Comparison of Carbonation Models I. Galan and C. Andrade Eduardo Torroja Institute IETcc-CSIC, Madrid, Spain ABSTRACT: In order to describe the CO 2 diffusion process into the concrete, several carbonation models have been proposed, being the simplest one based on the square root of time. Some authors, as Tuutti (1982), Bakker (1994) and Papadakis (1989), among others, have proposed models based on the CO 2 diffusion coefficient, that is, on the solution to Fick s second law. Parrott (1994) proposed a model based on the air permeability coefficient. Recently Castellote (2008) has also proposed a carbonation model based on the un-reacted core engineering model and also on the CO 2 diffusion coefficient. The aim of this work is to analyze and compare these carbonation models. The carbonation depth predicted by the models is also compared with the values experimentally measured in specimens exposed to natural carbonation in different environmental conditions. Finally, the main parameters in each model and their influence are discussed. 1 INTRODUCTION As a diffusion process, carbonation is governed by Fick s law (1): being C, in this case, the concentration of CO 2, D the CO 2 diffusion coefficient, t the time, and x the CO 2 penetration depth. Varying the initial and boundary conditions several general solutions to this equation are obtained. Based on these general solutions to Fick s law several carbonation models have been proposed. Most of these models consider a constant CO 2 diffusion coefficient and non steady state conditions. Here a selection of some of the most significant models is presented. (1) 2 MODELS DESCRIPTION 2.1 General model based on the square root of time The simplest model proposed is the one based on the square root of time, defined by the expression in Eq. (2): where x is the penetration depth (mm), k is the carbonation rate (mm/year 1/2 ), characteristic of the exposure environment and the type of concrete, and t is the time (year). (2)
Calculating the carbonation rate for a certain depth at a certain time, it can be used for predicting carbonation depths at different times. 2.2 Models based on the CO 2 diffusion coefficient 2.2.1 Tuutti Tuutti s model [1] is based on the mobile boundaries diffusion. This model estimates that the carbonation front advanced also proportional to the square root of time and it supposes that all CO 2 reacts with the solid phases, in such a way that beyond the carbonation front, the CO 2 concentration is zero, while above it, it is 100%. The reacted zone is well defined by a sharp front. The model proposed by Tuutti allows making predictions of the carbonation depth at different times using equation 3. (3) where C s is the CO 2 environmental concentration (kmol/m 3 ), C x is the CO 2 concentration bounded in the concrete (kmol/m 3 ), x is the carbonation depth (m), t is the time (s) and D is the CO 2 diffusion coefficient (m 2 /s). For calculating the CO 2 bound in the concrete, the following formula is used (4): where C a is the CaO concentration in cement (kg CaO/kg cement), HD is the hydration degree, which is a function of the w/c ratio, c is the cement quantity per m 3 of concrete (kg/m 3 ) and 56 is the CaO molecular weight. This model assumes that the diffusion takes place in a non-stationary state and that the diffusion coefficient D is constant and varies only as a function of the concrete humidity content. For estimating the CO 2 diffusion coefficient Tuutti provides diagrams for O 2 effective diffusion coefficient as a function of the water/cement ratio and the relative humidity (RH) for different type of cements and w/c ratios. It should be noted that the diffusion coefficients for O 2 and CO 2 are not identical, but as Tuutti states, since the moisture content of the concrete is the determinant parameter for CO 2 and O 2 diffusion a satisfactory relative measure can be obtained. 2.2.2 Bakker Bakker s model [2] considers that carbonation only progresses if the concrete is dry. Combining both processes, drying and carbonation, the model formulation is as follows (5): The effective carbonation time, t eff, is the sum of the dry periods minus the time it takes to dry out the concrete (6): (4) (5) (6) A y B are functions that define carbonation and drying rate, respectively, (7) and (8). (7) x n is the carbonation depth after the n th cycle (m), D C is the CO 2 diffusion coefficient at a given moisture distribution in the pores (m 2 /s), t dn is the length of the n th period (s), c 1 c 2 is the CO 2 (8) 42
concentration difference between air and the carbonation front (kg/m 3 ), a is the amount of alkaline substance in the concrete (kg CaO/ m 3 ) and it can be calculated with the expression (9): (9) where C a is the CaO concentration in cement (kg CaO/ 100 kg cement), HD is the hydration degree, c is the cement quantity per m 3 of concrete (kg/m 3 ) and M CO2 and M CaO are the CO 2 and CaO molecular weights respectively. D v is the effective diffusion coefficient for water vapour at a given moisture distribution in the pores (m 2 /s), b is the amount of water to evaporate from the concrete (kg/m 3 ), and c 3 c 4 is the difference in water vapour concentration at the drying front and outside the concrete (kg/m 3 ). 2.2.3 Papadakis Papadakis proposes a mathematical model based on the physiochemical processes of the carbonation phenomenon [3-7]. It is a complex model, in which some simplifications can be done. The assumptions imply the formation of a carbonation front and lead to a simple analytical expression for calculating the evolution with time of this front, depending on compositional parameters of the cement and the concrete, as well as on the environmental conditions. In this model, based on the square root of time, the proportionality constant can be calculated from the portlandite, C-S-H gel, C 2 S and C 3 S quantities, the CO 2 concentration and its effective diffusion coefficient. The general formula for calculating the carbonation depth is (10): (10) The denominator is the total CaO molar concentration, in form of carbonatable materials. In case the material is completely hydrated [C 3 S] and [C 2 S] are equal to zero. The CO 2 effective diffusion coefficient is calculated by Papadakis by means of the following expression (11): being RH the relative humidity and ε p the paste porosity. Papadakis proposes some simplified expressions for calculating the carbonatable constituents concentrations and the paste porosity. From those expressions, which consider a concrete with no air occluded in it, with common gypsum values and with common clinker composition, he proposes a formula to calculate the carbonation depth, using the water/cement and aggregate/cement ratios, their respective densities, the relative humidity and the CO 2 concentration (12). (11) (12) 2.2.4 Castellote Castellote s model [8] is based on the principles of the unreacted-core systems, typical of chemical engineering processes, in which the reacted product remains in the solid as a layer of inert ash, adapted for the specific case of carbonation. The model considers that the controlling 43
step in the carbonation rate is the CO 2 diffusion through the carbonated part of the sample. The first equation to be used for a cylindrical sample is the one for calculating the fractional conversion X S of the solid reactant s at time t (13). where r is the radius of the unreacted core (cm) and R is the radius of the cylinder (cm). After that, the time for complete conversion of the reactant τ (s), that is, for complete carbonation, can be calculated as follows (14). Finally, the diffusion coefficient D (cm 2 /s) can be calculated with the following equation (15). (13) (14) (15) where b is the stoichiometric coefficient for the reaction bs(s) + CO 2 ->., and ρ S is the molar fraction of the reactant in the solid (mol/cm 3 ). Castellote proposes simplified values for b and ρ S to be used in the model. The simplified relationship between ρ S and b is as follows (16): where RF is a reduction factor that takes into account the CO 2 concentration (17): If the samples are not cement paste, the factor CFP, Correcting Factor to Paste, must be used for correcting the amount of paste in the sample. The quantity of calcite formed when fully carbonating at 100% CO 2 is also calculated by Castellote and related to the amount of CaO in the binder. With all these values and simplifications equation 15 can be re-written as equation (18): (16) (17) (18) 2.3 Model based on the air permeability coefficient 2.3.1 Parrott Parrott s empirical model [9] is based on the air permeability coefficient, determined in concretes pre-conditioned al 60% RH. The estimation of carbonation is made from oxygen permeability measurements, adjusting them to the humidity content in the concrete for each environment. The proposed expression is the following (19): where K is the air permeability coefficient (10-16 m 2 ), x is the carbonation depth (mm), and t is the time (year). c is the alkaline material in the concrete, the CaO in the hydrated cement matrix that can react with the CO 2 and that delays the CO 2 penetration rate. This parameter depends on cement composition, concrete mix proportions, exposure conditions and proportion of cement reacted. It is expressed in CaO kg per m 3 of cementitious matrix. Parrott proposes values of c related to the type of cement and the RH. It can also be calculated with the following formula (20): (19) 44
(20) where C is kg of cement per m 3 of concrete, C a is CaO kg per kg of cement, w is kg water per m 3 of concrete, and HD the hydration degree. The different humidity content in the concretes is taken into account in two ways: by calculating the K air for other RH than 60% as a function of the RH, and by making the n exponent of time evolve also as a function of RH. Based on experimental measurements, Parrott proposes an expression to relate the permeability coefficient with the diffusion coefficient (21): (21) 3 EXPERIMENTAL Concrete specimens were fabricated with CEM I 42,5R cement, with 63,8% CaO. The w/c ratio for the mix was 0,6 and the cement content was 300kg/m 3. The specimens were cylindrical 7,5cm diameter and 15cm height. The specimens were subjected to three different exposure conditions: inside, outside sheltered from rain and outside not sheltered from rain. After one year and after 3,7 years the carbonation depth of the specimens was measured by means of the phenolphthalein indicator. During the exposure time the characteristic parameters of environmental conditions, i.e. temperature, RH, precipitation and CO 2 concentration were measured regularly inside and outside. 4 RESULTS AND PREDICTIONS The average RH measured during the exposure time was 45% inside and 60% outside. The average CO 2 concentration was about 650 ppm, and outside about 500 ppm. 4.1 Depth prediction In figures 1, 2 and 3 the predicted values from the models of the carbonation depth as well as the experimental results obtained after one and 3,7 years are represented. Figure 1 represents the values for the specimens that were kept outside sheltered from rain, figure 2 for the ones that were outside not sheltered from rain and figure 3 the ones that were inside. For the calculations the diffusion coefficients were calculated from Tuutti s diagrams, considering that the CO 2 coefficient is about 0,78 times the O 2 coefficient, that is 7,8. 10-8 m 2 /s for inside and 5,5. 10-8 m 2 /s for outside. The air permeability coefficient at 60% RH was measured, and the value obtained, 6,7. 10-16 m 2, was used for applying Parrott s model. In order to simplify the calculations, the rain-dry periods were approximate to 4 days of rain per month, that is, all rain days during one year were homogeneously distributed in 12 months. As it can clearly be seen in Fig. 1 the depth predicted for the specimens outside sheltered from rain is in most cases very similar to the experimental values, both for 1 and for 3,7 years. For 1 year Tuutti, Bakker and Parrott are the models that better predict the real values. For 3,7 years Parrot is the best model. Outside not sheltered from rain (Fig. 2) the values predicted for 1 year by Tuutti and Bakker are in the range of the measured ones. For 3,7 years the experimental value measured is almost the same as the one for 1 year, while the predicted values are much higher. Inside (fig. 3) all predicted values for 1 year are higher than the measured ones. For 3,7 years the standard deviation for the measurement is very high which makes that all predicted values are in the range of the experimental ones. 45
Figure 1. Carbonation depth predicted and measured outside sheltered from rain. Figure 2. Carbonation depth predicted and measured outside not sheltered from rain. Figure 3. Carbonation depth predicted and measured inside. 4.2 Diffusion coefficient prediction Using the experimental values of the carbonation depth, the corresponding diffusion coefficient were predicted with all models. Figures 4, 5 and 6 represent these predicted values as well as the ones obtained from Tuutti s diagrams for outside sheltered from rain, outside not sheltered and inside, respectively. In most cases, the D predicted by Tuutti and Parrott is very similar. The D 46
predicted by Papadakis and Castellote is very similar too. And the D predicted by Parrott is always much higher than the other predictions and than the one taken from Tuutti s diagrams. Figure 4. D predictions outside sheltered from rain. Figure 5. D predictions outside not sheltered from rain. Figure 6. D predictions inside. 47
5 DISCUSSION 5.1 Influence of the environment in the depth prediction Some of the models take into account the influence of the RH for predicting depths, but they all consider this parameter as external, that is, they consider the environmental RH. As this RH is the same for outside sheltered from rain and for not sheltered, the depth predictions are the same for both environments. In reality, the RH inside the concrete and outside is not the same which means that the RH for the environment sheltered from rain and for the not sheltered might not be equal. When considering the RH all models consider an average value, but the RH is continuously changing, which may have an effect in the carbonation depth advance. About the rain, as explained above, only Bakker considers dry-rain cycles and takes into account this effect of saturated pores during and after rain until they dry out. For using Bakker s model a lot of meteorological data are needed, cycles duration, water content, concentration of water vapour are variables not always available. About the CO 2 concentration all models allow to change this parameter except Parrott that include this value inside the proportionality constant. Changing this parameter might be useful when considering CO 2 concentrations very different from the atmosphere. 5.2 Diffusion coefficient for depth prediction All models consider the CO 2 diffusion coefficient as a parameter but no one has measured the this coefficient. Tuutti measured O 2 diffusion coefficients for different w/c ratios and RH and he proposed diagrams from which it can be correlated to the CO 2 diffusion coefficient. Papadakis proposed an equation to calculate it as a function of the porosity or as a function of densities and ratios. All models suppose a constant D CO2, which is not true due to the evolution of carbonation which implies porosity lowering. 5.3 Square root of time in depth prediction For Tuutti, Bakker, Papadakis and Parrott the square root of time equation applies. For Castellote it should apply as well, but as the model is proposed in cylindrical coordinates the relation is not direct. Outside sheltered from rain all models predict quite similar values to the experimental one. Inside and outside not sheltered from rain all models predict higher values than the experimental. 5.4 Diffusion coefficients prediction from depth measurements Tuutti and Bakker predict very similar diffusion coefficients. Papadakis and Castellote predict also very similar diffusion coefficients which are about half the ones predicted by Tuutti and Bakker. Parrott s equation gives values much greater than others. The diffusion coefficients calculated from Tuutti s diagrams are quite similar to the ones predicted for the sheltered from rain environment. Inside and outside not sheltered from rain the predicted coefficients are considerably smaller than the ones calculated from Tuutti s diagrams. 5.5 Alkaline material All models except Castellote suppose all alkaline material present in the hydrated cement phases may react with the CO 2 which might not always be true. 48
6 CONCLUSIONS If previous depth measurements are available the square root model is enough for making predictions at different ages. If no previous measurements are available any of the mentioned models can give approximate predictions. Each model has advantages and disadvantages. Depending on the parameters needed and the ones available, the most suitable model should be chosen. REFERENCES Tuutti, K. 1982. Corrosion of steel in concrete, PhD Thesis, Swedish Cement and Concrete Institute CBI, Stockholm, Sweden. Bakker, R. 1964. Prediction of service life of reinforcement in concrete under different climatic conditions at given cover, Corrosion and Protection of Steel in Concrete International Conference, Sheffield U.K., R.N. Swamy Ed., Papadakis, V. G., Vayenas, C. G. and Fardis, M.N. 1989. A reaction engineering approach to the problem of concrete carbonation, AIChE Journal 35 (10) p. 1639-1650. Papadakis, V. G., Vagelis, G., Vayenas, C. G., Costas, G. and Fardis, M.N. 1991. Physical and Chemical Characteristics Affecting the Durability of Concrete, ACI Materials Journal 88 (2) p.186-196. Papadakis, V. G., Vagelis, G., Vayenas, C. G., Costas, G. and Fardis, M.N. 1991. Fundamental Modeling and Experimental Investigation of Concrete Carbonation, ACI Materials Journal 88 (4) p. 363-373. Papadakis, V. G., Vayenas, C. G. and Fardis, M.N. 1991. Experimental Investigation and Mathematical Modeling of the Concrete Carbonation Model, Chemical Engineering Science 46 (5/6) p. 1333-1338. Papadakis, V. G., Fardis, M.N and Vayenas, C. G. 1992. Effect of composition, environmental factors and cement-lime mortar coating on concrete carbonation, Materials and Structures 25 p. 293-304. Castellote, M. and Andrade, C. 2008, Modelling the carbonation of cementitious matrixes by means of the unreacted-core model, UR-CORE, Cement and Concrete Research 38 p. 1374-1384. Parrott, L.J. 1994. Design for avoiding damage due to carbonation induced corrosion, CANMET Conference on Durability of Concrete, Nice, p. 283-298. Malhotra Ed. 49