To develop an explicit scheme to solve the advection-diffusion equation numerically, we can use a
finite difference approximation. Let's start by discretizing the equation in both time and space.
Discretization in Time:
We'll use a forward difference approximation for the time derivative:
𝜕𝑐/𝜕𝑡 ≈ (𝑐ᵢⱼ⁺¹ - 𝑐ᵢⱼ) / Δ𝑡
Discretization in Space:
We'll use a centered difference approximation for the spatial derivatives:
𝜕²𝑐/𝜕𝑥² ≈ (𝑐ᵢ⁺¹ⱼ - 2𝑐ᵢⱼ + 𝑐ᵢ⁻¹ⱼ) / Δ𝑥²
𝜕𝑐/𝜕𝑥 ≈ (𝑐ᵢ⁺¹ⱼ - 𝑐ᵢ⁻¹ⱼ) / (2Δ𝑥)
Now, substitute these approximations into the advection-diffusion equation:
(𝑐ᵢⱼ⁺¹ - 𝑐ᵢⱼ) / Δ𝑡 = 𝐷 (𝑐ᵢ⁺¹ⱼ - 2𝑐ᵢⱼ + 𝑐ᵢ⁻¹ⱼ) / Δ𝑥² - 𝑈 (𝑐ᵢ⁺¹ⱼ - 𝑐ᵢ⁻¹ⱼ) / (2Δ𝑥) - 𝑘𝑐ᵢⱼ
Where i represents the spatial index (along x-axis) and j represents the temporal index (along t-axis).
Now, rearrange the equation to solve for the unknown concentration at the next time step, cᵢⱼ⁺¹:
cᵢⱼ⁺¹ = cᵢⱼ + (Δ𝑡 / Δ𝑥²) * D * (cᵢ⁺¹ⱼ - 2cᵢⱼ + cᵢ⁻¹ⱼ) - (Δ𝑡 / (2Δ𝑥)) * U * (cᵢ⁺¹ⱼ - cᵢ⁻¹ⱼ) - Δ𝑡 * k * cᵢⱼ
Now, we can proceed with the numerical simulation using the given parameters and the derived
explicit scheme.
a) Steady Condition:
In the steady-state condition, the concentration does not change with time. We can solve the
equation iteratively until the solution converges to a steady state. In this case, we'll simulate until the
change in concentration between two consecutive time steps is below a certain threshold.
Initialize parameters and variables:
Length of the tank (L) = 10 m
