**4. Discussion**

The results of carbonation depth were used to formulate a model expressing the depth of carbonation *ht* (after a predetermined time *t* under assumed exposure conditions) as a function of the water-cement ratio (*w*/*c*) and the mass proportion of ash and cement in the binder (*p*/*c*), as follows:

$$h\ln(w|c;p|c) = a + b(w|c) + c(p|c) + d(w|c)^2 + e(w|c)(p|c) + f(p|c)^2\tag{5}$$

Response surfaces, including the total range of w/c ratio values from 0.35 to 0.55 and p/c values from 0.2 to 0.50 assumed in the tests, are presented in Figures 7–9, corresponding to selected exposure times *t*, i.e., 56, 70, and 90 days, with the determination coefficient in all cases significantly exceeding the value of 0.9.

**Figure 7.** Depth of carbonation as a function of w/c and p/c, after exposure time *t* = 56 days, *R*<sup>2</sup> = 0.93.

**Figure 8.** Depth of carbonation as a function of w/c and p/c, after exposure time *t* = 70 days, *R<sup>2</sup>* = 0.92.

**Figure 9.** Depth of carbonation as a function of w/c and p/c, after exposure time *t* = 90 days, *R<sup>2</sup>* = 0.95.

For a complete set of results, a general concrete carbonation model with fly ash was developed, taking into account both material variables (w/c ratio and p/c ratio) and the time of carbonation under specific exposure conditions. The general form of the model was assumed to take into account the finite nature of the carbonation process over time, consistent with the considerations presented in the introduction to this article. Finally, the following general form of the model was adopted:

$$h(\text{u}\diamond\text{c};\text{p}\diamond\text{c};\text{t}) = a + b\_1 \times \text{(u}\diamond\text{c)} + b\_2 \times \text{(p}\diamond\text{c)} + b\_3\text{(sqrt}(\text{t})\tag{6}$$

where: *h*—depth of carbonation, mm; *w*/*c*—water-cement ratio; *p*/*c*—powder to cement ratio; *a, b*1*, b*2*, b3*—material and technological coefficients, *t*—time of exposure, days.

The model does not take into account the essential technological factor, which is the time of early curing of concrete, because all tests were carried out with reference to samples hardening for up to 28 days in water. The possible practical use of models of the proposed type requires taking into account this factor in the form of an additional expression or limiting the validity of the equation to a specific regime of concrete curing. The detailed form of the proposed model determined by the curve fitting method, based on the experimental results, is as follows:

$$h(w/c; p/c; t) = 1.07 + 25.28 \times (w/c) - 3.53 \times (p/c) - 41.07/sqrt(t)\tag{7}$$

and is characterized by the determination coefficient 0.85. This indicates the good fit of the adopted model to the results of laboratory tests obtained under accelerated carbonation conditions in 4% CO2. The presented detailed model can be applied only to concrete with the range of constant and changed material and technological variables used in the research, but the method of its determination is an example that can be repeated in the case of various material and technological assumptions.

For further verification of model fitting to the experimental results, values calculated from the model and obtained from the experiments over 90 days of exposure in 4% of CO2 concentration were compared, as in Table 9. The accuracy of the prediction is ± 15%, which is an acceptable value from an engineering point of view.


**Table 9.** Comparison of experimental and calculated values of depth of carbonation.

Predicting the course of carbonation in a real structure using the developed model requires establishing an equation expressing the dependence of the progress of carbonation on the CO2 concentration in the research environment and in the exploitation environment. The dependence of carbonation depth on CO2 concentration is described in many publications [37,40,64], with a form similar to the following:

$$X\_1 = X\_2 \times \frac{c\_1 \times t\_1}{c\_2 \times t\_2} \tag{8}$$

where: *X*1—depth of carbonation after exploitation in c1 concentration after t1 time of exploitation; *X*2—depth of carbonation in accelerated conditions, *t*2—time of exposure in accelerated conditions; *c*2—CO2 concentration in accelerated conditions; *t*1—expected service life of construction; *c*1—CO2 concentration in exploitation conditions. Please check that intended meaning has been retained

Thus, having the developed detailed model in the general form (Equation (6)), i.e., a form that assumes the finite character of the carbonation process, we can calculate the limit of functions at *t* reaching to infinity, i.e., the ordinate of the model asymptote denotes the maximum possible depth of carbonation in the test conditions. In the analyzed case it means under accelerated conditions (CO2 concentrations equal to c2 = 4%). Assuming this value as *X*<sup>2</sup> in Equation (8), and as *t*<sup>2</sup> (the maximum test time under accelerated carbonation conditions), i.e., 90 days, one can determine the predicted carbonation depth after time *t*<sup>1</sup> (e.g., assumed service life of 50 or 100 years) at a given concentration of carbon dioxide *c*<sup>1</sup> (e.g., atmospheric concentration—approximately 0.04%). In Table 10 a simulation of the carbonation depth using the above dependence after 50 and 100 years of service life in the atmosphere with a concentration of 0.04% carbon dioxide is shown.



The values presented in Table 10 can be considered as the starting point for determining the required reinforcement cover in a construction made of concrete with given w/c and p/c ratios. These values should be increased by a safety factor, taking into account random factors and those related to the curing of concrete. This is an alternative approach to the standard procedure for determining the thickness of the reinforcement cover indicated by the EC2 standard [65], which is favorable due to the optimal method of determining the minimum safe thickness due to durability, taking into account the concrete material specification. The proposed procedure for determining the required cover is presented in the form of the diagram (Figure 10) for the concrete family. As a concrete family, a set of technologically similar concretes was defined, characterized by a constant type of cement, type of ash, chemical admixture, initial care method, and the following variables: any type of aggregate (except lightweight), any type of water (in accordance with PN-EN 1008 [66]), the specified range of variability of w/c and p/c ratios, and dosage of the same admixture in the amount necessary to the required consistency.

**Figure 10.** Proposed procedure for determining the thickness of the reinforcement cover with use of the developed model of carbonation.
