numerical simulation of moisture transport in concrete based on a pore size distribution model

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Numerical simulation of moisture transport in concrete based on a pore size distribution model Qinghua Huang 1 , Zhilu Jiang 1 , Xianglin Gu , Weiping Zhang 1 , Baohua Guo 1 Department of Structural Engineering, Tongji University, 1239 Siping Road, Shanghai, PR China abstract article info Article history: Received 23 December 2013 Accepted 14 August 2014 Available online xxxx Keywords: Transport property (C) Pore size distribution (B) Concrete (E) Modeling (E) This paper presents a numerical approach for predicting moisture transport in concrete based on pore size distri- bution represented by a multi-RayleighRitz model that includes gel pores, small and large capillaries, and microcracks. The comparisons between the predicted pore size distributions and the mercury intrusion porosimetry results indicate that several small capillaries with diameters less than 20 nm may be connected to smaller gel pores. Moisture transport experiments under different atmospheric conditions were conducted to verify the proposed approach. The comparison between the predicted and measured internal relative humidity values showed acceptable agreement. The simulation results showed that microcracks had greater inuence on the concrete drying rate than the other pore components. Results also revealed that more large capillaries caused a larger moving rate of the wet front and a larger moisture transport rate in the unwetted region. © 2014 Elsevier Ltd. All rights reserved. 1. Introduction Concrete durability has recently attracted much attention since it is essential in ensuring a long and durable service life of concrete struc- tures. Most processes of concrete deterioration, including carbonation and chloride penetration, are inuenced by moisture transport in concrete. In order to simulate the process of moisture transport, a multiphase (liquid water and water vapor) transport equation based on Darcy's ow and air diffusion has been commonly utilized [14]. Moisture transport coefcients vary with the moisture saturation of concrete; therefore, it is difcult to measure these coefcients directly and pre- cisely, although a number of reasonable methods have been proposed [58]. The relationships between moisture transport coefcients and mois- ture saturation expressed in terms of empirical parameters are widely used for predicting moisture transport in the concrete [1]. However, the applications of the empirical relationships are limited. Moreover, it is not possible to investigate the inuence of the pore size distribution on the moisture transport in the concrete by the empirical methods. Since moisture moves through the pores in concrete, the pore struc- ture directly determines the transport rate. Concrete contains several types of pores, including calciumsilicatehydrate (CSH) gel pores, thin and large capillaries, and entrapped air bubbles, in a wide range of pore sizes, according to Mindess et al.'s classication of concrete pores [9]. Experimental methods used to obtain the pore size distribution of concrete include mercury intrusion porosimetry (MIP), backscattered electron (BEI) and X-ray computed microtomography (micro-CT). Using the MIP method, a large scope of pores from 2.5 to 10 5 nm can be detected; however, the measured pore sizes need to be used cau- tiously due to the well-known ink bottle effects, and CSH gel pores smaller than 2.5 nm cannot be detected by this method. BEI is a com- monly used technique to observe the microstructure of the concrete, but the two-dimensional images are different from the real morphology of pores. Moreover, this method is not suitable for detecting pores of widely different sizes due to the limited observing range at xed magni- cation. The micro-CT is able to provide the three-dimensional images of pores but the resolution is limited, which is 0.5 μm even for synchro- tron X-ray instruments [10]. As the adsorption isotherm can be represented by mathematical integration of the pore size distribution as a function of unknown parameters, many researchers obtain the pore size distribution based on the adsorption isotherm of porous materials [1113]. However, the isotherm can only provide the pore size distributions of gel pores and capillary pores. Therefore, other methods should be considered to detect the larger voids in the concrete. A great deal of work has been done on the development of models for pore size distribution of cement-based materials. The existing pore models were recently reviewed and compared with test methods [14]. However, it is still difcult to explicitly establish a quantitative relation- ship between the pore size distribution and the mix proportion of the concrete. Cement and Concrete Research 67 (2015) 3143 Corresponding author. Tel./fax: +86 21 65982928. E-mail addresses: [email protected] (Q. Huang), [email protected] (Z. Jiang), [email protected] (X. Gu), [email protected] (W. Zhang), [email protected] (B. Guo). 1 Tel./fax: +86 21 65982928. http://dx.doi.org/10.1016/j.cemconres.2014.08.003 0008-8846/© 2014 Elsevier Ltd. All rights reserved. Contents lists available at ScienceDirect Cement and Concrete Research journal homepage: http://ees.elsevier.com/CEMCON/default.asp

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Page 1: Numerical simulation of moisture transport in concrete based on a pore size distribution model

Cement and Concrete Research 67 (2015) 31–43

Contents lists available at ScienceDirect

Cement and Concrete Research

j ourna l homepage: ht tp : / /ees .e lsev ie r .com/CEMCON/defau l t .asp

Numerical simulation of moisture transport in concrete based on a poresize distribution model

Qinghua Huang 1, Zhilu Jiang 1, Xianglin Gu ⁎, Weiping Zhang 1, Baohua Guo 1

Department of Structural Engineering, Tongji University, 1239 Siping Road, Shanghai, PR China

⁎ Corresponding author. Tel./fax: +86 21 65982928.E-mail addresses: [email protected] (Q. Huang), 10_j

(Z. Jiang), [email protected] (X. Gu), [email protected][email protected] (B. Guo).

1 Tel./fax: +86 21 65982928.

http://dx.doi.org/10.1016/j.cemconres.2014.08.0030008-8846/© 2014 Elsevier Ltd. All rights reserved.

a b s t r a c t

a r t i c l e i n f o

Article history:Received 23 December 2013Accepted 14 August 2014Available online xxxx

Keywords:Transport property (C)Pore size distribution (B)Concrete (E)Modeling (E)

This paper presents a numerical approach for predictingmoisture transport in concrete based on pore size distri-bution represented by a multi-Rayleigh–Ritz model that includes gel pores, small and large capillaries, andmicrocracks. The comparisons between the predicted pore size distributions and the mercury intrusionporosimetry results indicate that several small capillaries with diameters less than 20 nmmay be connected tosmaller gel pores. Moisture transport experiments under different atmospheric conditions were conducted toverify the proposed approach. The comparison between the predicted and measured internal relative humidityvalues showed acceptable agreement. The simulation results showed that microcracks had greater influenceon the concrete drying rate than the other pore components. Results also revealed that more large capillariescaused a larger moving rate of the wet front and a larger moisture transport rate in the unwetted region.

© 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Concrete durability has recently attracted much attention since it isessential in ensuring a long and durable service life of concrete struc-tures. Most processes of concrete deterioration, including carbonationand chloride penetration, are influenced by moisture transport inconcrete.

In order to simulate the process of moisture transport, a multiphase(liquid water and water vapor) transport equation based on Darcy'sflow and air diffusion has been commonly utilized [1–4]. Moisturetransport coefficients vary with the moisture saturation of concrete;therefore, it is difficult to measure these coefficients directly and pre-cisely, although a number of reasonable methods have been proposed[5–8].

The relationships betweenmoisture transport coefficients andmois-ture saturation expressed in terms of empirical parameters are widelyused for predicting moisture transport in the concrete [1]. However,the applications of the empirical relationships are limited. Moreover, itis not possible to investigate the influence of the pore size distributionon the moisture transport in the concrete by the empirical methods.

Since moisturemoves through the pores in concrete, the pore struc-ture directly determines the transport rate. Concrete contains severaltypes of pores, including calcium–silicate–hydrate (C–S–H) gel pores,

[email protected] (W. Zhang),

thin and large capillaries, and entrapped air bubbles, in a wide rangeof pore sizes, according to Mindess et al.'s classification of concretepores [9].

Experimental methods used to obtain the pore size distribution ofconcrete include mercury intrusion porosimetry (MIP), backscatteredelectron (BEI) and X-ray computed microtomography (micro-CT).Using the MIP method, a large scope of pores from 2.5 to 105 nm canbe detected; however, the measured pore sizes need to be used cau-tiously due to the well-known ‘ink bottle effects’, and C–S–H gel poressmaller than 2.5 nm cannot be detected by this method. BEI is a com-monly used technique to observe the microstructure of the concrete,but the two-dimensional images are different from the realmorphologyof pores. Moreover, this method is not suitable for detecting pores ofwidely different sizes due to the limited observing range atfixedmagni-fication. The micro-CT is able to provide the three-dimensional imagesof pores but the resolution is limited, which is 0.5 μm even for synchro-tron X-ray instruments [10].

As the adsorption isotherm can be represented by mathematicalintegration of the pore size distribution as a function of unknownparameters, many researchers obtain the pore size distribution basedon the adsorption isotherm of porous materials [11–13]. However, theisotherm can only provide the pore size distributions of gel pores andcapillary pores. Therefore, other methods should be considered todetect the larger voids in the concrete.

A great deal of work has been done on the development of modelsfor pore size distribution of cement-based materials. The existing poremodels were recently reviewed and compared with test methods [14].However, it is still difficult to explicitly establish a quantitative relation-ship between the pore size distribution and the mix proportion of theconcrete.

Page 2: Numerical simulation of moisture transport in concrete based on a pore size distribution model

32 Q. Huang et al. / Cement and Concrete Research 67 (2015) 31–43

According to thermodynamic theory, concrete pores with sizessmaller than a critical pore size are filled with condensate water, andthe larger pores are covered with layers of water molecules. Given thepore size distribution of concrete, it is possible to simulate liquidwater and water vapor transport in small and large pores, respectively.The pores have generally been assumed to be regularly shaped (cylin-drical [15], elliptic [16], or oblong [17]); and, the moisture transport co-efficients have been expressed in terms of pore structure characteristics,considering the laminar flow of liquid water and the transport of vaporin every single pore [15–21]. Wong et al. [22] used an effective mediumapproach to predict the permeability of cement pastes and mortarsbased on BEI images, in which stereological corrections were made toconvert the image data to the actual three-dimensional values, andthe hydraulic radius approximation was used to calculate the hydraulicconductance of each irregular pore. This is a feasible approach topredicting the permeability of the porous medium with pores of com-plex shapes. For the permeability of cementitious materials, the resultsare accurate enough since the permeability is dependent on larger cap-illary pores, while the moisture diffusivity is related to pores with awide range of sizes because at low relative humidity (RH), it is relatedto the vapor transport in larger pores; otherwise, the capillary watertransport in smaller pores will dominate. However, the BEI techniqueis not suitable for detecting pores with a wide range of sizes as statedpreviously.

The effects of the hysteresis of isotherms are due to different curva-tures of the liquid–vapor interface before capillary condensation. Inaddition, desorption and/or the ink bottle effect on moisture transporthave also been considered utilizing thermodynamic methods andassumed networks of pores [17–19]. Most of the moisture transportmodels directly based on the pore structure of the concrete were veri-fied under constant environmental conditions. However, the perfor-mance of these models under varied ambient RH conditions is stilluncertain and needs to be verified.

It is essential to understand the relationships between microstruc-tures and transport properties of concrete in a quantitative pattern.One possible approach to relating themicrostructure tomacrotransportproperty is a multi-scale technique, which homogenizes differentphases at lower scales for modeling at a higher scale. It is well knownthat concrete is a heterogeneousmaterial, consisting of different phasesat each scale. For example, it consists of bulk cement paste, aggregateand interfacial transition zone (ITZ) between them at the meso-scale.The multi-scale approach considers the heterogeneity of the concreteand it is also able to identify the influence of each phase at the interestedscale [23–25]. The approach can be complemented by analytical or nu-merical methods. The analytical methods apply appropriate homogeni-zation models to represent the heterogeneous materials, such asMaxwell models [26], differential effective-medium models [24,27]and self-consistent models [28]. It is convenient to obtain the homoge-nized transport property by using these models, but it is not accurateenough in some cases. For example, when these models are used atthe meso-scale of concrete, they do not consider shapes and randomdistributions of aggregate particles, which may have a great impact ontransport property. These factors could be taken into account by numer-icalmethods and thus the results could bemore accurate [29]. However,the numerical simulation at the pore scale becomes more complicatedand the development of the corresponding test methods is still neededfor characterization of microstructure of concrete [29].

This paper attempts to explore the influence of each component ofpores on the moisture transport through numerical simulations basedon a pore size distribution model. Such work is seldom a topic in con-crete technology references. The pore size distribution studied hereinwas represented by a multi-Rayleigh–Ritz (R–R) distribution modelfor a larger range of pores compared with previous studies [18], includ-ing the pore components of gel pores, small and large capillaries, andmicrocracks. For gel pores and capillary pores, a newmethod to predicttheir pore size distributionswas proposed given themix proportion and

curing time of the concrete. The proposed approach was verified bymoisture transport experiments under different atmospheric conditions.

2. Moisture transport model based on pore size distribution

Most transport processes (convective flow is not included) inconcrete are considered as being dependent on flow potentials. If thepotentials are substance concentration and fluid pressure respectively,the processes are diffusion [30,31] and permeation [32] corresponding-ly. When partially saturated concrete is exposed to liquid, the mecha-nism of transport is liquid absorption governed by capillary potential[33,34]. This paper focuses on the moisture transport in the concreteexposed to the atmospheric air. In this case, the mechanism of moisturetransport is considered to be combined permeation of water vapor andliquid water [18,20]. Water vapor flows through the unfilled pores dueto vapor pressure differences while the condensed liquid water flowsdue to the differences of capillary pressures within the concrete. Athigh RH, the liquid flow significantly accelerates the process ofmoisturetransport in concrete exposed to the atmospheric air.

The permeability of condensed liquidwater generally increaseswithincreasing of RH, since more liquid fills the pores and contributes to thetransport. If gravity and other external forces are neglected, the con-densed liquid water flow due to the capillary pressure pc (Pa) can beexpressed by extended Darcy's equation [35]:

JL ¼ −KL hð Þ ∂pc∂x ð1Þ

where JL is the rate of liquid water flow (kg·m−2 s−1), and KL (s) is thepermeability of liquid water dependent on relative humidity h.

According to Kelvin's equation based on the thermodynamic equilib-rium condition, the value of capillary forces is related to relative humid-ity h:

pc ¼ −ρLRT lnhMw

ð2Þ

whereMw is thewater molecularmass (kg/mol), ρL is thewater density(kg/m3), R is the gas constant (J·K−1 mol−1), and T is the temperature(K).

Water vapor flows through the pores not occupied by liquid water,due to the gradient of vapor pressure pv (Pa) [20]:

JV ¼ −KV hð Þ ∂pv∂x ð3Þ

where JV is the rate of vapor flow (kg·m−2 s−1), Kv is the permeabilityof vapor in unsaturated concrete (s), and vapor pressure pv is equal tothe production of the relative humidity h multiplied by the saturatedvapor pressure pvs (Pa).

After combining Eqs. (1) to (3) with the continuity equation, a one-dimensional moisture transport equation, including vapor and liquidtransport under isothermal conditions, is expressed as:

∂w∂h

∂h∂t ¼

∂∂x KL

ρlRTMwh

∂h∂x þ KVpvs

∂h∂x

� �ð4Þ

where ∂w/∂h is themoisture capacity, which is the derivative of adsorp-tion isotherms w(h).

In Eq. (4), two parameters (i.e., water vapor permeability and liquidpermeability) determine the rate of moisture transport in concrete.These parameters depend largely on the pore structure of concrete, par-ticularly the pore size and its distribution; therefore, they were directlycalculated based on the theoretical pore size distribution in this study.

Page 3: Numerical simulation of moisture transport in concrete based on a pore size distribution model

Fig. 1. Schematic illustration of pore components of concrete inspired from Jennings'model in [24].

Fig. 3. Isotherm adsorption of specimen A in Section 4 obtained from Eq. (7).

33Q. Huang et al. / Cement and Concrete Research 67 (2015) 31–43

2.1. Pore size distribution model

Typical concrete pores have a wide range of sizes, including C–S–Hgel pores smaller than 2.5 nm, small capillary pores between 2.5 and50 nm, large capillary pores between 50 and 104 nm and entrappedair voids larger than104 nm. Since entrapped air voids are usually closedpores, the former three components of concrete pores (as shown inFig. 1 [36]) are actually related to moisture transport. Since microcrackswith widths more than 1 μmmay have an impact on the transport pro-cesses of the concrete, the pore size distributionmodel should also con-sider the component of microcracks. Each component of pores was

(a) Before adsorption

(b) Before desorption

Fig. 2. Different liquid–gas interfaces before adsorption and desorption processes.

assumed to follow the Rayleigh–Ritz (R–R) model [37], which usestwo parameters, the fraction of poresϕi and the pore radius Bi (nm) cor-responding to the peak on a logarithmic scale, to represent the distribu-tion. Hence, the pore size distribution for the concrete pores is a sum ofall pore components' R–R distributions, i.e., multi R–Rmodel, expressedas:

ϕ rp≤r� �

¼ ϕt

X4i¼1

ϕi 1− exp − rBi

� �� �ð5Þ

where ϕt is the total porosity of concrete, and ϕi (i = 1, 2, 3, 4 denotinggel pores and the small, large capillary pores and microcracks, respec-tively) is the ratio of the porosity of component i to total porosity.

Pore size distribution is presented in the form of a probability densi-ty function fd that is equal to the pore radius derivative of porosity ϕ:

f d rð Þ ¼ ϕt

X4i¼1

ϕi

Biexp − r

Bi

� �: ð6Þ

Pore parameters ϕi and Bi (i = 1–3) in Eq. (6) can be determinedbased on the Brunauer–Skalny–Bodor (BSB) model, which is a predic-tion formula for adsorption isotherms. The BSB model is a three-parameter model applicable in the RH range from 5% to 100%, modifiedfrom the well-known Brunauer–Emmett–Teller (BET) model, whichcovers the RH range only from 1% to 10% for cement and concrete. Em-pirical functions for the BSB model's parameters have been established,i.e., water–cement ratio, curing time, type of cement, and temperature[38]. When the model is expressed in terms of water saturation Sr(h)(the ratio of the water volume to the pore volume of concrete), it be-comes a two-parameter formula:

Sr ¼h 1−kð Þ 1þ C−1ð Þk½ �1−khð Þ 1þ C−1ð Þkh½ � ð7Þ

where C and k are two parameters used in the BSB model.Parameters C and k can be approximately calculated, respectively,

as:

C ¼ exp855T

� �ð8Þ

k ¼ 1−1=nlð ÞC−1C−1

ð9Þ

where nl is the number of adsorbed layers at the saturation state.

Page 4: Numerical simulation of moisture transport in concrete based on a pore size distribution model

Fig. 4. Simulated pore size distribution from the isotherm adsorption model.

Table 1Mix proportions of the specimens used in the experiments (kg/m3).

Specimen Watercontent

Cementcontent

Sandcontent

Coarse aggregatecontent

A 168 420 718 1127B 210 420 607 1127C 252 420 496 1127D 294 420 384 1127

34 Q. Huang et al. / Cement and Concrete Research 67 (2015) 31–43

Xi et al. [38] established a formulation for parameter nl consideringthe effects of curing time tc, cement type ct, and water-to-cement ratiow/c:

nl ¼ Ntc tcð ÞNct ctð ÞNwc w=cð Þ ð10Þ

Ntc tcð Þ ¼ 2:5þ 15=tc ð11Þ

Nct ctð Þ ¼1:1 Type I1:0 Type II1:15 Type III1:5 Type IV

8>><>>: ð12Þ

Nwc w=cð Þ ¼ 0:33þ 2:2w=c ð13Þ

where Types I to IV are four commercial U.S. cements classified accord-ing to their compositions [38].

Due to Van der Waal's forces, a layer of water molecules is attachedover the pore walls. Capillary condensation occurs in pores with radiismaller than the critical radius rc, which is the sum of the capillary con-densate radius rk (m) and the thickness of absorbed water on the porewall ta (m):

rc ¼ rk þ ta: ð14Þ

Fig. 5. Schematic liquid water flow through pores a and b.

Therefore, the water saturation Sr can be obtained by the integrationof the volume of pores with sizes smaller than rc:

Sr ¼1φt

Z rc

0f d rð Þdr ð15Þ

The thickness of the absorbed layer ta (m) can be calculated usingthe BET model [17]:

ta ¼3� 10−10Ch

1−hð Þ 1þ C−1ð Þh½ � ð16Þ

where C is the BET constant obtained by Eq. (8).The interaction between solid pore walls is neglected for simplicity;

and, the capillary condensate radius rk of a cylindrical pore is given byKelvin's equation:

rk ¼ − ασMw

ρLRT lnhð17Þ

where σ is the surface tension of liquid water (N/m2) obtained from theregression of measured values [37] shown in Eq. (18); and, α denotes afactor considering different liquid–gas interfaces before adsorption (α isequal to 1) and desorption (α is equal to 2) processes, respectively, asshown in Fig. 2.

σ ¼ 7:5796� 10−2−1:45� 10−4 T−273ð Þ−2:4� 10−7 T−273ð Þ2 ð18Þ

The pore parameters of fd(r) in Eq. (6)were varied to achieve the op-timal curve for Eq. (15), which is the closest curve to the adsorption iso-therm obtained from Eq. (7) through a curve-fitting method [39].

The simulated probability density functions of each pore componentand the overall pores based on the isotherm adsorption (Fig. 3) wereplotted against pore radius and are shown in Fig. 4. The results indicatethat each pore component had a considerable range of pore sizes. This isreasonable, because various sizes of void spaces between randomly dis-tributed C–S–H needles or cement particles form the correspondingpore component of concrete, as shown in Fig. 1.

In Fig. 4, it shows that the calculatedmaximumpore sizewas smallerthan 1000 nm. This is reasonable because the adsorption isotherm isonly related to C–S–H pores and capillary pores whose sizes are gener-ally smaller than 1000 nm. Some larger pores will not be saturated dur-ing the isothermal adsorption in the humid air unless the concrete is indirect contact with water. In order to consider the component ofmicrocracks with a larger radius in the model, MIP tests werecomplemented and the fitting parameters of this component werethen obtained according to the measured values, which is discussed inSection 4.1.

Table 2Chemical composition of cement (percent by weight).

Tricalciumsilicate

Dicalciumsilicate

Tricalciumaluminate

Calciumaluminoferrite

61.24 16.92 7.86 10.76

Page 5: Numerical simulation of moisture transport in concrete based on a pore size distribution model

Fig. 6. Schematic diagram of specimen sealing.

(a) A plastic pipe used toaffix a humidity sensor

(b) A humidity sensorwithout coating

(c) A humidity sensorwith coating

Fig. 7. Treatment of humidity sensors.

35Q. Huang et al. / Cement and Concrete Research 67 (2015) 31–43

2.2. Water vapor permeability

When water vapor moves in concrete, there are two types of trans-port resistances: collision betweenmolecules and collision against porewalls. The Knudsen number lm/2re determines which type of resistanceis dominant. Therefore, the water vapor permeability of a single pore kvis dependent on the Knudsen number [20]:

kv ¼ Dva

1þ lm=2re

Mw

RT

� �ð19Þ

where lm is the mean free path of water vapor molecules (m); re is theeffective pore radius (m), which is the actual pore radius minus thethickness of the absorbed water layer; and, Dva is the diffusion coeffi-cient of water vapor in air (m2/s) given by:

Dva ¼ D0P0

PTT0

� �1:88ð20Þ

where D0 is the water vapor diffusion coefficient under pressure P0 andat temperature T0 (D0 = 21.6 × 10−6 m2/s, P0 = 11,325 Pa, T0 =273.16 K [20]).

According to Eq. (19), it is clear that thewater vapor permeability in-creases and becomes closer to the permeability in air as the effectivepore radius increases. Taking the tortuosity of the pores τ into account,the water vapor permeability of unsaturated concrete is obtained basedon the expression of integration as follows:

KV ¼Z ∞

rc

kvτ

r−tar

� �2f d rð Þdr: ð21Þ

2.3. Liquid water permeability

For simplicity, pores are supposed to be cylindrical andwell connect-ed with each other in the calculation of liquid water permeability. As-suming liquid water is transported through pores with no slip at theboundary, liquidwater permeability of every cylindrical pore can be cal-culated from the Hagen–Poiseuille equation [15]:

kL ¼ρL

ηr2

8

!ð22Þ

where η is the viscosity of liquid water (Pa·s) calculated by:

η ¼ η0 expGe

RT

� �ð23Þ

where Ge is the free activation energy of flow in excess of that requiredfor ideal flow conditions (J/mol), and η0 (Pa·s) is the viscosity underideal conditions given by the regression of measured values [37]:

η0 ¼ 3:57979� 10−11T4−4:88381� 10−8T3 þ 2:50218� 10−5T2−5:71236� 10−3T þ 0:49126: ð24Þ

The probability of liquid permeation through pores a and b dpab, asshown in Fig. 5, is calculated by:

dpab ¼ dAadAb ð25Þ

where dAa and dAb are the areas of pores a and b, respectively, normal-ized by the cross-sectional area.

Page 6: Numerical simulation of moisture transport in concrete based on a pore size distribution model

(a) Fixation of humidity sensors to a wooden bar

(b) Fixation of wooden bars to a mold

Fig. 8. Fixation of humidity sensors.

Fig. 10. Measured environmental conditions during experiments.

36 Q. Huang et al. / Cement and Concrete Research 67 (2015) 31–43

The liquid water permeability of concrete can then be calculated as:

KL ¼ρL

8τ2η

ZAc

0

ZAc

0

r2eqdAadAb ð26Þ

where Ac is the normalized area of the critical pore (m2). The equivalentradius req (m) (req2 = ra × rb) was obtained based on the equivalency ofpore volumes and the assumption of a proportional relationship be-tween the lengths and radii of pores. The detailed deduction of theequivalent pore radius can be seen in [40].

The normalized area is considered to be equal to the volumetric po-rosity. Eq. (26) can then be simplified as [25]:

KL ¼ρL

8τ2η

Z rc

0r f d rð Þdr

� �2: ð27Þ

Fig. 9. Specimen with humidity sensors.

2.4. Boundary conditions

When concrete is exposed to the surrounding air, moisture flowswithin the pores of the concrete and then transfers from the surface ofthe concrete to the surrounding environment by convective and/or dif-fusive processes [35]. As a result, the relative humidity (RH) of the am-bient environment is not necessarily equal to that of the concretesurface.

If the position of the concrete–environment boundary is consid-ered as x = 0, the continuity condition of the moisture transportrate j (kg·m−2 s−1) at the boundary can be expressed as:

jx¼0− ¼ jx¼0þ ð28Þ

where 0− and 0+ denote the ambient air and concrete sides of theboundary, respectively.

If the concrete is in a chamber, the wind velocity in the chamber isquite low; thus, the convective flow of water vapor from the surface ofthe concrete to the environment can be neglected. The diffusive flowof water vapor in the ambient air at the boundary can be approximatelycalculated as:

jx¼0− ¼ −DvaMwpvsRT

� hs−heδ

ð29Þ

where hs and he are the RHof the concrete surface and the environment,respectively; and, δ is the thickness of the air flow layer, which is equalto about 25 mm [41].

Themoisture transport rate at the concrete side of the boundary canbe expressed as:

jx¼0þ ¼ − KLρlRTMwh

þ KVpvs

� �∂h∂x : ð30Þ

Table 3Parameters of each specimen in the pore size distribution model obtained from the ad-sorption isotherm.

Specimen ϕ1 B1(nm)

ϕ2 B2(nm)

ϕ3 B3(nm)

Determinationcoefficient

Standarderror

A 0.431 0.99 0.424 5.02 0.134 32.76 0.99864 0.008923B 0.350 1.01 0.469 5.00 0.168 32.05 0.99932 0.007154C 0.289 1.02 0.500 5.09 0.196 32.39 0.99953 0.005996D 0.243 1.03 0.519 5.26 0.220 33.20 0.99965 0.005205

Note: 1. ϕi is the ratio of the porosity of the pore component i to total porosity.2. Bi is the pore radius at the peak of the pore size distribution on a logarithmic scale.

Page 7: Numerical simulation of moisture transport in concrete based on a pore size distribution model

Fig. 11. Predicted pore size distribution of the specimens' pore components.

37Q. Huang et al. / Cement and Concrete Research 67 (2015) 31–43

After Eqs. (28) to (30) are combined, the boundary conditions of themoisture transport equation are obtained as:

D hð Þ ∂h∂x ¼ f hs−heð Þ ð31Þ

D hð Þ ¼ KLρlRTMwh

þ KVpvs; f ¼DvaMwpvs

RTδ: ð32Þ

3. Moisture transport experiment procedure

In order to validate the proposed model, moisture transport experi-ments were conducted in a chamber with atmospheric environment.

(a) Specimen A

(c) Specimen C

Fig. 12. Comparison between predicted a

Humidity sensors were used to measure the internal RH, as well as theenvironmental RH, during the experiment.

3.1. Materials and specimen preparation

Mix proportions of four concrete specimens are shown in Table 1.Ordinary Portland cement was used, and its chemical composition isshown in Table 2. River sands and crushed granite stones were alsoused, with the sands' fineness modulus of 2.5–2.6 and crushed stonesizes of 4.76–16 mm.

Thedimensions of the concrete specimenswere 100×150×300mm.They were cured at 23 °C and 100% RH for 90 days. In order to achievemoisture transport in one dimension, their lateral surfaces were sealedby epoxy, leaving two opposite sides with dimensions of 150 × 300 mmexposed to ambient air after curing (Fig. 6). Due to the differences ofpore structures between the boundary surface and the core area of thespecimens, 2 mm boundary layers of the exposed sides were removed.

3.2. Method to measure the internal relative humidity

The internal RH of the specimens was measured by small humiditysensors (DB170: Dalian Beifang M&C Engineering Co., Ltd., China)with a width of 4mmand a thickness of 2mm. Themeasurement accu-racy of the sensors was ±1.8% RH in the ranges of 10–90% RH at 25 °C.Plastic pipes, as shown in Fig. 7(a), were used to wrap and affix the hu-midity sensors and then filled with epoxy to keep humid air fromleaking in and/or out through the pipes. In order to ensure measure-ment accuracy at high RH, the humidity sensors were coated with wa-terproof, air-permeable sheets to prevent liquid water penetration(Fig. 7). The humidity sensors were located 8, 13, 18 and 23 mm fromthe exposed area by affixing the pipes to certain positions of thewooden

(b) Specimen B

(d) Specimen D

nd measured pore size distribution.

Page 8: Numerical simulation of moisture transport in concrete based on a pore size distribution model

Table 4Modified parameters of each specimen in pore size distribution model.

Specimen ϕt1 B1 (nm) ϕt2 B2 (nm) ϕt3 B3 (nm) ϕt4 B4 (nm)

A 0.040 0.99 0.039 5.02 0.012 32.76 0.005 571.80B 0.042 1.01 0.056 5.00 0.020 32.05 0.012 571.80C 0.038 1.02 0.065 5.09 0.025 32.39 0.031 571.80D 0.039 1.03 0.082 5.26 0.035 33.20 0.039 571.80

Note: ϕti is the porosity of the pore component i.

38 Q. Huang et al. / Cement and Concrete Research 67 (2015) 31–43

bars on the top of the molds (Fig. 8). A typical cured specimen used forthe moisture transport experiments is shown in Fig. 9.

3.3. Environmental conditions

After curing, the concrete specimens were moved into an atmo-spheric environmental chamber where normal ranges of RH and tem-perature could be achieved. The environmental condition of thechamber was set to be 5%–70% RH and 20 °C. During the experiment,the environmental RH and temperature around the specimens weremeasured, with the results shown in Fig. 10. Due to technical problemsof the chamber, the environmental RH fluctuated at intervals of severaldays. Fortunately, the duration time of the fluctuation was relativelyshort compared with the whole experiment time. The measured envi-ronmental conditions were used for the simulation of moisture trans-port in concrete.

4. Validation of the model

Themeasured environmental RH values and temperatures shown inFig. 10 were used as the environmental conditions of the moisture

(a) Specimen A

(c) Specimen C

Fig. 13. Comparison between predicted and measured rela

transport equation. Eq. (4) was then solved with a finite differencemethod [42] with a time step Δt of 10 min. The time step was selectedin consideration of calculation accuracy and the frequency of data acqui-sition formeasured RH. Based on the pore size distributionmodel, mois-ture transport in the specimenswas simulated by the proposedmethod.

4.1. Pore size distribution

The parameters of the pore size distribution model for each speci-men as shown in Table 3, were obtained by the proposedmethod intro-duced in Section 2.1. The normalized probability density function (dϕ/d log D)/ϕt of each pore component of the specimens was plottedagainst pore diameter and is shown in Fig. 11. The results indicate thatas the water–cement ratio increased, the proportion of gel pores de-creased, and the proportion of the small and large capillary componentsof pores increased. This is in agreement with experimental results in[16] and [43]. After the measured pore size distribution of the concretesamples in [43] were normalized by their total porosity respectively, itshowed a similar trend compared with the authors' predictions. Withthe decrease of water–cement ratios from 0.45 to 0.27, an increase ofmicropores with radii less than 2 nm, which were in the range of C–S–H pores defined in this paper, was exhibited. By contrast, mesoporeswith radii between 2 nm and 10 nm decreased significantly, whichwere in the range of small capillary pores. With respect to the poreswith radii larger than 50 nm, the results of MIP tests for concrete of dif-ferent mix proportions in [16] also indicated that the pores of this com-ponent generally increased with increasing water–cement ratios from0.4 to 0.8. The relationship between the water–cement ratio and theproportion of each pore componentwas probably dependent on the dis-tances both between C–S–H gels and between cement particles. As the

(b) Specimen B

(d) Specimen D

tive humidity at different distances from the surface.

Page 9: Numerical simulation of moisture transport in concrete based on a pore size distribution model

Table 5Parameters of each specimen in pore size distribution model in the simulation.

Specimennumber

ϕt1 ϕt2 ϕt3 ϕt4 B1(nm)

B2(nm)

B3(nm)

B4(nm)

S 0.042 0.056 0.020 0.012 1.01 5.00 32.05 571.80S1-1 0.032 0.056 0.020 0.012S1-2 0.052S2-1 0.042 0.046 0.020 0.012S2-2 0.066S3-1 0.042 0.056 0.010 0.012S3-2 0.030S4-1 0.042 0.056 0.020 0.002S4-2 0.022

39Q. Huang et al. / Cement and Concrete Research 67 (2015) 31–43

water–cement ratio of the concrete increased, the amount of C–S–H gelwas smaller under the same hydration time while the distances bothbetween C–S–H gels and between cement particles were larger.

In order to verify the theoretical pore size distribution model, thepore size distributions of the specimensweremeasured by themercuryintrusion porosimetry (MIP) method, which could access pores withradii above 3 nm. The results of the MIP tests are shown in Fig. 12 andindicate that the specimens may include another pore component ofconcrete, the diameters of which were around 1000 nm. These poreswere probably microcracks and/or pores of the ITZ between the cementpaste and the aggregate. It is also worth noting that the component ofthe measured pores with the radius around 0.05 mm (the correspond-ing crack width was 0.1 mm) in Fig. 12 was probably the crack due tothe crushing of the test samples before the MIP test. These inadvertent

(c) Group S3

(a) Group S1

Fig. 14. Predicted moisture diffusivity ver

macro-scale voids in the MIP results should not be used in the analysisas recommended by Maekawa et al. in [37].

The widths of the microcracks in this paper were considered tobe around 1–8 μm (the corresponding pore radius was around500–4000 nm in Fig. 12), which was close to the results observedby Bisschop and van Mier using a microscopy method in [44]. As-suming that the sizes of microcracks also followed the R–R distribu-tion, the porosity of each component ϕti (ϕti = ϕt ⋅ ϕi) could beobtained, as shown in Table 4, using the MIP results and a constantproportion of the three pore components (gel pores and small andlarge capillaries).

The comparisons between the predicted andmeasured pore size dis-tributions of the specimens are shown in Fig. 12. It is not surprising thatthe measured volumes of pores with radii between 10 and 25 nmwerelarger than the predicted volumes because the volumes of measuredpores also contained larger pores that were connected to smallerpores leading to the ink bottle effect. On the other hand, the resultsalso show that the measured volumes of pores with diameters of lessthan 10 nm were smaller than the predicted volumes. This may havebeen because these pores were not connected to larger pores andwere only accessible through smaller gel pores. As modeling of the inkbottle effect still needs further investigation, it was not incorporatedin the simulation of moisture transport in concrete.

4.2. Comparison between predicted and measured results

The predicted and measured RH values at different distances fromthe concrete surface are shown in Fig. 13. During the experiments, the

(d) Group S4

(b) Group S2

sus relative humidity of each group.

Page 10: Numerical simulation of moisture transport in concrete based on a pore size distribution model

(a) Group S1 (a) Group S2

(a) Group S3 (a) Group S4

Fig. 15. Pore size distribution of each group and relationship between relative humidity and critical pore radius.

40 Q. Huang et al. / Cement and Concrete Research 67 (2015) 31–43

values of RH at distances of 8 and 13mm from the surface of specimen Bwere unfortunately not obtained, due to technical problems.

From the comparison of the predicted and measured results, it canbe concluded that their moisture transport rates were quite close. Thecomparisons also showed that the values of the measured internal RHremained unchanged for a longer time at the beginning of theexperiment.

The delay of the measured RH compared with the predicted valueswas in conformancewith the results of Li and Li [45]. This was probablydue to the time needed for the humidity balance of the humidity sensor,the air in the waterproof sheet and concrete. Moreover, aggregates mayhave been distributed around the humidity sensors, resulting in local ef-fects on the moisture transport.

Another reason for the delaywas probably the complex connectivityof pores, whichwas simplified in the calculation. The probability of per-meation through two pores was considered to be linear to their cross-sectional areas in Eq. (25); however, the actual connectivity of poresmay also depend on their individual length, orientation and spatial dis-tribution. Also, some larger pores may tend to be connectedwith small-er pores forming the ink bottle effect. The liquid water in the largerpores would be stuck at a high RH until the connected smaller pores be-came empty at low RH, leading to the delay of measured values.

The comparison in Fig. 13 also indicated that generally, the mea-sured values increased more significantly when the internal RH beganto increase slightly before 40 days. This could be explained by the de-creasedmoisture capacity comparedwith the value from the adsorptionisotherm, i.e., less moisture content increased given a small increase ofRH, since the liquid water trapped in the larger pores did not contributeto the increase.

At the time between 0 and about 40 days, the predicted RH de-creased more gently than the measured values for specimen A, while

it showed an opposite trend for specimens B–D in Fig. 13. This couldbe due to the determination of tortuosity of pores, which was given asa constant (π / 2)2 for a uniformly random porousmedium in the calcu-lation. However, more complicated networks of pores from 3D imagesof pore clusters were exhibited based on X-ray computedmicrotomography in [10]. Since the mix proportions of specimens A–D were different, their pore structures including the tortuosity couldalso be different. Another factor that contributed to the difference be-tween the predicted and measured values was the pore shape. In thecalculation, the pores were assumed as ideal cylinders; however, thepore geometry wasmuchmore complex in the real concrete. The selec-tion of pore shapes can have an impact on the prediction of moisturetransport. For instance, for the oblong shaped pores, the liquid flowrate formulas for the flattened parts are different from those for the cy-lindrical pores [18].

At low environmental RH values, both the predicted and measuredresults indicated that the drying rates of specimens C and Dwere great-er than those of the other specimens, probably due to the larger porosityof the microcracks of specimens C and D, shown in Fig. 12. However, athigh environmental RH values, the varying rates of the internal RH ofthe specimens were not significantly different. The moisture transportrates of the specimens mainly depended on their pore size distribution.The influence of each pore component's porosity on moisture transportwas studied systematically, the results of which are presented inSection 5.

5. Influence of pore size distribution on moisture transport

In order to investigate the influence of each component of concretepores on moisture transport, numerical simulations were conductedbased on specimens with different pore size distributions. The porosity

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(a) Group S1 (a) Group S2

(a) Group S3 (a) Group S4

Fig. 16. Relative humidity profiles of each group during drying and wetting processes.

41Q. Huang et al. / Cement and Concrete Research 67 (2015) 31–43

of each component of the control specimen S was taken as that of spec-imen B in Section 4. The other specimens varied one of the components'volumes (ϕt1–ϕt4), with the volumes of the others remaining constant,as shown in Table 5.

5.1. Simulated moisture diffusivity

For isothermal moisture transport in concrete, Eq. (4) can be rewrit-ten as:

∂h∂t ¼ ∂

∂x Dh∂h∂x

� �ð33Þ

where Dh is the moisture diffusivity (m2/s) given by:

Dh ¼ KLρlRT= Mwhð Þ þ KVpvs∂w=∂h : ð34Þ

The coefficient Dh reflects the rate of moisture transport; hence, de-termining its relationship with the pore size distribution of concretewas an important objective in this study. The moisture capacity ∂w/∂hhas a linear relationship with the probability density ∂φ/∂(log rc):

∂w∂h ¼ ρl

∂ logrc∂h

� � ∂φ∂ logrc

ð35Þ

where rc is the critical radius, and ∂logrc/∂h can be determined byEqs. (14), (16) and (17).

From Eqs. (21) and (27), liquid water permeability KL and watervapor permeability KV depend on the volumes of pores with radii small-er and larger than critical radius rc, respectively. As liquid water perme-ability increases linearlywith the square of the pore radius, its value alsodepends on the size of pores related to the liquid transport.

The simulated moisture diffusivity of each specimen in Table 5 isshown in Fig. 14. The steep increase of moisture diffusivity Dh at higherRH values resulted frommore poreswith larger radii contributing to liq-uid water transport. On the other hand, the diffusivity at lower RHvalues was dominated by vapor transport, which varied more gentlywith RH.

From the simulation results, the moisture diffusivity at the gentlyvarying stage increasedwith a decrease of the gel pore volume or an in-crease of the large capillary pores and microcracks. It is worth notingthat for gently dried concrete, a great amount of microcrack initiationor propagation was actually not expected although microcracks can in-crease the drying rate. The results can be explained by the pore size dis-tribution and critical pore radius versus RH, as shown in Fig. 15. If wetake an RH value of 60% as an example, the moisture capacity of speci-men S1-1, according to Eq. (35), was about 1.3 times that of specimenS1-2 at the corresponding critical radius; however, the difference inVr N rc, the volume of pores with radii larger than the critical radius,was relatively small. As a result, the moisture diffusivity of specimenS1-1 was larger than that of specimen S1-2 at 60% RH, due to a highermoisture capacity but a relatively close water vapor permeability.

On the other hand, the moisture capacity of specimens in groups S3or S4 was the same at 60% RH, and the pore volumes Vr N rc increasedwith the volume of the pore component. Consequently, the moisture

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42 Q. Huang et al. / Cement and Concrete Research 67 (2015) 31–43

diffusivity at 60% RH increased with the increase in the volume of com-ponents in groups S3 and S4.

The results of group S2 in Fig. 14 showed that the moisture diffusiv-ity at steeply varying stages increased with increases in the volumes ofsmall capillary pores due to differences in Vr b rc, the volumes of poreswith radii smaller than the critical radius at high RH values. It is clearlyshown in Fig. 15 that there was no difference in pore volumes Vr b rc ofgroup S4 at high RH values, leading tomoisture diffusivities very similarto the group at the steeply varying stage.

Although there were incredible differences in the pore volumesVr b rc of the specimens in group S1, as shown in Fig. 15, the differencesin theirmoisture diffusivities at the steeply varying stagewere small be-cause the sizes of the varying gel pores were relatively small. As forgroup S3, the moisture diffusivity at the steeply varying stage did notnecessarily increase with increases in large capillary pore volumes dueto the relatively small increase of permeability compared with the in-crease of moisture capacity.

5.2. Simulated internal relative humidity

In order to investigate the influence of pore size distribution onmoisture transport in concrete, the internal RH was simulated underdrying and wet conditions. The environmental conditions were set atRH values of 20 and 98%, with uniform initial internal RH values of100 and 20% of the specimens, respectively, corresponding to dryingand wet conditions. The specimens in Table 5 were dried for 90 daysor wetted for 45 days under a constant environmental temperature(20 °C).

The simulated results in Fig. 16 show that after drying, the RH pro-files h(x) of the specimens were similar except for group S4. For con-crete drying at a low RH, water vapor transport played a dominantrole; and, the drying rate largely depended on moisture diffusivity atthe gently varying stage. As the difference in moisture diffusivity ofgroup S4 at this stage was larger than the other groups, as shown inFig. 14, the RH profiles in group S4 were more distinct from each other.

In group S4, the influence of microcracks on concrete dryingmay beoverestimated since the microcracks are possibly not interconnectedbut connected to narrower pores, which impede the process of drying.Moreover, themicrocracks are randomly oriented, leading to a tortuousflow path. Bisschop and vanMier [44] also noted that the drying shrink-age microcracks due to aggregate restraint might have a less significanteffect on the drying rate than those due to substrate restraint.

For the concrete wetting process, the RH profile was usually steepfronted, with the front called the wetting front. Liquid water transportwas the dominant moisture transport pattern in the wetted region be-tween the wetting front and the exposed surface; whereas in theunwetted region, water vapor transport was the dominant pattern.

The simulated results of concretewetting of specimen groups S1 andS4 in Fig. 16 indicate that the RHprofiles in thewetted regionwere sim-ilar; however, they differed in the unwetted region due to the similarmoisture diffusivities at high RH values but distinct diffusivities at lowRH values, as shown in Fig. 14. The position of the wetting front of spec-imen S2-2 was a little farther from the concrete surface than that ofspecimen S2-1 due to the larger moisture diffusivity of specimen S2-2at a high RH. The simulated results also showed that the RH profiles ofgroup S3were distinct fromeach other in bothwetted andunwetted re-gions due to distinct moisture diffusivities at high and low RH,respectively.

6. Conclusions

A numerical approach based on a pore size distribution model wasdeveloped for the simulation of moisture transport in concrete. Thecomparisons between the predicted and measured pore size distribu-tions showed that the measured volumes of pores with diameters of20–50 nmwere larger than the predicted volumes due to the ink bottle

effect. The results also indicate that considerable small capillaries withdiameters of less than 20 nm may be connected only to smaller gelpores.

Moisture transport experiments were conducted to verify the pro-posed approach. The comparison results of the predicted and measuredRH at different positions in the concrete specimen indicated acceptableagreement. The simulation results of the internal RH of concrete speci-mens with prescribed pore size distributions showed that themicrocracks had a greater influence on the moisture transport thanthe other pore components under an isothermal drying process. It wasalso revealed that more large-capillaries result in both larger movingrates of the wet front and larger moisture transport rates of theunwetted region. Microcracksmay have a great impact on the moisturetransport of an unwetted region, but are not notable in the impact of awet front.

Acknowledgments

This research project was financially supported by the National Nat-ural Science Foundation of China (Grant Nos. 51320105013 and51109163).

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