Numerical Simulation on Carbonation Depth of Concrete Structures considering Time-and Temperature-Dependent Carbonation Process

School of Civil Engineering, Changsha University of Science and Technology, Changsha, Hunan 410114, China School of Civil Engineering, Hunan City University, Yiyang, Hunan 413000, China School of Civil Engineering and Architecture, Changsha University of Science and Technology, Changsha, Hunan 410114, China Department of Civil and Environmental Engineering, Louisiana State University, Baton Rouge, LA, USA


Introduction
CO 2 concentration is an important factor affecting carbonation in the atmospheric environment.When the CO 2 in the air permeates into concrete, the acid-base neutralization reaction happens between the CO 2 and the alkaline materials, which leads to the decrease of the concrete alkalinity and the change of chemical composition.When the carbonation front reaches the surface of steel bars, the passivation film on the surface of steel rebar will be destroyed, and then, the corrosion of steel rebars occurs.Overtime, along with the increase of the corrosion products, it is well recognized that the volume of corrosion product is 3∼8 times larger than that of the steel material.is will result in cover cracking and even spalling and also lead to a reduction of the structural load carrying capacity.ese phenomena affect the working performance and durability of concrete structures.
With the development of global economy and the sharp increase of population, the CO 2 concentration also increases rapidly.Before the second industrial revolution, the CO 2 concentration was about from 265 ppm to 290 ppm in the global atmosphere [1], was about 365 ppm in 2000 [2] (1 ppm � 0.00188 × 10 −3 kg/m 3 ), and reached 400 ppm in 2015, which broke the historical extreme.According to the prediction of International Panel on Climate Change (IPCC), the CO 2 concentration will exceed 1000 ppm by 2100 [2].e CO 2 concentrations under various different emission strategies are demonstrated in Figure 1 (A1F1, B1, and Best mitigation) [3].A1F1 describes a world of very rapid economic growth, a global population that peaks in midcentury, and rapid introduction of new and more ecient technologies that remain fossil intensive, and this is the worst CO 2 emission scenario.B1 describes a world with intermediate population and economic growth, emphasizing local solutions to economic, social, and environmental sustainability, and this is the lowest CO 2 emission scenario.Best mitigation describes that the CO 2 concentration is kept stable at 2010 levels (386 ppm) due to stabilization and reduction of CO 2 emissions.e climate change leads to the increase of CO 2 concentration, which accelerates the carbonation process of concrete structures, and it is seriously harmful to the structural durability and safety.In present experimental and theoretical studies, the e ects of the CO 2 concentration on the carbonation of concrete have already been explored by Visser [4], Cui et al. [5], and Castellote et al. [6]; however, the above scholars do not consider the time-dependent character of the CO 2 concentration, which will a ect the precision of the prediction of the carbonation depth, so it is necessary to consider the time-dependent character of CO 2 concentration resulting from climate change when investigating the carbonation progress.
e greenhouse e ect has become globalization problem in recent years.According to the prediction of atmospheric temperature from China's National Meteorological Center, the change of temperature in the Beijing district is shown in Figure 2. e average value, lower value and upper value of temperature are presented in Figure 2. Jianxin et al. [7] stated that the di usion coe cient of concrete carbonation will improve with the increase of temperature, so the temperature change should be considered when predicting the carbonation progress.However, in existing researches [8][9][10][11], the carbonation progress rarely considers the change of temperature.is leads to underestimate concrete carbonation behavior.erefore, the in uence of the temperature change on the carbonation progress should be considered in theoretical and experimental analysis.
Except the CO 2 concentration and temperature, the e ects of other environmental factors and the concrete material compositions on concrete carbonation have been investigated in-depth in existing researches.e environmental factor includes relative humidity [11][12][13][14], and the compositions include water-cement ratio [15], various supplementary cementing materials [16][17][18], aggregates [19,20], and mix proportions [21,22].In these studies, there are few researches about the carbonation situation on the carbonation surface.However, the carbonation degree at the di erent place on the carbonation face is di erent, which is meaningful for the research of spatial reliability [23], so it is necessary to carry out the related research.
In recent years, with the rapid development of computer science, many scholars have made numerical simulations of concrete carbonation.Saetta et al. [24,25] proposed a model to simulate all the phenomena based on carbonation process considering the multidimensional transport of moisture, heat, and carbon dioxide through concrete and taking in account the reduction of the porosity due to carbonation.Burkan Isgor and Razaqpur [10] built a nonlinear nite element approach for tracing the spatial and temporal advancement of the carbonation front in concrete structures with and without cracks.However, the above two models do not consider the time-and temperature-independence of the carbonation coe cient.Saetta et al. [11,12] established the di erential equation of the carbonation process considering the factors such as water, CO 2 di usion, and the variation of temperature and applied to practical engineering for predicting the carbonation depth of a prestressed RC beam, while the detail numerical simulation progress was not carried out for the control equation of the carbonation re ection.Peter et al. [26] proposed the system of reaction-di usion equations independent of the space dimension, while their numerical simulations only deal with the situations of one-dimensional di usion.Wang et al. [27] developed the numerical simulation model of the concrete carbonation and determined the value range of the factors of the CO 2 di usion coe cient and the reaction rate of carbonation.However, only two factors were considered in the model, and the model was realized by   According to Fick's di usion law, the di usion is the main transferring behavior for CO 2 in concrete.Considering only the di usion ux generated by the concentration gradient in various directions and ignoring the impact of the owing of CO 2 gas, the rate of CO 2 gas in ux in mass can be formulated as where Q CO 2 x , Q CO 2 y and Q CO 2 z are the di usion uxes in the x-, y-, and z-axis direction, respectively.eir units are all kg , and Q CO 2 z,z are the values of , and Q CO 2 z in the x-, y-, and z-axis direction, respectively, as shown in Figure 3. e corresponding quality rate of CO 2 gas e ux in mass in the microunit can be expressed as e rate of cumulative CO 2 in mass is where C CO 2 is the mass concentration of CO 2 .
If the rate of CO 2 mass consumption per unit volume is r CO 2 , the rate of CO 2 gas mass consumption in the microunit is M H r CO 2 dx dy dz. ( Substituting (3)-( 5) into (2), the equilibrium equation can be given as zC CO 2 zt dx dy dz Q CO 2 x,x dy dz + Q CO 2 y,y dx dz Taking the limit values for dx, dy, and dz as being closer to zero, (6) can be expressed as Assuming the materials in the concrete structure are isotropic, the di usion uxes of CO 2 in concrete can be calculated by Fick's rst law as where D AB (t, T) is the di usion coe cient and dC CO 2 /dx is the concentration gradient.e minus sign means the di usion direction is contrary to the concentration gradient.Advances in Materials Science and Engineering e CO 2 diffusion process in concrete is unsteady, and its concentration varies with the distance x and the diffusion time t.Combining ( 8) with ( 7), we get Most existing studies only consider the carbonation value of concrete under a certain apparent diffusion coefficient.In fact, with the development of carbonation and hydration, the porosity of the carbonation region becomes smaller, and the diffusion rate decreases.According to the research results from the Yoon et al. [29] and Jianxin et al. [7], the diffusion coefficient is time and temperature dependent.
erefore, this paper considers the time-and temperature-dependent diffusion coefficient, and the calculation model is where D 1 is the CO 2 diffusion coefficient after a year, n d is the age of CO 2 diffusion coefficient, E is the activation energy during the diffusion progress (40 kJ/mol), and R is the gas constant (0.008134 kJ/mol•K).
In practical engineering, the length of concrete structure is generally large.For convenience, the CO 2 diffusion problem can be simplified from three-dimensional to twodimensional, which sets dz � 1 in the control body and only considers diffusion along the x and y directions.When the mass transfer of CO 2 gas in concrete is dominated by diffusion, the reaction process ends in the carbonization region, and the rate of CO 2 mass consumption is zero (r CO 2 � 0).erefore, (9) can be simplified as Equation ( 11) is the diffusion equation of CO 2 gas in concrete, which demonstrates the diffusion law of molecules in the unsteady state.∇ is the Laplace operator.e solution of this equation is the CO 2 concentration function C CO 2 at time t. e concentration values at any time can be obtained by solving this equation.Finally, the concentration distribution of CO 2 gas in the concrete can be obtained.e calculation process can be achieved by using the Matlab program.

Finite Element Solution of Diffusion and Mass Transfer
Equations.
e main chemical reaction of concrete carbonation is that CO 2 penetrates into the pore of concrete and interacts with CH, C 3 S, C 2 S, and other basic hydration products produced by the hydration process of cement.Finally, calcium carbonate and other substances are formed.
Following conditions are generally known during the reaction and presented as (i) Initial condition: when t � 0, C CO 2 (0) is a certain value.(ii) Boundary condition: when x � y � 0, C CO 2 � C CO 2 (0) (i.e., the concentration of CO 2 is equal to the initial mass concentration).When x � α and y � β (α and β tends to infinity), C CO 2 � 0. (iii) According to Yoon's model [29], the carbonation front equals carbonation depth and the concentration of CO 2 is zero at the end of the carbonation depth, and this setting is applied in this paper.
e following differential equation is established considering boundary conditions: where b is the boundary (x � 0, y � 0) and C is the constant.is equation is a parabolic partial differential equation.
According to the variation principle and Euler equation, the corresponding function of ( 12) is Hence, solving the partial differential solution (13) can be converted to obtain the extreme value of function F in the carbonation region D, and the mass concentration of C CO 2 on the boundary b of the region D is known.
According to the function form, the region D is divided into several small triangular meshes, and a typical unit is taken out for analysis, which is the six-node triangular element [30] (Figure 4), and the node (a, b, c) on the edge of the triangular element is located at the middle point of the corresponding edge.Equation ( 13) can be expressed in the area of the unit as e mass concentration in the unit i is discretized into a concentration function which is only related to the 4 Advances in Materials Science and Engineering concentration at the node.e column vector of the mass concentration of node is A quadratic concentration function of a six-node triangular element is set as By using the area coordinate L i (i, j, and m), the element concentration function can be expressed by shape functions as e elements in the shape function matrix N { } can be expressed as where A is the area of the triangular element.Assuming that zC CO 2 /zt is only the function of the position (x, y), the functional variation at a particular time and the variation of time parameter t are considered.e variation calculation of ( 14) is e derivative of ( 17) with respect to each mass concentration of node is Considering the unsteady di usion process of CO 2 in the concrete, C k is a function of time parameter t. erefore, we have By combining (20) to (22) and term by term integration of ( 20), the following equations can be obtained: It can be obtained from the above equation as e coe cient matrices [H] and [R] are calculated as Other factors can be recursive based on the above equations.
Based on the forward di erence method, the concentration of CO 2 can be given as Figure 4: Six-node triangle element.
Advances in Materials Science and Engineering Substituting ( 26) into (27) gives According to the nite element method, the carbonation area D is discretized into nite triangular elements and n nodes, and the mass concentration of the corresponding elements is distributed to each node.erefore, it is necessary to solve the concentration values at each node and then solve the mass concentration distribution in the area D. F e i is the function of triangular elements, and the function of the entire region D is i .e function is a multivariate function to express the mass concentration of the unknown nodes.Combining with the variation principle, solving functional extremum can be converted to solve the extremum value of multivariate functions.e extreme condition for the function is e coe cient matrixes are synthesized as H H e and R R e , respectively, and the recursion equation can be obtained by combining (27) as where C CO 2 t is the initial mass concentration of CO 2 , and the concentration of the CO 2 value at any time interval (Δt) can be obtained through setting the initial concentration of CO 2 and solving (29).

Modeling of Carbonation Process.
e nite element method is implemented in the Matlab platform, and the ow chart of the program is shown in Figure 5. e carbonation surface of the test specimen is simpli ed (75 mm × 75 mm), and a simulation process at a carbonation age is shown in Figure 6.
e direction of arrow represents the di usion direction of CO 2 gas after penetrating into the concrete and along the straight line OA (Figure 6).Areas from red to blue represent the transition from the carbonated zone to the partially carbonated zone and to the noncarbonated zone, respectively.e L-shaped curves represent the distribution curve of CO 2 concentration in the concrete, and the values of curves are C CO 2 , 0.9C CO 2 , . . ., 0.1C CO 2 , respectively.Carbonation depth values at the corner and other positions are calculated for comparison, and the carbonation depth values at the corner (X j ) and other positions (X q ) for C25, C30, and C35 concrete are obtained under the condition of three kinds of CO 2 emissions strategy (A1F1, B1, and Best  Advances in Materials Science and Engineering mitigation) [3] for the next 100 years from 2000 as shown in Figure 7. e corner refers the line OA, and other positions are the upper triangular and lower triangular positions except the corner.It can be found that the ratio of the carbonation depth value at the corner (X j ) to that at the other positions (X q ) is between 1.3 and 1.4.erefore, when predicting the carbonation depth of concrete structures, it is necessary to consider the problem of different carbonation depths at different positions.

Material Properties.
e sizes of standard concrete blocks in this test were 150 mm × 150 mm × 150 mm.ree different concrete water-cement ratios (0.65, 0.55, and 0.45) were designed.e designed concrete compressive strengths in this experiment were 25 MPa, 30 MPa, and 35 MPa, respectively.
e mix compositions for concrete blocks are shown in Table 1.e blocks were taken out after the 28 days standard curing period and baked for 48 hours at 60 ± 2 °C and then put into a closed carbonation chamber (Figure 8).In addition, the test also used 1% phenolphthalein ethanol solution to determine the carbonation depth.

Test Instruments.
e test was carried out in both the high-and low-temperature carbonation chambers.e CO 2 gas supply device (including gas cylinders and pressure gauge) was used to maintain the CO 2 concentration in the chamber.e test block was broken by the universal testing machine.
e carbonation depths were measured by the digital carbonation depth detector (HC-TH01).

Test Parameters.
Based on the different concrete strengths, water-cement ratio (w/c), temperature, and test time, nine groups of the concrete samples were set up in the test.ere were totally fifteen samples in each group that is divided into five subgroups.In each subgroup, three concrete samples were tested for every test time.Table 2 presents the detailed setting of each parameter for each group in the test.

Test Progress.
One side of the block was tested for carbonation, while the remaining five surfaces were protected with epoxy resin, and the space between components was larger than 5 cm as shown in Figure 8. ese samples were placed in a chamber containing 20 ± 3% concentration of CO 2 at 20 (30 or 40) ± 2 °C and 70 ± 5% relative humidity.During the test, the temperature and humidity were measured and recorded every four hours, and the CO 2 mass concentration on the surface of the block in the chamber was kept constant.
ese blocks were taken out after designed time (Table 2) and were broken by a universal testing machine from the center of the uniaxial carbonation surface to measure the carbonation depth.After the test, the ear syringe was used to blow away the ash and slag on the concrete surface, and then the phenolphthalein ethanol solution was dropped on the concrete surface for about 30 seconds.e carbonation depth value of each measurement point whose space was set as 1 cm on the surface of concrete block was measured by the carbonation depth detector.e detailed carbonation schematic diagram is shown in Figure 9(a), the uniaxial carbonation surface is divided into corner areas (X j ) and other areas except corner areas (X q ) as shown in Figure 9(b).
When the coarse aggregate is just located at the dividing line of a certain measured point, the average carbonation depth of the tested block on the left and right sides of the coarse aggregate is taken as the depth of the point x i .e formula for the average carbonation depth of concrete at different ages is calculated as where x t is the average carbonation depth of the test blocks after t days and n is the total point numbers on two sides surface.concrete with three di erent water-cement ratios under the conditions of di erent temperatures.Due to the continuous reaction of the alkaline hydration product and carbon dioxide in the pore of concrete, which generates the difcult soluble substances CaCO 3 and blocking gas channel, the carbonation recreation is hindered.

Test Results and Discussion
is leads to a larger slope for the initial curve, while a smaller slope for the later carbonation depth curve.Under the condition of di erent water-cement ratios, the carbonation depth of concrete at the corner position is about 1.35 times larger than that at the other positions, which is in good agreement with the nite element results and veri es the correctness of calculation results.e reason of the difference between corner and other area is that the carbonation rate of each direction is di erent in the closed carbonation chamber, and the corner position is subjected to bidirectional carbonation corrosion on both sides of the corner position.

In uence of the w/c
Ratios. Figure 11 presents the relationship between the carbonation depth and watercement ratio at the age of 28 and 84 days, respectively.Compared with di erent water-cement ratios of concrete,   Advances in Materials Science and Engineering the carbonation depth of C25 concrete is less than that of the rest of the two other kinds of concrete under the same temperature and age. is is because the greater concrete water-cement ratio will result in the greater porosity which leads to the faster carbonation rate.rough the data tting for the tested result, the relationship between the carbonation rate coe cient k and the water-cement ratio of each group is linear, as shown in Figure 12. is is consistent with the experimental results reported by Jin et al. [31].

4.3.
In uence of Temperature.Figure 13 shows the relationship between the carbonation depth and temperature at the ages of 7 and 56 days, respectively.It can be found from Figure 6 that the carbonation rate increases as the temperature increases.Under the condition of the same humidity and CO 2 concentration, the carbonation depth would increase about 1.9 times with an increasing temperature from 20 °C to 40 °C for the same concrete grade, which is consistent with the data measured by Li et al. [32].According to the formula of temperature    10 Advances in Materials Science and Engineering in uence coe cient (k T e (8.748−(2563/T)) ) proposed by Uomoto and Takada [33], using Origin software to t the formula k T e [q((1/293)−(1/T))] , the values of tting coe cient q is given in Table 3.It can be observed from Table 3 that the tting coe cient q is basically consistent for the same concrete whether the carbonation depth is at the corner areas (X j ) or other areas (X q ).e values of the coe cient of carbonation rate for standard specimens at di erent temperatures can be calculated by the tting formula, as shown in Table 4.
It can be obtained from Table 4 that the coe cient of carbonation rate is proportional to the ambient temperature.In the range from 10 °C to 50 °C, the coe cient increases as the temperature increases.
e concrete with a smaller water-cement ratio is more sensitive to the temperature.

Comparison and Verification
e simulation results of concrete carbonation of C25, C30, and C35 three kinds of concrete are compared with that of the tested data, as shown in Figure 13.e tested results are   Advances in Materials Science and Engineering also compared with the predicted value (X q ) obtained by Fick's second law, as presented in Figure 14.Figures 14 and 15 show the carbonation depth value at the corner (X j ) and other positions (X q ), respectively, and the simulation results are basically in good agreement with the tested results.e maximum error is about 5%. is demonstrates that the developed numerical simulation method for predicting carbonation depth in this paper is valid for concrete structures under the atmospheric environment.Moreover, as compared with the predicted results obtained based on Fick's second law, the simulation results of the carbonation depth in this paper are more close to the tested results, which proves the correctness and reliability of the proposed method.
In order to verify the practicability of the nite element model in engineering, some carbonation data under the atmospheric environment from several references are used to compare with the nite element results.e environmental factors are listed in Table 5. e comparison results are shown in Table 6.
Figure 15: Comparative test results and simulation results (X q ) at other positions: (a) C25 concrete (X q ); (b) C30 concrete (X q ); (c) C35 concrete (X q ).It can be found from Table 6 that the error between the finite element value and practical measuring value is from −4.8% to 13.4%, and the mean absolute percent error (MAPE) is 5.73%.At the same time, the absolute error of the finite element value is lower than the reference values.In addition, the MAPE between reference values and practical measuring values is 7.67%, which is larger than finite element value.So the finite element model in this paper can more precisely predict the carbonation progress of concrete under the atmospheric environment compared with references.

Conclusions
is paper firstly builds a 2-D finite element model to predict the carbonation depth considering the time (t) and temperature (T) dependent carbonation coefficients and carbonation position difference for concrete structures.A two-dimensional mass transfer equation on concrete cross section is proposed, which is solved by the finite element method.e time dependence of CO 2 mass concentration in different positions of the concrete block is obtained.Secondly, a test program to investigate the diffusion process of CO 2 in concrete is carried out.Finally, the simulation results are compared with the tested results.Several conclusions are obtained as follows: (1) e tested results show that the influence factors such as temperature, time, and water-cement ratio have an obvious effect on the concrete carbonation depth.It was also found from the tested results that under the same humidity and CO 2 concentration, the carbonation depth increased about 1.9 times as the environmental temperature increased from 20 °C to 40 °C.(2) e carbonation depth curves obtained from the finite element method are consistent with the experimental results and the practical measuring value in engineering.is demonstrates that the numerical method can be applied to predict concrete carbonation of real concrete structures with a satisfactory accuracy.(3) e tested results and numerical results quantify the ratio of the carbonation depth value at the corner (X j ) to that at the other positions (X q ), which is from 1.3 to 1.4.It is necessary to consider the time dependence of CO 2 concentration and the concentration differences at different structural positions when predicting the carbonation depth of concrete structures.

Figure 3 :
Figure 3: Schematic diagram of mass transfer of element on the concrete.

Figure 8 :
Figure 8: Tested block in the carbonation chamber.

Figure 9 :
Figure 9: Carbonation schematic diagram: (a) carbonation measurement point and (b) the position of the corner areas and other areas.

Figure 11 :
Figure 11: Relationship between water-cement ratio and carbonation depth: (a) carbonation time for 28 days and (b) carbonation time for 84 days.

Figure 12 :
Figure 12: Fitting curves of the water-cement ratio and the carbonation rate coe cient.

Figure 13 :
Figure 13: Curves of temperature and carbonation depth: (a) carbonation time for 7 days and (b) carbonation time for 56 days.

Table 1 :
Mix composition and properties of concrete.

Table 2 :
Detailed setting of each parameter for each group.

Table 3 :
Results of the values of tting coe cient (q).
Note.K is the unit of the absolute temperature.

Table 4 :
Values of k T at di erent temperatures.

Table 5 :
Environmental factors in references.

Table 6 :
Comparison results between practical engineering and finite element model.