Effective thermal conductivity of lightweight porous concrete:theoretical models and application

Lightweight porous concrete blocks have become one of the preferred energy-saving envelope materials for modern buildings due to their good thermal insulation performance. This paper reviews the extensive literature on different theoretical models for the effective thermal conductivity of porous concrete blocks. The advantages and limitations of different models, including the pore structure, fractal theory, and thermal conductivity models, are summarized. The applicability and challenges of each lightweight porous concrete model are discussed in detail. It can be concluded that more research on lightweight porous concrete blocks is necessary to improve the theoretical thermal conductivity model by incorporating parameters related to indoor thermal and humid environments.


Introduction
With the development of modern society, population growth forms a sharp contrast to the depletion of natural resources.Thus, resource conservation becomes increasingly important.In most countries, 1/3 of energy consumption and 30% of greenhouse gas emissions come from building energy consumption [1,2], which largely depends on the thermal conductivity of building envelope materials [3].
Lightweight porous concrete is a porous silicate product formed by a physical or chemical foaming process with siliceous and calcareous materials as raw materials.Due to its porous characteristics and excellent thermal performance (effective thermal conductivity), lightweight porous concrete has become one of the preferred energy-saving envelope materials for modern buildings.Its large-scale production started with the establishment and commissioning of the Ytong factory in Sweden in 1929 [4].As of 1995, aerated concrete products, mainly used in walls and roofs, have been produced and applied in more than 50 countries and regions covering frigid, temperate and tropical zones, with an annual output of about 40-50 million cubic meters.By 2005, Ytong Technology has established 44 production lines in various countries, with a production scale of 11.84 million m 3 /year; Wehrhahn Technology has established 26 production lines worldwide; Durox has established 10 production lines in 6 countries, with a production capacity of 3.553 million m 3 /year; Hebel Technology has established 51 production lines in 22 countries, with a production capacity of about 8.5 million m 3 /year.According to statistics, by 2018, China has established about 2,000 production enterprises, with a total design capacity of about 200 million cubic meters and a total output of about 181 million cubic meters, accounting for 9% of the output of wall materials and 15% of new wall materials [4][5][6].The application scenarios of lightweight porous concrete are shown in figure 1.
Thermal conductivity is an important parameter for the thermal performance (effective thermal conductivity) of the envelope structure.As shown in figure 2, lightweight porous concrete has a porous structure inside, allowing not only heat conduction but also heat radiation and convective heat transfer.Thus, the heat conduction model is complicated.The low thermal conductivity of lightweight porous concrete is mainly caused by its porous structure.The thermal conductivity changes since the water absorption characteristics of the porous structure are related to the outdoor environment.Therefore, the study on the prediction model for the effective thermal conductivity of lightweight porous concrete has attracted the attention of many researchers.

Classical theoretical models for porous media
2.1.Pore structure thermal conductivity model Since lightweight porous concrete is a porous medium, its thermal conductivity model is similar to porous media.Many theoretical models of the thermal conductivity of porous media [7][8][9][10] have been established, as shown in figure 3. Maxwell-Eucken [11] believed that porous media, solid phase and fluid phase are irregularly dispersed and relatively independent, and bubbles do not form interconnected heat conduction paths.The effective thermal conductivity model is shown below: The model represents a uniform dispersion of one medium in another, with no connectivity in the pores in the dispersed phase.The volume of the continuous phase in the Maxwell-Eucken 1 model is higher than that in the Maxwell-Eucken 2 model.(whiterepresents the dispersed phase and black represents the continuous phase).
Based on the above model, the Laplace equation of electric field energy can be solved, and the prediction model for effective thermal conductivity of lightweight porous concrete can be obtained.
As shown in figure 4, the two components of the material in the effective medium model are randomly distributed, neither continuous nor dispersed between each phase.The formation of thermal conduction paths for each component depends on the number of components.The effective thermal conductivity of porous media can also be calculated by the effective medium theory equation (equation ( 3)) [12].
Wang et al [13] derived unified equations for five basic effective heat conduction structure models (including parallel, series, two forms of Maxwell-Euken, and effective medium theory).On this basis, a method was proposed for modeling complex materials as composite materials of these five basic structural models using simple combination rules based on structural volume fraction.
e j e j e j e j e j e j e j e j = +   EMT + Maxwell-Eucken 2 + series model.
( ) The combined models described above are superior to other general models because each of them has a unique physical basis, independent of any empirical parameters.The calculation of thermal conductivity by the Maxwell-Eucken 1 + Maxwell-Eucken 2 model is based on the conductivity of the composite material filled with randomly distributed spherical particles in a uniform continuous medium.The thermal conductivity of the continuous phase in the Maxwell-Eucken 1 model is greater than that of the dispersed phase.The thermal conductivity of the dispersed phase in the Maxwell-Eucken 2 model is greater than that of the continuous phase.The parallel + Maxwell-Eucken 2 model is based on the calculation of circuit thermal resistance and the electrical conductivity of filled composites.The model is not only applicable to materials with larger dispersed phases than continuous phases (high-porosity lightweight porous concrete) but can also be used to study the formation of different components of heterogeneous materials.The Maxwell-Eucken 1 + EMT model and the Maxwell-Eucken 2 + EMT model are based on the random distribution of two components of the material, where each phase is neither continuous nor dispersed.Calculations are performed separately when the volume of the dispersed phase is larger than that of the continuous phase and when the volume of the continuous phase is larger than that of the dispersed phase.The EMT + Maxwell-Eucken 2 + series model is based on the random distribution of components, with each phase neither continuous nor dispersed.The model is deduced by the electrical conductivity calculation method.It can be used when the thermal conductivity of the dispersed phase is larger than that of the continuous phase and the structure is constructed by layers of homogeneous materials.
Gong et al [14] regarded porous materials as two-phase systems, and the phases were considered as small spheres dispersed in hypothetical homogeneous media with a thermal conductivity of k .
m Based on this theory, a new effective medium model was proposed by unifying the above combination model.
In the above prediction model, only the effect of block porosity on the thermal conductivity of concrete was considered, without considering the remaining parameters of the pore structure.For porous media, bubbles of different shapes and sizes existed inside the specimen.At this time, it is not accurate enough to calculate the thermal conductivity by only considering the porosity.The effect of bubbles on the thermal conductivity inside the specimen should also be considered.Therefore, Loeb proposed a new prediction model based on the study of Russel [15].
Ordonez-Miranda et al [16] believed that the internal pores of porous media are spherical and considered the random arrangement and aligned distribution of pores.Based on the Bruggeman differential effective medium theory, the effect of pore shape on the thermal conductivity of porous media was studied.Hasselman et al [17] studied the effect of spherical pore size on the thermal conductivity of materials, further improved the model, and derived the Hasselman model.Through this model, the effect of the pore size inside the lightweight porous concrete on thermal conductivity can be investigated.Based on the Maxwell model, Hamilton Crosser proposed the Hamilton-Crosser model by considering the shape of pores in porous media.This model can be used to analyze the relationship between pore shape and thermal conductivity.
When the pore is spherical, y = 1, = n 3, and the model can be simplified to the Maxwell-Eucken model.

Analog circuit thermal conductivity model
The heat transfer inside concrete is mainly heat conduction.The series and parallel models proposed above are to determine the upper and lower bounds of the effective thermal conductivity of composite materials, respectively.The boundary is called the Wiener boundary [18].The series model is used for the heterogeneous material of different components in a layered structure, with heat flow passing through each layer from top to bottom.The parallel model is used to study heterogeneous materials formed by the superposition of different The models can be calculated as follows.
For series model: For parallel model: In series and parallel models, lightweight porous concrete can be regarded as a composite material of a solid phase (concrete) and fluid phase (air).The theoretical maximum and minimum thermal conductivity of lightweight porous concrete can be obtained through the above model.However, these two models can only determine a range and cannot derive the exact thermal conductivity.Chaudhary and Bhandari [19] argued that the heat flow in porous materials not only flows through each layer from top to bottom but also flows through different materials from top to bottom.Therefore, based on the effective thermal conductivity calculation model of the series and parallel models, a series-parallel hybrid model was proposed, as shown in figure 6: From the above literature, it can be seen that the contribution of series-parallel in the series-parallel thermal conductivity model of lightweight porous concrete still needs further research, and there is no uniform standard yet.

Fractal heat conduction model
In recent years, Pitchumani [20] and Taoll [21] introduced fractal theory into the effective thermal conductivity prediction model, However, despite certain practical applications, the process of determining the local fractal dimension of porous materials in their model is complicated.As shown in figure 7, Yang et al [22] proposed an effective thermal conductivity model of loose particles based on percolation theory and radiation contribution.Additionally, they introduced a dimensionless parameter(The ratio of the heat conduction length to the characteristic dimension of the pore.) for the heat conduction mechanism of the fluid in the pores, which implies that the pores are effective.
Giorgio Pia and Ulrico Sanna [23] proposed a thermal conductivity model of porous materials with different pore size distributions at the same porosity.The three pore diameter distribution diagrams are as follows: The equations of the above three pore diameter distributions are as follows.
The principle of the above fractal analog circuit model is to determine the contribution of solid-phase thermal conductivity and fluid-phase thermal conductivity to the thermal conductivity of lightweight porous concrete through fractal theory.
On this basis, Feng [24] established an effective heat conduction model of two-phase porous media with an n-order Sierpinski structure, as shown in figure 8.By controlling the porosity within 0.14-0.60, the general model of thermal conductivity of two-phase porous media was extended to the effective thermal conductivity of three-phase porous media.
The figure above shows the heat conduction model of 0-level and 1-level Sierpinski carpets.Ma et al [70].assumed one-dimensional heat flow and believed that the lateral contact resistance t has little effect on thermal conductivity.The calculation model for the effective thermal conductivity of porous media was obtained by iteratively calculating the thermal conductivity model of the 0-level Sierpinski carpet.Crane and Vachon [25] as well as Yu and Cheng [26] also performed similar calculations as Ma et al.  3. Estimation of the effective thermal conductivity of lightweight porous concrete 3.1.Theoretical models for lightweight porous concrete Lightweight porous concrete is a porous medium, and its theoretical thermal conductivity models include the pore structure model [27], analog circuit model [28], fractal theory model [29] and the thermal conductivity model of experiments [30,31].

Pore structure thermal conductivity model of lightweight porous concrete
A large number of pores exist inside the lightweight porous concrete block.The parameters of the internal pore structure of the block include porosity, pore size and its distribution, pore shape and its distribution, and connectivity between pores [32].The pore structure thermal conductivity model of lightweight porous concrete is different from the traditional porous medium model.Based on the Maxwell model, the pore shape, pore size and distribution characteristics were considered to make the model closer to reality.The thermal conductivity model is shown in figure 9: Based on the effect of porosity on effective thermal conductivity, Russel also proposed a more complicated effective thermal conductivity prediction model [34].
In the above model, lightweight porous concrete is regarded as a composite material mixed with air and concrete.The solid phase in the equation is the base material of lightweight concrete, and the fluid phase is air.
Wang integrated the Hasselman and Hamilton-Crosser models and considered the joint effect of both the pore shape and size on the effective thermal conductivity of the porous medium in the prediction model.The prediction model is expressed as follows [17]: The above literature shows that it is a common method to modify and improve the Maxwell-Eucken equation for thermal conductivity prediction models using pore structure parameters.However, for the pore structure parameters, only the porosity is easy to obtain in the testing phase, while the remaining pore structure parameters need to be obtained by complex characterization methods such as image processing and mercury intrusion method.
The comparative analysis of the above pore structure heat conduction model shows that the heat conduction model of lightweight porous concrete expresses the internal pore structure factors more clearly, and its result is closer to the actual thermal conductivity.However, the pore structure factors in lightweight porous concrete are more difficult to obtain, and the model is more complex.

Analog circuit thermal conductivity model of lightweight porous concrete
The analog circuit model is widely used for lightweight porous concrete [35,36].She et al [33] retained the basic characteristics of the experimental material using a random generation method and modeled its twodimensional microstructure.Additionally, based on the two-dimensional image, they introduced the resistance network analogy method to predict the effective thermal conductivity of the material numerically.
As shown in figure 10, Wang et al [37,38]  By comparing and analyzing the analog circuit thermal conductivity model of porous media and the analog thermal conductivity model of lightweight porous concrete, it can be seen that series and parallel models accounts for half of the traditional porous medium model.However, this is not true for lightweight porous concrete.The calculated thermal conductivity of the parallel model is closer to the actual result.Therefore, if an analog circuit thermal conductivity model is adopted, the proportion of the series-parallel model should be clearly defined, which is often complicated.

Fractal heat conduction model of lightweight porous concrete
Fractal theory has been fully developed and applied since its introduction in the mid-1970s [39].Its development has been accelerated since the establishment of the effective thermal conductivity model of lightweight porous concrete [40].
Jin et al [41] proposed a two-phase fractal model of dry aerated concrete block samples, as shown in figure 11.Considering the water phase in the pores of unsaturated and wet specimens, they constructed the structure of aerated concrete blocks using a self-similar Sierpinski carpet.In addition, a fractal model for predicting the effective thermal conductivity of the porous structure of wet aerated concrete blocks was proposed.Finally, the three-phase fractal model was extended, and the relationship between thermal conductivity, moisture content and porosity was proposed.
Giorgio et al [42] constructed a three-phase fractal structure based on the multi-stage Sierpinski structure and derived an effective thermal conductivity model that can be used to predict the thermal conductivity of lightweight porous concrete with moisture.The comparison between the calculated results and the experimental results showed that the second-order three-phase fractal model can effectively predict the thermal conductivity of wet lightweight porous concrete.
Tian [43] constructed the porous structure of aerated concrete based on the self-similar Sierpinski carpet model.Through the derivation of the two-phase model, the dimensionless effective thermal conductivity of k 0 ( ) at the initial stage can be expressed as: Then, it was expanded to a three-phase model by considering the water phase in the pores.The dimensionless effective thermal conductivity of k n ( ) after the nth iteration can be calculated by the following equation: i.e., the two-phase model represents the simplified case of the three-phase model.
Li [44] used a series-parallel structure to supplement the physical structure of water-containing porous materials and built a prediction model for the effective thermal conductivity of multi-stage three-phase porous materials.Huai et al [45] simulated the heat conduction in these structures using the finite volume method (FVM) and established a fractal model of porous media structure.The effects of solid-phase and fluid-phase thermal conductivity, porosity, pore size and spatial distribution on effective thermal conductivity were analyzed in detail.The research results showed that the effective thermal conductivity, solid thermal conductivity and fluid thermal conductivity conform to the power function, and the effective thermal conductivity and porosity conform to the exponential function.Using a similar method, Zhang et al [46] also found that the temperature distribution and heat flow distribution of the specimen have self-similarity in the fractal structure model.Through numerical simulation, Li and Shi [47] found that it is difficult to reflect the effect of pore structure on the internal heat transfer with only two parameters, porosity and area fractal dimension.The flow characteristics in the material were used to characterize the heat flow conduction process of the material.The calculation result showed that the effective thermal conductivity and the solid-phase volume ratio of the porous material are approximately in an exponential function [48][49][50].Using a parametric method, Tao et al [51] generated a large number of randomly distributed ellipsoidal bubbles and produced numerical specimens of three-dimensional foam concrete with 50% porosity.The thermal conductivity calculated by the numerical prediction model with 50% porosity agreed well with the experimental thermal conductivity by the model with 49% porosity.
Chen et al [52] proposed a calculation method for the effective thermal conductivity of porous concrete considering the two-dimensional heterogeneous structure.An image-assisted method was used to randomly generate the three-phase (aggregate, cement paste and air) microstructure model of porous concrete.By simulating the steady-state heat transfer process of porous concrete, a finite element calculation model of the thermal conductivity of porous concrete was established.Additionally, the thermal conductivity of porous concrete and its components were measured experimentally.Based on five known average field homogenization methods for porosity, Miled et al [53] first derived various analytical equations for calculating effective thermal conductivity.These equations are useful for calculating the thermal conductivity of lightweight concrete with low porosity.Their result is very close to the actual test value.However, as the porosity increases, the error of the result becomes larger.She et al [54] extended the random generation method of foam concrete microstructure from two-dimensional to three-dimensional.The effective thermal conductivity of the foamed concrete was obtained by solving the energy transfer equation of the two-phase coupled heat transfer in the porous structure using FVM.
To sum up, the fractal heat conduction model of lightweight porous concrete is not much different from the fractal heat conduction model of traditional porous media.Both are calculated based on analysis theory using analog circuit models, while the model of lightweight porous concrete is also based on fractal theory.The internal pore structure is reconstructed, and two-dimensional models are extended to three-dimensional models.The result of this method is closest to reality, but the model is the most complicated and requires the most computational effort.
Based on the thermal conductivity model of concrete by Kook-Han Kim, Ulykbanov [59].analyzed the thermal conductivity of lightweight porous concrete blocks and established a prediction model for the thermal conductivity of non-autoclaved lightweight porous concrete blocks based on performance prediction. = where Age is the curing days of the sample, and w b / is the ratio of water to glue.Wang et al [31].studied the thermal conductivity of concrete blocks mixed with aerogel at different temperatures and humidity.Additionally, the experimental results were fitted to obtain the relative humidity and the effective thermal conductivity model of concrete blocks.The humidity and the effective thermal conductivity satisfied the quadratic function condition.The thermal conductivity gradually increased as the temperature increased.
Emsettin et al [60]  where e k e represents the equivalent thermal conductivity when the microscopic porosity is e, and e i represents the equivalent porosity.
According to experimental data, Zhao et al [64] derived the relationship between the dry bulk density of steel slag lightweight porous concrete blocks and the thermal conductivity: Yang et al [65] compared and analyzed the test results using the thermal conductivity model of the two-phase system material, with the EMTmodel as the basic model.They referred to the improved Maxwell model by Wang [28], where the Hamilton-Crosser model and Hasselman model were combined to determine the pore size and the shape of pores were considered.The sphericity y was used as the influencing parameter of the thermal conductivity of the continuous phase (cement paste).Through fitting, the square root of the average pore size was used as the influencing parameter of the dispersed phase (air).The new improved equation of EMTand the shape of pores were verified.This equation is suitable for the case with a dry density below 300 kg m −3 .

Model evaluation and current challenges
At present, the thermal conductivity models of lightweight porous concrete are mainly the above four types, and there are still some problems.It is a common method to use the pore structure to solve the thermal conductivity of lightweight porous concrete.However, in practical applications, more accurate pore structure, such as pore size and pore shape, is often required, which demands very complicated pore structure characterization methods.Parameters such as hole diameter and hole type need to be considered in the effective thermal conductivity model, which is often inconvenient to solve.
For the series-parallel model, the upper and lower limits of the thermal conductivity of lightweight porous concrete can be obtained, and the expression is relatively simple.However, when calculating the thermal conductivity of lightweight porous concrete, there is still a lot of controversy about the contribution of series and parallel.Accordingly, the calculation results of the series-parallel model are also controversial.Therefore, it is necessary to re-determine the contribution of the series-parallel model when the analog circuit model is adopted to calculate the thermal conductivity of different lightweight porous concrete types.
A research method to model the thermal conductivity of lightweight porous concrete using fractal theory is to simplify the internal bubbles to ellipsoid, spherical or other regular shapes while obtaining the twodimensional or three-dimensional fractal dimension; then, the fractal dimension is introduced into the composite heat conduction model.Another method is to use the Sierpinski structure or improved Sierpinski structure model to calculate the internal structure of lightweight porous concrete; then, the numerical value, simulation or thermoelectric analogy principle can be used to derive the calculation equation of effective thermal conductivity.The application of fractal theory in the prediction model of thermal conductivity of lightweight porous concrete provides new ideas.Its modeling process is the most complicated compared with the above three models.
The thermal conductivity prediction model fitted by experimental data requires a large amount of experimental data and is limited to experimental materials.The experimental fitting models for different kinds of lightweight porous concrete vary greatly.Thus, their scope of application is narrow.

Conclusions and prospects
The lightweight porous concrete block has the advantages of light weight, high strength, heat preservation, heat insulation and easy processing.It has become one of the preferred energy-saving envelope materials for modern buildings.Therefore, the accurate prediction for effective thermal conductivity is crucial in practical applications.For example, in the application of lightweight porous concrete in building envelopes, attention should be paid to the different requirements for internal and external walls because the external wall is subjected to more radiation and water vapor.Many thermal conductivity models have been proposed for porous materials.However, these two factors are not considered in the current heat conduction models.The solidphase thermal conductivity of lightweight porous concrete varies with the changing moisture content and is related to the ratio parameters of the preparation process.
In addition, neural networks can also be used to predict the thermal conductivity of lightweight porous concrete.It is also necessary to consider the effect of outdoor environment climate on the thermal conductivity of external lightweight porous concrete.Due to the influence of building external and interior walls on indoor thermal and humid environments, it is essential to improve the theoretical thermal conductivity model by incorporating the parameters related to the indoor thermal and humid environments Thermal conductivity of the material calculated in the previous stage e i Volume rate of each phase k Ratio of the thermal conductivity of the concrete material to air k 7d Thermal conductivity of the concrete after curing for 7 days

Figure 1 .
Figure 1.Application of lightweight porous concrete in high-rise buildings.

Figure 2 .
Figure 2. Surface and internal microstructure of lightweight porous concrete.
components.Different from the series model, the heat flow in the parallel model flows through different materials from top to bottom.The schematic diagram of the models is shown in figure 5.

Figure 5 .
Figure 5. Schematic diagram of the parallel model and series model.
derived a general model by deriving series and parallel models, the Maxwell-Eucken model and the effective medium theoretical model.the above equation is a series model; when = ¥ d i and = k k , i the above equation is a parallel model; when = d 3 i and = k k , cont the above equation is a Maxwell-Eucken model; when = d 3 i and = k k , e the above equation becomes the effective medium theoretical model again.

Figure 10 .
Figure 10.Calculation principle of the effective thermal conductivity of lightweight porous concrete and sketch of the resistance network.Reprinted from [37], Copyright (2006), with permission from Elsevier.
above equation can be simplified by setting b = 0, n ( )

Figure 11 .
Figure 11.Self-similar pore structure of lightweight porous concrete and the thermal-electrical analogy of the Sierpinski carpet.Reprinted from [41], Copyright (2016), with permission from Elsevier.

3. 1 . 4 .
Experimental fitting model of lightweight porous concrete Fitting a thermal conductivity model through experiments is the most common and simple method.Many researchers have obtained the thermal conductivity model by fitting data experimentally [55-57].Kook-Han Kim et al studied the effect of the curing temperature ( =  T 20 C), volume fraction of coarse aggregate ( = AG 0.7), moisture content ( = Rh 1), water-solid ratio ( = W C 0.4 / ), and volume fraction of fine aggregate ( = S A 0.4 /) on the thermal conductivity of concrete.Finally, a nonlinear thermal conductivity prediction model of concrete was established[58].
Yang et al[65] simplified the model on this basis and found that the thermal conductivity increases with the increase of dry density, which can be expressed by a linear equation: Zhu et al[66] fitted the regression equation between the thermal conductivity and dry density of lightweight porous concrete and found that it satisfies an exponential relationship, which is different from the result obtained by the linear model of Yang et al.Zivcovaic and Xu et al[67,68] proposed an exponential relationship to describe the thermal conductivity of porous materials with microstructures.Their conclusions are consistent with those of Zhu et al.Peng experimentally measured the thermal conductivity of lightweight porous concrete and used the exponential function, Fourier series function, polynomial function and power function to fit.The result was compared with the result obtained by the primary exponential model of Zivcova and Xu as well as Zhu et al.It is found that the secondary exponential model had the highest test accuracy[69]: [62]ied the relationship between porosity and thermal conductivity through experiments:Yao[61]studied the effective thermal conductivity of wet lightweight porous concrete with a porosity of 75% to 85%.The thermal conductivity of the specimen was measured by the transient plane heat source method.The subsection prediction equation of its effective thermal conductivity was fitted with mass moisture content and porosity.Through experiments and simulations, Shi et al fitted the relationship between the porosity of porous media and the effective thermal conductivity[62]