Advances in Materials Science and Engineering

Volume 2015, Article ID 682940, 10 pages

http://dx.doi.org/10.1155/2015/682940

## A Thermo-Hygro-Coupled Model for Chloride Penetration in Concrete Structures

^{1}Civil Engineering Program, School of Engineering, University of Phayao, Phayao 56000, Thailand^{2}Department of Civil Engineering, Faculty of Engineering, Prince of Songkla University, Songkla 90112, Thailand^{3}Department of Civil, Environmental, and Architectural Engineering, University of Colorado Boulder, Boulder, CO 80309, USA

Received 12 April 2015; Revised 26 June 2015; Accepted 29 June 2015

Academic Editor: Rui Wang

Copyright © 2015 Nattapong Damrongwiriyanupap et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Corrosion damage due to chloride attack is one of the most concerning issues for long term durability of reinforced concrete structures. By developing the reliable mathematical model of chloride penetration into concrete structures, it can help structural engineers and management agencies with predicting the service life of reinforced concrete structures in order to effectively schedule the maintenance, repair, and rehabilitation program. This paper presents a theoretical and computational model for chloride diffusion in concrete structures. The governing equations are taking into account the coupled transport process of chloride ions, moisture, and temperature. This represents the actual condition of concrete structures which are always found in nonsaturated and nonisothermal conditions. The fully coupled effects among chloride, moisture, and heat diffusion are considered and included in the model. The coupling parameters evaluated based on the available material models and test data are proposed and explicitly incorporated in the governing equations. The numerical analysis of coupled transport equations is performed using the finite element method. The model is validated by comparing the numerical results against the available experimental data and a good agreement is observed.

#### 1. Introduction

Corrosion of reinforcing steel is one of the main causes of deterioration process for reinforced concrete structures exposed to chloride-rich chemicals which are generally from deicing salts or sea water. The structures that are potentially threatened by the chloride attack are bridge decks, parking lots, and marine structures in splash and tidal zones. The corrosion of reinforcing steels eventually causes spalling, cracking, and delamination of concrete cover thus causing poor load bearing capacity and reducing aesthetic feature of the structures. The corrosion process can be divided into two periods, including an initiation period during which chloride ions penetrate into the concrete cover and initiate the corrosion of steel and a propagation period during which the rust deposits in the surrounding porous cement paste and results in cracking of concrete cover. The initiation period is usually much longer than the propagation period, because once the corrosion starts, it will take only a few years for the rust to crack the concrete cover. Therefore, improving the penetration resistance of concrete to various chloride sources is an important and challenging task in concrete and construction industries and equally important is to predict the chloride penetration process into concrete structures by theoretical and numerical models. The model prediction will provide useful information for structural engineers, owners, and management agencies to make correct decisions.

One of the theoretical models for predicting chloride ingress into concrete structures is based on Fick’s second law, leading to a simple solution of one-dimensional linear ordinary partial differential diffusion equation [1–3]. The main problem of the simplified linear models is that the apparent diffusion coefficient in the model, , has to be considered as a constant. In fact, it is not a constant and it varies by more than one order of magnitude and depends on many factors such as material characteristics and surrounding exposure conditions of structures. As a result, we cannot use the constant diffusion coefficient of concrete to reliably predict the service life of structures which have been exposed to a corrosive environment. Also, the apparent diffusion coefficient of concrete for a given location cannot be applied to use for other circumstances [4]. Furthermore, if the variations of initial and boundary conditions such as moisture and temperature surroundings were to be taken into account, the prediction error of using Fick’s second law could be even larger. It was emphasized by Marchand and Samson [4] that Fick’s model can only be appropriately used for concrete structures in saturated condition. However, the chloride exposure conditions of actual concrete structures are often found to be nonsaturated and nonisothermal. To solve this problem, some Fick’s models have been modified to account for moisture and temperature effects [5–7]. This can be done either by using moisture and temperature dependent transport parameters or by incorporating the coupling terms in the chloride flux equation, which is adopted for the present study.

In saturated and isothermal conditions, chloride ions diffuse in concrete mainly due to its own concentration gradient. This means that there are no moisture and temperature effects on chloride penetration mechanism. On the contrary, the coupled diffusion process of chloride and moisture was evident in nonsaturated condition where the rate of chloride diffusion was accelerated by moisture gradient. Similarly, moisture flow was affected by the chloride transport mechanism [8, 9]. For nonisothermal condition, chloride transport mechanism in concrete is driven by not only its concentration gradient but also temperature variation gradient. It was emphasized by Isteita [10] study that, by increasing temperature, the rate of chloride ingress into concrete can be increased.

Over the years, modeling of chloride transport in nonsaturated concrete has been studied by many researchers. Saetta et al. [6] proposed a numerical model of chloride penetration in partially saturated concrete. The model included the coupling effect of moisture movement on chloride diffusion. The transport equations of chloride and moisture were then solved using the finite element method. In 2000, Nilsson [11] developed a chloride diffusion model for the concrete exposed to seawater splash or deicing salts. It was found from the study that diffusion and convection mechanisms of chloride could be explained as a function of moisture content in concrete. Ababneh et al. [5] presented a mathematical model of chloride ingress into concrete in nonsaturated condition. The coupling term for the moisture movement was incorporated in the diffusing equation of chloride. The two coupled partial differential equations of chloride and moisture were analyzed using the finite difference method. The numerical results are validated against the available test data and the verification showed that the model could be used to simulate and predict chloride diffusion in nonsaturated concrete satisfactorily. Nielsen and Geiker [12] proposed a simplified Fick’s model to examine chloride penetration into partially saturated concrete and studied the relationship between chloride diffusion coefficient and degree of saturation. Their findings showed that the chloride diffusion coefficient depends on degree of saturation of concrete. Conciatori et al. [13] presented a comprehensive model called “TransChlor” for simulating chloride diffusion associated with heat transfer, liquid and vapor movement, and carbon dioxide transport in concrete. Chloride profiles predicted by this model were validated with the experimental data and a good agreement was observed. Recently, Lin et al. [14] established a systematic and numerical model for predicting service life of concrete structures subjected to chloride attack. The effect of moisture transport on chloride diffusion was included in the chloride flux equation which was represented as the degree of water saturation. The numerical results obtained from the model showed that the service life of structures exposed to drying and wetting circumstances is significantly different from that under the saturated condition. Based on these studies, it can be concluded that moisture variation has a remarkable influence on degradation of concrete structures caused by chloride attack.

As noticed from the above review, most of the mathematical models have been focused on chloride diffusion in nonsaturated concrete, and the temperature effect on chloride transport has not been well investigated. The temperature variation is very significant and must be included in the mathematical model. A recent experimental study conducted by Isteita [10] showed that the rate of chloride penetration is accelerated by temperature gradients. The temperature variation has a significant effect on not only the chloride transport but also moisture movement. This can be described by Khoshbakht et al.’s [16] study where moisture diffusion in masonry walls is significantly influenced by heat transfer. However, there is no available mathematical model that takes into account the influences of temperature variation on simultaneous chloride and moisture diffusion in concrete. Therefore, in this study, a transport model based on Fick’s law taking into account the fully coupled chloride, moisture, and heat flow in nonsaturated and nonisothermal concrete is developed. The coupling parameters related to chloride, moisture, and heat transport in concrete are characterized and explicitly incorporated in the governing equations. The governing equations are then solved numerically by the finite element method. The validation is performed by comparing the numerical results obtained from the present model with the available test data and a good agreement is observed.

#### 2. Basic Formulation of Governing Equations

The flux of chloride ions through porous concrete depending on the concentration gradient is described by Fick’s law as follows:in which is the flux of chloride ions, is the diffusion coefficient of chloride ions, and is the free chloride concentration.

The total amount of moisture contained in concrete, the so-called moisture content, is generally represented by water content () or by pore relative humidity (). In the present study, the moisture content in concrete is presented by pore relative humidity which consists of amount of liquid water and water vapor existing in concrete pores [17]. The moisture flux () can be described in terms of the gradient of pore relative humidity given bywhere is the humidity diffusion coefficient. Heat flow in concrete is described by the well-known Fourier’s law of heat conduction giving the heat flux in function of gradient of temperature:where is the heat flux, is the thermal diffusivity of concrete, and is temperature. As indicated earlier, in order to describe the coupled transport processes involving chloride, moisture, and temperature, the above listed three governing equations can be modified by adding new terms expressed explicitly in terms of the gradients of the state variables, and as a result, the three governing equations will become fully coupled and then these coupled equations must be solved simultaneously. For example, the flux of chloride ions (), (1), in nonsaturated and nonisothermal concrete can be written asin which and are the coupling parameters corresponding to the effect of moisture and temperature variation on the chloride diffusion, respectively. Similar to chloride flux, the coupling terms are also included in moisture and heat flux. Then, (2) and (3) can be rewritten aswhere , , , and are coupling parameters. In general, coupling parameter represents the effect of process on the process.

The mass balance equations of chloride, moisture, and heat transport in concrete are given by (6), (7), and (8), respectively:in which , , and represent the chloride binding capacity, moisture capacity, and heat capacity, respectively. The coupling parameters corresponding to the coupled transport mechanisms among chloride, moisture, and temperature will be explained later.

#### 3. Material Models

To solve chloride, moisture, and heat transport in nonsaturated and nonisothermal concrete, the material parameters in the governing equations, (6), (7), and (8), need to be modeled first. This is because these material parameters are usually not constants but depend strongly on concrete mix design parameters, age of concrete, and concentrations of the transport variables, and therefore, material models must be developed to take into account the variations of influential parameters.

In the governing equations, the material parameters are chloride diffusion coefficient, chloride binding capacity; moisture diffusion coefficient, moisture capacity; thermal diffusivity, heat capacity, and coupling parameters. In this study, the influential parameters taken into account to characterize the coupled transport characteristics in concrete are concrete mix design parameters, that is, curing time, age of concrete, diffusivity of cement paste, and aggregate. One of the major improvements of the present mathematical model is that several material models based on available experimental data are proposed and incorporated in the governing equations. The determination of chloride diffusion coefficient, chloride binding capacity, moisture diffusion coefficient, and moisture binding capacity was once discussed in Damrongwiriyanupap et al. [18, 19]. For reader’s convenience, we will summarily introduce the previously developed transport parameters and the coupling parameters will then be explained in detail.

##### 3.1. Chloride Diffusion Coefficient ()

The diffusion coefficient of chloride ions in concrete can be evaluated using the multifactor equation given byin which represents a factor taking into account the influence of water cement ratio and curing time of concrete (). Porosity of concrete is directly affected by water cement ratio. The higher the water cement ratio, the higher the porosity and the higher the diffusion coefficient. A formulation for was proposed by Xi and Bažant [20] and expressed asThe second factor is to consider concrete as a composite material comprising cement paste and aggregates. Thus, based on the composite action, the diffusivity of concrete can be evaluated simply by including the diffusivity of cement paste and aggregates. This factor can be calculated by using the three-phase composite model developed by Christensen [21]: where is the volume fraction of aggregates in concrete and and are the diffusivity of aggregates and cement paste, respectively. These two parameters can be determined by using the model proposed by Martys et al. [22]:in which is the porosity, denotes the surface area, and is the critical porosity (the porosity at which the pore space is first percolated). When (12) is used for the diffusivity of cement paste, , , , and are considered as the parameters for cement paste. The critical porosity may be taken as 3% for cement paste [22]. As previously proposed by Xi et al. [23], the surface areas of cement paste, , can be defined by the monolayer capacity, , of adsorption isotherm of concrete which is proportional to . The porosity, , can be estimated by adsorption isotherm, at saturation . and are the weight of pore solution and concrete, respectively. The explanation of adsorption isotherm of concrete can be found in Xi et al. [23, 24]. It is mainly due to the fact that pores in aggregates are discontinuous. Thus, the diffusivity of aggregates, , can be considered as a very small value and simply taken as a constant with a proposed value in the literature 1 × 10^{−12} cm^{2}/s [5, 20].

The third factor, , takes into account the dependence of chloride diffusion coefficient on the effect of internal relative humidity. A model developed by Bažant and Najjar [17] is used for this factor given byin which is the critical humidity level at which the diffusion coefficient drops halfway between its maximum and minimum values ( = 0.75). When chloride ions transport in nonsaturated condition, (13) corresponds to the influence of coupling effect between the moisture and chloride diffusion.

To consider the effect of temperature on chloride diffusion in concrete, the fourth factor, , is included in the equation. This can be calculated by using Arrhenius’ law:in which is the activation energy of the diffusion process, is the gas constant (8.314 J mol^{−1} K^{−1}), and and are the current and reference temperatures, respectively, in Kelvin ( = 296 K). As reported by Collepardi et al. [25] and Page et al. [26], the activation energy of the diffusion process is a function of water cement ratio, , and cement type which is illustrated in Table 1.