## 2 Circular Diffusion Model for a Cross-Section of an RC Circular Column  
### Theoretical Derivation  
The transport mechanisms for chloride penetration into concrete are rather complicated. In general, diffusion, viewed as the primary avenue, is considered. Chloride diffusion through a circular concrete cross-section is described by [PERSON]'s second law in polar coordinates as:  
\[\frac{\partial C(\rho,t)}{\partial t}=D(\frac{1}{\rho}\frac{\partial C(\rho,t) }{\partial\rho}+\frac{\partial^{2}C(\rho,t)}{\partial\rho^{2}}) \tag{1}\]  
where \(D\) is the diffusion coefficient and \(C\) (\(\rho\), \(t\)) represents the chloride concentration at a distance \(\rho\) from the center of the circular section at an instant \(t\).  
The initial boundary conditions before chloride diffusion are the following: (a) the chloride concentration is assumed as constant on the external surface of the column \(C(R,t)=C_{0}t\in[0,+\infty]\), and (b) the initial chloride concentration inside the concrete at time zero is assumed at zero \(C(\rho,0)=0\)\(t\in[0,R]\).  
As the concrete servicing age increases for the situation neglecting the effect of the surrounding, \(D\) is expressed as follows [23]:  
\[D(t)=D_{\text{ref}}\bigg{(}\frac{t_{\text{ref}}}{t}\bigg{)}^{m}=D_{\text{i}}t ^{-m} \tag{2}\]  
where \(D_{\text{ref}}\) is the diffusion coefficient at the referenced time \(t_{\text{ref}}\) and \(m\) represents the exponent coefficient.  
According to the adaptation of variable separation methodology (\(C(\rho,t)=F(\rho)T(t)\)) and replacing \(D\) with \(D(t)\) according to Equation (2), Equation (1) is updated as:  
\[\frac{\partial(F(\rho)T(t))}{\partial t}=D_{\text{i}}t^{-m}\bigg{(}\frac{1}{ \rho}\frac{\partial(F(\rho)T(t))}{\partial\rho}+\frac{\partial^{2}(F(\rho)T( t))}{\partial\rho^{2}}\bigg{)} \tag{3}\]  
Equation (3) is transformed as:  
\[\frac{T^{\prime}(t)t^{m}}{D_{\text{i}}T(t)}=\frac{1}{F(\rho)}\bigg{(}\frac{1} {\rho}F^{\prime}(\rho)+F^{\prime}(\rho)\bigg{)} \tag{4}\]  
The left side of Equation (4) contains a function of \(t\) only, and the right side of Equation (4) involves a function of \(\rho\) only, with constant \(\lambda\) making Equation (4) true. Therefore, Equation (4) is transformed into a mutually independent unary differential equation expressed as:  
\[\frac{T^{\prime}(t)t^{m}}{D_{\text{i}}T(t)}=\lambda \tag{5}\]  
\[\frac{1}{F(\rho)}\bigg{(}\frac{1}{\rho}F^{\prime}(\rho)+F^{\prime}(\rho) \bigg{)}=\lambda \tag{6}\]  
The solution of Equation (5) is obtained as:  
\[T(t)=A\exp\bigg{(}\frac{1}{1-m}t^{1-m}\lambda D_{i}\bigg{)} \tag{7}\]  
Equation (6) is viewed as a Bessel differential equation, and thus, its solution is expressed as:  
\[F(\rho)=BJ_{0}(\lambda\rho)+CY_{0}(\lambda\rho) \tag{8}\]where \(J_{0}\) represents the Bessel's first function with order zero and \(Y_{0}\) denotes the Bessel's second function with order zero. The parameters \(A\), \(B\), and \(C\) are undetermined coefficients, and on the basis of Equations (7) and (8), the specific formulation of \(C(\rho,t)=F(\rho)T(t)\) is:  
\[C(\rho,t)=A\exp\biggl{(}\frac{1}{1-m}t^{1-m}\lambda D_{t}\biggr{)}(BJ_{0}( \lambda\rho)+CY_{0}(\lambda\rho)) \tag{9}\]  
Furthermore, boundary conditions (a) and (b) are utilized, and the circular diffusion model is expressed as follows:  
\[C_{\text{cir}}(x,t)=C_{0}\Biggl{\{}1-\frac{2}{R}\sum_{n=1}^{\infty}\frac{1}{ \alpha_{n}}\frac{J_{0}((R-x)\alpha_{n})}{J_{1}(R\alpha_{n})}e^{-(\frac{1}{1-m} \alpha_{n}2D_{t}t^{1-m})}\Biggr{\}} \tag{10}\]  
where \(C(x,t)\) represents the chloride concentration at distance \(x\) from the external surface at time \(t\), \(R\) denotes the radius of the RC circular column, \(J_{1}\) is [PERSON]'s first function with order one, and \(\alpha_{m}\) depends on the solution of \(J_{0}(R\alpha_{m})=0\).  
Assuming that the circular section is oversimplified as a slab element directly, the corresponding slab diffusion model is given as [23]:  
\[C_{\text{slab}}(x,t)=C_{0}\Biggl{[}1-erf\Biggl{\{}x/\Biggl{(}2\sqrt{\frac{1}{ 1-m}D_{t}t^{1-m}}\Biggr{)}\Biggr{\}}\Biggr{]} \tag{11}\]