frac{1}{{R_{3} }} + \cdots + \frac{1}{{R_{2m + 1} }} = \frac{{k_{c} }}{\text{L}}\left( {A_{1} + A_{3} + \cdots + A_{2m + 1} } \right),$$
(17)

and mixed zone is also given as follow:

$$S_{\text{e}} = \frac{1}{{R_{2} }} + \frac{1}{{R_{4} }} + \cdots + \frac{1}{{R_{2m} }} = \frac{m}{{R_{2} }}$$
(18)

To obtain R2, the formula is derived sequentially:

$${\text{R}}_{2} = {\text{R}}_{2,1} + {\text{R}}_{2,2} + {\text{R}}_{2,3} + \cdots + {\text{R}}_{{2,2{\text{n}} + 1}} ,$$
(19)
$${\text{R}}_{{2,{\text{o}}}} = {\text{R}}_{2,1} + {\text{R}}_{2,3} + \cdots + {\text{R}}_{{2,2{\text{n}} + 1}} = \frac{L - 2rn}{{k_{c} A_{2} }}$$
(20)
$${\text{R}}_{{2,{\text{e}}}} = {\text{R}}_{2,2} + {\text{R}}_{2,4} + \cdots + {\text{R}}_{{2,2{\text{n}}}} = {\text{nR}}_{2,2}$$
(21)

Integration is conducted over 2r to obtain R2,2 on k2,2; the process is the same as in Sect. 2.1, so details are omitted. Thus, R2 can be expressed as

$${\text{R}}_{2} = \frac{L - 2rn}{{k_{c} A_{2} }} + \frac{{4nr^{2} }}{{k_{s} \left( {\pi r^{2} L} \right) + k_{c} \left( {4r^{2} {\text{L }} - \pi r^{2} L} \right)}}$$
(22)

Therefore, keff can be expressed by deriving equation of Rtotal (23). Calculation details are omitted.

$$\therefore k_{eff} = k_{c} \left[ {1 - \left( {\frac{{4m\phi_{s} }}{n\pi }} \right)^{1/2} } \right] + \frac{m}{{ \frac{1}{{k_{c} }}\left[ {\left( {\frac{mn\pi }{{4\phi_{s} }}} \right)^{1/2} - {\text{n}}} \right] { + }\frac{4n}{{\pi \left( {k_{s} - k_{c} } \right) + 4k_{c} }}}} \left( {\because \phi_{s} = \frac{{m \cdot n \cdot \pi r^{2} }}{{L^{2} }}} \right).$$
(23)

Because 2mr and 2nr must both be < L, ΦS must simultaneously be < nπ/(4 m) and < mπ/(4n).

3 Numerical Simulations

The numerical simulations to analyze the keff of reinforced concrete as ΦS and Nre were performed using ANSYS CFX16.2 in the steady-state 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).

Table 1 Thermal properties of reinforced concrete.
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The numerical domain is a cube with side length 0.1 m. Thirty test cases are conducted with various ΦS and Nre. The rebars have diameters of 6–42 mm, following KS R 3504 for practical and actual uses of reinforced concrete (Fig. 5), with ~ 1% ≤ ΦS ≤ ~ 14% (Table 2).

Fig. 5
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Numerical domain in reinforced concrete as volume fraction and the number of rebars.

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Table 2 Test cases for the numerical simulations of keff in reinforced concrete.
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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.

Fig. 6
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Boundary conditions for the numerical simulations in reinforced concrete.

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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 keff, additional numerical analysis was conducted that considered heat flux inside the concrete, and heat transfer coefficient and temperature of the outside air.

4 Results and Discussion

keff of reinforced concrete containing multiple round rebars was numerically determined versus ΦS, and Nre and θ. The average temperature gradually decreases as ΦS increases at the left side of concrete wall (Figs. 7, 8); this trend means that keff of reinforced concrete increases as ΦS increases.

Fig. 7
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Temperature distributions of reinforced concrete arranged in the normal heat transfer direction at the center cross sections of x–y plane in the volume fraction of minimum and maximum.

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Fig. 8
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Temperature distributions of reinforced concrete arranged in the heat transfer direction at the center cross sections of x–y plane in the volume fraction of minimum and maximum.

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As the number m of rows of rebars increases, temperature field becomes uniform. Increase in m affects the reduction of keff at the same ΦS (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 keff at the same ΦS (Fig. 8).

To evaluate how ΦS and Nre affected the keff 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 keff as increasing m.

Fig. 9
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Effective thermal conductivity of reinforced concrete versus volume fraction at the normal to heat transfer direction.

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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 ΦS. 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 keff depending on n due to different shape and assumption of regular arrangement.

Fig. 10
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Effective thermal conductivity of reinforced concrete versus volume fraction at the heat transfer direction.

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keff of reinforced concrete containing rebars is strongly affected by ΦS at the same Nre. keff decreases as Nre increases if the rebars are in the θ⊥orientation, but increases as Nre increases if they are in the θ∥ orientation. For reinforced concrete with Nre = 1, a correlation from FVM is expressed as a function of ΦS as

$$k_{eff} = 3.457^{2} + 2.925 + 1.598,\quad (0 \ll 0.15)$$
(24)

Experimental data, FVM results, and keff models predictions are represented over ΦS (Fig. 11). Thermal conductivity of concrete varies with conditions such as density, w/c ratio and its types; therefore, non-dimensional keff 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; ΦS is 1.7 and 2.7%. In Zhao’s experiment, keff is 1.4313 W/(m K) in the bare concrete, 1.5145 W/(m K) in the reinforced concrete with ΦS = 1.7%, and 1.5524 W/(m K) in the reinforced concrete with ΦS = 2.7%. The mathematical model predicts Zhao’s experimental data well, with error of 3.38% in the reinforced concrete with ΦS = 1.7 and 4.53% in the reinforced concrete with Φs = 2.7%. Overall, the mathematical model matches the results of FVM well, with average standard deviation of 0.52%.

Fig. 11
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Comparisons of models, experimental data, and FVM results for the reinforced concrete containing multiple round rebars.

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5 Conclusion

This paper presents a model based on the thermal network concept and Fourier’s law, a model to predict the effective thermal conductivity(keff) 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 heat-flow 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 keff 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.