Characterization and simulation of microstructure and thermal properties of foamed concrete

Construction and Building Materials 47 (2013) 1278–1291

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Characterization and simulation of microstructure and thermal properties of foamed concrete She Wei a,b,⇑, Chen Yiqiang b, Zhang Yunsheng a, M.R. Jones b a b

Jiangsu Key Laboratory for Construction Materials, Southeast University, Nanjing 211189, PR China Concrete Technology Unit, Division of Civil Engineering, University of Dundee, Dundee DD1 4HN, Scotland, UK

h i g h l i g h t s 3

 Foamed concretes with different densities (300–1700 kg/m ) have been fabricated.  The microstructures were characterized in 3D by the X-CT.  2D microstructures were modeled by a random generation method.  A resistor network analogy numerically method was used to predict their ETC.

a r t i c l e

i n f o

Article history: Received 19 February 2013 Received in revised form 14 May 2013 Accepted 17 June 2013 Available online 10 July 2013 Keywords: Foamed concrete Effective thermal conductivity X-ray computerized tomography Random generation method Resistor network analogy method

a b s t r a c t Foamed concretes are composed of cement or mortar mixed with small size foams (0.1–1 mm). They exhibit good thermal insulation properties that are suitable for the insulating construction industry. To facilitate the design and development of this material, simulation of their thermal properties is essential. In this paper, foamed concretes with a large range of densities (300–1700 kg/m3) have been fabricated by the pre-forming method. The corresponding microstructures were characterized in 3D by the X-ray computerized tomography. A random generation method was introduced to efficiently model the 2D microstructures that retained the essential features of the experimental materials. Based on the reproduced 2D images, a resistor network analogy method was then introduced to numerically predict the effective thermal conductivity of this material. Finally the predictions were compared with the experimental data and other existing models. It is show that the 2D numerical predictions obtained for porosity less than 35% give very good agreement to the experimental data and the Hashin–Shtrikman upper model. The underestimation of the 2D numerical predictions mainly comes from the difference between the 2D image and the 3D structure of the real system. The radiation heat transfer is also a non-negligible factor for thermal transfer in foamed concretes for high porosity cases, and the radiation influence is diminishing as the porosity decreases. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction An increasing interest has been focused on building insulation to relieve the energy crisis and ecological problems, especially in developing countries, where the large-scale construction of urbanization is proceeding. Various types of insulation materials [1–6] have been developed by many researchers to solve the problems of energy conservation, in addition to guaranteeing that their structures satisfy both strength and stability requirements. Among such materials, those obtained by mixing cement paste or mortar with air or gas in the form of small bubbles (usually 0.1–1 mm in ⇑ Corresponding author at: Jiangsu Key Laboratory for Construction Materials, Southeast University, Nanjing 211189, PR China. Tel.: +86 25 52090640. E-mail address: [email protected] (S. Wei). 0950-0618/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.conbuildmat.2013.06.027

diameter), are particularly interesting for various reasons. First, it is expected that construction elements in foamed concrete can be fabricated on construction site. This is an important advantage with respect to other insulation materials such as autoclaved cellular concrete, whose preparation process is rather complex and energy consuming. In addition, large amount of industrial wastes (up to 70% fly ash) can be used in foamed concrete production without dramatically changing its mechanical properties, thus providing a means of economic and safe disposal of these waste residues [7]. Furthermore, it is possible to design the properties of foamed concrete by varying material parameters such as cement paste composition, foam size and volume friction. According to their densities and mechanical performances, these materials are capable of being used as either insulating or both semi-load bearing and insulating elements.

S. Wei et al. / Construction and Building Materials 47 (2013) 1278–1291

The literatures on foamed concrete are mostly devoted to characterizing the mechanical properties of these materials [8–12]. It has been shown that foamed concrete can be designed to have any density within the range of 400–1600 kg/m3, which processes self-compacting, light weight, thermal insulation, low strength (e.g. between 1 and 10 MPa) and fireproof properties. It is known that the macroscopic properties of a cellular material depends on its composition and structure, therefore, studies [13–16] have attempted to link mechanical properties of this material with its microstructural parameters, which include the porosity, the geometrical distribution of the phases and the size and distribution of the pore structure. The problem of estimating the effective thermal conductivity of two-phase materials is a classical one. There have been a significant number of analytical models [17–20] which can be used to describe the effect of pore volume fraction as a variable on the thermal conductivity of a porous material. In each case, the expression is based on a geometrical simplification of the microstructure concerning the spatial distribution of the two phase system (see Table 1). For example, the Hashin and Shtrikman [18] expressions give the most restrictive upper and lower limits of the ETC for a two-phase system where spherical inclusions are placed in a continuous matrix. Landauer [19] derived a practical expression in which the connectivity of the phases is taken into account. This approach is also called ‘‘Effective medium percolation theory’’ (EMPT). These approaches become limited when the pore volume fraction increases and the isolated pores become connected. Recently, owing to the rapid developments in computer and computational techniques, some numerical models have also been used to predict the thermal conductivity of porous materials [21–26]. Coqurad and Baillis [25] calculated the ETC of two-phase heterogeneous materials by using a numerical finite volume method. The work of Wang and Pan [26] who solved the energy transport equation through random open-cell porous foams using a high-efficiency Lattice Boltzmann method can also be cited. Compared with analytical models, the advantages of the numerical methods are that they can give a fine description of the complex structure of the media, so the results may not be dependent upon the empirical or semi-empirical parameters, which are often difficult to obtain. However, for the existing numerical models, both the geometry complexity and the conjugate heat transfer boundary conditions require extremely fine grid, often causing extremely highly computational cost and thus limiting the wide utilization of these models. Therefore, a more efficient modeling method is necessary. Based on the thermal resistance theory in the thermal transfer [27], a thermal analysis model was developed in which the thermal resistance was treated as the electrical resistance in circuit conduct, and the calculation efficiency could be greatly increased [28,29]. The objective of this work is to investigate the possibility of predicting the effective thermal conductivity (ETC) from the point of microstructure reconstruction and numerically simulation. Foamed concretes with different densities (300–1700 kg/m3) have been fabricated by the pre-forming method. Their microstructures have also been characterized and analyzed in 3D by X-ray tomography. Next, a random generation method was introduced to reproduce the 2D models that captured the main features of the actual experimental material. The effective thermal conductivity (ETC) was then predicted by a resistor network method. Finally, the proposed model was validated by the comparison with a series of experimental data and other existing models, and the factors causing the deviations between the predicted and experimental results were also discussed.

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2. Experimental investigation 2.1. Constituent materials Combinations of the following constituent materials were used to produce the foamed concrete mixes. (1) Portland cement (PC): 80%, with the compressive strength of 64.5 MPa at 28 days, conforming to BSEN 197-1 type I cement [30]. (2) Fly ash (FA): 20%, with a median particle size of 35 lm, loss on ignition (LOI) of 5.0% and conforming to BS EN 450 [31]. (3) Water (W): 40%, this percentage was fixed in order to satisfy both the workability criterion and the CLSM [32] recommendations for the insulation materials. (4) Polycarboxylic type of superplasticizer (SP) conforming to BS EN 934-2 [33]. The dosage was kept at 0.1 wt.%, of the total binders. (5) Surfactant (a commercial, protein based foaming agent), used in a 5% aqueous solution and foamed to a density of 50 kg/m3 (note: this is typical of industry practice) by an air-compressed foam generator. The properties of cement and fly ash used in this study are presented in Table 2 and corresponding size distribution is shown in Fig. 1.

2.2. Mix proportions There are no standard methods for proportioning foamed concrete. However, considering the w/c ratio, free water and fly ash content and maintaining a unit volume, the specified target wet density can become a prime design criterion [34]. It should be noted that it is difficult to design foamed concretes according to their dry density, as foamed concrete will desorp between 50 and 150 kg/m3 of the total mix water, depending on the concrete wet density, early curing and subsequent exposure conditions. The mix proportioning method used in the study is that developed at University of Dundee [12] and thoroughly described below. Assuming a given target plastic density (D, kg/m3), water/binder ratio (w/b) and fly ash/binder ratio (f/b), the total mix water (W, kg/m3), cement content (c, kg/m3), fly ash content (f, kg/m3) and foam volume (Vfoam, m3) were calculated from Eqs. (1)–(3) as follows.

Target wet density; D ¼ b þ w

ð1Þ

where b = c + f

Free water content; W ¼ ðw=bÞ  ðc þ f Þ

ð2Þ

Foam volume; V foam ¼ 1  c=3150 kg=m3  f =2400 kg=m3  W=1000 kg=m3

ð3Þ

where 3150, 2400 and 1000 kg/m3 are the densities of cement, fly ash and water, respectively. Various volume fraction of foam (0–84.2%) have been introduced in order to cover a large range of densities, just as shown in Table 3. The different foamed concretes produced by varying the percentage of foam have been denoted from A to J, respectively, where A represents the matrix sample with 0% foam and so on.

2.3. Specimen preparation It is important to establish the characteristics of the mixing procedure because they will exert a significant influence on the density, distribution, shape and size of the cells created. In our experiment, the foamed concretes were prepared by the pre-forming method. The Portland cement and fly ash were firstly dry-mixed for 1 min in a vertical mixer. The total quantity of water was then added (along with the superplasticizer) and mixed with the dry materials until a homogeneous mortar without lumps of undispersed cement was obtained. The pre-formed foam was then produced by the foam generator and the approximate quantity (calculated by the mix proportions) was added to the mix, immediately after preparation. This was combined with the mortar for at least 2 min, until all foam was uniformly distributed and incorporated in the mix. The plastic density of the mix is then measured in accordance with BS EN 12350-6 [35] by weighing a foamed concrete sample in pre-weighed container of a known volume. A tolerance on wet density was set at ±50 kg/m3 of the target value, which is typical of industry practice for foamed concrete production. If the density was higher, additional foam was added incrementally until the target value is achieved, followed by further mixing. Mixes with densities lower than the range of acceptable values were rejected and repeated. Three representative samples of each mix were tested and a mean value was calculated. The foamed concretes with different densities in this study along with their physical and mechanical characteristics are also listed in Table 3.

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Table 1 The five analytical effective thermal conductivity (ETC) models for two phase systems. Model

Structure schematic

ETC equation

Ref.

Parallel model

k = k1t1 + k2t2

[17]

Hashin bounds-high (k1 = continuous phase, k2 = dispersed phase)

2 k ¼ k1 þ 1=ðk2 k1tÞþð t1 =3k2 Þ

[18]

Russel’s model

2

k ¼ k1

[20]

2

k2 t32 þð1t32 Þk1

2

2

k2 ðt32 t2 Þþð1t32 þt2 Þk1

EMPT model

k k t1 kk11þ2k þ t2 kk22þ2k ¼0

[19]

Hashin bounds-low (k1 = dispersed phase, k2 = continuous phase)

1 k ¼ k2 þ 1=ðk1 k2tÞþð t2 =3k1 Þ

[18]

Series model

k ¼ ðtk11 þ tk22 Þ

[17]

1

Table 2 Chemical composition and physical properties of raw materials. Raw materials

Cement Fly ash

Physical properties

Chemical composition (%)

Density (kg/m3)

Blaine surface area (m2/kg)

CaO

SiO2

Al2O3

Fe2O3

Na2O

K2O

MgO

SO3

3150 2400

350.5 361.8

64.8 6.09

21 49.96

6.16 34.02

4.01 4.52

0.1 0.66

0.4 0.98

1.94 1.17

1 0.62

2.4. Measurement of the effective thermal conductivity (ETC) The thermal conductivity test was done using a Rapid-K Thermal Conductivity instrument (R-K). After 28 days’ sealed curing, the slabs (30 cm  30 cm  3 cm) were oven dried at a temperature of 60 °C until constant mass (approximately 4 days). This was done to eliminate any moisture retained

in the slabs, as it would have had effect on the conductivity results [14]. The reason why the temperature of the oven was chosen to be below boiling point is to avoid cracking in the sample, creating a path for heat to flow through and making thermal testing useless. The sample was placed between the two plates of the instrument and upper and lower temperature limits were chosen at 40 °C and 0 °C respectively.

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100 PC

Percentage passing (%)

FA 80

60

40

20

0 0.1

1

10

100

1000

Particle diameter, μm

Fig. 2. Schematic diagram of the X-ray computed tomography system.

Fig. 1. Particle size distribution of cement and fly ash.

(25003 voxels) is thus able to completely describe a concrete specimen with volume up to 64 cm3 (cube with an edge of 4 cm). An example of 3D image is presented in Fig. 3.

Table 3 Foam volume fraction and characteristics of the foamed concretes studied. Mix

A B C D E F G H I J

Target density (kg/m3)

Vfoam (%)

Actual density (kg/m3)

Dry density (kg/m3)

r28 (MPa)

Thermal conductivity (W m1 K1)

1900 1700 1500 1300 1000 800 600 500 400 300

0 10.47 21 31.54 47.33 57.87 68.4 73.67 78.9 84.20

1903 1677 1503 1265 987 802 608 497 408 298

1870 1636 1461 1201 948 757 570 453 374 252

45.5 14.5 12.2 11.2 8.7 5.2 3.2 1.89 0.63 0.42

0.5 0.423 0.363 0.282 0.217 0.165 0.124 0.091 0.08 0.065

3. Characterization of air void system 3.1. Porosity According to Cebeci [40], air entraining agents introduce large air voids without altering the characteristics of fine pore structure of hardened cement paste appreciably. Therefore, by knowing the solid density of matrix paste (without foam), one can easily predict the porosity of foamed concrete of any other density using the following equation:

e¼1

Heat was allowed to flow between the two plates until the system stabilized. The maximum time allowed for the samples to stabilize was about 3 h. Then the thermal conductivity was calculated using the Fourier heat flow equation:

kexp ¼

Q D A  DT

ð4Þ

where kexp is thermal conductivity of the tested sample. Q is the time rate of heat flow. D is the thickness of the tested sample. A and DT are cross-sectional area and temperature difference across the sample, respectively. 2.5. 3D imaging of microstructure X-ray computerized tomography (X-CT) coupled with 3D image processing is a three-dimensional imaging technique that uses a series of radiographic images to reconstruct a map of an object’s X-ray absorption [36]. Fig. 2 shows the general scheme of the X-CT system. In the XCT, a detector measures the resulting intensity of known unidirectional (x-axis) X-ray beam intensity after absorption by the tested material, and for different directions of irradiation (h), called projections [37]. According to Beer–Lambert law, for each angle of projection h, the resulting intensity I(y, z, h) on each pixel of the detector (coordinates y, z) is given by:

 Z Iðy; z; hÞ ¼ I0  exp 



lðx; y; z; hÞdx

ð5Þ

where I0 is the intensity of the beam before the sample and l(x, y, z, h) for a great number of angles h is named the Radon transform of the attenuation coefficient of the sample. The projection-slice theorem ensures that, for a sufficient specimen sampling, the 3-D map of l can be reconstructed. In our case, image reconstruction is achieved using the three-dimensional FBP (filtered back-projection) algorithm of Feldkamp [38,39]. The attenuation coefficient mainly depends on the local density and chemical composition of the investigated specimen. In this study, all samples were examined visually with Y.CT Precision System (YXLON, Germany). The detector type is flat panel Y.XRD1620. The effective voxel size within the reconstructed image can be chosen between 10 and 100 lm. The object rotation angles are 360°. Due to the relatively low density of this kind of concrete, a 80 kV value of the X-ray tube potential was chosen. A voxel size of 15 lm was selected for the reconstructed images of those specimens. The corresponding 3D image format

qdry qsolid

ð6Þ

where e is the porosity, qdry is the dry density of foamed concrete and qsolid is the solid density of cementitious paste (without foam). It should be noted that an average solid density of cementitious paste (qsolid) of 1870 kg/m3 was established through experiment (see Table 3). Table 4 shows the variation of porosity with the volume of foam added computed from Eq. (6) and through 3D-XCT image analysis in foamed concretes. Since the pores inside foamed concrete were created due to addition of foams, the foam volume added and the measured porosity of foam concrete is close. Beside the percentage volume voids measured through 3D-XCT images is marginally lower as compared to volume of voids calculated by Eq. (10), due to the fact that the local resolution of 3D-XCT equipment was limited to 10 lm and thus the pores finer than this size cannot be accurately detected or measured. 3.2. Average air void size The relationship between average air-void size and plastic density is shown in Fig. 4. Although the property of the foam introduced into the foamed concrete was uniform for all mixtures, it can be observed that the average size of the voids varies with the density of the concrete. According to the SEM images of foamed concretes with the density of 500 kg/m3 and 1000 kg/m3 as shown in Fig. 5, concrete with higher air content tends to result in larger air voids because of the proximity of the air voids, which leads to higher incidence of void coalescing forming larger irregular air voids. This observation is more significant in concrete with plastic density less than 900 kg/m3, where the corresponding foam content is 52%. It is apparent that when the paste content is less than 48%, the average air-void size increases significantly because there is not enough cement paste to prevent the air voids from coalescing. However,

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Fig. 3. Extraction and 3D-visualization of pore structure in J-300 foamed concrete. (a) An 8-bit grayscale image of the representative slice (4  4 cm2) of mixture J-300. (b) The volume of interest (VOI = 2103 voxels or 43 mm3) extracted from the centre of representative slice. (c) Cluster multiple labeling of image in (b) resulting a 3D image where in the solid voxels are imaged as gray and the pore cluster is imaged as blue. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Table 4 Characterization of air void system of foamed concretes studied.

A B C D E F G H I J

Target density

1900 1700 1500 1300 1000 800 600 500 400 300

Vfoam (%)

0 10.47 21 32.54 47.33 57.87 68.4 73.67 78.9 84.20

Porosity of air voids based on Eq. (3) (%)

3D-XCT (%)

– 12.51 21.87 35.78 49.30 59.52 69.52 75.78 80.00 86.52

– 12 21.39 35.5 47.24 59 69.22 75.45 79.19 84.17

– 0.104 0.113 0.122 0.173 0.263 0.59 0.7 0.8 0.956

as the plastic density increases, the average air-void size asymptotes toward a threshold value of 0.1 mm, which would be the average air-void size of the foam introduced into the concrete. 3.3. Air-void sizes distribution The frequency distribution of pore sizes in Fig. 6 shows that the majority of the voids are of uniform size. There are a few bigger sized pores present and their number also increases with a decrease in plastic density, which is also attributed to the possibility of merging and overlapping of pores at higher foam concrete. For the cumulative frequency distribution considered in Fig. 7, at high density the pore size distribution is more uniform than at low density. It should be noted that, as expected from the literature [41], the size distribution of pores in foamed concrete nearly follows a log–normal distribution. A probability density function denoted by fx(x; l, r) as follows, was used to fit the size distribution [42]

fx ðx; l; rÞ ¼

ðln xlÞ2 1 pffiffiffiffiffiffiffi e 2r2 ; xr 2p

x>0

Average air-void size (mm)

ð7Þ

Parameters of fitting log–normal curves Mean value l

Standard deviation r

– 5.259 5.312 5.347 5.514 5.693 6.132 6.232 6.587 6.952

– 0.457 0.323 0.544 0.465 0.235 0.342 0.343 0.212 0.122

1.0

Average air-void size, mm

Mix

0.8 0.6 0.4 0.2 0.0 200

400

600

800

1000 1200 1400 1600 1800

Plastic density, kg/m 3 Fig. 4. Relationship between plastic density and average air-void size.

where l and r are the mean and standard deviation of the variable’s logarithm, respectively. The correlation coefficients of fit curves are respectively 0.8, 0.87 and 0.96 for foamed concretes with the density of 1700 kg/m3, 800 kg/m3 and 300 kg/m3. For the foamed

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25

1700kg/m3 LogNormal fit

20

R2 =0.8

20

Frequency,%

Frequency,%

Fig. 5. SEM images of foamed concrete (left: 500 kg/m3, right: 1000 kg/m3).

15 10

800kg/m3 Lognormal fit R2 =0.87

15

10

5

5

0

0 100

200

300

400

500

600 700 Air-void diameter, μm

800

900

200

400

600

800

1000

1200

Air-void diameter, μm

(b) 800 kg/m3

(a) 1700kg/m3 20

300kg/m3

18

Lognormal fit

16

2

R =0.96

Frequency,%

14 12 10 8 6 4 2 0 500

1000

1500

2000

2500

Air-void diameter , μm

(c) 300 kg/m3 Fig. 6. Air-void size distribution of foamed concretes (a: 1700 kg/m3, b: 800 kg/m3, c: 300 kg/m3).

concretes prepared in this study, the corresponding fitting values of l and r are also summarized in Table 4.

Considering the two-phase porous structure of foamed concrete shown in Fig. 3, the pores are described as spheres randomly distributed in the solid matrix. The generation process for such porous structures is described as follows:

4. Modeling approaches 4.1. Random generation method for air void structure As mentioned in the introduction, the existing analysis models are based on the geometrical simplification of the microstructure. In such models, the stochastic natures of the porous material are neglected. Here we propose a random generation method to reproduce the 2D microstructure of foamed concrete with the pores forming randomly based on some main structural parameters.

(1) Randomly locate the pore centers based on a uniform distribution probability, c, which is defined as the probability of a point in a given area to become a center of the pore. Each point will be assigned one random number uniformly distributed within (0, 1), thus ensuring that the pore centers are uniformly distributed in the given area. (2) Randomly generate the pore diameters with a mean value, l, and a standard deviation parameter, r, which are based on the fitting values of the log–normal distribution function

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Cumulative frequency,%

100

80 3

1700kg/m 3 1300kg/m 3 1000kg/m 3 800kg/m 3 500kg/m 3 300kg/m

60

40

20

0

0

500

1000

1500

2000

2500

Air-void diameter, μm Fig. 7. Cumulative frequency distribution of air-void sizes of foamed concretes with different density.

as mentioned above. In this paper, it should be noted that the pore diameters in the range of l ± 3r are chosen to make the structure more realistic. (3) Search N target pores with the diameters generated above. To make the searching process more understandable, here we describe and sketch it in Fig. 8. The searching process is moving outward from an original pore center O with a radius of ro. Pores around the pore O will be a potential target such as A, B and C, based on an overlap ratio, d, which is defined as the area friction of the overlap section to the new circle or the chosen circles. Only those whose overlap ratio is lower than a set value will be chose and recorded. (4) Add the target pore number (N) until the porosity (e) or the solid volume fraction (1  e) attains the given value. The generation process can also be simply illuminated by the flow chart shown in Fig. 9. Thus the generated structures are controlled by four parameters (l, r, d and e). We fixed d = 0.4 in the work of this paper and then the porosity was only a result of dm and r. Fig. 10 demonstrates two examples of thus generated two-dimensional microstructures on 500  500 grids. The dark area represents the voids structure and the white the solid phase. The stochastic characteristics appear clearly and realistically from the structures. In addition, by changing log–normal parameters of pore size distribution and the overlap ratio at certain porosity, the effect of pore size and clustering can be studied systematically in this model.

B o

r

Fig. 9. Flow chart of the pore generation method.

4.2. Calculating model of effective thermal conductivity Generally speaking, the heat transfer in any cellular material is the result of a contribution of three different mechanisms: conduction, convection and radiation, therefore the overall heat transfer in foamed concrete compose by heat conduction in cementitious matrix, heat conduction, heat convection and heat radiation in pore. Skochdopole [43] conducted of an experiment by reversing the hot and cold plates of a modified guarded hot plate unit in order to maximize and minimize convection, and showed that heat transfer by convection does not exist for cell diameters smaller than 4 mm. Most foam concretes in this study possess closed cells smaller than 4 mm, therefore, heat transfer due to convection can be negligible. In this paper, we assume that the heat flux transfer from the left side to the right side of the sample without internal heat generation. The piece considered is a slab of foamed concrete with perfectly insulated sides which is sandwiched between two slabs of non-porous solid maintained at temperatures Tcold and Thot (see Fig. 11). Then thermal conductivity can be obtained as

r

A

knum ¼

r

Q tot l hðT hot  T cold Þ

ð8Þ

C r

Fig. 8. Sketch illumination of searching target pores.

where l and h are the thickness and height of the slab respectively. Qtot is the total heat flow through the slab. Based on the reconstructed 2-D images, we can mesh the area with m  n elements (m = n in this paper, see Fig. 11). To be specifically, when the area is pore, this cell is determined as a black one. Otherwise the cell is determined as white one for solid base. In this

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(a) 500 kg/m3, ε =75.45%, µ= 6.232 and (b) 800kg/m3 , ε =59 %, µ= 5.693 and σ=0.343 σ=0.235 Fig. 10. Generated 2D microstructures using different parameters for the foamed concretes with the densities of 500 and 800 kg/m3, respectively.

Fig. 11. 2D illustration of the principle of the ETC computation.

way, the whole image will be divided into two kinds of cell: pore as black and solid base as white (see Fig. 12). The simplified thermal resistance and nodes structure of cell is shown in Fig. 13. Every lattice center is represented as a ‘node’, and the thermal resistance between neighboring nodes is represented as ‘resistor’. The network is governed by the conservation law of energy. Namely

Fig. 13. Sketch of the resistance network.

at any node in this network, the sum of heat flow towards that node is equal to that away from it. A relation similar to Kirchhoff’s electric current law can be stated as bm X

Q jk ¼

k¼b1

bm X Tk  Tj k¼b1

Rjk

¼0

ð9Þ

where the node are labeled as j = 1, 2, . . . , n2, (n2 is the total number of nodes in the model), b is the number of nodes adjacent to node j with a maximum value h (h 6 4 in this structure), Qjk is the heat flow from node k to j, and Tj is the temperature at node j. Rjk is the thermal resistance between node k and j. The thermal resistance between the adjacent lattices is Rs and Rf for the same materials, otherwise it is

Q

R¼

Fig. 12. Schematic diagram of the meshed 2D image.

RS þ Rf 2

ð10Þ

Eq. (9) is suitable for all the nodes except those in the left and right surfaces. The nodes in the left surface are labeled from 1 to Nleft, where Nleft is the node number in the left surface, and the temperature at these nodes is Tj = Thot. And the nodes in the right surface are set from n  Nright + 1 to n with Tj = Tcold, where Nright is the node

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number in the right surface and Nleft = Nright. There will be n equations for the n unknown variables Tj as

8 T j ¼ T hot > > > > bm
j ¼ f1; 2; . . . ; Nleft ; g

j

¼ 0 j ¼ fNleft þ 1; . . . ; n  Nright g Rjk > > k¼b1 > > : T j ¼ T cold j ¼ fn  N right þ 1; . . . ; ng

P ð11Þ

By solving the Eq. (11), the temperature of each lattice can be obtained, and the heat flow can also be determined. The calculating process is implemented within MATLAB and the equations were solved by the Gauss–Seidel method. The criterion of convergence is set as

X jT ðnþ1Þ  T n j i

T ðnþ1Þ

After the temperature distribution is obtained, knum and the heat flux vectors, q can be obtained by Eqs. (13) and (14), respectively.


ð12Þ

where i denotes the serial number of all lattices, superscript n denotes the step of the iteration and b the preset small number.

knum ¼

j in right

Pbm

k¼b1 Q jk

T hot  T cold

q;j ¼ kj gradðT j Þ

ð13Þ

ð14Þ

The significance of introducing the 2D resistor network approach lies in the dramatic reduction in computational cost so that a large quantity of the random distribution patterns can be considered. The typical run times for the longer simulations discussed below require 20 min on a 3 GHz Core (2) PC having 4 GB of memory. Then the statistic average can be obtained reasonably. Fig. 14 shows the three examples of the numerical calculated temperature distribution and heat flux vectors for foamed concretes of different density.

Fig. 14. The numerical calculated temperature distribution and heat flux vectors for foamed concretes of different plastic density. (a) 300 kg/m3, (b) 500 kg/m3,(c) 1300 kg/m3, and (d) 1700 kg/m3.

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Fig. 14 (continued)

5. Results and discussion

Unlike the models cited in the introduction of this paper, the proposed model does not need a representative unit to calculate ETC. The porous structures generated in this paper show remarkable stochastic characteristics, thus resulting in fluctuations around an averaged value for each case with given parameters. After studying such instability, we found that the fluctuation is strongly dependent on the cell number and slightly affected by the solid volume fraction. A larger cell number and higher solid volume fraction will lead to smaller fluctuations. A value taken from literature for the thermal conductivity of air equal to 0.025 W m1 K1 was chosen [1]. An experimental value of 0.5 W m1 K1, determined for a full dense sample, was used for the thermal conductivity of the solid phase and this is consistent with other literature values [14,44]. Fig. 15 shows an example of the stability with respect to the cell number n by using the simulation parameters as e = 47.24%, l = 5.693 and r = 0.235, and the relative deviations were calculated according to the eight numerical results. The results indicate the following: (i) the cell number in the image has to be at least over a certain number (n = 40 in this

Relative deviation,%

5.1. Stability analysis

35 30 25 20 15 10 5 0 -5 -10 -15 -20 -25 -30 -35 10

20

30

40

50

60

70

n Fig. 15. The relative deviation of the predicted ETC of foam concrete versus n. The simulation parameters are ks = 0.5 W m1 K1, kg = 0.025 W m1 K1, e = 0.485.

case) to stabilize the influence of the random structure on the ETC and (ii) once the cell number is large enough, the relative deviation of predicted ETC becomes stable and the maximum relatively deviation roughly smaller than 3% in this case. A larger n will lead

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Thermal conductivity, W/mK

0.5 Numerical results Parallel model Series model

0.4

0.3

0.2

0.1

0.0 0.0

0.2

0.4

0.6

0.8

1.0

Porosity Fig. 16. Comparison between the numerical predictions and the theoretical solutions for two hypothetical structural cases: parallel model and series model.

to a more accurate prediction yet require a higher computational cost. In the following simulations, n is normally set from 40 to 100 from foamed concretes with different densities. 5.2. Benchmarks To validate the algorithm and the codes, the numerical predictions are compared with the theoretical solutions for two hypothetical structural cases: parallel model and series model. Fig. 16 shows our numerical predictions of ETCs as a function of porosity compared with the solutions made from the analytical models. It is clear that our predictions agree perfectly well with the theoretical solutions with the maximum deviation less than 5%. The results validate the proposed algorithm and the boundary condition processing. 5.3. Comparisons with experimental data and other analytical models Fig. 17 shows the measured ETC compared with the predictions made from numerical model and other analytical models, over the full range of composition. A schematic of the material structures assumed by each model is also shown. The component thermal conductivities used in the simulations are ks = 0.5 W m1 K1 and kair = 0.025 W m1 K1. It can be clearly seen that both the experimental results and numerical predictions lie in the region bounded above by the high HS bound model in which the gaseous component is the isolated phase, and below by the EMPT model, in which the two components are distributed randomly. Carson et al. [45]

referred to this region as ‘‘internal porosity region’’, where the condensed phase forms continuous conduction pathways. Base on the magnitudes and directions of the heat flux vectors shown in Fig. 14, it can be seen that the heat flow transfer through the solid phase easily avoiding through the air phase as much as possible. This kind of optimal heat conduction pathway decreases as the porosity increases. The striking feature of the simulated values and HS bound-high predictions for porosity below 35% are the close agreement with the experimental results. In fact this physically makes sense because both approaches concern 2D calculations on isolated inclusions. As the porosity increase above 35% the HS bound-high model is evidently no longer appropriate since the system becomes even more heterogeneous once the pores become connected. Further examination of the simulate values shows that their general trend follows most closely EMPT model. This is because both EMPT model and the numerical model concern the stochastic characteristic of pore structure. To further investigate the difference between numerical and experimental results, the ratios between the experimental data and numerical predictions of ETC as a function of porosity are shown in Fig. 18. The 2D numerical predictions are always smaller than the experimental results unless the porosity equals zero and unit. The value of kexp/knum varies with the porosity of the foamed concrete. The curve firstly keeps constant around unity until the porosity come up to 35%. With the porosity increases, the curve increases up to maximum value when the porosity is around 70% and then decreases to unity. There are several reasons that can explain the numerical calculation underestimates ETC. First, we noticed that the mean temperature was 20 °C when performing the experimental measurements. Therefore the underestimation of the ETC could result from the neglected thermal radiation. To verify our speculation, we have added the thermal radiation contribution to our calculation using the existing models for radiative heat transfer. Yuan [46] have proposed a simple relationship between the radiation contribution krd to the thermal properties and the temperature T for foams as:

krd ¼ 3n8de rT 3

where T is the mean temperature, r is Stefan–Boltzmann constant (5.67  10–8 W/m2 K4), and de is the cell diameters. Fig. 18 also shows the ratio between the experimental data and numerical predictions of ETC with the radiation modification at 293 K. After the radiation modification the value of kexp/k2D decreases especially in foamed concretes with the high porosity (P70%). The results show that the importance of radiation

0.5

1.7 Experimental data Numerical data

Parallel

Numerical results Numerical results+effect of air radiation EMPT EMPT+effect of air radiation

1.6

0.4 1.5

Russel's model

0.3

kexp /knum

Thermal conductivity,W/mK

ð15Þ

HS bound-high

0.2

HS bound-low

EMPT

1.4 1.3 1.2 1.1

0.1 1.0

Series

0.0

0.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.2

0.4

0.6

0.8

1.0

Porosity

Porosity Fig. 17. Comparison of ETCs predicted by various models with experimental data.

Fig. 18. The ratios of experimental to 2D predicted ETCs for different porosities of foamed concrete.

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Fig. 19. Cross sections of 3D-XCT image (a) and the 2D reconstructed image of foamed concrete with 84.17% of pore volume fraction (300 kg/m3).

contribution increases with an increasing porosity; e.g., the radiation contribution accounts for almost 6% when the air volume fraction is over 85% for 293 K and standard air pressure conditions. While at low porosity the effect of air radiation can be negligible. Second, compared with 2D cases, there are more degrees of freedom of heat flux in 3D heat transfer, resulting in the 3D real thermal conductivity being larger than the 2D thermal conductivity. In Fig. 19, both the 3D-XCT cross sections and the 2D simulation microstructure representing the highest pore volume friction samples in this study can exhibit continuous porous zones crossing the entire study area. Another interesting situation was found that many solid areas are completely isolated by the pore phase. Actually, completely separated solid phase cannot appear in 3D real structures. The optimal heat conduction pathways through solid phase are completely blocked in this situation (see Fig. 14a), which decreases the calculated ETC significantly. When the porosity is very high (>70%), the main heat conduction pathway is through the air phase. So the influence of decreased connectivity of slid phase in 2D images on the numerical results became smaller and smaller. Grandjean et al. [47] and Wang et al. [48] found similar phenomenon as well. According to Grandjean’s research [47], up to 20% of pore volume fraction in porous ceramics, image-based simulations agreed well with analytical expressions. Wang et al. [48] also compared the ratios of ETC between 2D and 3D as the function of porosity. He found that the 2D predictions are always

smaller than the 3D ones and the ratio reaches a minimum value when the porosity approaches almost 0.7, which is in very close agreement with the results we got in this study. In order to try to seek a simple analytical method and compare with the numerical method, the EMPT solutions with and without the radiation modification are also shown in Fig. 18. It can be seen that both EMPT and numerical solutions for the porosity below 20% are close to the experimental results. As the porosity increase above 20% the values of kexp/knum of EMPT model are always larger than that of the numerical model, which show that the numerical model in this paper is closer to experimental results than EMPT model in large porosity case. However, when the porosity is small (620%), the EMPT model can be used as a simple analytical method. 6. Conclusions Foamed concretes with a large range of densities (from 300 to 1700 kg/m3) were prepared and characterized with respect to their microstructures and thermal conductivities. After analysis of 3DXCT images, we found that the size distribution of pores in foamed concrete nearly follows a log–normal distribution. Based on these structural parameters, a random generation method has been developed to reproduce the 2D microstructures of foamed concretes by computer algorithms. The energy transport equations

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through the porous structure are solved using a resistor networks method. The effective thermal conductivities of foamed concretes are then numerically studied. Comparing the experimental results with 2D numerical and other analytical model, the following conclusions can be drawn: (1) Both the experimental results and numerical predictions lie in the region bounded above by the upper HS bound model and below by the EMPT model, which proved that foam concrete is a typical kind of ‘‘ internal porosity material [42]’’, in which the solid phase is continuous. (2) Examination of the 2D numerical values shows that their general trend follows most closely EMT model. This is because both EMPT model and the numerical model physically concern the stochastic characteristic of pore structure. When the porosity is small (620%), the EMPT model can be used as a simple analytical method. (3) Up to 35% of pore volume fraction in foamed concrete, the numerical predictions agreed well with experimental data. However, the numerical predictions are always smaller than the experimental results unless the porosity equals zero or unity. The ratio of the two reaches a maximum value when the porosity approaches almost 0.7. Two reasons can be used to explain the underestimation of the ETC by 2D numerical prediction. First, the radiation heat transfer is a non-negligible factor for thermal transfer in foamed concretes for high porosity cases, and the radiation contribution decreases as the solid volume fraction increases. Second, the random sampling of the porous structure by the 2D image inherently affects the physical representation of the real system by the connectivity of solid phase decreased in the simulation. From this perspective, simulation on a 3D structure based on micrographs would get around this obstacle. However the risk is to generate high computation cost. Therefore, the emergent idea is to develop a quick calculation method to solve the energy transport equations through 3D complex structure of this kind of material. For this evaluation more investigations will be performed and analyzed in future work. Acknowledgements Authors gratefully acknowledge the financial support from open projects from State Key Laboratory of High Performance Civil Engineering Materials (2010CEM002), China National Natural Science Fund of China (51178106, 51138002), Program for New Century Excellent Talents in University (NCET-08-0116), 973 Program (2009CB623200), Program sponsored for scientific innovation research of college graduate in Jiangsu province (CXLX_0105). Thanks are also due to the Concrete Technology Unit, University of Dundee for providing facilities and equipments. References [1] Bouvard D, Chaix JM, Dendievel R, Fazekas A, Létang JM, Peix G, et al. Characterization and simulation of microstructure and properties of EPS lightweight concrete. Cem Concr Res 2007;37:1666–73. [2] Goual MS, Bali A, Queneudec M. Effective thermal conductivity of clayey aerated concrete in the dry state: experimental results and modeling. J Phys D 1999;32:3041–6. [3] Skujans Juris, Vulans Andris, Iljins Uldis, Aboltins Aivars. Measurements of heat transfer of multi-layered wall construction with foam gypsum. Appl Therm Eng 2007;27:1219–24. [4] Almanza O, Rodríguez-Pérez MA, de Saja JA. The thermal conductivity of polyethylene foams manufactured by a nitrogen solution process. Cell Polym 1999;18:385–401. [5] Kim HK, Jeon JH, Lee HK. Workability, and mechanical, acoustic and thermal properties of lightweight aggregate concrete with a high volume of entrained air. Constr Build Mater 2012;29:193–200. [6] Lu TJ, Chen C. Thermal transport and fire retardance properties of cellular aluminium alloys. Acta Mater 1999;47:1469–85.

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