Dr Jerzy J. Jasielec

Fick+Reaction Solver

Online simulation of motion of chloride ions through a concrete sample. The numerical solution of Fickian diffusion and binding according to Freundlich isotherm.

Online simulation of transport of chloride ion through a concrete sample in contact with chloride solution. Fickian diffusion and binding according to Freundlich isotherm is assumed inside the sample. Method of Lines with uniform grid is used to transform partial differential equations (PDEs) to ordinary differential equations (ODEs). Resulting set of ODEs is then solved using explicit Euler's method.

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Time-dependent diffusion-reaction model - basic equations

Diffusion and binding

After including the information about porisity of concrete and chloride binding in solid phase, the mass ballance equation takes the form:

$$ \left\{ \begin{array}{l} \phi \frac{\partial c}{\partial t} = D \frac{ \partial^2 c}{\partial x^2} - r \\ (1 -\phi) \rho_s \frac{\partial C_b}{\partial t} = r \end{array} \right. $$

where `c` represents the concentration of free chlorides in porous liquid, `C_b` represents the concentration of bound chlorides, `\phi` represents porosity, `D` is the diffusion coeffisient, `x` is space, and `t` represents time.

The reaction term, `r` is calculated using Freundlich isotherm:

$$ r = k [c - (C_b \text{/} K_b)^{1\text{/}\eta}] $$

where `k`, `K_b` and `\eta` are reaction constants.

The density of the solid phase in concrete, `\rho_s` is calculated using the weighted sum:

$$ \rho_s = \frac{\rho_c - \phi \cdot \rho_w}{1 - \phi} $$

where `rho_c` oznaczarepresents total density of concrete, and `rho_w = 1000 {kg}/m^3` is the density of water.

Initial and boundary conditions

We assume the constant concentrations at the boundaries (Dirichlet boundary conditions):

$$ \left\{ \begin{array}{l} c(0,t) = c_L = const \\ c(l,t) = c_R = const \end{array} \right. $$

At the beginning of the process, there is no chlorides in the sample:

$$ \left\{ \begin{array}{l} c(x,0) = 0 \\ C_b(x,0) = 0 \end{array} \right. $$

The total concentration of chlorides

The total concentration of chlorides (rescaled to `kg`/`kg` of concrete) is a sum of free and bound chlorides:

$$ C_{t o t}(x,t) = C_{f r e e}(x,t) + C_{b o u n d}(x,t), $$

where `C_{f r e e}(x,t) = \frac{\phi \cdot c(x,t)}{\rho_c}` and `C_{b o u n d}(x,t) = \frac{(1- \phi) \cdot \rho_s \cdot C_b(x,t)}{\rho_c}`.

Diffusion-reaction model - Numerical solution

Numerical solution of the above equations is based on the Method of Lines (MoL) and on the explicit Euler method.

Method of Lines

We divide the considered segment into `n+1` sub-segments, each having the length `h`, to obtain the discrete points `x_0` to `x_{n+1}`. Let's then define `c_i \equiv c(x_i)` and `C_{b,i} \equiv C_b(x_i)`. The second derivative can be now substituted with the finite difference. That leads to the following set of equations:

$$ \left\{ \begin{array}{lll} \phi \frac{dc_i}{dt} = \frac{D}{h^2} (c_{i+1}-2c_i+c_{i-1}) - k [c_i - (C_{b,i} \text{/} K_b)^{1\text{/}\eta}] & & \text{ for } i=1..n \\ (1 -\phi) \rho_s \frac{dC_{b,i}}{dt} = k [c_i - (C_{b,i} \text{/} K_b)^{1\text{/}\eta}] & & \text{ for } i=0..n+1 \end{array} \right. $$

Eulers Method

If `c_{i}` and `C_{b,i}` denote the concentrations at time `t_j`, while `\overline c_{i}` and `\overline C_{b,i}` denote the chloride concentrations at time `t_{j+1}` then:

$$ \left\{ \begin{array}{ll} \overline c_{i} = c_{i} + dt \cdot \frac{dc_{i}}{dt} \\ \overline C_{b,i} = C_{b,i} + dt \cdot \frac{dC_{b,i}}{dt} \end{array} \right. $$

After inserting the time derivative, one obtains:

$$ \left\{ \begin{array}{ll} \overline c_{i} = c_{i} + \alpha \left( c_{i+1}-2c_i+c_{i-1} \right) - \beta \bigl( c_i - (C_{b,i} \text{/} K_b)^{1\text{/}\eta} \bigr) \\ \overline C_{b,i} = C_{b,i} + \gamma \bigl( c_i - (C_{b,i} \text{/} K_b)^{1\text{/}\eta} \bigr) \end{array} \right. $$

where `\alpha = \frac{dt \cdot D}{\phi \cdot h^2}`, `\beta = \frac{dt \cdot k}{\phi}` and `\gamma = \frac{dt \cdot k}{(1 -\phi) \rho_s}`.