## 3 Evaluation of Service Life Using Different Design Approaches  
### Design Parameters and Governing Equations  
External conditions like temperature and humidity have a great effect on the diffusion coefficient. For humidity, it is greatly reduced below 75% for chloride ions [25], so that the humidity inside is implemented as a function of saturation in the thermodynamic coupling model [13,14]. Considering the long-term service life, the average saturation and the external humidity become similar, so the change in humidity becomes a relatively small problem in the case of full saturated conditions like sea water. For the temperature effect on ion diffusion, it significantly varies with mixing proportions, materials to be used, coarse aggregate use, and test method types [26, 27, 28, 29, 30]. Figure 1: Schematic diagram for deterministic and probabilistic method. Figure 2: Deteriorated concrete from sulfate ion intrusion. In this study, service life evaluation on sulfate erosion was performed focusing on the evaluation techniques and the temperature effect in sulfate ion diffusion. The technique of deterioration depth is listed in Equation (1) [16; 17]. \[R=\frac{Z_{p}}{t}=\frac{EB^{2}C_{0}D_{i}\chi\Phi_{Al2}O_{3}}{0.10196\alpha\gamma (1-\
u)} \tag{1}\]  
where \(R\) is the deterioration rate of concrete by sulfate ions, \(Z_{p}\) is the predicted sulfate penetration depth, \(t\) is the exposure period, \(c_{0}\) is the external sulfate concentration, \(D_{i}\) is the sulfate diffusion coefficient in concrete, \(E\) is the elastic modulus, \(\alpha\) is the roughness coefficient of the area where performance deterioration occurs, \(B\) is the stress of 1 mol of sulfate ions that reacts in 1 m\({}^{3}\) of concrete, \(\gamma\) is the required energy for the concrete surface fracture, \(\
u\) is [PERSON]'s ratio, \(x\) is the cement content of the target structure, and \(\varphi_{Al2O3}\) is the aluminum oxide content in the used cement [17]. The diffusion coefficient in Equation (1) can be modified with the effect of temperature, which can be explained by Arrhenius, as in Equation (2) [12; 30]. \[D(T)=D_{ref}\cdot exp\left[\frac{U}{R}\cdot\left(\frac{1}{T_{ref}}-\frac{1}{T} \right)\right] \tag{2}\]  
where \(D(T)\) is the sulfate coefficient at temperature \(T\); \(D_{ref}\), which can be assumed as \(D_{i}\) in Equation (1),  
The specified method for the evaluation of sulfate diffusion was not proposed, so the natural diffusion cell test suggested from previous studies [15; 29; 30] was adopted, which is listed in Equation (5). \[D_{i}=\frac{V\Delta Q}{\Delta\Delta T}\times\frac{L}{C_{1}-C_{2}} \tag{5}\]  
where \(D_{i}\) is the diffusion of sulfate ions, V is the volume of the diffusion cell, \(\Delta Q\) is increased sulfate ions in distilled water (kg/m\({}^{3}\)), A is the surface area of concrete sample (m\({}^{2}\)), \(\Delta T\) is the duration period of the test (s), L is the thickness of the concrete slice (m), \(C_{1}\) is the initial sulfate ion concentration (kg/m\({}^{3}\)), \(C_{2}\) is the initial sulfate ion concentration in distilled water (kg/m\({}^{3}\)). The increasing ion concentration in the opposite cell was evaluated through IC (ion chromatography). The detailed dimensions of the equipment are shown in Figure 3, and photos for the diffusion cell and IC equipment are shown in Figure 4.  
To test the elevation of temperature, the previously performed test was referred to. In the previous test [30], almost 13.3 h was taken for raising 20 \({}^{\circ}\)C to 40 \({}^{\circ}\)C in a control chamber, and 8.3 h was taken for elevating 5 \({}^{\circ}\)C. The diffusion cell and the sample were preheated independently, and the test started from \(-\)5 \({}^{\circ}\)C to the target temperature. The temperature changes in the diffusion cell and test setup are shown in Figure 5. The diffusion test results are summarized in Table 3, where the mean and COV were calculated from three samples. The test result and probabilistic patterns are shown in Figures 6 and 7, assuming a normal distribution.