simulation of humidity fields in concrete: experimental

Simulation of humidity fields in concrete: experimentalvalidation and parameter estimationMateus A. Oliveira, Miguel Azenha Paulo B. Lourenço,

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Characterization and Modeling of Pores and Surfaces in Cement Paste: Correlations to Processing andPropertiesHamlin Jennings, Jeffrey W. Bullard Jeffrey J. Thomas, , Jose E. Andrade, Jeffery J. Chen, George W. SchererJournal of Advanced Concrete Technology, volume ( ), pp.6 2008 5-29

Enhanced Shrinkage Model Based on Early Age Hydration and Moisture Status in Pore StructureYao Luan , Tetsuya IshidaJournal of Advanced Concrete Technology, volume ( ), pp.11 2013 360-373

Hygrometric assessment of internal reative humidity in concrete: Practical application issuesJosé Luís Granja , Miguel Azenha Christoph de Sousa, , Rui Faria, Joaquim BarrosJournal of Advanced Concrete Technology, volume ( ), pp.12 2014 250-265

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Journal of Advanced Concrete Technology Vol. 13, 214-229, April 2015 / Copyright © 2015 Japan Concrete Institute 214

Scientific paper

Simulation of Humidity Fields in Concrete: Experimental Validation and Parameter Estimation Mateus A. Oliveira

1, Miguel Azenha

2* and Paulo B. Lourenço

3

Received 2 January 2015, accepted 29 March 2015 doi:10.3151/jact.13.214

Abstract The moisture content in concrete structures has an important influence in their behavior and performance. Several vali-dated numerical approaches adopt the governing equation for relative humidity fields proposed in Model Code 1990/2010. Nevertheless there is no integrative study which addresses the choice of parameters for the simulation of the humidity diffusion phenomenon, particularly in concern to the range of parameters forwarded by Model Code 1990/2010. A software based on a Finite Difference Method Algorithm (1D and axisymmetric cases) is used to perform sensitivity analyses on the main parameters in a normal strength concrete. Then, based on the conclusions of the sensi-tivity analyses, experimental results from nine different concrete compositions are analyzed. The software is used to identify the main material parameters that better fit the experimental data. In general, the model was able to satisfactory fit the experimental results and new correlations were proposed, particularly focusing on the boundary transfer coeffi-cient.

1. Introduction

The properties, performance and durability of cement–based materials such as concrete depend strongly on the moisture content (Xi et al. 1994b; Mehta 1997; Zhang et al. 2009; Maekawa et al. 1999; Gawin et al. 2003). The moisture content affects the material in different ways, such as in the creep behavior, shrinkage, carbonation process, chloride and sulfates ingress, evolution of the hydration, freeze-thaw resistance, durability, compres-sive strength, elastic modulus and others.

The influence of the humidity on the concrete me-chanical aspects and response was studied by different authors (Loukili et al. 1999; Yuan and Wan 2002; Baroghel-Bouny et al. 1999; Cadoni et al. 2001; Grasley et al. 2006), who studied the shrinkage or the developed stresses in the material during the drying process. Also, Bažant and Chern (1985) and Benboud-jema et al. (2005) have analyzed the relation between the humidity and the creep behavior. Regarding material

durability, the carbon dioxide diffusion and the carbona-tion processes depend on the humidity (Ferretti and Bažant 2006; Papadakis et al. 1991; Saetta et al. 1995), as well as the chloride ingress process (Oh and Jang 2007; Saetta et al. 1993; Lindvall 2003) and other multi–physics processes such as the ingress of sulfates into concrete (Nehdi and Hayek 2005). It is also known that there is a critical degree of pore saturation above which freeze-thaw cycles can be especially deleterious and generate spalling in concrete (Persson 1997; Neville 1995; Nehdi and Hayek 2005). As an example of com-mon practical application in construction, knowledge of the internal humidity level in concrete is particularly relevant to support decisions on the instant for applying slab coatings (floor coverings or resinous coatings) for industrial floors, in order to prevent future problems associated to moisture migration and consequent debonding of the coating layer (ASTM 2011; Kim and Lee 1998).

Therefore, knowledge of the moisture distribution within concrete structures since construction and throughout service life can assist a better understanding their actual performance (Conciatori et al. 2014), and even support measures to prevent damage and extend the service life. Thus, moisture diffusion in concrete has been studied by many authors in view of several differ-ent circumstances and final objectives such as: evaluat-ing the impact of material composition (Bažant and Najjar 1971; Bažant 1972; Kang et al. 2012; Persson 1996; Mjörnell 1997; Nilsson 2002), time (Bažant and Najjar 1971; Bažant 1972; Kang et al. 2012; Persson 1996; Mjörnell 1997; Nilsson 2002), governing equa-tions or simulation formulations (Zhang et al. 2009; Bažant 1972; Kim and Lee 1999; Ishida et al. 2007), choice for driving potential for measurement/simulation (e.g. internal humidity (Kim and Lee 1999) or actual

1PhD Student, ISISE - Institute for Sustainability and Innovation in Structural Engineering, School of Engineering, Department of Civil Engineering,University of Minho, Guimarães, Portugal. 2Assistant Professor, ISISE - Institute for Sustainability and Innovation in Structural Engineering, School of Engineering, Department of Civil Engineering,University of Minho, Guimarães, Portugal. *Corresponding author, E-mail: [email protected] 3Full Professor, ISISE - Institute for Sustainability and Innovation in Structural Engineering, School of Engineering, Department of Civil Engineering, University of Minho, Guimarães, Portugal.

M. A. Oliveira , M. Azenha and P. B. Lourenço / Journal of Advanced Concrete Technology Vol. 13, 214-229, 2015 215

water content (Janoo et al. 1999; Klysz and Balayssac 2007), and other aspects (Nilsson 2002; Roels 2000; Zhang et al. 2014).

The drying process in concrete is a complex mecha-nism, as different aspects are coupled and involved on the transport of water in porous materials such as con-crete. As water is present in the porous matrix under the form of gas and liquid, several simulation approaches have explicitly considered both states of water in their modelling assumptions and governing equations (Gawin et al. 1996; Whitaker 1977). In spite of such complexity, it has already been shown that the simulation of con-crete drying can be simplified and reduced to a single diffusion equation, based on the assumption that the drying of weakly permeable materials is mainly achieved by the transport of moisture in its liquid form (Mainguy et al. 2001). In another study Baroghel-Bouny et al. (1999) define internal relative humidity of concrete as the relative humidity (h) of the gaseous phase in equilibrium with the interstitial liquid phase in the pore network of the material. This internal relative humidity can also be used within the framework of a single diffusion equation that simulates the process of drying and lumps together the implicit consideration of transport of liquid and gaseous water. The Partial Dif-ferential Equation (PDE) that models the humidity (Kim and Lee 1999; Azenha 2009; CEB–FIP 1993; CEB–FIP 2010; Bažant and Najjar 1972) is typically solved through the Finite Difference Method (FDM) (Kang et al. 2012; LeVeque 2007) or the Finite Element Method (FEM) (Di Luzio and Cusatis 2009b; Di Luzio and Cu-satis 2009a).

Bažant and Najjar (1971) used a numerical formula-tion based on internal concrete humidity as the driving potential for moisture movements, which included a specific model for the corresponding diffusivity coeffi-cient. This approach has been adopted by Model Code 1990 (MC90) (CEB–FIP 1993) and Model Code 2010 (MC2010) (CEB–FIP 2010), and will also be used in this paper. In fact, according to FIB bulletin 70 (FIB 2013), the parameters proposed by MC1990/2010 seem to have been solely derived with basis on diffusion ex-periments (e.g. the cup-method) and no validation of the Model Code was found to focus on the humidity profil-ing of concrete specimens. It is worth to remark that neither MC90 nor MC2010 provide any recommenda-tion on how model the boundary conditions for the hu-midity field simulation, which nonetheless have an im-portant influence on the results.

This paper presents a set of simulations of moisture movement in hardened concrete, assessing and critically evaluating the predictive capacities of the models of MC90/2010. First, a review of the literature is given focusing on moisture, humidity, relevant concepts, pos-sible formulations and numerical implementation. Then, the algorithm in FDM (1D and axisymmetric) (LeVeque 2007) is used to perform sensitivity analyses and to re-produce the behavior of nine different experiments us-

ing distinct concrete compositions (Kang et al. 2012; Persson 1997; Kim and Lee 1999). Finally, the inte-grated analysis of results allows issuing recommenda-tions about the diffusivity and boundary transfer coeffi-cients, together with the main conclusions of this work.

2. Numerical simulation of moisture fields in concrete

This section presents the general aspects regarding nu-merical simulation of moisture fields in concrete, focus-ing namely on the driving potential (absolute water con-tent of average pore humidity), governing equations, diffusion properties and boundary condition considera-tion. 2.1 General considerations and governing equations Water in concrete (W) is usually classified in two main categories (Taylor 1977; Powers and Brownyard 1948): evaporable water (We) and non–evaporable water (Wn).

Evaporable water corresponds to the parcel of the to-tal water that is available for transport and may ulti-mately evaporate to the outer environment. Evaporable water comprises interlayer water between the layers of reacted material, adsorbed water at the pore surfaces and capillary condensed water in the pores. The rest of the water in the cementitious material, which is either chemically combined or with strong physical bonds to the material, is called non–evaporable water (Neville 1995; Granger 1996). The total water (W) corresponds to the sum of We and Wn. It is however known that dur-ing early ages, cement hydration causes transformation of significant parts of We into Wn. Mass balance equa-tions for moisture content in concrete since early ages should therefore account for this phenomenon (Azenha 2009; Taylor 1977), as follows:

( ( ))e e nW div Dgrad W W= − (1)

where: D is the diffusion coefficient [m2s–1] and the super-script sign ‘.’ stands for the first derivative in time. Other formulations based on water content, with spe-cific multi–phase models that consider liquid and vapor water, as well as convection transport within the porous medium are available (Jennings et al. 2008; Granger et al. 1997; Pel et al. 2002).

Alternatively to the water concentration modeling, some authors propose formulations based on internal relative humidity h as the driving potential. Internal humidity of the material can be defined as the relative humidity of the gaseous phase in equilibrium with the interstitial liquid phase in the pore network of the mate-rial (Baroghel-Bouny et al. 1999; Azenha 2009). Ac-cording with Xi et al. (1994a) and Roncero (2000), boundary conditions are easier to express in terms of h than in terms of W. Furthermore, internal profiling of moisture fields through non–destructive and quantitative methods is more feasible through h than W (Kang et al.


2012; Kim and Lee 1999; Baroghel–Bouny 1996; Zhang et al. 2012; Xi et al. 1994b). This is an important aspect when validation of simulation models is envis-aged. Authors that model moisture fields for shrinkage prediction also defend the advantages of considering internal h due to the simplicity of the relationship be-tween h and local shrinkage strains (Azenha 2009). Also, for usual w/c ratios the drop in h due to chemical hydra-tion of cement is relatively small (less than 3%), and thus it can be neglected even if hydration reactions have not ceased. This is not the case when W is used as a po-tential, as Wn and We vary significantly during hydration.

It is possible to plot the relationship between internal h and W (expressed in mass) for a specific concrete at constant temperature. This relationship is known as moisture isotherm (adsorption/desorption) (Baroghel–Bouny 1996), as qualitatively depicted in Fig. 1. It can be seen that the adsorption and desorption curves are different, highlighting the hysteretic behavior of con-crete in regard to water retention (Azenha 2009; ACI 2006). This hysteresis is usually explained with the so–called ink–bottle effect (Brunauer 1943; Bazant and Bažant 2012).

Bearing in mind that the slope of the moisture iso-therm (moisture capacity) can be expressed by dW/dh, that the self-desiccation can be expressed by dhs/dt, and that humidity diffusion can be expressed by the term Dh, Eq. (1) may be transformed into a format based on in-ternal relative humidity:

1

( ( )) sh

hh W div D grad ht h t

− ∂∂ ∂⎛ ⎞= +⎜ ⎟∂ ∂ ∂⎝ ⎠ (2)

The transformation of Eq. (1) into Eq. (2) implies the factor 1( / )W h −∂ ∂ at the right–hand side of the equation. This factor is the reciprocal of the slope of the moisture isotherm W = f(h). Some authors defend that moisture capacity of cementitious materials at usual environ-mental relative humidity (h > 50%) is fairly constant (Roncero, 2000), thus motivating a further simplifica-tion on Eq. 2, by lumping terms 1( / )W h −∂ ∂ and Dh into a single term called *

hD . This simplification is even more reasonable if solely desorption processes are en-visaged (frequently valid in sheltered concrete subjected to drying). It is however remarked that other authors such as Baroghel-Bouny (2007) have reported quite distinctive slopes of 1( / )W h −∂ ∂ at high humidity ranges, thus limiting the validity of the above-mentioned sim-plification. In coherence with the varying-slope assump-tion for the moisture isotherm, Xi et al. (1994a) have proposed a mathematical model to predict experimental adsorption isotherms for cement pastes. Nonetheless, no general validated model was found in the literature in regard to the prediction of moisture isotherms in con-crete (Azenha, 2009). Therefore, moisture isotherms in concrete are usually obtained experimentally (Baroghel-Bouny 2007; Hansen 1986).

The simplification of considering a constant slope for

the moisture isotherm (that allows the use of a single term *

hD ) is adopted in this work, both because of the lack of a general model of the moisture isotherm for concrete, and due to the inherent simplicity of applica-tion of such approach, already adopted in MC1990/2010.

Additionally, some considerations can be made in re-gard to the term /sh t∂ ∂ of Eq.(2), as it can be consid-ered negligible outside the scope of the early ages, dur-ing which most of the hydration reactions occur. This implies that /sh t∂ ∂ can be assumed negligible in the study of hardened concrete, if the effect is taken into account through the initial conditions of the problem, e.g. initial humidity of 95% at 28 days to consider inter-nal reductions due to self-desiccation. As a result of the reasoning above, Eq. (2) can be transformed into Eq.(3) below, which coincides with the formulation proposed by MC90/MC2010 (Bažant and Najjar 1972; Kim and Lee 1998; CEB–FIP 2010; CEB–FIP, 1993; Kim and Lee 1999).

* ( ( ))hh divD grad ht

∂=

∂ (3)

2.2 Diffusion coefficient When considering h as the driving potential for the moisture field, the diffusion coefficient has been defined as a non-linear function of the local relative humidity or of the moisture content by authors such as (Bažant and Najjar 1971; Mjörnell 1997; Mensi et al. 1998; Martinola and Sadouki 1998; Sadouki and van Mier 1997; Christensen 1979; Suwito et al. 2006). However the most widespread formulation for moisture diffusiv-ity in concrete, using h as the driving potential, was proposed by (Bažant and Najjar 1971) and was included in MC90 and also in MC2010. For isothermal condi-tions the diffusion coefficient can be expressed as a function of the pore relative humidity 0 < h < 1 (Bažant and Najjar 1971; CEB–FIP 2010; CEB–FIP 1993):

11

1 (1 ) /(1 )h nc

D Dh hαα

⎡ ⎤−⎢ ⎥= +⎡ ⎤+ − −⎢ ⎥⎣ ⎦⎣ ⎦

(4)

Fig. 1 Typical shape of a moisture sorption isotherm for concrete (adapted from Azenha (2009)).


where: D1 is the maximum Dh for h =1 [m2/s]; D0 is the minimum Dh for h = 0 [m2/s], α = D0 / D1 = 0.05, hc the relative pore humidity at

1( ) 0.5hD h D= (hc = 0.80) and n is an exponent (n=15). According to MC90 and MC2010, D1 is defined as a function of the mean compressive strength of concrete fcm expressed in MPa as:

1, 0

18

cm

DD

f=

−

8 2 2

1, 0 1, 01 10 / 864 /D m s or D mm day−= × =⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦ (5)

These parameter values were also used by Kim and Lee (1999).

2.3 Boundary conditions In regard to the boundary conditions to apply for the humidity field modelled according to Eq. 3, two stan-dard strategies can be used. The first strategy corre-sponds to imposing the value of the environmental hu-midity to the concrete surface (Dirichlet boundary con-dition) (Ferretti and Bažant 2006; Zill 2012; Crank 1979). The alternative strategy corresponds to the appli-cation of Neumann’s boundary condition, through the use of a proportionality factor (fboundary) between the exposed surface flux and the humidity difference be-tween the environment (hen) and the concrete surface (hsurf) (Azenha 2009; Shimomura and Maekawa 1997):

( )h boundary en surfs

hD f h hx∂⎛ ⎞ = −⎜ ⎟∂⎝ ⎠

(6)

where x is the coordinate in a direction perpendicular to the boundary surface.

Alternatively to the above-cited approaches, Bažant and Najjar (1972) dealt with the boundary condition issue by assuming an additional thickness to the speci-men (i.e., the equivalent surface thickness). Through a comparison between analytical and experimental results, they recommended an equivalent surface thickness of 0.75 mm (Bažant and Najjar 1972; Kim and Lee 1999) at the end of which a Dirichlet condition would be ap-plied. Nonetheless, no further research works were found to share this modelling strategy for boundary conditions.

The MC90 and the MC2010 do not provide recom-mendations for modelling the boundary condition for humidity diffusion. Research works that use the govern-ing Eq. 3 have either used Dirichlet (Ferretti and Bažant 2006) or Neumann conditions (Kim and Lee 1999).

If an analogy is made to thermal field simulations, it is easily acknowledged that Dirichlet and Neumann conditions apply in quite distinct situations (LeVeque 2007; Zill 2012). In fact, the imposition of a given tem-perature in the boundary of a solid is rare, whereas boundaries that correspond to direct contact with the surrounding environment lead to the consideration of Neumann boundary conditions (Shimomura and Maekawa 1997). The temperature in the surface of a solid in contact with the atmosphere is systematically

distinct from that of the boundary surface. The surface boundary coefficient for thermal models is furthermore dependent on the wind speed in the vicinity of the sur-face, which can increase the intensity of thermal ex-changes. It has also been demonstrated that the surface humidity is distinct from the environmental one (Nilsson 2002) and that the moisture exchange intensity can be dependent on wind speed at very early ages when a wet film is still present on the cementitious material (Azenha et al. 2007a, 2007b).

In coherence with this acknowledged relevance of us-ing Neumann boundary conditions for the simulation of moisture fields, both Sakata (1983) and Akita et al. (1997) have obtained experimental correlations between the boundary coefficient and the water to cement ratio (w/c). However, their findings were contradictory: Sa-kata claims that an increase in w/c leads to an increase of the boundary coefficient, whereas Akita et al. (1997) observed an opposite trend.

Theoretical approaches to the moisture diffusion problem by Shimomura and Maekawa (1997), Yiotis et al. (2007), and Zhi et al. (2010) also acknowledge the importance of considering Neumann boundary condi-tions, namely through the adoption of a surface factor related to porosity.

Taking into account the above reasoning, the ap-proach adopted in this paper focuses on the adoption of Neumann boundary conditions for surfaces in contact with the environment.

3. Numerical implementation

This section presents a brief description of the numeri-cal implementation in Finite Difference Method (FDM) in 1D to solve the humidity diffusion equation. This shows the simplicity of the implementation for 1D prob-lems, which is suitable for design purposes through rela-tively simple spreadsheets. Details about the mathe-matical background of the FDM can be found in Incrop-era et al. (2007) and Özisik (2002).

The mathematical equations are developed for an in-finite slab, symmetrical in regard to its middle plane. The corresponding discrete model has a finite number of nodes, starting from node 1 in the vicinity of the bound-ary, and progressively numbered until the extremity node at the at the symmetry plane. The notation adopted herein considers that “i” represents the ith node, and “n” the nth time step. Therefore, the humidity at node i and time step n is denoted by the following set of super-script/ subscript: i

nh . The time and space discretization for the FDM can be

assumed for small intervals of time (dt = Δt) and length (dx = Δx). The implementation considers the field equa-tion shown in Eq. (3), which can be adapted through the application of the chain rule and transformed in Stewart (2007):


2h h

h D h D ht

∂= ∇ ∇ + ∇

∂ (7)

The second term of the right-hand side of Eq.(7) can be neglected as it represents a second order derivate and its value in the simulations presented herein was always ~10-6 times smaller than the other terms of the equation. Some intermediary steps are herein omitted in the de-velopment of the implementation, for sake of brevity. More information regarding omitted steps can be ob-tained in LeVeque (2007) and Thomas (1995).

In FDM, the final equation for inner nodes (nodes not located on boundary neither symmetry) can be re-written in a more compact format as shown in Eq. (8) and (9).

11 1 1(1 2 )i i i i i

n n n nh r r h h−+ + ++ − = (8)

, 11 2( )

ih ni

n

tDr

x+

+

Δ=

Δ (9)

Accordingly, the boundary flux at the extremity node can be expressed as defined in Eqs. (10), (11) and (12).

11 12 (1 2 ) 2i i i i i

n n nh r h r h rβ γ−+ += − + + − (10)

, 1

1 boundaryih n

f xD

β+

Δ= + (11)

, 1

en boundaryih n

xh fD

γ+

Δ= (12)

The node that pertains to the symmetry plane has null flux, and its corresponding equation can be written as:

11 1 1 1(1 2 ) 2i i i i i

n n n n nh r r h h−+ + + ++ − = (13)

By assembling Eqs. (10) and (13) for the extremity nodes (boundary and symmetry) and Eq. (8) for the set of internal nodes, it is possible to express the set of equations in matrix form, as shown in Eq. (14) (for sim-plicity of representation this set of equations pertains to a set of 6 nodes).

1 1

1

2 2

1

3 3

1

4 4

1

5 5

1

6 6

1

0 0 0 0 0

1 2 0 0 0

0 1 2 0 0

0 0 1 2 0

0 0 0 1 2

0 0 0 0 2 1 2

n n

n n

n n

n n

n n

n n

h h

r r h h

r r r h h

r r r h h

r r r h h

r r h h

γ β

γ+

+

+

+

+

+

+

− + −

− + −× =

− + −

− + −

− +

⎡ ⎤ ⎡ ⎤⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥

⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ (14)

The humidity in each step is calculated in an incre-mental/iterative process. The h value obtained on time step “n” is adopted as the first trial value for step “n+1”, particularly in regard to the estimation of Dh (implicit backward-Euler formulation) (Holmes 2007). This is a

typical non-linear process, because of the dependence of Dh on h.

The Newton–Raphson method is used to solve the non–linear system of equations (Kelley 1987).

For each iteration a residuals vector ( )1ˆ inψ + is calcu-

lated as: 1

1 1 1ˆ ˆˆ

j j

n n nh hψ+

+ + +⎡ ⎤ ⎡ ⎤= −⎣ ⎦ ⎣ ⎦ (15)

where: the symbol (^) denotes the vector and the super-script ( j )denotes the iteration.

The residual vector is calculated with the difference of humidity values in two consecutives iterations. The convergence criterion was based on the comparison between the norm of the residuals vector with the maxi-mum tolerance, which was considered equal to 0.0001. More information about the calculation of the residual vector and the overall procedure can be found in Azenha (2009).

4. Sensitivity analyses

4.1 General considerations This section aims to analyze the influence of the main parameters involved in the calculation of moisture fields according to Eqs. (3), (4) and (6). The relevant parame-ters are the boundary transfer coefficient fboundary and the diffusion coefficient Dh, which depends on several other parameters itself. As a starting point, for calculation of Dh parameter, the proposal of MC2010 for a Normal Strength Concrete (fcm = 30 MPa) is considered. There-fore, the following starting parameters for calculation of Dh were adopted: α = 0.05, hc = 0.80, D1 = 39.3 mm2/day and n = 15 (CEB–FIP 2010; CEB–FIP 1993). In regard to the boundary transfer coefficient, the initial value for these sensitivity studies is adopted as fboundary = 3.0×10-4 m/day which consistent and in the same order of magnitude of the corresponding coefficient used by Kim and Lee (1999), fboundary = 5.02×10-4 m/day.

The selected example for the sensitivity analysis con-sists in an infinite concrete slab with 60 cm thickness, symmetric in regard to its middle plane and in contact with the same environment at both surfaces. Concrete is considered as hardened and initially fully saturated (h = 100%), in correspondence to many practical situations of exposure after an adequate curing time. The sur-rounding environment was considered to have a con-stant temperature of T = 20 ºC and constant environ-mental humidity henvironment = 50 %. Self-desiccation of concrete at early ages has been disregarded.

Even though the sensitivity analyses focused in a set of 5 simulation parameters (α, hc, n, D1 and fboundary), the obtained results have shown that D1 and fboundary are by far the most influential parameters on humidity fields. Therefore, for the sake of brevity, the presentation of sensitivity analyses will be limited to these two parame-ters. In all analyses, constant time step and constant spatial discretization are applied as Δt = 1 day and Δx =


1 cm, respectively. It is noted that similar studies were made involving slabs with distinct thicknesses (20 and 100 cm) and concretes with distinct compressive strengths (15 and 50 MPa). The general conclusions of such parametric analyses were quite similar to the ones presented next.

4.2 Sensitivity analyses regarding fboundary The influence of the fboundary on the humidity diffusion process is studied though evaluation of the effects in increasing it or decreasing it by a factor of ten times in regard to the reference values. Further to these scenarios, two additional simulations were made: one considering Dirichlet boundary conditions (i.e. prescribed humidity at the boundary), and another considering the approach of a fictitious additional length of 0.75 mm as proposed by Bažant and Najjar (1972). The results of simulation for the depth of 20 cm from the surface are depicted in Fig. 2, whereas the humidity profiles for the ages of 500 days and 50 years are shown in Fig. 3.

Figure 3 presents the results using the symmetrical condition of the analyzed slab, showing calculation re-sults for 50 years on the left hand side, and calculation results for 500 days on the right hand side.

Based on the observation of Fig. 2 and Fig. 3, it is relevant to mention that, when the value of the fboundary is the highest (fboundary = 30×10-4 m/day), the drying behav-ior approaches the one that is obtained with Dirichlet conditions (fixed surface humidity). However, the de-crease in fboundary by a factor of 10 and 100 in regard to the maximum value led to significantly different results. This shows that the boundary coefficient can be quite relevant in the final results of a humidity simulation. It is also worth to note that the fictitious additional thick-ness proposed by Bažant and Najjar (1972) allowed obtaining a behavior that differs from the Dirichlet boundary condition, and feasibly resembles an interme-diate behavior between that of fboundary = 3×10-4 m/day and fboundary = 0.3×10-4 m/day.

4.3 Sensitivity analyses regarding D1 For evaluation of the impact of the D1 factor in the simulation results, a similar strategy is adopted, cen-tered on the reference value recommended by MC2010 for the applicable concrete strength, with D1 = 39.3 mm2/day and considering two alternative values of D1: one of them 10 times higher and another 10 times lower than the reference value. The computed humidity at 20 cm depth along time is shown in Fig. 4, whereas Fig. 5 shows the humidity profiles at the ages of 500 days and 50 years

The observation of Fig. 4 and Fig. 5 for the age of 50 years confirms the expectable tendency of faster drying when higher D1 coefficients are considered. As a matter of fact, the analysis that considers D1 = 393 mm2/day leads to a full humidity equilibrium with the outer envi-ronment (h = 50%) at the end of the period of analysis. The effect of D1 is in fact very relevant as a decrease of

D1 from the reference value to 3.93 mm2/day causes the humidity calculated at 50 years age in the symmetry to be increased by almost 20%.

There is an interesting aspect to remark regarding the results at 500 days age plotted in Fig. 5: the humidity

Fig. 4 Parametric analyses of the effect of D1 (box value is the value for 50 years) on the humidity computed at 20cm depth ([D1]= mm2/day).

Fig. 2 Parametric analyses of the effect of fboundary [m/day] (box value is the value for 50 years) on the hu-midity computed at 20 cm depth.

Fig. 3 Humidity profiles at the ages of 500 days and 50 years: parametric study regarding the boundary condi-tions (box caption with fboundary (10-4) [m/day] – age). Given the symmetry only half of the profile is given for each age.


computed within the first ~5 cm near the surface is higher when D1 = 393 than when D1 = 39.3 mm2/day. Even though this would seem surprising at first sight, it is easily explained by the significant flow of humidity that is migrating towards the surface in the case of D1 = 393 mm2/day. Therefore, even though the humidity dif-fusion coefficient is very high and would theoretically lead to lower surface humidity when compared to the other cases, it ends up leading to higher humidity due to the intense transport occurring from the inner regions.

5. Applications

This section describes the numerical simulations of hu-midity fields carried out for three sets of experiments with concrete specimens, in which moisture profiling with embedded humidity sensors was performed. The basic intent is to test the performance of direct applica-tion of MC2010 for diffusivity, while trying to assess the boundary condition coefficient based on Neumann’s formulation. It is noted that all case studies presented were considered with the values of α, hc and n recom-mended by MC2010.

5.1 Simulation of concrete specimens tested by Kim and Lee (1999) The experiments conducted by Kim and Lee (1999) considered three different concrete compositions using 10 cm×10 cm×20 cm specimens. After an initial period in which the specimens were kept inside their mold, they were submerged in water from the age of 1 day until the age of 28 days. At 28 days, the specimens were removed from water, their surfaces were sealed accord-ing to the scheme Fig. 6, and placed in a climatic cham-ber with T = 20 ± 1 ºC and henv = 50 ± 2%. As evapora-tion could only take place through a 10 cm×10 cm sur-face, these specimens endured a one–dimensional mois-ture flow, similar to that of an infinite slab of 40cm thickness exposed in both surfaces (as all sealed sur-faces act as symmetry planes in terms of moisture flow). Humidity sensors were placed at three distinct depths measured perpendicularly to the evaporating surface: 3 cm, 7 cm and 12 cm.

The cement used in the experiments was ordinary Portland cement (ASTM Type I), with river sand as fine aggregate and crushed granite gravel passing the 19 mm sieve as coarse aggregate. Detailed mix proportions of the three studied concrete specimens (H, M and L), as well as their corresponding compressive strength are given in Table 1.

Simultaneously to the mentioned experiments, Kim and Lee (1999) performed measurements on sealed specimens as to infer the humidity decrease associated to self-desiccation. At the age of 28 days, the recorded value was stabilized at approximately 95% for mixes H and W, whereas the value for mix L was 99%. These values were used as a starting conditions for the humid-ity of concrete in the simulations herein.

An initial simulation attempt was made by strictly following the MC2010 recommendations for diffusivity and enforcing Dirichlet boundary conditions (i.e. pre-scribed humidity on the evaporating surface). Figure 7 shows the comparison between such numerical simula-tion and experimental results for the mix M. The results show a relatively reasonable coherence for the depths of 3 cm and 7 cm, but the humidity values at 12 cm depth are being clearly overestimated at all ages.

Fig. 5 Humidity profiles at the ages of 500 days and 50 years: parametric study regarding the maximum diffusiv-ity (Box caption with [D1 mm2/day] – age). Given the symmetry only half of the profile is presented for each age.

Fig. 6 Geometry and size of test (adapted from Kim and Lee (1999)).


Due to the inability of MC2010 together with pre-scribed boundary conditions in satisfying the experi-mental results, a strategy of evaluating the fitness of considering Neumann boundary conditions has been carried out. The results of solely adjusting the boundary coefficient did not provide significant improvements in the agreement with the experimental results. Therefore, a combined strategy of searching for the adequate fbound-

ary, together with slight adjustments of the D1 parameter (originally considered as 16.1 mm2/day) has been pur-sued. As a result, the results of the best combination of fboundary and D1 values are shown in Fig. 8. It can be no-ticed that a change of D1 from 16.1 mm2/day to 32 mm2/day and the introduction of fboundary = 3.2×10-4

m/day provided simulation results that approximated the experimental results quite satisfactorily, with particu-larly relevant improvement at the depth of 12 cm. Com-plementarily to the informed guess strategy mentioned above, it is remarked that all fitting processes mentioned in this paper were further performed for a wide range of values for both D1 and fboundary parameters, as to evaluate the uniqueness of the initially obtained solution. In fact, D1 was studied in the range 0.1 mm2/day to 200 mm2/day with increments of 0.5 mm2/day. Simultane-ously, fboundary was varied in the range 0.1×10-4 m/day to 100×10-4 m/day. These ranges took into consideration the recommendations of MC90/2010 and work of Kim and Lee (1999).

For all the subsequent simulations (specimens H and L), the proposed value for D1 given by the equation pre-sented on the MC2010 and imposed boundary condi-tions, were tested and the results are similar, showing

the same tendency to the one showed on the Fig. 7. For the concrete specimen L, after observing that the com-bined use of MC2010 parameters with a prescribed boundary conditions did not lead to acceptable agree-ment with the experiments, the same procedure was adopted. The inverse fitting process had a starting point in the D1 value proposed by MC2010 for this concrete (D1 = 39.3 mm2/day) and the fboundary obtained for con-crete specimen M (fboundary = 3.2×10-4 m/day). The best–fit set of parameters did not significantly deviate from the initial values, with D1=52 mm2/day and fboundary = 4.8×10-4 m/day. The corresponding results are shown in Fig. 9, where a fairly good agreement with the meas-ured values can be observed.

A similar overall strategy was applied for specimen H, in which again the direct application of MC2010 with

Table 1 Concrete compositions by Kim and Lee (1999).

Unit Weight (kg/m3) Specimen w/c S/G Water (w) Cement (c) Sand (S) Gravel (G) S.Pa (c %) f’c (MPa)

Hb 0.28 38 151 541 647 1055 2.0 76 Mb 0.40 42 169 423 736 1016 0.5 53 Lb 0.68 45 210 310 782 955 – 22

a. Superplastizer. b. H, M, and L denote high, medium, and low-strength concrete, respectively.

Fig. 8 Specimen M – Experimental humidity data (Kim and Lee, 1999) and numerical simulation – (D1 = 32 mm2/day / fboundary=3.2×10-4 m/day).

Fig. 7 Specimen M – Experimental humidity data from Kim and Lee (1999) and numerical simulation (D1 =16.1 mm2/day / Fixed boundary condition) (no self-desiccation was considered).

Fig. 9 Specimen L – Experimental humidity data from Kim and Lee (1999) and numerical simulation (D1 =52 mm2/day / fboundary = 4.8×10-4 m/day) – w/c = 0.68 – Ex-periments start after 28 days.


Dirichlet boundary conditions did not yield good coher-ence with the experimental measurements. The starting point for the trial and error procedure involved D1 = 11.4 mm2/day (as proposed by MC2010) and fboundary = 3×10-4 m/day (value used in specimen M). The best–fit also involved a slight increase in D1 to 20 mm2/day, as it also had occurred in the cases of specimens M and L. The fboundary value was decreased from 4.8 ×10-4 to 2.0 ×10-4 m/day. The results of this best–fit combination of parameters are shown in Fig. 10.

This process resulted in best fit scenarios for the stud-ied depths of measurement that always matched those that had been predicted by the ‘informed guess’ strategy, thus confirming the uniqueness of the obtained solution, Such uniqueness can be further assessed by observation of the R2 values of each calculated pair of D1, fboundary, (w/c=0.28) which is graphically represented in Fig. 11. The peak of R2 that surpasses 0.99 is indeed confined to a very limited region of the diagram.

Two final remarks are given in regard to the fit-ting strategy/results for the three concretes. Firstly, all models needed to use slightly higher values of D1 in regard to those proposed by MC2010. Secondly, the best fit for fboundary seems to follow a tendency of higher val-ues of fboundary for concretes of lower compressive strength. This could be an indication that fboundary might be proportional to the surface porosity (which is closely related to compressive strength).

5.2 Simulation of five concrete specimens by Persson (1996)

The experiments conducted by Persson (1996) consisted in casting circular slabs of 1 m diameter and 0.1 m thickness, schematically shown in Fig. 12. The speci-mens were sealed with thick layers of epoxy resin on their top and bottom flat surfaces to impose only radial moisture transport. Even though Persson’s experiments have been conducted with exposure of concrete to dry-ing at the age of 3 days, and the MC1990/2010 ap-proach does not encompass any specific correction for exposures earlier than 28 days, it was decided to per-form the simulation of Persson’s experiments anyway. This decision was supported on the fact that Persson’s monitoring depths were the deepest ones found in the literature (max. 35cm deep measurement), thus adding to the interest of their simulation. Also, the experiments of Kim and Lee (1999) performed simultaneously for specimens exposed at 3 and 28 days have demonstrated very similar results for the largest monitored depths (e.g. 12 cm).

Even though the environmental conditions during the experiment (temperature and relative humidity) were not constant throughout the entire period of testing, their variation observed by monitoring was limited (Persson, 1996), with average values of 21.5 ºC and 32.6 % (h). These average values were considered in the numerical

Fig. 11 R2 coefficient simulation of concrete H (w/c=0.28) and D1 and fboundary.

Fig. 12 Schematic representation of the geometry and size of test specimens (adapted from Persson (1996)).

Fig. 10 Specimen H – Experimental humidity data (Kim and Lee, 1999) and numerical simulation (D1=20 mm2/day / fboundary= 2×10-4 m/day) – w/c = 0.28 – Ex-periments start after 28 days.


simulation. Relative humidity measurements were ob-tained through cast–in plastic probes placed at depths of 5, 15 and 35 cm from the exposed surface (Persson, 1997, Persson, 1998, Persson, 1996). Companion specimens under totally sealed conditions were also cast in order to assess internal humidity decreases associated to self-desiccation.

Five concrete mixes have been studied, here termed as Mix 1 and Mix 2, Mix 3, Mix 4 (with silica), Mix 5 (with silica). The detailed concrete composition for these five mixes is shown in Table 2. It is remarked that Persson had tested a total of 8 mixes. Nonetheless, three of those mixes had very low w/c ratio (≤0.25). They were not analyzed because they tend to be outside the scope of derivation of the diffusion coefficients pro-posed by MC1990/2010 (FIB, 2013).

These concrete mixes had average compressive strengths of fcm, Mix1 = 80 MPa and fcm, Mix2 = 37 MPa, fcm, Mix3 = 57 MPa, fcm, Mix4 = 67 MPa, fcm, Mix5 = 91 MPa. The numerical simulation of these experiments de-manded slight adaptations to the FDM simulation be-cause an axisymmetric algorithm needed to be imple-mented. This was a simple and straightforward task, and specific information on the adaptations in regard to a 1D formulation can be found in Özisik (2002). The effects of self-desiccation were experimentally assessed in sealed specimens and corresponding humidity decreases were imposed in the numerical simulation, based on the data provided by Persson (1996). Such humidity de-creases in the sealed specimens are presented as [day, humidity], for Mix 1 [28, 0.92; 90, 0.88; 446, 0.86], Mix 2 [28, 0.96; 90, 0.96; 440, 0.96], Mix 3 [28, 0.97; 90, 0.95; 446, 0.88], Mix 4 [28, 0.95; 90, 0.88; 446, 0.83], Mix 5 [28, 0.88; 90, 0.81; 446, 0.76].

Again, the direct application of the parameters of MC2010 for D1 (D1,Mix1 = 12.2; D1,Mix2 = 29.8 mm2/day; D1,Mix3 = 17.6 mm2/day; D1,Mix4 = 14.6 mm2/day; D1,Mix5 = 10.4 mm2/day) and assumption of Dirichlet boundary conditions, led to inadequate agreement between nu-merical results and monitored h. In pursuit for better agreement, D1 was kept a very similar value to the ini-tial value: D1,mix1 = 12.0 mm2/day; D1,mix2 = 30 mm2/day, D1,mix3 = 8 mm2/day, D1,mix4 = 13 mm2/day, D1,mix5 = 8 mm2/day, while the fitted values of fboundary for Mix 1, 2, 3, 4 and 5 were respectively 1.4×10-4 m/day, 3.0×10-4 m/day, 1.0×10-4 m/day, 1.5×10-4 m/day and 0.8×10-4 m/day, which are similar to the values reported for simi-lar strength classes in the previous example. The agree-ment between experimental and numerical simulation results for the studied specimens is shown in Fig. 13 to 17.

5.3 Simulation of one concrete specimen by Kang et al. (2012) Kang et al. (2012) performed experiments on prismatic concrete specimens with dimensions 100 mm × 100 mm × 300 mm. These specimens were kept under sealed conditions until 1 day age, and at such age, the surfaces of 100 mm× 100mm were exposed to drying as shown in Fig. 18. The drying environment had constant tem-perature and humidity of T = 20 ± 1 ºC and henv = 50 ± 1 %. The adopted concrete mix is shown in Table 3. Additionally to the drying specimens, one specimen was sealed hermetically in order to measure self-desiccation due to hydration, with recorded internal humidity (re-sults presented as [day, humidity]): [32, 0.97; 37, 0.97; 44,0.97; 53, 0.96; 61,0.96; 74,0.96]. This self-induced internal humidity consumption was considered in the

Table 2 Composition of the concrete mixes tested by Persson (1996).

Unit Content (kg/m3) Specimen Quartzite Gravel Cement Silica fume Superplasticizer w/c Aggregate content

Mix 1 1214 723 400 - 3.35 0.33 0.731 Mix 2 1145 812 299 - - 0.57 0.738 Mix 3 1150 845 303 - 3.01 0.465 0.753 Mix 4 1153 825 298 30 2.13 0.483 0.746 Mix 5 1158 730 389 39 3.07 0.358 0.712

Fig. 13 Experimental humidity data from Persson (1996) and numerical simulation (Mix 1) (D1 = 12 mm2/day / fboundary = 1.4 ×10-4 m/day) (time from casting).

Fig. 14 Experimental humidity data from Persson (1996) and numerical simulation (Mix 2) (D1 = 30 mm2/day / fboundary = 3×10-4 m/day) (time from casting).


calculations using the strategy highlighted before. The authors (Kang et al. 2012) do not provide infor-

mation about the compressive strength of the tested concrete. However, by analyzing the mix composition in terms of cement content and water-to-cement ratio, the authors infer that this concrete was bound to have fcm values at 28 days within the range 40~50 MPa. However, at the age of exposure to drying (1 day), the value of fcm would be clearly lower. If the expectable value of fcm at 28 days is taken into account in the simulation (that spans ~75 days), the initial guess according to MC2010 for D1 would be D1, mix1 = 23 mm2/day (considering fcm = 45 MPa). The direct application of such value of D1

together with Dirichlet boundary conditions led to un-satisfactory results. Reasonable coherences could only be attained when D1 was dramatically increased to 55.0 mm2/day and fboundary was set to 5.5×10-4 m/day – see results in Fig. 19. The elevated change in the diffusion coefficient is most likely attributed to the fact that con-crete was exposed to drying at a very early age (1 day), with much higher porosity and lower strength than at 28 days age. The diffusion parameters of this concrete ended up being comparable to those of Mix L shown in section 5.1, which had a much higher w/c and much lower cement content, albeit being only exposed to dry-ing at the age of 28 days.

Fig. 15 Experimental humidity data from Persson (1996) and numerical simulation (Mix 3) (D1 = 8 mm2/day / fboundary = 1×10-4 m/day) (time from casting).

Fig. 17 Experimental humidity data from Persson (1996) and numerical simulation (Mix 5) (D1 = 8 mm2/day / fboundary = 0.8×10-4 m/day) (time from casting).

Fig. 18 Geometry and size of test specimens (adapted from Kang et al. (2012)).

Fig. 19 Results Experimental (Kang et al., 2012) and numerical analyses (D1 = 55 mm2/day / fboundary = 5.5× 10-4 m/day).

Fig. 16 Experimental humidity data from Persson (1996) and numerical simulation (Mix 4) (D1 = 13 mm2/day / fboundary = 1.5×10-4 m/day) (time from casting).


5.4 Integrated analysis of results Table 4 summarizes the results obtained in the preced-ing sections, with emphasis on the original values of D1 according to MC2010 and the pairs D1 / fboundary found after the inverse fitting procedures. The uniqueness of the D1 / fboundary pairs obtained was confirmed for all cases through extensive simulations of possible combi-nations of both parameters within wide variation spectra. It is interesting to observe that the post–fitting results provided similar or slightly higher values of D1, when compared to the initial estimates of MC2010. Further-more, the fboundary values are coherent with each other in the sense that: (i) they all keep the same order of magni-tude; (ii) there is an increase in fboundary as compressive strength decreases (i.e. surface porosity increases).

Based on the results of Table 4, the values of fitted D1 were graphically plotted together with the predic-tions of D1 according to MC2010 in Fig. 20, as to facili-tate comparison between these two approaches. It is noticeable that the values of D1 fitted in this research work are similar (equal or slightly higher) to those pre-dicted by Model Code 2010, thus confirming the feasi-bility of MC2010 diffusivity predictions.

The fitted values of fboundary are plotted in Fig. 21 as a function of the compressive strength of concrete as to infer possible correlations. This leads to the observation that there seems to be a correlation between fboundary and fcm, thus corroborating the feasibility of the surface fac-tor theory suggested by Zhi et al. (2010). It should be remarked that this observed correlation solely pertains

Table 4 Summary of the simulations and results.

Designation fcm (MPa)

Mean Compressive Concrete Strength

fck (MPa) (calculated as

fcm-8MPa)

2

1mmDday

⎛ ⎞⎜ ⎟⎝ ⎠

CEB-FIB

2

1mmDday

⎛ ⎞⎜ ⎟⎝ ⎠

Best-fit simulation

4(10 )boundaryf − (m/day) Best-fit

simulation Kim and Lee (1999)

H 84 76 11.4 20 2.0 M 61 53 16.3 32 3.2 L 30 28 39.3 52 4.8

Persson (1996) Mix 1 80 72 12.0 12 1.4 Mix 2 37 29 29.8 30 3 Mix 3 57 49 17.6 8 1 Mix 4 67 69 14.6 13 1.5 Mix 5 91 83 10.4 8 0.8

Kang et al. (2012)

Kang et al. 55 5.5

Fig. 21 Correlation between the boundary coefficient and the concrete compressive strength.

Fig. 20 Correlation between the diffusivity coefficient and the concrete compressive strength.

Table 3 Concrete composition from Kang et al. (2012). Unit Content (kg/m3)

w/c S/G (%) Water Cement Sand Gravel Admixture AE Admixture WR 0.50 48 199 398 842 912 0.020 0.20


to the effect of fcm on fboundary, neglecting the potential influence of wind speed on the boundary coefficient. Bearing in mind that the present work is targeted at hardened concrete, this is considered a plausible simpli-fication. In fact, the works conducted by Azenha et al. (2007a) and Azenha et al. (2007b) have shown that cementitious materials exposed to drying after the ages of 7 days tend to show little or no sensitivity to wind speed in terms of drying velocity (i.e. mass loss).

The range for the best fitted results for fboundary are co-herent with the value used by Kim and Lee (1999), but impossible to compare with the values forwarded by Akita et al. (1997) or Sakata (1983) because of the fact that they were expressed for an absolute water content field (no moisture isotherm provided).

Finally, Fig. 22 provides the correlation between the fboundary and the D1 for the studied concretes. Whereas the correlation shown in Fig. 21 is relatively poor, in Fig. 22 the R2 coefficient is close to one, providing at excellent fit.

6. Conclusions

This paper addresses a unifying and validated set of recommended parameters to be used together with the approach for humidity diffusion simulation proposed in Model Code 2010. After a brief literature review on the background of the diffusion equations and their corre-sponding parameters, focus has been given here to a simplified 1D implementation through the finite differ-ence method, suitable for use in design offices that may wish to integrate this kind of simulation in their analysis procedures. It is nonetheless remarked that the humidity diffusion model of MC2010 has underlying simplifica-tions that need to be considered with care upon interpre-tation of its results: (i) it considers that the slope of the moisture isotherm is constant upon drying; (ii) it relies on a single driving potential, the internal relative humid-ity (rather than the pressure of vapor or liquid water), which is only actually valid in scenarios of constant pressure and temperature. It was nonetheless observed

that the equation proposed by Model Code and adopted herein, with the consideration of a boundary coefficient was able to obtain reasonable results.

Based on the implemented simulation model, para-metric analyses have been conducted, in order to sup-port sensitivity studies of the relevance of each model-ling parameter involved in the diffusion equation. It has been found that the most influential parameters in the simulation results are the diffusion coefficient D1 and the boundary transfer coefficient fboundary, being the latter not mentioned in Model Code 2010.

Based on the lessons learned from the sensitivity studies, the simulation model has been applied to a set of nine specimens, monitored for humidity profiles in the scope of three distinct publications. The direct appli-cation of MC2010 diffusion parameters together with prescribed surface humidity did not yield satisfactory results, and thus an inverse–fitting process was made to evaluate plausible values for fboundary and corresponding D1. Such process led to the conclusion that very good agreement between the simulations and experiments can be achieved by using D1 values that are similar to those proposed in MC2010, together with fboundary values that range from 0.8 to 5.5 × 10-4 m/day) in the studied con-cretes. The uniqueness of the obtained solutions was confirmed by extensive calculations of pairs of D1 and fboundary within wide ranges.

It was further found that, within the studied speci-mens and analyses, there seems to be a roughly linear relationship between fboundary and the compressive strength of concrete fcm, which is provided in this paper. The fboundary and D1 presented a clearly linear correlation, which is also provided in this paper.

Acknowledgements Funding provided by the program “Ciência sem Fron-teiras” supported by Brazilian National Council of Technological and Scientific Development (CNPq) and by the Portuguese Foundation for Science and Technol-ogy (FCT) to the Research Project PTDC/ECM/099250/2008. The contribution of the re-viewers to the scientific content of the paper is also gratefully acknowledged. References ACI (2006). “Guide for Concrete Slabs that Receive

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List of symbols and abbreviations For the sake of clarity, the description of each notation or symbol is made upon its first appearance in the text. The following list is presented in alphabetic order and does not include symbols or notations of a secondary nature Roman Letters D Diffusion coefficient Dh , Dh

* Humidity diffusion coefficients D0, D1 The minimum and maximum values for Dh fcm Mean compressive concrete strength fboundary Proportionality factor used to simulate the

boundary effect h Internal relative humidity of concrete hc Relative humidity for which Dh = 0.5×D1

(Model code 2010 and Model code 1990 ap-proaches)

hen Environmental humidity hs Internal humidity decrease associated with

concrete hydration – self-desiccation hsurf Concrete humidity in the surface i Denotes the analyzed node n Exponent used to calculate Dh ; nth time step t Time T Temperature W Total water concentration We Evaporable water concentration Wn Non-evaporable water concentration x Abscissa along the FDM model; vector normal

to the boundary

Greek Letters α Parameter used to calculate the diffusion coef-

ficient D0 / D1 (Model code 2010 and Model code 1990 approaches)

Δt Time difference ∇ Nabla operator ψ̂ Residual vector

simulation of humidity fields in concrete: experimental

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