Peculiarities of the porous structure and their influence on thermal conductivity

In article [1] formulas were proposed for the calculation of the thermal conductivity coefficient of cellular concrete (hereinafter – aerated concrete), obtained on the basis of a mathematical model representing aerated concrete, the inner space of which is divided into equal tightly packed cells, having the form of a rhombic dodecahedron, in the centre of which there is one spherical pore of diameter d_P, surrounded by a shell of cement-silicate stone, filling the space between the pores. The proposed formulas did not take into account the dependence of the thermal conductivity coefficient of aerated concrete on the pore size, which was confirmed by experimental data.

To improve the proposed formulas for the former approach of dividing the aerated concrete space into cells shaped like rhombic dodecahedrons, in addition to the previous assumptions we consider that the space between the pores is filled with microporous cement stone (hereinafter called MCS stone) which consists of micropores surrounded by cement-silicate stone (hereinafter called CSS stone). These micropores consist of gel pores and capillary pores. The gel pores occur where the water is bound in the hydration process of silicates, and the capillary pores are created by evaporation of excess mixing water [2], [3]. The indicators that characterize the porous structure of aerated concrete, such as its density, porosity, pore size, pore number, the distance between the pores, are related to each other by definite relations. These parameters influence the physical-technical (thermal conductivity, water vapour permeability, etc.) and the physical-mechanical (compressive strength, tensile strength, modulus of elasticity, etc.) properties of aerated concrete [4].

In determining these indicators of the porous structure of aerated concrete, the following assumptions are accepted:

1) macro- and micropores have the form of spheres with diameters d_P and d_PM, respectively;

2) the diameters of macro- and micropores relative to each other are consistent;

3) the interior space of cellular concrete (aerated concrete) is divided into identical densely packed macro- and micro-cells;

4) each macro- and microcell of aerated concrete consists of one pore located in its centre, surrounded by shells that fill the space between the pores of the aerated concrete, consisting of microporous cement stone (MCS) around macro-cells and of cement-silicate stone (CSS) around micro-cells.

Uniform filling of the space by macro-cells and micro-cells with absolutely dense packing can be obtained if the cells are shaped like rhombic dodecahedrons [5]. A rhombic dodecahedron is a duodecahedron all sides of which are parallel in pairs and look like identical rhombuses. The volume of rhombic dodecahedron V_D is equal: 

V_D=0.707d ³_d                    (1)    

Where d_d is the transverse dimension of the dodecahedron, equal to the distance between its pairwise parallel sides, (cell size), [6, 7].

The density of aerated concrete R_G, (kg/m³), determines the amount of cement-silicate stone (CSS) without micropores in a unit volume of aerated concrete.

The density of cement-silicate stone (CSS) in aerated concrete is R_Z (kg/m³).

The density of microporous cement stone (MCS) in aerated concrete R_K, consists of cement-silicate stone (CSS) and micropores, (R_K<R_Z)(kg/m³)

The aerated concrete density coefficient K_Z= R_G/R_Z determines the proportion of the volume of cement-silicate stone (CSS) per unit volume of aerated concrete.   

The microporous cement stone (MCS) density coefficient K_S = R_K/R_Z – determines the proportion of microporous cement stone (MCS) volume per unit volume of aerated concrete.

In a unit volume of aerated concrete, the porosity coefficients are determined: P_G = (1 - K_Z) – its total porosity, consisting of macropores and micropores; P_MK = (K_S - K_Z) – its fraction of macropores; P_MP = (1 - K_S) – its fraction of micropores.

The volumes of macropore V_P and micropore V_PM with diameters d_P and d_PM, having a spherical shape, are equal: V_P = 0.523*d_P³ and V_PM = 0.523*d³_PM.

Macropores in aerated concrete are surrounded by microporous cement stone shells (MCS shells). Their number per unit volume of aerated concrete is calculated by the formula n_MK = (K_S - K_Z) / 0.523d³_P.

Micropores are surrounded by cement-silicate stone shells (CSS shells). Their number per unit volume of aerated concrete is calculated by the following formula: n_MP = (1 - K_S) /0.523d_PM³.

The volume of aerated concrete in this unit of volume per macropore, which includes, in addition to the macropore, the surrounding microporous cement stone shell, is V_P= 0.523d³_P /(K_S - K_Z).

The volume of aerated concrete per one micropore surrounded by a shell of cement-silicate stone (CSS) is V_PM=0.523d³_PM/(1 - K_S).

By equating the volumes of dodecahedrons V_D and V_DM with the volumes V_P and V_PM in formula (1), we obtain:

0.707 d³_D = 0.523 d³_P/(K_S – K_Z)

0.707 d³_DM    = 0.523  d³_PM/( 1 – K_S)                         (2).

From formulae (2) it follows that:

(K_S – K_Z)  =   0.74 (d_P³/d_D³)                            (3),

(1 – K_S)    =  0.74 (d³_PM/d_DM³).                      (4).

A dodecahedron in aerated concrete is a pore cell, with transverse dimensions d_D and d_DM, consisting of a pore of diameter d_P or d_PM and a layer of microporous cement stone (MCS) or cement-silicate stone (CSS) surrounding it, of thickness Δd_P/2 and d_PM/2, the doubled value of which is the distance between the pores.  As a result, the cross-sectional dimensions of the pore cells are d_D = d_P + Δd_P and d_DM = d_PM + Δd_PM, and formulas (3), (4) will be as follows:

(K_S  -  K_Z) = 0.74 (d_P³/(d_P + Δd_P)³),                             (5),

(1 - K_S) = 0.74 (d_PM³/(d_PM + Δd_PM) ³).                         (6).                 

From formulas (5), (6), it follows that the coefficients of density of aerated concrete K_Z and of microporous cement stone (MCC) K_S determine the ratio values between the volumes of macro-, micropores and macro, micropore cells and the values of product of these ratios, regardless of the size of the diameter of these pores. As the thickness of the shells of the pore cells changes, the pore diameters change proportionally.

After transformation of formulas (5), (6) the following formulas for the calculation of the diameter of macro- and micropores are obtained:

d_P =  Δd_P/S_P,                                                                 (7),

d_PM =Δd_PM/ S_PM,                                                           (8),

where:                 S_P = (1/(1.106  (K_S – K_Z )^1/3 ) -  1;                 K_Z = R_G/R_Z ,   R_K < R_Z ,

                              S_PM = (1/(1.106 ( 1 – K_S ) ^1/3 ) - 1                K_S = R_K/R_Z.     

The coefficients S_P and S_PM determine the influence of the density of the aerated concrete, the cement-silicate stone (CSS) and the microporous cement stone (MCS) on the porosity and the strength of the aerated concrete. The distance between the macropores is calculated by the formula Δd_P = Δd_PM + d_PM where Δd_PM depends on the grain size of the raw materials used in the production of aerated concrete. The average grain size can be assumed to be 0.020mm.

From formulas (7, 8) it follows that as the volume of microporous cement stone (MCS) increases its density (coefficient K_S) decreases and as a consequence the coefficient S_P increases while coefficient S_PM decreases. As a result the diameters of the macropores (d_P) are decreasing while diameters of the micropores (d_PM) are increasing. It follows that the diameters of the macropores are reduced by increasing the micropore diameters and the thickness of the microporous cement stone (MCS) layer is also changed, as the overall density and porosity of the aerated concrete, however, do not change.

The number and size of the micropores influences the changes in the density of the microporous cement stone (MCS) R_K ; the formation of the micropores depends on the amount of free water during the preparation of the aerated concrete mixture. The more water in the mixture, the more micropores are formed in the microporous cement stone (MCS) [2]; the micropores are filled with the free water, which subsequently evaporates, leaving behind the empty micropores. As a result, the volume of the microporous cement stone (MCS) increases while its density decreases. It should be noted that the density of the microporous cement stone (MCS) R_K may be equal to R_Z only theoretically, in reality R_K will always be less than R_Z as there is always free water in the aerated concrete mixture, hence there will always be micropores and as a result there will always be R_K < R_Z.

The coefficient of thermal conductivity of aerated concrete depends on the thermal conductivity of the microporous cement stone (MCS) and the thermal conductivity of the air in the pores.

In aerated concrete with a density R_G, the proportion of the volume of cement-silicate stone (CSS) increases due to the formation of micropores in it up to the volume of microporous cement stone (MCS) with a density R_K. As the amount of cement-silicate stone (CSS) in the aerated concrete does not change, its density decreases with its increasing volume and this decrease in density is taken into account by the coefficient K_M = 1/(2 - (R_K/R_Z)).

As the density of the cement-silicate stone (CSS) decreases, its coefficient of thermal conductivity λ_K decreases in proportion to the coefficient K_M. As a result, the thermal conductivity of the cement-silicate stone (CSS) with micropores (microporous cement stone, MCS) is  λ_ZG = K_Mλ_K. In aerated concrete, as the density of cement-silicate stone (CSS) decreases, its amount determined by the coefficient K_Z=R_G/R_Z does not change. Consequently, the thermal conductivity coefficient of the cement-silicate stone with micropores in aerated concrete is λ_KZ = (R_G/R_Z) K_Mλ_K, W/(m^oC).

The porosity coefficient of aerated concrete (1 - R_G/R_Z) takes into account the proportion of macro- and micropores in its unit volume. The coefficient of thermal conductivity of all pores is calculated according to the formula:

λ_VG = (1 - R_G /R_Z )λ_V, W/(m^oС),

where λ_V = 0.0259 W/(m^oС) is the thermal conductivity coefficient of the air at a temperature of t=20 ^oC. 

The thermal conductivity coefficient of aerated concrete λ_G is the sum of the thermal conductivities of the layer of microporous cement stone (MCS) and the air layer. As a result, the formula for calculating the thermal conductivity of aerated concrete takes the form of:

λ_G = λ_ZG+ λ_VG = K_ZMλ_Z  +  (1 – K_Z) λ_V ,      W(m^oС),    (9),

where: K_ZM =K_Z/(2 – K_S) = (RG/RZ)/(2 - (R_K/R_Z)), K_Z=R_G/R_Z.

For a more exact experimental determination and calculation of a thermal conductivity coefficient of aerated concrete according to Russian Governmental Standard GOST 7076-99, it is necessary to preliminary determine the quantity of macropores, micropores, density and thermal conductivity of cement-silicate stone (CSS) according to Russian Governmental Standard GOST 12730.4-78, and also the density of aerated concrete according to Russian Governmental Standard GOST 12730-78. It is necessary to know density and thermal conductivity of the cement-silicate stone (CSS), as its thermal conductivity depends on quantity of initial components used for the manufacture of aerated concrete and their mineralogical compositions.

The following example calculation shows the change in the thermal conductivity of aerated concrete depending on the above-mentioned values.

The calculation has been carried out using formulae (7), (8), (9).

The results of the calculation are shown in table 1.

Input data:

Density of the cement-silicate stone (CSS) is R_Z = 2,200 kg/m³ and its thermal conductivity coefficient is λ_Z = 0.5 W/m^oC. (Accepted according to [8] taking into account densities and thermal conductivity coefficients of cement-lime-gypsum-sand mortars and hydrated water in silicates).

Density of aerated concrete R_G=500 kg/m³;

Density of microporous cement stone R_K < R_Z kg/m³.                                         

Thermal conductivity coefficient of the air λ_V= 0.0259 W/m^oC

Calculation results:  K_Z= R_G/R_Z= 500/2,200 = 0.227

Table 1 Change in the thermal conductivity of aerated concrete depending on the quantity of macropores, micropores, density and thermal conductivity of cement-silicate stone (CSS), and the density of aerated concrete.

The example shows that the thermal conductivity coefficient of aerated concrete decreases with decreasing density of the microporous cement stone (MCS) and at R_K=1,900 kg/m³ its value λ_G = 0.12 W/m^oC is equal to the value given in Russian Governmental Standard GOST 31359-2007. 

Thus, diameters of macropores decrease and diameters of micropores increase, as a result the distance between them decreases and this leads to a decrease in density of the microporous cement stone (MCS), and consequently to a decrease in its thermal conductivity.

From formulas (8), (9) also follows that with decreasing the density of the microporous cement stone (MCS) R_K and with the constant value of the density coefficient of aerated concrete R_G/R_Z, the value of the coefficient K_M and, therefore, the calculated coefficient of thermal conductivity of aerated concrete λ_G, decrease.

The experimental determination of the thermal conductivity coefficient of aerated concrete confirms the dependence between the size of the macropores and the thermal conductivity coefficient of the aerated concrete. For example, in the data given in the article [9]:

R_G= 370 kg/m³ average, d_P=1.17mm, λ_G= 0.102 W/m^oС.

R_G= 366kg/m³,      ---         d_P=0.97mm, λ_G =0.098 W/m^oС,

R_G= 368 kg/m³,     ----        d_P=0.64mm,  λ_G=0.088 W/m^oС,

Thermal conductivity coefficients were calculated using formulas (7), (8), (9) and their corresponding pore diameters were determined, in order to compare them with the experimental data. For the calculation, R_Z=2,200 kg/m³ was assumed. The following input data was taken as the initial values for the calculation: R_Z=2,200 kg/m³, R_G= 370 kg/m³, d_P=1.17mm, λ_G=0.102 W/m^oC. λ_K=0.5 W/m²C. The resulting calculations are shown in Table 2:

Table 2 Calculation of thermal conductivity coefficients using formulas (7), (8), (9) and determination of their corresponding pore diameters

From the calculation results given in table 2, it follows that with a decrease in density of the microporous cement stone (MCS) R_K the thermal conductivity coefficient and the diameters the of macropores (d_P) of aerated concrete decrease, while the diameters of the micropores d_PM increase. The obtained calculation results correlate well with the experimental data: the deviation in pore diameters is 7.8 %, in thermal conductivity coefficients it is 2.2%.

Conclusions   

1. A model is proposed of the porous structure of aerated concrete, allowing to take into account its characteristic indicators (coefficients of density, porosity, pore diameter and the distance between them), as well as the dependence of these indicators on the coefficient of density of aerated concrete. 

2. The dependence is confirmed and a simplified formula is given for calculating the thermal conductivity coefficient of aerated concrete depending on the density of microporous cement stone (MCS) as well as on the thermal conductivity coefficient of cement-silicate stone (CSS) and of air.

3. It is shown that at invariable density of aerated concrete its calculated thermal conductivity coefficient decreases with decreasing density of microporous cement stone (MCS) at the expense of an increase of the quantity of micropores in the aerated concrete.

4. It is suggested, at experimental determination of the thermal conductivity coefficient of aerated concrete, to receive more exact results, by determining in advance according to the Russian Government Standard GOST 12730.1-78 the volume of the large pores, the total amount of pores, density and thermal conductivity coefficient of cement-silicate stone (CSS). 

5. It is shown that the size of grains obtained as a result of grinding of initial components of aerated concrete affects the minimum distance between pores.

6. Formulas show that the distance between pores is in direct proportion to the size of the diameters of macropores, and the smaller their diameter, the smaller is the distance between the pores, and the lower is the thermal conductivity coefficient of aerated concrete.

7. It has been found that the thermal conductivity coefficient of aerated concrete and the size of its macropores depend on the density of the microporous cement stone (MCS), for example, with decreasing density of the MCS, the size of macropores also decreases.

8. It is obtained that increasing the amount of mixing water of aerated concrete, can increase the porosity of the microporous cement stone (MCS) and consequently reduce its thermal conductivity coefficient.

9. It should be noted that in reality macropores and micropores have different shapes compared to those assumed in the model, but the trend of the influence of the porosity of the microporous cement stone (MCS) on the thermal conductivity of aereated concrete will remain.

10. In continuation of the present work, using the proposed model of the porous structure of aerated concrete and obtained formulas, it is possible to make numerical dependencies of optimal pore sizes and distances between the pores taking into account the grain size of raw materials of aerated concrete, to estimate the influence of the density of the cement-silicate stone (CSS) on the thermal conductivity of aerated concrete depending on its mineralogical composition, and to determine the effect of porosity on the strength properties of aerated concrete.

References

[1]          Vylegzhanin, V. P.; Pinsker, V. A.: Influence of porosity of autoclaved aerated concrete on its thermal conductivity and ways to change it by improving the selection of raw materials (Vliyaniye poristosti avtoklavnogo gasobetona na yego teploprovodnost i puti yeyo ismeneniya za sshchet sovershenstvovaniya podbora syryevykh materialov), in: Stroitelnye materialy, 2019, No. 8, pp. 36-38

[2]          Makridin, N. I.; Maksimova, I. N.: Structure and mechanical properties of cement dispersion systems (Struktura i mekhanitcheskiye svoystva tsementnykh dispersnykh sistem), PGAUS, Penza, 2013, p. 340.

[3]          Pinsker, V. A.; Vylegzhanin, V. P.: The theory of strength and composition of the aerated concrete mix (Teoriya prochnosti i podbora sostava gazobetona), in: Proceedings ‘Celular concretes in modern construction’ of the 2^nd International Scientific and Practical Conference  (Sbornik statey ‘Yacheistye betony v sovremennom stroitelstve’, 2-ya Mezhdunarodnaya nauchno-prakticheskaya konferentsiya), St. Petersburg, 2005.

[4]          Russian Industry Standard 501 – 52-01-2007 (STO 501 – 52-01-2007) Design and erection of the envelopes of residential and public buildings using aerated concretes in the Russian Federation (Proyektirovaniye i vozvedeniye ograzhdayushchikh konstruktsiy zhilykh i obshchestvennykh zdaniy s primeneniyem yacheistykh betonov v Rossiyskoy Federatsi), Moscow, 2008.

[5]          Fyodorov, Ye. S.: The beginning of the teaching of shapes (Nachalo ucheniya o figurakh). Moscow, Publishing house of the Academy of Sciences, USSR, 1953.

[6]          Vylegzhanin, V. P.; Romanov, V. P.: Reinforcement structure of fibre concrete and its effect on boundary breaking loads (Struktura armirovaniya fibrobetona I yeyo vliyaniye na predelnye znacheniya razrushayushchikh nagruzok), LenZNIIEP, Collection of scientific papers, Calculation and design of spatial structures of civil buildings and structures (Raschet i proyektirovaniye prostranstvennykh konstruktsiy grazhdanskikh zdaniy i sooruzheniy), Leningrad, 1975.

[7]          Pinsker, V. A.: Some questions of the physics of aerated concretes. Collection of papers “Residential buildings made of aerated concretes”. Moscow, Gosstroyizdat, 1963.

[8]          Russian Industry standard 00044807-001-2006 (STO 00044807-001-2006), Thermal properties of building envelopes, Standardinform, 2006.

[9]          Avdeyev, Ye.: What does the thermal conductivity of concrete depend on: Influence of density and aggregates, classification of concrete, construction (Ot chego zavisit koeffitsiyent teploprovodnosti betona: Vliyaniye plotnosti i zapolniteley, klassifikatsiya betonov, stroitelstvo),  www.masterabetona.ru, 2015.

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Vylegzhanin, Viktor P.

graduated from the Faculty of Civil Engineering of the Petersburg State University of Infrastructure Construction Alexander I (LIIZhT).

He worked as a Senior Scientific Associate and as the Head of the Scientific Department of the Leningrad Zonal Scientific and Research Institute for Experimental Design (LenZNIIEP). In 1984, he received his PhD in "Building Design" and "Building Mechanics".

Since 2001, Vylegzhanin has been director of the Scientific Centre for Aerated Concretes (TsYaB), which was founded on the basis of LenZNIIEP by decision of the Russian Construction Authorities and is now the leading institute in this field in Russia. The task of TsYaB is to develop the production of aerated concrete and to promote its application in the construction industry.

Vylegzhanin's scientific areas of interest are aerated concretes, fibre concrete and building construction.

Petrova, Tatyana M.

graduated from the Faculty of Civil Engineering of the St. Petersburg State University of Infrastructure Construction Alexander I (LIIZhT), where she also obtained her PhD and currently works as a lecturer.

She received her doctorate in 1997 in the field 05.23.05 "Building materials and products" and was appointed professor. Since 1999 she has been head of the chair "Building Materials and Technologies".

Petrova's scientific areas of interest are concretes, mortar mixes based on Portland cement and cement mixes, including admixtures from industrial waste products.